Partitional Information Revelation under Renegotiation∗

(Job Market Paper)

Weixin Chen†

December 3, 2019

Most recent version

Abstract

Consider a long-term relationship between a seller and a buyer whose valuation

(for a per-period service or a rental good) is private. Both the valuation and the

trade quantity are continuous variables. Contracts are long-term and subject to rene-

gotiations. In such a dynamic game, a new dimension of mechanism design, namely

intertemporal type separation, arises as its induced belief-updating affects the rent

extraction–efficiency tradeoff. I derive properties of any PBE: at each history, the

seller partitions the posterior support into countable intervals and offers a pooling con-

tract to each of these intervals. The set of equilibrium outcomes of any PBE of a

finite-horizon converges to the set of MPE outcomes of an infinite-horizon game as the

horizon goes to infinity.

Keywords: Asymmetric information, commitment, partial pooling, dynamic contracting

∗I am immensely grateful to Ilya Segal, Gabriel Carroll, and Takuo Sugaya for their invaluable guidanceand continuous support. I thank Daniel Barron, Alex Bloedel, Rodrigo Carril, Juan Camilo Castillo, JackFeng, Brett Green, Matthew Jackson, Fuhito Kojima, Seunghwan Lim, Eric Maskin, Shunya Noda, AndySkrzypacz, and Robert Wilson for helpful feedback in different stages of this project. All errors remainmy own. Financial support from the Forman Fellowship through a grant to Stanford Economics and Sum-mer Research Fellowship through a grant to John M. Olin Program in Law and Economics is gratefullyacknowledged. See supplemental material here.†Department of Economics, Stanford University, Email: weixinc@stanford.edu.

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1 Introduction

Trade and economic activity takes place over time, opening up opportunities to renegoti-

ate. Long-term relationships see revisions of trade terms from time to time, with earlier

agreements revealing parties’ private information. This perspective can change parties’ in-

centives from the outset of negotiations. Incorporating the revealed information into future

agreements changes the incentives for information revelation today. The resulting informa-

tion elicitation changes the way one reacts to incentives and introduces extra complexity

of the dynamic mechanism design.1,2 This paper sets forth to study the gradual revelation

of asymmetric information and the associated incentive provision when the ability to refine

contracts is (almost) unrestricted.

Take Software-as-a-Service (SaaS) as an example. The service provider (i.e., software

company) and the buyer interact over time. The buyer has private information about his

valuation of the service, and the company had a commitment issue. The company profits

from price discriminating buyers with distinct valuations along with different service quality.

However, to boost long-term profit, the company will encourage buyers to upgrade to higher

quality service, mitigating the preset discrimination.

Generally, in a repeated adverse-selection problem where the parties cannot commit to

not renegotiate an inefficient agreement, how information is revealed determines the in-

tertemporal tradeoff between rent extraction and efficiency. It is notoriously hard to ana-

lyze mechanism design problems with such limited commitment. Auxiliary assumptions are

placed either on the class of contracts the designer can choose from, as in Maestri (2017) and

Strulovici (2017), or on the length of the horizon, as in Hart and Tirole (1988) and Skreta

(2006). Moreover, less is known beyond the binary-type case or a unit-sale case.

Richer environments complicate the question for the following reasons: (i) Unlike unit-

trade, any agreement may induce further negotiations, which requires keeping track of the

1Empirically, Gagnepain et al. (2013) study French urban transport procurement where only fixed priceand cost-plus contracts are in use. They demonstrate that the lack of commitment prevents those partiesfrom reaching the incentive efficient (Holmstrom and Myerson 1983) solution. Their counterfactual analysisconcludes that “welfare gains from improving commitment would be significant but would accrue mostly to[the agent].” A key underlying force that retains efficiency in such process is the speed up of informationrevelation.

2Joskow (2014) finds related evidence in the post-privatization United Kingdom. The UK regulates itsutility sectors commonly through a de facto fixed price contract with a ratchet every five (or so) years whenthe level of the price cap is reset to reflect the current realized (or forecast) cost of service. Regulated firmsappear to make their greatest cost reduction efforts during the early years of the price cap period and thenexert less effort at reducing costs as the date of the price review proceeding approached. This observationsuggests that the dynamic attributes of the regulatory process and how regulators use information aboutcosts revealed by the regulated firm’s behavior over time have significant effects on the incentives and hencebehavior of the regulated firm.

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(infinite-dimensional) posterior belief, and (ii) Payoffs may be nonlinear in the trade terms,

which disables a simple aggregator (summary statistics) of the intertemporal allocation.

With binary types, belief is a one-dimensional object that records the probability of type

being high. With unit-sale, there is no scope of payoff non-linearity.

For concreteness, I use a repeated Mussa–Rosen framework to present the results.3 In

such a setup, the seller is the principal and the buyer is the agent. Even though this is a

(dynamic) signaling game, all equilibrium outcomes can be characterized in a seller mech-

anism design problem (§8). The seller lacks commitment powers to stick to the ex-ante

second-best trade quantities at high prices and consequently has an incentive to delay the

type separation. If she fully separated all types today, then her future self, who knew the

type by then, would renegotiate towards a first-best quantity, leaving a small markup for the

future. The main result is that the seller optimally partitions types in the current posterior

into multiple intervals (Theorem 1) and offers each interval of types an agreement specifying

the future delivery quantities and the required payments. Therefore, the characterization

of equilibrium hinges on how these intervals are determined. To facilitate the analysis of

the seller’s incentives, I define an “aggregate loss function” of a mechanism: its differen-

tial form captures how much profit an equilibrium trade profile loses relative to the static

second-best benchmark. I demonstrate the incentive tradeoffs in information revelation us-

ing such reduced-form loss function.4 I derive analytically for an example the equilibrium

gradual separation of types as well as a dynamic mechanism that implements the equilibrium

outcomes with simple contracts.

Back to the Software-as-a-Service (SaaS) example. The typical “freemium” model adopted

by the company serves the purposes of maintaining potential recurring customers and en-

couraging them to upgrade their plans. It offers different levels of service, in which a free

but limited version is provided as well as various premium versions with better service and

additional features. The adoption of a freemium pricing strategy aims to create value from

the start via personalized pricing tiers for specific user profiles. This structure differenti-

ates users and may even offer a loyalty discount program to recurring customers. The more

flexible customization for the higher-value customers and the relative limited options for

3The results apply to other principal-agent setups. In many of these relationships, the principal poolsagents of different types in groups, and treats those in each group the same despite their inherent differences.Manufacturers (like Toyota) place their suppliers into a small number of categories that receive differentialtreatment. A pyramid structure in these relationships manifests the gradual separation of types: the best(and scarce) suppliers get distinguished and receive more discriminating contracts over time.

4The equilibrium allocation can be implemented by relatively simple contracts when exploiting equilib-rium renegotiations. This resonates with the key insight “renegotiation-proof contracts can be replicatedby simple contracts that are then renegotiated in equilibrium” from the more general contract renegotiationliterature (Aghion et al. 1994).

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the lower-value customers resonate with the countable interval pooling result in Theorem 1.

More broadly, one focus of marketing (customer success team) is to upsell to the existing

customers. Indeed, upselling accounts for a significant fraction of revenue. Its empirical

association with high profit margin is consistent with our prediction of more intense renego-

tiation in earlier negotiation phases as there is a high markup. A common practice is to limit

what the customer gets in a single transaction, which allows for later upselling new features

and capabilities. For other examples, see car rental companies and Microsoft Dynamics 365.5

The key insight behind Theorem 1 is that the renegotiation weakens the force of the

standard sorting condition from a full separation to pooling close types and separating distant

ones. In view of the seller mechanism design problem, the interval partition comes from two

properties: the buyer plays pure strategy in the seller-optimal equilibrium, and the posterior

support is always convex. Intuitively, having the buyer mix works against the spirit of

sorting. It is a less effective way to counteract the renegotiation-induced force that drives

out rent-seeking distortions: it inflates the present value of quantities to low types. Similarly,

a non-monotone pooling (of disjoint sets of types while leaving out some intermediate ones)

also works against the sorting heuristics. Ignoring the change in total cost, violating either

property required by the interval partition decreases the total price for a given total quantity

and results in lower profit due to a worse rent extraction.

I provide a characterization (Theorem 2) of the equilibrium boundaries under regularity

conditions. The tradeoff involved in raising a boundary is a decrease in the aggregate loss

of the interval below the boundary type at the expense of an increase in the differential

loss of the boundary type. The regularity conditions ensure that for the relevant range of

boundary types, such marginal gain decreases and marginal cost increases in the boundary.

Therefore, if monotonicity constraint were binding in equilibrium (corresponding to a low

marginal cost), a profitable deviation is to increase the boundary to reduce the associated

shadow cost. Therefore, the equilibrium has slackness in monotonicity constraints, implying

that quantities are assigned based on (pooling-adjusted) virtual values. Building on such

simple characterization, one can derive comparative statics results to gain insights on the

optimal way to elicit information in long-run relationships.

The rest of this paper is organized as follows. Section 2 compares results with the

related literature. Section 3 develops the model. Section 4 formulates the mechanism design

problem. For a finite horizon, Section 5 presents the main results. For a seller-optimal

PBE, Section 6 derives the necessary conditions of such equilibrium, and Section 7 verifies

5Car rental companies such as Hertz push customers to voluntarily buy-up to higher grade products byoffering upgrades at attractive prices at the time of fulfillment. Microsoft Dynamics 365, software for cus-tomer relationship management, supports and promotes among its small- and medium-sized user enterprisesthe use of loyalty programs and preferential discounts for the enrollees of such programs.

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these are equilibrium properties by a straightforward implementation. Section 8 extends

these results to any PBE (Theorem 1). Section 9 illustrates the dynamic programming via

a relaxed first-order approach and provides regularity conditions that validate this relaxed

approach (Theorem 2). For an infinite horizon, Section 10 introduces some extra notions

of the solution concept and illustrates the extension of the results. Section 11 discusses an

extension to general dynamic mechanisms and the limitation of two forms of contracting

that are common in the literature.

2 Related Literature

This paper builds upon and contributes to the literature on contract negotiation with private

information. The previous literature studies Coasian dynamics under either a finite horizon,

binary types, or unit-trade. Laffont and Tirole (1990) considered two-period procurement

with renegotiation, establishing that full separation in the first period is not optimal. Hart

and Tirole (1988) investigated the case of unit-trade. Maestri (2017) and Strulovici (2017)

studied the binary-type case for more general outcomes and utility functions. The former

proposed a recursively-formulated renegotiation-proof refinement of PBE and analyzed such

equilibria while the latter allowed renegotiation and studied all PBEs of the negotiation

game. The common finding is that the equilibrium outcomes converge to efficiency quickly

over time. The principal may benefit from proposing multiple almost-identical contracts as a

communication device to emulate cheap talk, or, equivalently, she can use arbitrary message

spaces in her mechanisms. Strulovici (2017) addressed this possibility in the binary-type

case.6 I allow communication via non-substantive moves by the principal (in particular

having agent mixing) and demonstrate this form of signaling in the main discussion.7

Due to the binary structure (in types or outcomes), this literature finds results remi-

niscent of the cream-skimming property in bargaining games (Fudenberg et al., 1985). In

a durable-good monopoly setup, the seller trades earlier with higher-value (unit-demand)

consumers. In contract renegotiation, the principal attempts to benefit from discrimination;

the single-crossing condition again yields a sequential separation resembling the skimming

process. However, the literature’s focus on binary types creates extensive equilibrium ran-

domization of the efficient type’s choice and prolongs the bargaining procedure. See Hart

and Tirole (1988) for an example on price paths for a durable-good monopoly. Moreover,

results were unknown beyond the binary-type case: the randomization can lead to com-

6Doval and Skreta (2019b) revisit the classical durable good monopoly problem while addressing generalcommunication, and confirm the optimality of the posted-price.

7In contrast, the principal may maintain certain reputation in infinite horizon by using substantive moves.The main focus is on (weak) Markov perfect equilibria, which rules out the reputation channel.

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plex belief dynamics. This paper picks up such a task of exploring equilibrium features. I

first identify a general but weaker form of skimming property, dubbed as refining property,

in the continuum-type model, by which one can retrieve (the intuition and) the result on

binary-type as well as finite-type cases. For finite types (more than two), the implication of

my results rules out skimming property; instead, nearby efficient types are pooled together,

and a boundary type of each pooling randomizes its selection of contracts. The seminal

work by Dewatripont (1989) looks at a similar problem in a labor contract setup, but his

focus on pure-strategy equilibria evades the above complexity in belief dynamics. In this pa-

per, I address the aforementioned contractual aspects and the related difficulties altogether.

This paper contributes by studying which informational and strategic aspects are endoge-

nously relevant to the principal’s problem and how the optimal strategy is characterized in

terms of these aspects. The results imply that the general message space (richer than the

type space) and randomization are redundant from a mechanism design perspective for the

double-continuum setup.8 It thus provides a way to reduce the mechanism design program to

a tractable problem for the continuum-of-type case: In a principal-optimal equilibrium, the

posterior support at any on-path history is convex, and the agent only uses pure strategies.

Even in the durable-good setup (but with multiple types and flexible quantity or quality),

much is to be explored. The result of this paper does not automatically extend to such

setup, despite an isomorphism between a durable-good bargaining model and a contract

renegotiation model with constant contracts. The chance to renegotiate (any) agreement is

crucial. Wang (1998) showed that a different renegotiation protocol (as opposed to Strulovici

2017) can overturn the conclusion. In terms of a buyer-seller analog, he considered binary

valuation and bargaining processes that continue upon rejection and end as soon as a sale

occurs. The result is that the monopolist can implement the commitment solution.

A broader group of literature studies the incorporation of information in contract design

under different contracting modes. They consider different types of imperfect commitment:

no commitment, i.e., spot contract; short-term commitment (Laffont and Tirole 1993, chap-

ter 10; Rey and Salanie 1996; Laffont and Martimort 2002, chapter 9); commitment with

renegotiation (also known as limited commitment). They deduce that the incorporation

of asymmetric information only occurs at a slow pace in the absence of perfect commit-

ment. The ratchet effect literature (Laffont and Tirole 1988) has shown that under spot

contracting, (partial) pooling is necessary for feasibility. With relational contracting, Mal-

comson (2016) obtained the same conclusion. Section 11 explains the relative advantage of

8Hart and Tirole (1988) conjecture that partial separation (via the agent’s mixing) does not play a bigrole in mechanism design under many types. Theorem 1 confirms this observation in the continuous-typelimit. The main analysis here also gives a characterization of the speed and the intertemporal pattern oftype separation.

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long-term contracts: it offers a lower bound on agent’s future continuation utility.

In a regulatory framework, Baron and Besanko (1987) studied a closely related two-

period agency problem. They adopted partial commitment for the second-period regulatory

contract, which comes from a fairness restriction. Such restriction coincides with the notion

of commitment with renegotiation on the equilibrium path in a two-period model. They

looked at two types of equilibria in search of an optimal mechanism – full revelation and

finite interval pooling contract in the first period. With that assumption, they derived

equilibrium properties consistent with my preliminary findings in Section 5.

From a mechanism design perspective, the state-of-art is Bester and Strausz (2001) and

Doval and Skreta (2019a). These two papers provide sufficient conditions for a revelation

principle in limited-commitment environments and showcase the dynamic programming for

the principal’s problem. However, their analogs of the revelation principle do not apply here:

the former deals with finite-type space and the latter deals with spot contracts. Nonetheless,

the interval-partition property reduces the necessary message space to the type space.

3 Model of Contract Renegotiation Game

I consider two risk-neutral parties, a seller (she) and a buyer (he), who have the opportunity

to trade in each of the periods t = 0, 1, · · · , T . The buyer purchases some quantity of

good in each period from the seller. They share a common discount factor δ ∈ (0, 1). The

relationship can have either a finite horizon (T <∞) or an infinite horizon (T =∞).

3.1 Fundamentals

For a one-period trade of quantity q, the buyer derives a surplus S(θ, q), and the seller incurs

a cost C(q), where θ ∈ [θ, θ] encodes the buyer’s valuation of the good. Denote the type

space [θ, θ] as Θ, which is a bounded subset of R+. The seller has a commonly known prior

of θ, which is an atomless distribution F (with density f). The cost function C(q) is smooth,

increasing, and convex:9 C(0) = 0, and C ′, C ′′, C ′′′ ≥ 0.10 For simplicity, take S(θ, q) = θq.

For a payment p, buyer and seller obtain quasilinear utilities θq − p and p− C(q).

Furthermore, suppose C satisfies the following bounded curvature condition:

Condition 1 (bounded curvature). qC ′′(q)/C ′(q) ∈ (r, R) for some R > r > 0.

For example, a power function C = xα for α ≥ 2 satisfies the above curvature condition

and other conditions imposed on the cost function.

9Back to the SaaS example, the marginal cost in providing the base service free version is typically closeto zero, while the premium versions cost more in customer support and quality maintenance.

10This is a standard set of assumptions in incentive theory, see e.g., Laffont and Tirole (1986).

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3.2 Contracting Game

The seller and the buyer play a contracting game to reach temporary agreements. An

agreement is a long-term contract selected by the buyer. Each long-term contract at time t

specifies a trade proposal starting in period t, which is an infinite array of payments and

non-durable outputs (pts, qts)s≥t.11 The continuation of a period-t agreement (pts, qts)s≥tis (pts, qts)s≥t+1.

The initial agreement, at t = 0, is no trade: q ≡ 0 and p ≡ 0. The buyer privately

observes his type θ before t = 0.

In each period, the following events occur in sequence. First, the seller offers a menu of

contracts to the buyer, including the continuation of the recent agreement (reached in the

previous period). Second, the buyer chooses a contract, which becomes the new agreement.

Lastly, a trade occurs according to the new agreement, and the seller updates her posterior

belief of the buyer’s type accordingly.

As any buyer type may be mixing, for a generic contract, generally there is a mass of

types (subset of the total distribution F ) choosing it. We will refer to the notion of mass

later in the discussion.

3.3 Assessment

An assessment 〈σS, σB, µ〉 consists of a strategy profile and a belief system. Denoted by

ht, the public history in period t records the proposed menu of contracts and the selected

contracts in the periods before t. A strategy of the seller, σS, is a complete contingent plan

of the menu of contracts to offer given a public history. In particular, each trade proposal

in period t is contingent on ht. A strategy of the buyer, σB, is a complete contingent plan

of which contract to select given the public history and the current menu. A belief system

µ maps history ht to a posterior belief, i.e., a probability distribution over Θ.12

Let Fht(θ) = F (θ|ht) denote the seller’s beliefs about the buyer’s valuation type at history

ht. I use µ(ht) as shorthand notation for such distribution. Sometimes, when referring to a

particular history, I will simplify the notation as Ft = Fht . Define the set of types consistent

with history ht by htB. Let us say a history is non-degenerate if the posterior belief at such

history is non-degenerate.

The strategy profile of an assessment (not necessarily an equilibrium) implies a trade

11This notion of contracts does not allow signaling through general messages. Such choice is for exposi-tional purpose. Section 11 demonstrates that allowing contracts with identical trade proposal but differentmessages does not change the result.

12The richness of the contractual space allows cheap talk, e.g., communication through payoff-irrelevantinformation.

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profile (pσs , qσs )s≥t at each history ht that will take place if both players follow the respective

strategy. This particularly takes into account of the renegotiation over the trade proposal

specified in any accepted agreement. More precisely, if we denote the trade proposal accepted

(according to the assessment) in period t as (pts, qts)s≥t, then the corresponding trade profile

satisfies pσs = pss and qσs = qss for each s ≥ 0. For period t, I will call qtt as quantity today.

Because of potential mixing, qσs (θ|hs) (and pσs (θ|hs)) may depend on hs (the past contracts

that θ has selected); I will simply write qs(θ) (and ps(θ)) unless there is a need to clarify.

The seller and the buyer derive quasilinear utilities from such (randomized) trade profile as

V +t ≡ E[N(δ, T − t)

T∑s=t

δs−t(pσs − C(qσs ))|ht, σ], and

U+t ≡ E[N(δ, T − t)

T∑s=t

δs−t(S(θ, qσs )− pσs )|ht, σ, θ].

In the above expression, the normalization factor is N(δ, s) = (1− δ)/(1− δs+1).

To study incentives, it is helpful to consider reduced-form outcome variables, forward

quantity and payment of a trade profile, i.e., the present values of future quantities and

payments, as:

q+t (θ) = N(δ, T − t)

T∑s=t

δs−tqs(θ), and p+t (θ) = N(δ, T − t)

T∑s=t

δs−tps(θ).

I use an analogous shorthand notation for the forward quantity of a recent agreement and

for the discounted sum of the seller’s cost

qt−1,+t (θ) = N(δ, T − t)

T∑s=t

δs−tqt−1s (θ), and c+

t (qss) = N(δ, T − t)T∑s=t

δs−tC(qs).

3.4 Solution Concepts

In this paper, the minimum requirement of an assessment is Perfect Bayesian Equilibria

(henceforth PBE) of the contracting game.

Definition 1. A Perfect Bayesian Equilibrium is an assessment 〈σ∗S, σ∗B, µ∗〉 such that for

each ht:

1. Given belief µ∗t (ht), σ∗S(ht) is sequentially rational given σ∗B.

2. Given σ∗S(ht), σ∗B(θ, ht) is sequentially rational for each θ ∈ htB.

3. µ∗(ht) is calculated by Bayes rule whenever possible (htB has non-zero measure).

4. The beliefs following a selection of outside option satisfies Condition 3.

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Denote the implied trade profile of a PBE by (p∗s, q∗s)s and associated forward quantity

by q+,∗s . In the PBE definition above, players are sequentially rational in the sense that at

each history one is to move, one maximizes his/her expected utility given beliefs at that

history. The belief off-path is also pinned down by Bayes’ rule. In particular, the contin-

uation play after the seller’s deviation must be a PBE of the corresponding continuation

game. This extra criterion of belief consistency, compared to weak PBE, is essential for my

characterization.

Remark 1. My choice of belief updating rule is consistent with Strulovici (2017) but different

from Doval and Skreta (2019a). The latter assumes that the principal does not update her

beliefs about the agent following a deviation by the principal.

Note that the belief following an observed buyer deviation is left unspecified. I impose the

following two conditions. The first is an extension of the “Never Dissuaded Once Convinced”

condition in Osborne and Rubinstein (1990, p. 96):

Condition 2 (NDOC). The seller views a buyer’s deviation as coming from types within

the posterior support.

The second regards belief after rejection (i.e. sticking with the previous contract):

Condition 3 (No update following rejection). If the equilibrium specifies that no type within

the current support selects the previous agreement, then the seller does not update belief if

she observes the buyer chose the continuation of the recent agreement.

Section 7 shows that there is no loss in generality for the seller to offer equilibrium trade

profiles only, in which case each option in the menu is selected with a positive probability

(except the outside option at the very beginning of the relationship). In such case, the above

assumption only restricts belief following a selection of “no-trade” in period 0.

Finally, the sequential rationality for the finite-horizon game particularly implies that an

equilibrium shall be characterized in backward induction. This essential observation leads

to our inductive definition of a seller-optimal equilibrium of horizon-T game:

Definition 2. A seller-optimal equilibrium of a horizon-T game is a PBE that maximizes

profit such that in period 1, the continuation play maximizes expected profit among all con-

tinuation PBE of the horizon-(T − 1) game.

The above defines profit maximization in a recursive (i.e., backward induction) manner.

Section 8 establishes that any PBE is outcome-equivalent to a seller-optimal equilibrium.

Therefore, to study the on-path properties, it suffices to characterize the seller-optimal PBE.

Based on this observation, from now on, an “equilibrium” refers to a seller-optimal PBE,

while “PBE” is reserved for general PBE.

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3.5 Static Benchmarks

Let us introduce a few allocation rules that will recur in the discussion. The first-best output

level q fb(θ) defined by:

C ′(q fb(θ)) = θ.

Denote the second-best output level conditional on posterior beliefG by q sb(θ|G), and call the

associated profit as commitment profit. For general type space (or the support of posterior

distribution), one can define the inverse of the mapping of q sb(θ|G), denoted by θ(q|G), as

follows.

θ(q|G) ≡ inf

θ

∣∣∣∣∣∫ θ

θ

(v − C ′(q))dG(v)−∫ θ

θ

(1−G(v))dv ≥ 0,∀θ ∈ [θ, θ]

.

To perform the inverse operation, q sb(θ,G) = supq|θ(q|G) ≤ θ. Effectively, θ(q|G) works

as Myerson ironing to produce monotone control (quantity): the weak inequality ensures

that bunching any interval of types with lower boundary as θ(q|G) gives a positive virtual

surplus at quantity q, or equivalently, the average virtual value over such range is above the

marginal cost. This result particularly characterizes the allocation in period T . It is easy to

see that q sb(θ,G) ≤ q fb(θ) as θ(q|G) ≥ C ′(q).

When the posterior belief is the prior truncated above at θ, and the associated conditional

virtual value v(θ|θ), is well-defined, i.e., v(θ|θ) ≡ θ − [F (θ) − F (θ)]/f(θ),13 and increasing,

the conditional second-best output level q sb(θ|θ) can be defined by

C ′(q sb(θ)) = v(θ|θ),

More generally, when such virtual value is well-defined (e.g. when posterior distribution has

density and continuum support), one can replace v(θ|θ) in the above equality by its ironed

counterpart.

Example 1 embeds a canonical Mussa-Rosen setup into a two-repetition buyer-seller

game. I present the intertemporal allocations and welfare results of a few candidates of

seller-optimal equilibrium in the two-period renegotiation game. These candidates satisfy

the properties of Theorem 1 and are sequentially rational at each on-path history except

perhaps for the period-0 seller. I also include the commitment solution as a reference.

Example 1. Consider a two-period (T = 1) linear-quadratic setup, where C(q) = q2/2 and

13Recall that the unconditional virtual value is v(θ) ≡ θ − [1− F (θ)]/f(θ).

11

S(θ, q) = θq, and θ ∼ U [1, 2].14 The bounded curvature holds: One can take the curvature

bounds R, r to be arbitrary close to 1. Let us assume for the moment (which will be made

clear in Section 9) that each interval [θl, θr] pooled in period 0 is assigned a pooling quantity

q0 = θl + θr − 2. In any candidate equilibrium, each type gets fully separated by period 1 as

it is the end of the game, and assigned q1 = 2θ − θr for type θ ∈ [θl, θr].

Figure 1 compares the trade terms under a candidate equilibrium (“3-pooling”) that pools

three intervals of low types and screens the remaining high types in period 0, and those under

a candidate equilibrium (“screening”) that screens all types in period 0. The left panel plots

the period-0 and period-1 quantities for different types against the first-best and (ex-ante)

second-best allocation benchmarks: the black solid lines for the 3-pooling candidate and the

green and blue dashed lines for period-0 and period-1 assignment of the screening candidate.

The right panel plots the total price for total quantities over the two periods: black curve

for 3-pooling, and blue for the screening candidate. The total price is lower under 3-pooling

for the following reasons. Types receiving pooling contracts have lower total quantities. For

a fixed total quantity, 3-pooling assigns it to type(s) higher than those who get the same

assignment under screening. Hence, 3-pooling charges a higher price.

valuation

quantity

first-best

period-0 quant.

period-1 quant.

second-best

total quantity

total price

3-pooling

screening

Figure 1: Quantities and average prices when there are three pooling intervals.

The following table records the utilities of the seller and the buyer under different candi-

date equilibria alongside the performance of the commitment (second-best) solution. These

are profit-maximizing among the candidate equilibria exhibiting the same trait in period-0

separation: “1-pooling”, “∞-pooling”, and “screening” refer to the traits where the candi-

date equilibrium induces one pooling interval, arbitrarily many pooling intervals, and full

separation in period 0 respectively. The commitment solution has the lowest ex-ante welfare

and highest ex-ante profit while the screening candidate has the opposite. I normalize these

extreme values of welfare and profit to be 0 and 1 to show the comparison.

14The model is isomorphic to the regulatory framework of Baron and Besanko (1987).

12

Table 1: Welfare performance of different constrained-optimal mechanisms.

Limited commitment

1-pooling ∞-pooling Screening Commitment

Normlized welfare .702 .732 1 0Normalized profit .444 .462 0 1Average quantity

period 0 1 1 1 1period 1 1.278 1.268 1.5 1

Model specification: δ → 1, linear-quadratic utilities, F = U [1, 2].

This simple example already illustrates the main tradeoff highlighted in this paper: pool-

ing results in lower profit in the current period because trade terms are not responsive to

buyer valuation; it reduces information revelation and hence reduces information rent left

to the buyer in the future period(s). Therefore, to characterize the equilibrium by studying

a relevant mechanism design problem, it is inevitable to take into account the gradual type

separation. Moreover, compared to the screening case, employing 1-pooling of the lowest

types already recovers a big fraction of the loss in profit. Increasing the number of allowed

pooling intervals further recovers profit. A final observation (made clear later in the dis-

cussion) is that the sorting condition yields separation in equilibrium. I will show that this

interval partition pattern of information revelation is indeed optimal, and confirm the con-

jecture in Baron and Besanko (1987) that seller-optimality requires infinitely many pooling

intervals.

3.6 Interpretation

The aforementioned is a Mussa–Rosen paradigm: the buyer has private information of his

type, and the seller offers the menu of contracts in each period. More generally, the trade

can deliver services, and the variable q represents quality. An example of such a paradigm

is a licensing agreement that specifies a quality or quantity level for a short pilot phase

and another level for the future mature phase. Another example is the labor contract in

Grossman and Hart (1983) and Dewatripont (1989).

An alternative is a Baron–Myerson paradigm, where the principal–agent relationship

can represent one between a consumer and a supplier whose cost parameter is his private

information. The baseline payoff functions are S(q) and C(θ, q) = θq. Main sorting and

regularity conditions are S(q) being smooth, concave, and F (θ)/f(θ) increasing in θ. One

example is the supply relationship, where a manufacturer and its supplier(s) sign a long-

term agreement of procuring intermediate goods. Others include procurement, financial

13

restructuring, and regulation of natural monopoly, all of which involve tasks over a relatively

long horizon.

4 Formulation of the Seller’s Problem

To characterize a seller-optimal equilibrium, I follow the tradition of formulating a mecha-

nism design problem to maximize the ex-ante expected profit subject to implementability

constraints, which is known as the seller’s problem. As local incentive constraints are

binding in equilibrium, we will mostly ignore the payment dimension of the problem as it

is determined by the buyer’s utility, or equivalently, the quantity allocation. Just as in the

contracting game, such a problem is inherently dynamic, because belief updating changes the

mechanism designer’s (in this case, the seller’s) perspective. The seller needs to decide what

information to elicit at each history, i.e., what posterior distributions to induce, and what

quantity allocation to use to elicit such information. The former determines the process

of type separation, i.e., a martingale of posterior beliefs over time (describing how types

separate intertemporally), and the latter determines the buyer’s incentives. They impose

constraints on the quantity allocation: the per-period quantity should be measurable with

respect to the type separation, i.e., the mass choosing the same contract in some period t

share the same qt; the reduced-form quantity outcome, i.e., forward quantity, should respect

the buyer’s incentive constraints.

I break down the characterization of the optimal mechanism into two steps. First, I

characterize the optimal quantity allocation given the process of type separation. Second, I

plug in that solution to derive the optimal process of type separation.

For the validity of this characterization strategy, let us see the following diagram, which

summarizes the feasibility dependence between the programs of quantity allocation and type

separation.

separation today separation in the future

quantity today quantities in the future

Figure 2: Dependence Diagram

In the dependence diagram, the tail of an arrow depends on the head of it. For example,

quantity assignment today should be measurable with respect to separation induced today:

the mass of types pooled together should obtain the same trade term. Quantity assignment

today depends on future quantities as the implementability constraints involve the latter.

The essence is that incentives only depend on forward quantities, and therefore, future

14

incentive constraints do not depend on today’s quantity. One way to see this is via an

implicit state variable in the above diagram – the outside option (from the continuation of

the recent agreement). Future incentives to renegotiate only depend on such continuation

instead of the quantity today.

A trade profile is implementable if it is the equilibrium allocation for some sequentially

rational mechanism. There are two ingredients of such implementability: (i) the feasibility of

the allocation rule, and (ii) the sequential rationality of the mechanism. To be more precise,

(i) comes from the sequential rationality of the buyer, and (ii) from that of the future selves

of the seller.

The observation in §3.4 that sequential rationality gives rise to a recursive characteriza-

tion of the seller-optimal equilibrium carries over to that of the seller’s problem. It requires

a specification of the mechanism at any possible history, where the seller can possibly induce

a belief support htB with arbitrary shape. In the following formulation and the subsequent

analysis, I will exploit such backward induction and only describe for one generic period t

for some generic support htB.

4.1 Implementable Allocations

The following arguments provide a reduction of the description of the seller’s problem and

the constraint sets. A priori the posterior may not have a convex support. I extend the

incentive problem (and its constraints) to the convex hull of the posterior support, reducing

the incentive problem to the classical framework where the type space is convex and making it

easier to compare the incentive problems at different histories.15 Building on such reduction,

the mechanism design problem in period t will be isomorphic to the period-0 problem, albeit

a shortened horizon, and a belief Fht , (extended) type space ht

B = [θt, θt] (convex hull of htB),

outside option different from the prior F , Θ, and no trade.

With such reduction, I first characterize the set of feasible allocations given a process of

type separation (a sequence of htB). This feasible set is constrained by the usual buyer’s par-

ticipation and incentive compatibility constraints, written in forward quantities and defined

for posterior Fht and its support htB. The allocations can be randomized. In the following

discussion, q+t (θ) refers to q+

t (θ|ht) for a specific history ht.16 In the following constraints,

the subscript t refers to the dependence on ht. I will suppress such subscript when there is

no confusion.

15The formal proof adapts the arguments in Skreta (2006).16Presumbly, q+t (θ) can depend on the whole path of type separations. In the above definition, we only

consider a representation that depends only on the type separations in the past. In supplemental material,I show that such representation does not incur loss of generality.

15

Participation constraints (PCt). Any buyer type θ is willing to participate in the mech-

anism game in period t: U+t (θ) ≥ θqt−1,+

t (θ)− pt−1,+t (θ) for any θ ∈ htB.

Note that the above ensures the buyer’s participation not only in period 0 but at all on-path

histories, i.e., he prefers the equilibrium trade profile to the outside option, i.e., the contin-

uation of an old agreement.

Incentive compatibility (ICt). Any buyer type θ finds it optimal to follow the suggested

sequence of contracts (i.e. to comply with the expectation on his own type): U+t (θ) =

maxθ U+t (θ) + (θ − θ)q+

t (θ) for any θ ∈ htB.

Apart from the above (PC) and (IC), the seller needs to ensure that a type θ has no incentive

to pretend to be another type and stick with certain agreement of the mimicking type (in

its history of renegotiations). I have taken a relaxed approach in the above formulation.

I first ignore the set of these incentive constraints, and ex post verify an implementation

of the optimal mechanism that satisfies these constraints. I also only consider (PC) of the

lowest type within the support and furthermore, characterize quantity alone. I will also

demonstrate the implications of these ignored constraints on the agreement quantities of the

implementation. For instance, optimality and (PC) of other types imply that renegotiation

only goes upward in equilibrium (Proposition 4). Moreover, I only impose local (IC). For

this purpose, it is important that the validity of the relaxed approach in characterizing the

optimum extends to this dynamic contracting environment. More precisely, it is not ben-

eficial to structure today’s agreement in a way that reduces slackness of future incentive

constraints. See discussion in Section 7.

An immediate implication of (IC) is that q+0 (θ) is monotonically increasing in θ. Moreover,

within htB, q+t (θ) is also increasing. By linearity of the buyer’s utility, the local (IC) binds by

a standard argument.17 As renegotiation only goes upward, (PC) of the lowest type binds

under optimality. With this, we can further get rid of the payment dimension: it is fully

specified by the incentives and, consequently, the buyer’s payoff is given by envelope formula

(Milgrom and Segal 2002):

U+0 (θ) = U+

0 (θ) +

∫ θ

θ

q+0 (θ)dθ.

For future reference, I will refer the following monotonicity constraint as local (IC).

Monotonicity (Mt). q+0 (θ) increases in θ; q+

t (θ) increases in θ for θ ∈ htB.

17The argument extends so long as the buyer’s utility is single-crossing in his type and some discountedsum of (transformed) quantities, e.g., S = θu(q) for some increasing u. More general buyer preferencescannot be summarized by forward quantity alone.

16

The above constraints entail the sequential rationality of the buyer. Now, let’s state the

seller quantity allocation program. To express her objective, the expected profit at ht, we

use the binding local (IC) constraints and the binding (PC) of the lowest type:∫ θtθt

(θq+t (θ)−

c+t (qss≥t) − U+

t (θt))dFht(θ) −∫ θtθtq+t (θ)(1 − Fht(θ))dθ, where U+

t (θt), via binding (PC),

coincides with the promised utility offered by agreement in period t − 1. As renegotiation

goes upward in equilibrium, neglecting U(θt) in the objective incurs no cost. Note q+t =

N(δ, T − t)qt + (1−N(δ, T − t))q+t+1. Conditional on the type separation, the future seller’s

problem has no dependence on quantity qt today. Hence, in period t, the seller takes future

quantities qs for s > t as given, and maximizes the following simplified seller’s objective

Π(qt, ht) =

∫ θt

θt(θq+

t (θ)− c+t (qss≥t))dFht(θ)−

∫ θt

θtq+t (θ)(1− Fht(θ))dθ,

over qt, which is a function of θ. One can further simplify the objective as

Π(qt, ht) =

∫ θt

θt(θqt(θ)− C(qt(θ)))dFht(θ)−

∫ θt

θtqt(θ)(1− Fht(θ))dθ.

Therefore, the quantity allocation program of ht is to maximize Π(qt, ht) over qt (on the

relevant ht+1) subject to (M) for given q+t+1. The latter q+

t+1’s are pinned down by the

sequential rationality of the future selves of the seller:

Sequential rationality constraints (SRCt): At any future hs (s = T, · · · , t+1), the seller

chooses a quantity allocation rule qs for a pooling mass hs+1 (associated with [θs, θs] = hs+1B )

that maximizes Π(qs, hs) subject to (PCs), (ICs), and (SRCs).

In the above definition of (SRC), recall that the forward quantity of a trade profile q+0 (θ)

is the present value of quantities qs that θ is assigned following a particular sequence of

equilibrium pooling. Therefore, I formulate the optimal quantity allocation program

as the sequence of allocations that recursively maximizes (posterior) expected profit among

the feasible set. Such relaxed program is a one-stage problem:

Optimal quantity allocation program of ht: maximizes Π(qt, ht) over qt such that qt is

constant on each pooling mass ht+1B and satisfies (Mt), and q+

t+1 is pinned down by (SRCt).

This formulation strengthens the observation made in the beginning of this section (§4.1):

under optimality, the seller’s problems at different history only differ by different posterior

belief (along with induced separation) and length of remaining horizon. Equivalently, the

solution (in trade quantities qss≥t on types htB) of history ht is identical to a solution

(qss≥0) of the period-0 problem of T +1− t horizon over type space ht

B restricting onto htB.

17

Such equivalence strengthens the recursive formulation of the seller’s problem by allowing a

concrete solution method via backward induction.

Remark 2. One way to interpret the above recursive formulation of the relaxed program is:

the opportunity to renegotiate implies a sequential rationality that once the information has

been revealed, the seller cannot further commit distortion relevant to such information.

Here are three examples of a PBE-implementable allocation for a contracting game start-

ing with belief G. The payment rule is determined using the envelope formula.

Example (efficient). qt(θ) = q fb(θ) for all t and induces full revelation in period 0.

Example (instant-screening). q0(θ) = q sb(θ,G), qt(θ) = q fb(θ) for t ≥ 1, and induces full

revelation of types in period 0.

In view of (SRC), one can reinterpret the instant-screening contract of horizon T as a

second-best allocation followed by an efficient contract of horizon T − 1. Particularly, under

full information (a result of full revelation), the efficient contract is optimal. In the same

spirit, one can construct a seller-optimal mechanism of horizon T from that of horizon T − 1

and belief as G:

Example (full-pooling). q0(θ) = q fb(θ) and qt(θ) = qt−1(θ) for t ≥ 1, where (qs)s is a seller-

optimal mechanism of horizon T − 1 under the belief G.

The seller-optimal mechanism, by definition, induces a profit-maximizing allocation among

those PBE-implementable. When choosing the process of type separation, the seller accounts

for its effect on profit, i.e., she works with Π instead of Π. To highlight the dependence on

type separation, denote Π(ht+1|ht) as Π(qt, ht) at the optimal qt subject to the induced poste-

rior belief as ht+1 (a martingale). (Future qt+s for s > 0 are specified below). Therefore, one

writes the seller’s problem as choosing a process of type separation in a backward induction:

Optimal type separation program of ht: maximizes Π(ht+1|ht) over ht+1 such that

for s > 0, qt+s is pinned down in the optimal quantity allocation program of ht+s, and for

s > 1, ht+s is the solution of optimal type separation program of ht+s−1.

In such formulation, the seller is allowed to involve the buyer in playing mixed strategies.

She is also allowed to mix her play. The earlier analysis of this section fully characterizes

the optimal quantity allocation. The rest of the paper focuses on characterizing the optimal

process of type separation.

5 Main Results

Our first objective is to find an assessment that is a profit-maximizing PBE. In order to solve

such a mechanism design problem, we search for an allocation that maximizes expected profit

18

among all allocations that are implemented by PBE assessments. At history ht, the seller’s

goal is to maximize her expected profit given her posterior belief, the strategy of the buyer,

and the current agreement. The above notion corresponds to the seller-optimal equilibrium

in which the seller recursively chooses her best PBE continuation in each history.

The problem is effectively a design of the process of type separation. When the mecha-

nism designer behaves sequentially rationally, information revelation is difficult because the

mechanism designer, the seller in this case, cannot commit not to use it and exploit the

buyer in the future. Bester and Strausz (2001) showed a remarkable result that tackles such

complexity: when the principal faces one agent whose type space is finite, she can, without

loss of generality, restrict attention to mechanisms where the message space has the same

cardinality as the type space. Unfortunately, this result does not apply to our problem

because the type space can be a continuum. Another difficulty in applying their result to

the finite-type case is that one has to check the binding incentive compatibility constraints.

Under limited commitment, constraints may be binding “upwards” and “downwards”.18

Nonetheless, restricting to seller-optimal equilibria proves useful in closing the gap in both

dimensions of technical and conceptual difficulties. The optimality induces a simple form

of equilibrium posteriors, which is equivalent to partitioning the type space using cutoffs.19

In this sense, direct mechanism suffices. Moreover, incentive compatibility constraints only

bind locally at optimum when the type space is a continuum.

The first equilibrium property is that the seller separates types in a monotonic manner.

Definition 3. A process of type separation satisfies refining property if the belief in any

non-terminal period has a convex support.

The following two plots demonstrate the type separation and quantity allocation of a

long horizon, i.e., there are more than 2 periods.20 The left panel shows the type separation

and pooling quantities for the first two periods, where the black step function gives the

allocation for period 0, and the green step function for period 1. The right panel shows

the type separation and quantity allocations for three specific types. They start off pooled

together, then the relatively high type (blue) gets separated, and later on, the other two

types (red and green) are separated from one another. With a sufficiently long horizon, for

each type, its equilibrium quantity assignment converges to first-best level.

18Moreover, the dynamic programming in the finite-type case lacks tractability for a long horizon.19The partitional structure gives the mechanism-selection game a revelation principle for free: the direct

mechanism involves treating report of any type within an element of the partition as choosing the strategy forsuch element. Without the partitional structure, there is no known result of an analog of revelation principle.Doval and Skreta (2019a) include a revelation principle for short-term contracts and continuum-of-type caseif imposing monotonic expectational differences on the utility function.

20A complementary view is provided in supplemental material in terms of posterior distribution.

19

θ

qt

q0

q1

t

qtq fb

q fb

q fb

θ1 > θ2 > θ3

We provide a sketch of the proof for the refining property (under seller optimality) in

Section 6. The proof uses the buyer’s incentive compatibility and the seller’s revealed pref-

erence. Intuitively, this property follows from (i) giving a relatively higher expectation of

the current payoff (by sorting condition: “total surplus has increasing differences in θ and

q”) and (ii) inducing a continuation game with posterior distribution more aligned with the

prior. An alternative intuition is based on revealed preference in information revelation: if

the seller finds it beneficial to pool two disjoint intervals, then the gain in rent extraction

due to pooling outweighs the respective loss in efficiency. Pooling them altogether with the

types setting them apart inflates the loss by a smaller scale relative to the boost in gain.

The second equilibrium property is that a generic buyer type plays pure strategy, i.e.,

he selects a certain contract with certainty. Section 6.4.2 provides a sketch of a purifica-

tion argument: for any candidate equilibrium that involves buyer’s mixing, there is a way

to weakly increase the lower types’ intertemporal distortionary levels, while reducing the

inefficiency of pooling. This means that a strategy profile involving buyer’s mixing cannot

be seller-optimal. The intuition is that to maintain the overall low types’ distortion level,

one needs to control the separation speed of these low types from some intermediate types.

With continuous type distribution, one can adjust such speed by continuously varying the

pooling boundary. These arguments are not true in the finite-type case, especially for the

boundary type of a convex pooling.21,22

Finally, observe that instant screening and full pooling are two special cases of type

separations that satisfy both properties. However, neither is compatible with the seller

optimality. The key insight is that the sorting condition provides an incentive to separate

21To benefit from delay of separation, one needs to pool some types in early periods. In the finite-typecase, to flexibly adjust the rate of separation, one needs to partially separate some types.

22The earliest discussion is in Hart and Tirole (1988), where the authors contrast their results, allowingthe buyer to randomize, with those of Dewatripont (1989) that restricts the buyer to play a pure strategy.For binary-type, the former generates multiple-period renegotiation and the latter forces the renegotiationto halt at the very first period. See also Laffont and Tirole (1990) §7 for a binary-type two-period discussion.More recently, Maestri (2017) studies contract renegotiation for binary types, and proves that the selleroptimality requires semi-separation of the high type. See also Strulovici (2017).

20

and the bounded curvature condition provides an incentive to delay (a full) separation (of

any type). The following theorem summarizes the above main results.

Theorem 1. Any PBE satisfies refining property, pure strategy property, and involves count-

able interval partition of the posterior support. The buyer type is revealed in period T .

When the buyer plays a pure strategy, the posterior supports following different selections

of contracts form a partition of the type space. Equivalently, the collection of sets of buyer

types htB for all on-path histories ht at time t forms a partition of Θ. In such case, the

refining property23 implies that the posterior supports followed from different buyer choice

of contract are ordered in the strong set order.

Definition 4. A set of closed intervals are ordered if for any two of such intervals, I1 and

I2, max Ii ≤ min Ij for some i, j = 1, 2.

As mentioned above, the two properties in Lemma 1 and 2 imply

Corollary 1. An optimal mechanism uses type separation in which the posterior supports are

ordered, i.e., it induces the buyer’s pure strategy and type separation as an interval partition.

When posterior supports satisfy the strong set order, it is easy to describe the incentive

constraints and reduce the dimensionality of the optimization (mechanism design) problem.

To specify the equilibrium requires a characterization of the cutoffs, i.e., boundaries of inter-

vals. Generally, the solution of the seller’s problem may involve infinite binding monotonicity

constraints, which greatly complicates the cutoff characterization. With some extra regu-

larity condition, one can fully characterize the equilibria: Imposing monotone hazard rates:

[1−F (θ)]/f(θ) is monotonically decreasing in θ yields slack (M) in equilibrium. Under such

regularity condition, the separation of the seller-optimal PBE can be characterized iteratively

by the first-order condition of a M-relaxed program where (M) is ignored. The result is

a simple characterization via a first-order approach and dynamic programming (Theorem 2

in Section 9).

The proof of the main result (Theorem 1) takes the following steps. I first prove the

results for all seller-optimal equilibria, and then show that any PBE is outcome-equivalent

to some seller-optimal PBE. To see the second step, consider there is a unique seller-optimal

PBE outcome. One can show that there exists an equilibrium implementing such outcome

in all continuation equilibria. In other words, given the seller’s period-0 menu of contracts,

there is no other continuation equilibrium. The same argument implies that at any history,

the seller is able to induce the seller-optimal continuation equilibrium. This last observation

23Also known as partitional equilibrium in Laffont and Tirole (1988).

21

implies that in a finite horizon, all PBE are outcome-equivalent: they share the same

trade profile.

Let us talk about a roadmap to prove the results for seller-optimal equilibria.

countableintervals

separation poolingcountableintervals

intervalpartition

T-1:

T:

induction

(c)

(b)(a)

(c)

bootstrap

(d)

(d)

Figure 3: Roadmap for proofs.

The proof builds on the recursive formulation and uses an inductive argument on the

horizon T . The logical order of the proof is provided in Figure 3, where the first row refers to

horizon T − 1 and the second to horizon T . Each block in the figure refers to an equilibrium

property of the respective horizon: partial separation, interval partition property, partial

pooling, and countable interval partition respectively. Each arrow represents the use of

former result in proving the latter. Downward arrows refer to the use of induction hypothesis.

For each induction step, (a) I show the seller’s incentives to separate types, i.e., Example

(full-pooling) is not consistent with seller optimality (§6.1). Using such result, I show the

interval partition (§6.4). Independently, I show the seller’s incentive to pool types, i.e.,

Example (instant-screening) is not consistent with seller optimality (§6.2). Lastly, combining

the result of interval partition and bootstrapping (made clear in the later discussion) the

pooling argument gives the countable interval partition (§6.3). Finally, with finite-horizon,

backward induction ensures the existence of an equilibrium.

Section 6 provides the key intuitions and the proofs for the case T = 1.24

6 Equilibrium Properties

Let us start with some basic incentive implications of a seller-optimal equilibrium implied by

the PBE-implementation. A first simple observation is that at any time and onwards, the

concavity of the social surplus forces the equilibrium trade profile to be deterministic. For

24Appendix A.2 provides the remaining proofs. Technical details of the proofs of Lemma 1 and 2 aredelegated to the same Appendix.

22

simple notation and exposition, let us assume for the moment the seller plays pure: I will

work with deterministic menu and confirm a purification result at the end of this section.

Next, the seller’s sequential rationality implies that following a full revelation, the equilib-

rium continues with first-best quantities for the remaining periods. Under full information,

no matter what the current agreement is, a renegotiation towards first-best quantity exploits

the residual efficiency and creates a Pareto improvement. Note that the seller is not able to

extract full rent of such first-best trade: to incentivize truthful reports, she needs to provide

incentives based on (equilibrium) forward quantities.

With the above pre-requisites, let us follow the diagram in Figure 3 to show the necessary

properties of a seller-optimal equilibrium.

6.1 Incentives to Separate

Why not pool all the time? No discrimination at all implies substantial allocation inefficiency.

I will show that the seller does not fully pool in period 0. Consider a static scenario. Under

full pooling, the seller has to offer a contract that satisfies the participation constraints of

the least efficient type, which means that the seller can get a surplus of at most θ q fb(θ) −C(q fb(θ)), which is the same as her surplus in an efficient contract. The intuition is that

under the efficient contracts, the seller effectively sells off the relationship and makes the

buyer the residual claimant. To ensure such relationship “sale” (not to be confused with

sale or trade of good), she is made indifferent as if she is facing the worst type of buyer;

therefore, she obtains the full-information profit of trading with the lowest type of buyer.

Back to the repeated game. When T = 1, following full pooling in period 0, the seller

screens in period 1. Consider a deviation of instant-screening in period 0. The former

yields a pooling profit in period 0, and a commitment profit in period 1; the latter yields

a commitment profit in period 0, and an efficient profit in period 1 (which equals to the

pooling profit by the above profit-equivalence result). As the commitment profit is strictly

higher and δ < 1, the deviation is profitable. Appendix A.2 extends this welfare analysis

of delayed separation to general T < ∞ using inductive argument and establishes that the

seller partially separates types in period 0. This completes the proof of (a) in Figure 3.

Any equilibrium involves some (partial) separation. Suppose to the contrary that the

separation involves no quantity variation, i.e., q+0 (θ) is the same for all θ. Clearly, this is not

the case when types are fully separated in period 0: the q sb in period 0 and q fb in future

periods are weakly and strictly increasing respective. When some types are pooled initially,

the induction hypothesis implies some partial pooling until the second-to-last period. For

types pooled in such period, the earlier supposition implies they share the same forward

quantity q+T−1(θ), or equivalently, the same qT (θ). This contradicts with full separation in

23

the last period. To resolve this contradiction with equilibrium assumption:

Proposition 1. An equilibrium has type separation and variation in the forward quantity,

i.e., for any ht, there exist types θ, θ′ ∈ htB such that q+t (θ|ht) 6= q+

t (θ′|ht).

6.2 Incentives to Pool

Why not screen all at once? Earlier we established that following a full revelation, future

trades stand at the Pareto frontier. For δ 0, this gives too much rent to the efficient types.

Following Figure 3, we consider an interval posterior support for the following analysis to

establish that the seller optimality requires some pooling.

There is a clear-cut tradeoff of pooling: it reduces future information rents at the expense

of less responsive trade terms (prices and quantities) and hence reduced welfare and profit in

the current period. To prove some pooling is desirable, we will show that an instant-screening

mechanism, taken as the status quo, is suboptimal. The analysis encompasses cases whether

ironing is involved or not (left and right charts in Figure 4).

θ

q

θ θ

f.b.

s.b.

θ

q

θ θ

f.b.

s.b.

Figure 4: Gap between first-best and second-best allocations: two configurations.

Consider the following thought experiment: pooling the lower end initially and screening

these types in the next period (denote the size of pooling, in terms of cdf, by ∆) while

screening higher types right away. Such pooling decreases allocation efficiency by o(∆2) and

rent loss by O(∆2).25

For intuition, think of the linear-quadratic case where the loss is quadratic. From the

viewpoint of the seller, the buyer’s type is a random variable. The status-quo screening (in

period 0) reveals its realization while pooling keeps some uncertainty. The average loss for

types in the pooling mass under the uncertainty equals to the variance of the latter, which

25Note that this analysis provides a lower bound for the effect of pooling: the equilibrium under tail-pooling involves tail pooling for future periods, which further increases the seller’s surplus as of today.

24

is O(∆2). If the seller screened following the period-0 pooling, then on average, the earlier

pooling would allow for ∆/2 period-1 distortion for the types in the pooling interval: the

amount of distortion of type θ in the pooling interval is ∆/f(θ). Finally, aggregating such

effect over the pooling ∆ gives O(∆3) loss in period 0 and O(∆2) gain in period 1.

Analogously, for general convex costs, pooling the lower end reduces allocation efficiency

by at most Ω∆3 and rent loss by at least δω inf(1 − F (θ))∆2 (where the infimum is taken

over the pooling interval). When the pooling size is small, the latter dominates.

The above thought experiment describes the continuation of the case T = 1 but violates

the seller’s sequential rationality for T > 1. Appendix A.2 shows a similar construction yields

a valid profitable deviation for the seller: pool the same mass ∆ at the lower end initially and

continue with a seller-optimal continuation. The intuition is that for such continuation to

outperform the instant screening continuation, it must involve more future distortion. This

is because screening yields the highest allocation efficiency among all implementable mecha-

nisms in the next period (i.e., not considering the seller’s sequential rationality of tomorrow).

As the current-self observes more payoff externality of low types on high types, such devi-

ation yields a higher profit from the perspective of the current self than the hypothetical

equilibrium in the thought experiment.

The above argument particularly implies that at any non-degenerate history (except

period T ), the seller will not screen the lower end, establishing (b) in Figure 3.

Proposition 2. Any equilibrium involves some pooling. No equilibrium screens a neighbor-

hood of θ in a non-terminal period.

Indeed, no equilibrium screens a neighborhood of an interior type so long as pooling such

neighborhood does not violate (M).

6.3 Recursions

The pooling argument can be applied to an interval posterior in each period, thereby creating

a recursive pooling intertemporally. It only requires that the first-best quantity curve and the

posterior-based second-best quantity curve have a gap at the lower end, which holds true so

long as the type has not been fully revealed. We call such recursive pooling intertemporal

recursion. This applies to non-terminal periods.

The observation above Proposition 2 gives rise to another type of recursion for pooling.

The seller can bootstrap the desirability of lower-end pooling: the fact that she finds pooling

an interval of the lower end improving her surplus would imply that truncating such pooling

region, she finds pooling a new interval of the lower end in the truncated type space also

25

surplus-improving. Let us call this process bootstrapped recursion and the associated

pooling as bootstrapped pooling. Now, let us prove such recursion.

Suppose refining property and pure strategy hold in equilibrium. Proposition 2 says

the tail must be involved in some pooling and that the tail of the non-truncated region

(which is non-trivial by §6.1) must be pooled so long as the pooling satisfies (M). Parallel

to the argument of §6.2, a small-scale tail-pooling in the non-truncated region is a profitable

deviation: pooling at a small scale does not violate (M); it also eases (M) in the non-truncated

region. Such recursion implies countable pooling intervals. Otherwise, there is an interval

containing the highest type of the support. The derivation of Proposition 1 implies that some

separation within such interval gives a profitable deviation. Therefore, for an equilibrium

that satisfies the refining property and pure strategy, the posterior support is partitioned

into infinitely many intervals at each history, establishing (d) in Figure 3.

Proposition 3. Suppose an equilibrium satisfies the interval partition. Then the equilibrium

involves countable pooling intervals in each non-terminal period.

6.4 Interval Partition

Now, it remains to prove (c) in Figure 3 by elaborating the proof sketch laid out in Section

5. Using the recursive formulation of Section 4, here we prove an induction result for horizon

T , assuming that any seller-optimal equilibrium of horizon l (l = 0, · · · , T − 1) satisfies

the interval partition. The base case l = 0 is already demonstrated in Section 3. Section

6.4.1 shows that certain violations of refining property are not even incentive feasible. The

argument builds on the sequential rationality of the seller’s future selves, hence relies on PBE

implementability rather than seller optimality. Section 6.4.2 shows that buyer randomization

and (other) violations of refining property are suboptimal. I show that a seller’s deviation

that includes either non-convex support or buyer’s mixing decreases the profit.

Lemma 1. The optimal mechanism exhibits refining property.

To show the refining property, let us suppose to the contrary that in equilibrium, there

is some earliest period t <∞ that the seller pools θl and θr (and perhaps some other types

outside [θl, θr]) without the inside interval (θl, θr).

...θl θr

Figure 5: Violation of refining property

26

I break down the proof into two parts: (1) Show that the inside interval [θl, θr] involves

no inside pooling or cross pooling in equilibrium (§6.4.1); (2) Show that instantly screening

the inside interval is suboptimal (§6.4.2).

A remaining case where pooling the inside interval with other types is incorporated as a

case of buyer’s mixing, for which we are going to establish

Lemma 2. The optimal mechanism does not involve buyer’s mixing.

6.4.1 Refining Property (Pooling)

We show that if a type in the inside interval is pooled with some other type, then it induces

a contradiction.

For types in [θl, θr] that are pooled prior to t, q−t (θ) is the same. We will focus on two

critical types θl and θr. Denote their equilibrium quantities following the period-t pooling

contract by (qls)s≥t and (qrs)s≥t. The (IC) of types θ ∈ [θl, θr] imply the monotonicity that

q+0 (θ) is increasing in such domain, and so is q+

t (θ). (IC) of type θr implies that it weakly

prefers the equilibrium pooling than pretending to be a type θ−r . Using the envelope formula

for U(θr) further gives [θr− θl]q+0 (θl) ≥

∫ θrθlq+

0 (θ)dθ. Combining these two observations gives

q+t (θ) constant on [θl, θr]. Alternatively, this can be derived by a binding downward (IC) of

θr to masquerade as θl: such incentive constraint binds as the two types are adjacent in the

future posterior support.

Consider two pooling cases.

θl θr θl θrαl αr θ′

Figure 6: Pooling Cases

(1) The seller pools some types within [θl, θr]. With non-degenerate period-(t + 1) belief in

such pooling, Proposition 1 implies that the highest and lowest types of such pooling have

different forward quantities. This contradicts with q+t (θ) being constant on [θl, θr]. This

concludes that inside pooling is infeasible.

(2) The seller pools some type θ ∈ [θl, θr] with some other types θ′ /∈ [αl, αr], where [αl, αr]

denotes the convex hull of types pooled with θl and θr. First, consider θ′ > αr: it is the

closest to θr among such types. Then (IC) of θ′ implies that q+t (θ) = q+

t (θ) for θ ∈ [θr, θ′).

The (IC) of θr implies q+t (θ) = q+

t (θl). This contradicts with the following claim.

Claim 1. In any seller-optimal equilibrium, q+t is strictly increasing near the highest type of

a pooling interval.

27

For a finite-horizon, the above claim follows from the definition of q sb(θ,G), which implies

that q+t is strictly increasing on (αr − ε, αr).26

For θ′ < αl, one can use the fact that inside pooling is infeasible to claim that αl must

be the highest type of some pooling interval. Again, Claim 1 implies that q+t is strictly

increasing on (αl − ε, αl). An analogous analysis as above implies that (IC) of θ is violated.

6.4.2 Refining Property (Screening) and Pure Strategy

The inductive proof that the remaining non-monotonic patterns are suboptimal further uses

an induction on the type space. In particular, the seller-optimality implies that the lowest

type lies in a pooling interval that satisfies both refining and pure-strategy properties. The

same argument applies to a new lowest type after truncating the pooling interval of the

previous lowest type, and a transfinite induction establishes the proof. We first combine

redundant contracts: If two contracts yield the same quantity today and same posterior

tomorrow, then we merge the two contracts. After such operation, the following gives a

sketch of the proof.

By induction hypothesis on T , one only considers such violation in period 0. Suppose in

period 0, v < θ is the highest type such that all types below involve in some interval-partition

pooling. Consider a contract z selected by v with a positive probability in equilibrium.

Denote the posterior belief after a selection of contract z as F1, and the convex hull of its

support as [v, v]. By Proposition 2, seller-optimality implies that F1 is non-degenerate and

v > v: the interval-partition result of horizon (T − 1) implies partial pooling of horizon T .

Consistent with earlier plan, we consider two cases in period 0: (1) no inside interval of

[v, v] is screened, and (2) an inside interval of [v, v] is screened. Case (1) deals with buyer’s

randomization: if the inside interval is screened, then there is no scope of buyer’s mixing.

First, consider case (1). By the induction hypothesis, a strictly positive measure of

types within such interval selects another contract z with a positive probability. Denote the

induced posterior by F1. We focus on the case of non-degenerate F1; the degenerate case

is trivial as F is atomless. Denote the probability of each type θ ∈ [v, v] choosing either

contract as m(θ); conditional on that, the probability of choosing the first contract is β(θ),

and the second is 1− β(θ).

Step 1. Under buyer’s randomization, one can construct an implementable allocation that

yields a higher total price at each given total quantity, lower forward quantities, and a weakly

lower expected total cost.

26For the infinite-horizon case, the claim follows from the seller’s sequential rationality and the validityof bootstrapped pooling. Applying Proposition 1 locally to αr retains the claim.

28

One can use Bayes’ rule to express F1 and F1 in terms of β, m and F . The two contracts

specify the same forward quantity for types in the overlap of F1 and F1 supports.27 When

m = 1, one can replace z and z with some new contract(s) for which the total prices are

higher and forward quantities lower than those under F1. Moreover, the period-1 belief

following the deviation is a truncated prior. For general measurable m, one replaces z, z and

other contracts sharing non-trivial overlap in support in a similar manner.

Step 2. With lower forward quantities, the above construction gives a profitable deviation.

The welfare bound of the construction in step 1 uses an indirect comparison similar to

the proof of Proposition 2. In particular, by the seller’s revealed preference, the constructed

deviation (with its continuation equilibrium) yields a higher expected profit from the per-

spective of period-1 seller (whose belief is a truncated prior). The lowered forward quantities

imply that the period-0 seller also prefers the continuation equilibrium of the deviation to

the original continuation. As she also prefers the period-0 trade term of the deviation, the

seller finds the above construction a profitable deviation. This construction ignores the po-

tential violation of (M). Finally, one needs to ensure that even when the potential increased

distortion makes (M) more stringent, the seller can re-adjust the interval cutoff (between the

current and the lower interval) to benefit from the increase in distortion.

Now, consider case (2). A profitable deviation is to pool these types with the mass

choosing contract z. Consequently, the forward quantities decrease, the total prices increase

and the cost decreases. The last result comes from the fact that under instant screening, the

inside interval has decreasing q0, yielding substantial allocation inefficiency; the increase in

pooling loss is relatively small compared to the reduced loss in profit over the inside interval.

Similar to derivation of Proposition 2, the last step is to show that the actual continuation,

which is preferred by the future self of the seller, is also preferred by the current self.

The above analysis implies a convex support of the posterior at any history on-path or

via a single deviation by the seller. A non-convex support can only be reached from a joint

deviation of seller and buyer. In particular, at least one of them is strictly worse off from

such joint deviation, which is why we do not need to consider such history. Together with the

sub-optimality of having an interval of types indifferent between two contracts, one obtains

an interval-partition structure of the seller-optimal equilibrium.

Remark 3. The proof shows that after merging redundant contracts (or equivalent messages),

the optimal equilibrium exhibits interval partition in type separation.

Now, let us establish that the seller also plays pure strategy in equilibrium using backward

27Note that making a continuum of types indifferent between two contracts (and their respective renego-tiation) may not even be feasible. When proving the pure strategy result, I do not rule out such possibility;however, the constructed seller’s deviation strategy is pure.

29

induction. For today’s perspective, only randomizing future menus (not today’s) could

help relax the constraints. However, the last period shows screening in the equilibrium.

Anticipating continuation of such, the above results show that the seller chooses one of the

seller-optimal equilibria in the second-to-last period. Any randomization over this set does

not change the incentives in the previous period, which gives a purification result.28

Corollary 2. The optimal mechanism does not require seller’s randomization.

Let me highlight some main differences from Dewatripont (1989) and Skreta (2006) in

terms of a buyer–seller model.

Remark 4. Here are some differences from Dewatripont (1989) that deals with finite types.

First, I allow for discounting and the buyer’s mixing. Second, Dewatripont’s argument relies

on the assumption that downward (IC) binds, which is generally not the case for finite types

(see discussion in Doval and Skreta 2019a). Moreover, his use of the Pareto improvement

argument in constructing the seller’s profitable deviation assumes no change in the future

separation pattern when including an extra type into the current pooling.

Remark 5. The analysis in Skreta (2006) makes use of the observation that in a durable-

good monopoly with unit sale, PBE implementability implies a step function type of quantity

allocation among low types. In Skreta’s setup, offering a contract that specifies an upper

bound of price on the future allows the seller to replicate a bang-bang commitment solution

for high types. In contrast, general contract renegotiation (with convex cost and multiple

quantities) rules out the possibility of such replication. Despite the extra difficulty due to

the convex cost, there are gains in terms of results. Compared to Skreta (2006), I am able to

show a stronger result (Section 8) for the identified desirable properties: interval partition

is necessary in any PBE. Furthermore, I am also able to partially extend the results to an

infinite horizon (Section 10), given the necessity of interval partition for a finite-horizon PBE.

7 Implementation and Slack Constraints

Section 6 gives a partial characterization of the solution of the seller’s problem that tries

to capture the seller-optimal equilibrium. We now verify the existence of an equilibrium

coinciding with the seller’s problem in type separation and quantity allocation. The seller

can implement the optimal mechanism by offering the following menu: at each time t, and

28Unlike the durable-good monopoly case (§2 of Gul et al. 1986; §3.1.1 of Ausubel et al. 2002), an equi-librium does not involve randomization off-path. The former requires randomization to make the boundarytype indifferent when the price function is not continuous with respect to such boundary and the seller’sdeviation price is within the gap of such discontinuity. With quantities as choice variables, one can createindifference using incentives through forward quantities instead.

30

on-path history ht, a trade proposal designated for new pooling ht+1B specifies equilibrium

quantities q∗s(v)s≥t where v is the minimum element of ht+1B .

The monotonicity constraint (M) implies that renegotiation only goes upward. There-

fore, only (PC) of the lowest type of the support binds. For other implementation of the

same allocation, we will show its converse result that (PC) of higher types implies upward

renegotiation. We start by discussing two omitted slack constraints:

1. Any type θ has no incentive to stick with an old agreement different from the new one.

2. Any type θ has no incentive to masquerade as another type θ′ and stick with one of the

latter’s old agreement.

Constraint (1) is just (PC) of higher types. Denote constraint (2) by (SC), which is a

combination of reporting and obedience deviations.

Denote by (SRC’) when introducing these incentive constraints into the seller’s problem

defined in (SRC). For the identified mechanism to be optimal, it is crucial that a relaxed ap-

proach in characterizing the optimum remains valid in a dynamic game with renegotiations.

Lemma 3. The optimal revenue under (PC), (IC), (SC) and (SRC’) is the same as that

under (PC), (IC) and (SRC).

The reason is that a constraint today only affects the quantity allocation today but not

the future quantities. Intuitively, for the seller’s optimality, bygone (past agreement and

past belief) is bygone. I formalize these insights in Appendix A.3.

Since constraints (PC) of higher types and (SC) are not binding, there is generally some

degree of freedom in the contract design. An agreement in period t can have slightly lower

forward quantity than the equilibrium q+t for v if (M) does not bind locally. In particular,

the only constraint on the initial agreement for the lowest pooling contract is that it specifies

the one-period equilibrium quantity for period 0.

Now, let us explore the limits of such contractual flexibility. For this purpose, denote by

q+t−1 and q+

t the forward quantities of agreements in period t− 1 and period t respectively.

Denote by q+t,t−1 the forward quantity of period t from the period-(t− 1) agreement. Denote

by qls, plss≥l the agreement achieved in period l, where l is either t or t− 1.

We claim that to incentivize the buyer not to stick with previous agreement, optimally,

the seller proposes a new agreement that has a weakly higher forward quantity than the

continuation of the previous agreement.

To gain some insight, we first demonstrate a slightly weaker result: at optimality qt−1s s≥t

should give a weakly lower forward quantity than q∗,+t (v). The seller has a profitable deviation

strategy that replaces qt−1s s≥t by q∗s(v)s≥t when latter gives a lower forward quantity.

Associated is a decrease in the total price by v(q+t,t−1 − q

∗,+t (v)) (and some other adjustment

in other incentive prices).

31

The formal proof is by contradiction. The key step is to construct a deviation strategy

that replaces qt−1s s≥t by qtss≥t when latter gives a lower forward quantity (and some

associated in incentive prices). The final step is to show that this is a profitable deviation

for the seller when some new agreement specifies a renegotiation downwards.

The claim particularly implies that the equilibrium forward quantity in period t is always

weakly greater than the promised amount from previous agreement. Here is an alternative

intuition of such implication that exploits the linearity of utility.

type

utility

Figure 7: Promised utility of different types

The equilibrium utility is continuous, increasing and convex. The utility lower bound

(red in Figure 7) provided by the previous agreement is steeper than that (black in Figure

7) provided by the envelope formula using the equilibrium quantities. The optimal equilib-

rium forward quantity is increasing. If pricing via local (IC), i.e., providing rent based on

equilibrium quantities instead of the higher outside option, there will be a range of types

choosing the outside option (previous agreement), violating the equilibrium assumption.

The above analysis provides an upper bound on the forward quantity of any agreement

along the equilibrium-path. Now, let us consider its lower bound. By above result, (IC)

implies that a type has no incentive to mimic a lower type and stick with its old agreement.

Therefore, the remaining concern of (SC) is to mimic a higher type and stick with its old

agreement. An agreement has a forward quantity weakly higher than the equilibrium forward

quantity of the highest type of the lower adjacent pooling.

The proof is analogous to that of the upper bound. When the forward quantity of

some agreement qts, ptss≥t is less than what a lower type obtains in equilibrium, the price

reduction for such agreement (required by the higher-type information rent) makes it a

desirable option for lower types. To preserve (IC), the equilibrium total price for the lower-

type contracts should drop. Consider the following deviation strategy for the seller: increase

the future quantities of agreement qts, ptss≥t and adjust the price so that the boundary type

v is indifferent to the change. By (IC), total price increases for the lower-type contracts yet

32

all types stick with their designated contracts.

Note that the above two results pin down necessity conditions on the implementation

rather than on the equilibrium quantities. They ensure that the only incentive constraints

for consideration are the ones sketched out in the program introduced in Section 4.

Proposition 4. An agreement has a forward quantity weakly higher than

(i) that of the continuation of the recent agreement, and

(ii) the equilibrium forward quantity of the highest type of the lower adjacent pooling.

The upper-bound result implies the equivalence between the original problem and the

relaxed problem: the outside option, being lower in forward quantity than the equilibrium

analog, does not introduce extra constraints on the optimization of quantities alone.

Specifying equilibrium quantities that exhibit the full dynamics seems complicated. There

is an alternative implementation that delegates the complexity of the rich dynamics to future

renegotiation: it uses mostly constant quantities. Again, we use q∗s(v)s≥t to denote the

equilibrium future quantities of the lowest type v of some ht+1B . For such a pooling, one can

specify qss≥t where qt = q∗t (v), and qs = q+,∗t+1(v) for s > t. The above two lemmas validate

the effectiveness of such a contract.

Remark 6. The seller-optimal equilibrium requires a menu of countable contracts in each

on-path history. The contract of the highest type is the limit of a sequence of contracts.

8 Perfect Bayesian Equilibrium Outcomes

The next goal is to establish a certain uniqueness of the equilibrium outcomes identified in

the previous section. This novel result can be surprising given the vast space of available

strategies. One can view the continuation of the game starting at the node of a buyer’s

action as a signaling game. The classical plethora of equilibria does not appear as the seller

can restrict the set of actions (i.e., long-term contracts) available to the buyer. The goal is

to show with such restrictive set of actions, the buyer cannot induce a PBE with outcomes

different from seller-optimal ones.29

Suppose the seller’s problem outlined in Section 4 has a unique solution (in on-path

separation and quantities). A proof sketch of the uniqueness of PBE outcome is as follows.

First, prove that following the equilibrium proposal (in taxation principle flavor), the buyer

cannot induce any other continuation equilibrium. Next, observe that at any history, the

seller can always implement her optimal continuation equilibrium. Her sequential rationality

29One can consider the different buyer types as “agents”. By showing that these simultaneously movingagents cannot induce continuation equilibrium inconsistent with seller optimality, we effectively reduce thegame to one with sequential moves, for which the PBE is generically unique.

33

implies that any PBE will reduce to the scenario of the seller’s problem. In this sense, the

above result and the seller’s sequential rationality force a unique equilibrium outcome.

To see the first step, start with a simple observation. Offering a menu with equilibrium

trade profiles rules out the possibility of screening (in the current period). In equilibrium,

the buyer will not self-select into non-monotonic pooling, for otherwise, some types can find

a profitable deviation (by the argument in §6.4.1). Now, the only possibility different from

the seller-optimal continuation is that types coordinate in a different pooling pattern.

The intuition to disprove such possibility when the buyer plays a pure strategy is as

follows. After a type separation, the seller no longer needs to adjust the total price or the

distortion level to respect the (IC) relevant to types outside the posterior support. However,

the information rent guaranteed by the original menu only purports incentives under the

type separation of the seller-optimal equilibrium. Under a different type separation (which

has different interval boundaries), some boundary types will discover a profitable deviation

by mimicking some boundary type of another pooling interval.

The formal proof deals with the buyer’s randomization. In view of §6.4.2, the additional

ingredient in the proof is the observation that buyer’s mixing is not compatible with the

seller optimality. To satisfy (IC) when the buyer mixes, the seller has to incur extra loss

in profit. Intuitively, after a (partial) type separation different from equilibrium, the seller

would not respect the previous (IC).

Finally, applying the same argument to any history, one concludes that the seller always

chooses her best continuation equilibrium. Now back to the general case where there might be

multiple seller-optimal equilibria. If these equilibria have different on-path posteriors, then

the above argument implies that following any period-0 menu of a seller-optimal equilibrium,

there is a unique continuation equilibrium. Finally, even if some equilibria have the same

posteriors at some history, applying the same argument to such history shows that the

continuation equilibrium has to be a seller-optimal one. This completes the proof of Theorem

1 when combining with the optimality results in Section 6 and 7.

9 First-Order Characterization

Theorem 1 establishes the general result of interval partition. From this, one can identify

the search of an optimal mechanism with the search of an optimal interval partition. I will

impose an extra regularity condition in order to fully characterize the intervals.

To simplify notations, I introduce L as the loss function relative to the ex ante second-best

profit, which is the seller’s objective function to minimize. As in standard incentive theory,

I focus on the notion of virtual value v(θ), and that of the posterior-based version v(θ|θ) (for

34

which the posterior is a truncated prior on some interval with θ as upper bound)30 as belief

evolves over time. The following derivation takes a M-relaxed approach by ignoring (M)

and provides sufficient conditions under which such an approach is valid.

9.1 Loss Function

Using the second-best quantity exhibited in Section 3, q(θ|G), and the pooling quantity ex-

hibited in Section 5, one can derive the virtual surplus of type θ. For the next two paragraphs,

super- and subscripts of a refers to the posterior interval at making an action/decision; those

of e refers to the posterior interval at evaluating the action. On-path, the posterior G is a

truncated prior on some [θa, θa]. Given such posterior and a partition of its support, I will

use the notation q(θ|θa) to denote the pooling quantity qp that θ is assigned to in a non-

terminal period or q(θ|θa) in period T . Denote by v(θ|θa) the associated (pooling-adjusted

or ironed) virtual value. The welfare evaluated at a belief as truncated prior on [θe, θe] is:

J(θ, θe, θa) ≡ v(θ|θe)q(θ|θa)− C(q(θ|θa))

= J(θ, θe, θe) +

∫ v(θ|θa)

v(θ|θe)

q fb(γ)dγ︸ ︷︷ ︸efficiency deviation

− q fb(v(θ|θa)))(v(θ|θa)− v(θ|θe)

)︸ ︷︷ ︸

rent deviation

= J(θ, θe, θe)−∫ v(θ|θa)

v(θ|θe)

(q fb(v(θ|θa))− q fb(γ)

)dγ.

The first equality uses the envelope theorem and the optimality of quantity q with respect

to virtual value v. I take the benchmark surplus level as J(θ, θe, θe)31 and consequently a

differential loss function l as:

l(v(θ|θe), w) ≡∫ v(θ|θe)+w

v(θ|θe)

(q fb(v(θ|θe) + w)− q fb(γ)

)dγ.

To study the optimal interval partition in period t, we need to evaluate the loss for

all future actions. Specifically, given an implementable intertemporal partition starting in

history [α, α] (with a truncated prior belief), we take its induced trade profile and evaluate

the aggregate loss under belief α. For s ≥ t, denote by θs(θ) and θs(θ) the lower and upper

bounds of the hsB that contains θ. So α is θe in the previous discussion, and each θs(θ) is a

30On the equilibrium path, the posterior belief is a truncated prior, in which case, the conditional virtualvalue is parametrized by the upper bound of the posterior support.

31Note that in the definition of J , v(θ|e) is not ironed. Generally, J(θ, θe, θe) may not be achievable evenunder the commitment solution if the virtual value is not monotone.

35

θa. The seller’s profit in period t is

π([α, α]|α) =

∫ α

α

J(θ, α, α)dF (θ)− L([α, α]|α),

where for any v > v as two boundaries of (perhaps different) intervals partitioned in period

t, we define the aggregate loss L

L([v, v]|α) ≡ N(δ, T − t)∑s≥t

δs∫ v

v

dF (θ)l(v(θ|α), v(θ|θs(θ))− v(θ|α)

).

To study the optimal boundary of one interval, decomposing its marginal effect on various

parts of the partition gives a structure of dynamic programming if the loss of these various

parts are independent of one another.

9.2 Variational Approach

To decompose the loss function into various parts, I further introduce two notions: L([v, v]) =

L([v, v])|v) and allocation inefficiency of a pooling interval:

lp([v, v]|α) =

∫ v

v

dF (θ)l(v(θ|α), v(θ|α)− v(θ|α)

).

By definition, L and lp are continuous and differentiable in the boundary arguments.

θ1 θ2 θ3α = θ0 θ∞ = α...

For an equilibrium partition shown as above, the loss can be decomposed into three

components relevant to a boundary θi: a loss today of the pooling interval having θi as

upper bound (and θi−1 as lower bound), a future loss of this interval, and a loss of all

intervals above θi. Specifically, when (M) is slack, the loss of intervals below θi−1 does not

depend on θi.

In the following discussion, we use θi to denote the equilibrium boundary, and θi as the

choice variable. A superscript + or − refers to the right or left limit of a type. The loss

minimization program must satisfy the following variational inequalities

∂

∂θi

[L([θi, α])︸ ︷︷ ︸

loss of the top

+ (1− δ)lp([θi−1, θi]|α)︸ ︷︷ ︸loss of pooling today

+ δL([θi−1, θi]|α)︸ ︷︷ ︸future loss

]< 0 if θi → θ−i

> 0 if θi → θ+i

.

36

The set of variational inequalities entails that the marginal change in loss for boundary

type θi increases when increasing it across the equilibrium θi: By envelope theorem, the

leading cause (first-order term) of the increase of the derivative on the left-hand-side is due

to such marginal change in the differential loss l associated with boundary type θi.

Results in Section 6 ensure the existence of an interior solution to the set of variational

inequalities, or equivalently, a solution of the first-order condition. Under interval partition,

the equilibrium tail-pooling (Proposition 2) implies that the inequality has sign > for θi close

to α; the equilibrium separation (Proposition 1) implies that the inequality has sign < for θi

close to α. Then using the differentiability of L and lp and applying the intermediate value

theorem, one establishes the existence of a solution.

The first-order approach sets the above marginal effect (l.h.s of the variational inequal-

ity) to 0. We want such first-order approach to provide a dynamic programming recursively

solving the optimal interval partition problem. Here, dynamic programming has two dimen-

sions of recursion: apart from the backward induction, it also solves the partition problem

recursively on the type space. For this purpose, it is essential that the shadow costs of the

monotonicity constraints (M) do not factor into these loss functions so that the components

for different parts of the type space are independent. In this sense, the dynamic program-

ming is equivalent to a (recursive) M-relaxed first-order approach, or simply as relaxed

first-order approach, in characterizing the interval boundaries.

9.3 Relaxed First-Order Approach

For validity of the recursive relaxed first-order approach, I impose monotone hazard rates.

This particularly implies that the virtual value v(θ) is increasing. Suppose the posterior

is the prior truncated on an interval with an upper bound of α. The monotonicity of

hazard rates implies that of the conditional hazard rate [F (α)− F (θ)]/f(θ). Therefore, the

conditional virtual value v(θ|α) also increases in θ. The pooling-adjusted virtual value for

a subinterval [θl, θr] is the average of the conditional virtual value over such range: vp =

θl−(θr − θl

)(F (α)− F (θr))/(F (θr)− F (θl)). The optimal one-period pooling quantity, qp,

for a subinterval [θl, θr], when there is no concern of violating (M), is pinned down by a

pooling-adjusted virtual value: C ′(qp) = vp.

By envelope theorem, the marginal effect of an increase in θi on the loss of the higher

interval is second-order at θi = θi: with the same preference32, changes in the optimizer

of the objective do not contribute to the change in the objective function. The first-order

marginal effects with respect to an increase in θi are therefore a change from discounted sum

32Under monotone hazard rate, the preference is parametrized by the upper boundary alone.

37

of l of θ+i to that of θ−i and a change in future loss (of the lower pooling interval). The latter

reduces loss as a higher support cutoff tomorrow yields a belief more aligned with today’s.

Therefore, at the equilibrium θi, θ−i has a higher discounted sum of loss than that of θ+

i .

The following gives the key steps of the proof based on analysis of the net marginal effects.

The first step is to show that an increase in θi enhances the slackness of (M) at the

boundary type. Consequently, there is a threshold type θi such that (M) is slack when

θi ≥ θi. The second step is to show when θi ≤ θi, the net marginal effect of an increase in θi

is a reduction in the aggregate loss. Lastly, by the argument at the end of last subsection,

there is a solution to the variational inequality in the range θi > θi.

Theorem 2. When F has monotone hazard rates, the solution of the M-relaxed program

coincides with the seller-optimal equilibrium: the equilibrium forward quantities q+0 (θ) defined

using pooling-adjusted virtual values is increasing in θ. Such program can be solved by using

dynamic programming on the first-order conditions:

∂

∂θi

[L([θi, α]) + (1− δ)lp([θi−1, θi]|α) + δL([θi−1, θi]|α)

]∣∣∣∣θi=θi

= 0.

Under the regularity conditions, Theorem 2 implies that proposing the posterior-based

second-best quantity for each (equilibrium) pooling interval is robust optimal for the seller.

A backward-induction inspection33 of the above first-order condition gives the following

comparative statics:

Proposition 5. For a higher δ, or a uniformly lower hazard rate, or a uniformly higher

C ′′′, separation is faster among high types and slower among low types: the lowest pooling

interval grows in size and the high pooling intervals shrink.

It is consistent with the following intuitions. A higher δ means a higher future stake

relative to current stake. Therefore, in the main tradeoff behind tail-pooling, one is more

willing to reduce allocation efficiency in return of better rent extraction tomorrow. On the

other hand, for a uniformly lower hazard rate f/(1 − F ), the higher types becomes more

important in the seller’s problem, therefore, the seller is more willing to sacrifice allocation

efficiency among the low types in return of better rent extraction among the high types.

Another intuition is that under lower hazard rate, there is a greater wedge between first-

best and (ex-ante) second-best allocations among low types. Therefore, long-run distortion

becomes more important for profit maximization.

33See supplemental materials.

38

Example 2. We again use the uniform linear-quadratic setup in Example 1, except that now

we consider general T < ∞.34 As in the two-period case (T = 1), the optimal solution has

posterior support as an interval, and given the posterior, it further divides the interval into

infinitely many (pooling) subintervals. Furthermore, the boundaries of these intervals are

defined linearly recursively, which follows from the bootstrapped recursion and the homo-

geneity of the loss function L in the size of the [α, α]. More precisely, the linear recursion gives

a geometric sequence in the size of the intervals: for period t, θ∞− θi = (1− λt)(θ∞− θi−1),

where [θ0, θ∞] = [α, α] denotes the belief support at the beginning of period t. Indeed, one

can solve the constant λt recursively.

Let us first revisit T = 1 case. In such case, period 1 involves screening (as in Example

1) and period 0 pools each interval [θi−1, θi]. Due to the linear-quadratic utilities, the loss l

associated with a type θ in period 1 is quadratic in the difference between the upper bounds

of posterior and prior beliefs. The pooling loss in period 0 is effectively a variance. Given

the geometric sequence in the size of the intervals, one can apply the techniques in telescope

and sum of geometric series to express L([θ, θ]) as a function of λ1. For example, for δ → 1

(the case of Table 1), L turns out to be 1/2[1−λ21(1−λ1)/(1− (1−λ1)3)]. Minimizing such

loss gives the equilibrium λ1 = 0.634 for δ → 1.

For general T , the periods t = 1, 2, · · · , T − 1 partition the posterior interval using

λT−1, · · · , λ1 respectively, and period T screens. Again, one can write down the quantity

allocation for all future periods as a function of the boundaries (which are determined by

variable λT as well as the λt obtained in the previous iterative steps), and then write the

differential loss l with respect to prior belief. Again, one can apply the same techniques

to find an analytic (but more complicated) expression for the aggregate loss for horizon T .

Finally, minimizing such loss with respect to the argument λT gives the equilibrium period-0

linear rule for the T -horizon game.

Remark 7. The characterization in Example 2 relies on the linear recursive structure. Gen-

erally, a numerical method implements the relaxed first-order approach. To do so, solve

the optimal interval partition when restricting the number of intervals to be some finite k.

Increase k and use the convergence of the respective θ1’s as the solution θ1. Iterate this

process on [θ1, v] (where v denotes the upper bound of the current belief support) to find the

other interval boundaries, say until some θk such that [θk, v] < ε for some ε small enough.

34One can fully derive the solution for power distribution, but with slightly more involved algebra.

39

10 Infinite-horizon Setup

Now, let us consider the infinite-horizon game (T =∞). As we will see, for some cases, the

stationarity of this game simplifies the task of characterization. There are two approaches to

tackle an infinite horizon. One is to take the result from the finite-horizon analog, and obtain

the limiting result by letting T go to infinity. The other is to directly study the equilibria of

the infinite-horizon game. I will take the latter approach and show that if focusing on weak

Markov equilibria alone, it agrees with the former.

For this section, I will first introduce some new ingredients in the solution concepts, and

then demonstrate the steps to extend results on equilibrium properties (with some caveats

in doing so). Lastly, I will present an example with an explicit constructive equilibrium.

10.1 Conceptual Differences

With infinite horizon, the richness of the contractual space opens up the possibility of rep-

utation. In such case, one appealing class of equilibria has strategies contingent mostly on

payoff-relevant information. As demonstrated earlier, one can specify a trade proposal in

terms of future quantities and a discounted total sum of payment for that stream of quan-

tities. To motivate, note that fontloading or backloading payments does not affect either

party’s incentive as they have quasilinear utilities and share the same discount factor. One

related property I find desirable is an invariance in the buyer’s strategy.

Condition 4. The buyer’s strategy has invariance property: backloading payment for each

contract in a menu uniformly by a certain amount does not change the buyer’s (probabilistic)

selection of a contract.

Therefore, I motivate the following definition of (weak) Markov perfect equilibrium, and

henceforth MPE.

Definition 5. A Markov Perfect Equilibrium is a PBE in which the buyer’s selection of

contract from any given set of contracts at any history ht only depends on µ(ht), and it

satisfies invariance property.

The above definition of MPE still allows for certain signaling through payoff-irrelevant

information. Under limited commitment and anticipating future renegotiation, the buyer’s

utility depends on the sum of discounted payments, the quantity today, and the sum of dis-

counted future quantities, and the seller’s posterior belief.35 Such belief can vary across two

35In contrast, if there were perfect commitment, there won’t be any renegotiation. In such case, thebuyer’s utility depends on the sum of discounted payments and the sum of discounted quantities specifiedin the selected contract.

40

contracts with the same total payment, quantity today, sum of discounted future quantities

but different composition of these future quantities.

An alternative solution concept involves reputation.36 In those equilibria, the seller main-

tains a reputation of committing to a sequence of (separating) contracts until her first devi-

ation from it, which reverts the play to a MPE or a worst PBE. Typically, the reputation

encourages information revelation among higher types. It also gives rise to a much richer

set of equilibrium outcomes. However, extra difficulties arise in a full characterization of a

seller-optimal reputational equilibrium. My focus on MPE sidesteps these difficulties, and

retains the equivalence of the equilibrium outcomes to those of the seller-optimal MPE.

Some notations are simplified in the infinite-horizon case because of the stationary struc-

ture. The normalization factor N becomes (1− δ).

10.2 Differences in Results

Observe that results in Section 4 naturally extend to the seller-optimal MPE in the infinite

horizon. Moreover, Proposition 1 and 3 can be applied to obtain asymptotic learning.

Definition 6. The buyer’s type is asymptotically learned if there exists a vanishing sequence

εss such that for each history ht and posterior Ft, there exists some θ such that Ft(θ) −limv→θ− Ft(v) ≥ 1− εt.

Let us first discuss the case of MPE. A passing to the limit argument builds a clear

correspondence between the sets of PBE in finite horizon and MPE in infinite horizon. With

such connection, the (recursive) optimization problem in characterizing an MPE is the limit

of a sequence of the relevant problems in characterizing a PBE for finite horizon. I formally

prove the correspondence when the relaxed first-order approach is valid, which connects the

two approaches of studying MPE in the infinite horizon.

Remark 8. When (M) is not slack in equilibrium for sufficiently long horizon or in a MPE

in the infinite horizon, the analysis needs to take into account how the shadow cost of

such constraints vary with the horizon parameter T . If one can establish that such shadow

cost converge smoothly, then one can establish the general case, i.e., without imposing the

regularity conditions.

When the correspondence holds (established in the supplemental material), one can se-

lect a sequence of the relevant solutions that converges to the Markovian solution. This

particularly implies that a violation of refining property and pure strategy at any history

36See Ausubel and Deneckere (1989), Abreu and Gul (2000) for durable-good monopoly, and Atakan andEkmekci (2012), Mailath and Samuelson (2006) for more general discussion.

41

will lead to a profit loss. Moreover, one can straightforwardly extend the infeasibility of

inside and cross pooling. This observation also implies the outcome equivalence of the MPE:

any MPE is outcome-equivalent to a seller-optimal MPE.37 Suppose the optimization pro-

gram outlined in Section 4, simplified using the properties in Section 5, has a unique solution

(on-path quantities) for each finite horizon. Then Section 8 implies that there is a unique

PBE outcome for each finite horizon. Therefore, as the limit of the sequence of such PBE

outcomes, the MPE outcome is unique. Section 8 also provides arguments to deal with cases

of multiple seller-optimal MPE. Therefore, one has

Theorem 1*. Under monotone hazard rates, any MPE satisfies refining property and pure-

strategy property. Moreover, the buyer’s type is learned asymptotically, and the quantity

distortion converges to zero.

Theorem 1* also implies that any MPE has asymptotic learning. Therefore, with the

mild regularity conditions and the innocuous restriction on belief-updating (Condition 3)

and invariance behavior (Condition 4) that respect the nature of a Markovian structure of

equilibrium, one obtains a Coase conjecture result for free for all MPE: equilibrium achieves

efficiency in the frequent-interaction limit.

Remark 9. The case of a seller-optimal PBE is more involved. One cannot apply Fuden-

berg and Levine (1983) to establish the relevant equilibrium properties: they use ex ante

ε-optimality, which is not well-suited for a backward induction argument. The supplemental

material proposes one way in drawing the connection of PBE of finite-horizon games having

certain extra commitment with a seller-optimal PBE of the infinite-horizon game.38 Com-

pared to Skreta (2006), this argument of passing to the infinite-horizon limit relies on the

non-trivial loss when violating either property.

10.3 A Linear-Quadratic Example

Consider a linear-quadratic paradigm with a uniform type distribution. Take C(q) = q2/2

and S(θ, q) = θq, where θ ∼ U [1, 2]. Under such specification, first-order approach applies

and the monotonicity constraints hold when setting per-period quantity as one associated

with the pooling-adjusted virtual value. One can take the curvature bounds R, r to be

37For general PBE, reputation can sustain other equilibrium outcomes and separation patterns.38Observe that an MPE is the worst PBE: the seller, as a Stackelberg leader in each period, can guarantee

a profit at the MPE level by the finite-horizon uniqueness argument. Therefore, to sustain the seller-optimalPBE, one can use a grim trigger strategy (reverting to a MPE when the seller deviates). Since the off-pathcontinuation is an MPE, neither has an incentive to deviate. The on-path best-response property requiresthat at each such history, the seller’s benefit from consuming and pocketing the reputation to be dominatedby the continuation benefit of reputation.

42

arbitrary close to 1. With uniform distribution, the virtual values become v(θ|θ) = 2θ − θ,and vp([θl, θr]|θ) = (θl + θr) − θ. In particular, q sb(θ|θ) = 2θ−θ, qp([θl, θr]|θ) = (θl + θr)−θand q fb(θ) = θ.

Given a uniform belief on [θ, θ], the equilibrium employs a linear recursive rule, depicted

in Figure 8, to partition the support for the current period: [θ, θ] is partitioned into infinitely

many intervals, with end points θi∞i=0, where θ0 = θ, θ∞ = θ, and θ∞ − θi = (1− λ)(θ∞ −θi−1) for some constant λ ∈ (0, 1).

θ1 θ2 θ3θ0 θ∞

θ θθ + λθ

...

Figure 8: Linear recursive cutoffs.

Types in each interval are pooled at the same current quantity. Figure 9 depicts the pool-

ing quantities of period-0 partition, and the induced equilibrium separation and pooling

quantities of period 1.

θ

q

q fb

cutoff = 2

qsb

t = 0

θ

q

q fb

new cutoffs

qsb

t = 1

Figure 9: Quantity assignment under the linear recursive rule.

Claim 2. (M) is slack in the uniform linear-quadratic setup.

One can check the claim in a brute-force manner or apply Theorem 2: quadratic cost

function satisfies the bounded curvature condition and the uniform distribution satisfies the

monotone hazard rates.

Similar to the finite horizon analysis, the linear recursion of the cutoffs come from the

homogeneity of L. Moreover, as L is now time-invariant, λt = λ is also time-invariant. I

will exploit such linear recursive structure to back out the equilibrium λ by optimality and

consistency of the cutoffs. More precisely, to minimize L([1,2]), the first-order condition

43

gives: f ′(θ) = 0, where θ = λ in equilibrium, and

f(θ) = (2− θ)3L([1, 2]) +1− δ

12(θ − 1)3 + δL([1, θ]|2).

We use the fact that L is homogeneous of degree 3 in the size of the posterior support.

Appendix B includes an expansion of L in details terms. The consistency of the equilibrium

rules requires f ′(θ) = 0 at θ = λ. Numerically, when δ → 1, the equilibrium λ is close to

0.7; for lower δ, λ is still close in range, e.g., when δ = 0.7, λ is 0.66. When δ increases, θ

increases. This is consistent with the intuition that as future becomes more important, the

seller is more willing to sacrifice profit today in return for less information cost in the future.

Remark 10. Naively maximizing V as a function of λ (ignoring the equilibrium consistency

between the two) yields a hypothetical λ much higher than the equilibrium λ. When δ = 0.7,

λ is 0.77; when δ → 1, λ is roughly (1+δ)/2. The big gap between λ and λ suggests that the

seller-optimal PBE strictly dominates any MPE and uses reputation to maintain a linear rule

λrep > λ using MPE as grim-trigger. For such solution, instead of an equilibrium consistency

equation, one imposes a weaker self-enforcing equation for reputation maintenance.

11 Contracting

This section discusses the canonical form of contracting as in the mechanism design literature

(§11.1) and two natural yet restrictive forms of contracting (§11.2 and §11.3). In the former,

the canonical dynamic mechanism yields the same equilibrium profit: there is redundant

degree of freedom in terms of information channel. In the latter case, the restriction on the

form of contracting limits the seller’s ability in incentivizing the buyer.

11.1 General Dynamic Mechanisms

In the main text, each contract specifies a distinct trade proposal. General mechanism design

allows different signaling of contracts with the same trade proposal. In terms of providing

expected utility to the buyer (not taking renegotiation or the induced future posterior into

account), two contracts are equivalent if they specify the same current trade, discounted

future trade, and the same total payment. The main analysis (especially in the proof of

Lemma 2 tackles general signaling of these “equivalent” contracts.

We have sidestepped the technical difficulty in general type space in a manner similar to

Skreta (2006), i.e., by focusing on a subclass of mechanisms (exhibiting refining and pure

strategy properties) consistent with the seller optimality. The induced interval partition

implies that there is no loss in restricting message space to type space itself.

44

There are two formulations of general mechanisms. One is that the mechanism today only

specifies outcome contingent on previous messages, but not on future messages. The other

allows specifying the dependence of future trade terms on future messages. For the former,

there is an exact equivalence to the main framework. For the latter, the optimality results in

Section 6.4.2 extend while the feasibility results in Section 6.4.1 do not: the downward (IC)

across a gap in the support generally does not bind under this more general contracting.

This means that under the most general dynamic mechanisms, the set of equilibrium welfare

(profit and buyer utility) is the same as what is pinned down in this paper, but the equilibrium

type separation can be different: refining property is consistent but not necessary with seller

optimality.

11.2 Spot Contracting

A spot contract only specifies the trade term for the contracting period, i.e., its trade proposal

indicates no trade for future periods. In the spirit of Proposition 4, when the seller is only

allowed to offer spot contracts, she is strictly worse off. These contracts are not desirable in

providing lower bounds on the buyer’s info rent, leading to restricted information revelation.

Consequently, the ratchet effect (Laffont and Tirole 1988) kicks into play: screening is not

only undesirable but also infeasible.

Indeed, spot contracting is compatible with the skimming property in the following sense.

Under skimming property, at each non-degenerate history, there is a single pooling to further

separate later, and it consists of the lowest types. These types do not require an info rent as

they are not separated from the worst type. Therefore, one can offer a spot pooling contract

for them.39 When there are multiple pooling intervals, intervals separated from the lowest

type requires an info rent, which makes spot contracting an incompetent option.

11.3 Constant Contract with Renegotiation

Another seemingly natural class is the constant contracts. A constant contract specifies the

same p and q for all future periods. Unlike spot contracting, a constant contract can provide

utility lower bound for the buyer. However, the renegotiation alone does not provide enough

flexibility to implement the solution of the seller’s problem. Indeed, constant contracts are

not compatible with slack constraints. Therefore, restricting to the class of constant contracts

will significantly decrease the seller’s profit. Section 7 demonstrates that introducing an extra

degree of freedom, as in the mostly constant contracts, resolves such issue.

39This is precisely why Maestri (2017), who studies the binary-type case, uses spot contracts for the lowtype in the implementation.

45

In the main text, I adopt a framework of repeated trades. Alternatively, one can consider

renegotiating a one-time trade of a durable good, a la Strulovici (2017). Each t denotes a

round for renegotiation. At the end of each round, there is a probability (1− δ) chance that

the renegotiation ends exogenously. This modeling captures the observation that high stake

deals involve time-consuming negotiations before reaching a final agreement. Such framework

is isomorphic to restricting to constant contracts in the repeated-trade framework.40

12 Conclusion

This paper provides a framework to study contract negotiation in a long-term relationship

with adverse selection. It shows that under mild conditions, the mechanism designer can

focus on a relatively simple design making use of a much lower dimensional object (cutoffs

of pooling intervals) than the otherwise arbitrarily complicated posterior beliefs. Such a

model of simple design is suitable for applied works (e.g., industrial organization) that study

contracts under limited commitment. Moreover, the paper predicts upward renegotiations

in equilibrium, which is consistent with the empirical evidence when there is no significant

ex-post uncertainty.

The sorting condition gives the seller an incentive to separate types, the chance of rene-

gotiation creates a need to pool the tail, and the bounded curvature (or roughly concavity of

the total surplus) gives an incentive to pool locally across the entire type spectrum. These

three together imply that the refining property is in line with profit maximization. The cur-

vature is important for the optimality of the refining property as opposed to the skimming

property. For linear C and a bounded range of quantity q, e.g., restricting q ∈ [0, 1], the

linearity in total surplus creates a bang-bang solution in equilibrium. The latter becomes

identical to a classical durable-good monopoly setup.

For future work, one natural extension is a full characterization of an optimal reputational

equilibrium (infinite-horizon) perhaps under some extra regularity conditions. Another di-

rection to explore is the high-stakes model, which provides a complementary narrative of

contract negotiation under asymmetric information.

40Another equivalence is between the durable-good model of Strulovici (2017) and the repeated-trademodel of Maestri (2017), both of which study binary-type case. To see this, in the repeated-trade PBE, theclassical no-distortion-at-top holds: the continuation outcome when a high-type separates is first-best. Thecanonical implementation highlighted in Maestri (2017) assigns the low-type a spot contract. Nonetheless,as participation constraint of the low type binds and renegotiation goes upward, one can replace each on-path spot contract with a constant contract that repeats its trade term in each future period. With suchimplementation with constant contracts, one establishes the equivalence of the two models under binary-type.

46

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49

Appendix

A. Omitted Proofs for Auxiliary Lemmas

A.1 Proofs of Section 4

Lemma A.1. The seller’s problem is equivalent to identify the optimal quantity allocation

given a type separation process, and identify an optimal type separation process.

Proof. Follows from the dependency diagram (Figure 2) and Lemma A.9.

Lemma A.2. The seller’s problem is equivalent to one in which the incentive constraints

are required to hold on the convex hull of the belief ’s support.

Proof. In this proof, I use the term “type space” to refer the posterior support.41 Here, I deal

with the isomorphic program with shortened horizon, but neglect the index for remaining

horizon.

Denote the original type space as hB, and its convex hull as hB. Denote the posterior

belief as Fh. The seller’s objective under hB is to maximize

∫ θ

θ

p+0 (θ)dFh(θ) =

∫hB

p+0 (θ)dFh(θ).

Using envelope formula, one can write the program under hB as

Π(qss, hB) =

∫ θ

θ

(θq+0 (θ)− c+

0 (qss))dFh(θ)−∫ θ

θ

q+0 (θ)(1− Fh(θ))dθ.

subject to (PC), (IC), (SC), (SRC’) imposed on hB. Call this as program A and its solution

as solution A.

The program that we have replaced the type space with its convex hull hB is to maximize

the same objective function subject to (PC), (IC), (SC), (SRC’) imposed on hB. Call this

program B and its solution as solution B.

We prove the equivalence of the two programs by an induction on T . For T = 0, one can

get rid of (SC) and (SRC’). Clearly, program B has a weakly lower value than program A,

as it shares the same objective function with the latter but has extra constraints. Suppose

to the contrary that program B has a strictly lower value, we will derive a contradiction.

We will construct from the solution of program A a feasible mechanism that outperforms

the solution of program B. Take an interval (θl, θr) ∈ hB\hB, and let v ∈ (θl, θr) be the type

41This proof follows the sketch in Skreta (2006). I include it for completeness.

50

indifferent between choosing the contracts specified for θl and those specified for θr:

θq+0 (θl)− p+

0 (θl) = θq+0 (θr)− p+

0 (θr),

when θ = v. Such type exists as the above equality becomes “>” when θ = θl and “<” when

θ = θr; a continuity argument guarantees that v exists.

Now extend the allocation rule of solution A onto (θl, θr) by assigning types in (θl, v) the

allocation of θl and types in (v, θr) the allocation of θr. Apply the analogous procedure to all

other intervals in hB\hB. Note that such extension does not introduce any new contract. By

single-crossing property, such extension respects (IC) for types in hB\hB. Therefore, (IC)

holds for all types in the convex hull hB. Moreover, as the lowest type θ ∈ hB, the fact that

its (IR) holds together with (IC) of other types imply that (IR) holds on hB. Hence, such

extension of the solution A is a feasible mechanism under the constraints of program B. As

the extension has the same profit as solution A, it yields a higher profit than solution B,

contradicting the optimality of solution B.

For the induction step, suppose the equivalence holds for horizon T = 0, 1, · · · , n. Then

apply the hypothesis to the program in (SRC’) to extend to the relevant convex hull ht.

As a result, for program A, nothing changes on the set of constraints described in (SRC’)

when convexifying the relevant posterior support. Observe that (SC) being slack in solution

B (Lemmas A.9 and A.10) and that including these no-density type does not change the

equilibrium separation pattern (along the inductions). Then the proof of T = 0 together

with these two observations implies that values of the two programs are the same.

A.2 Proofs of Section 6

This section deals with proofs for the induction step from T periods to T + 1 periods. The

induction hypothesis is that for a repeated game with less than or equal to T periods, the

properties of equilibrium separation, equilibrium pooling, and interval partition hold. The

base case is established in the main text.

Lemma A.3. Any equilibrium has a deterministic trade profile.

Proof. Start with any mechanism and its implied equilibrium trade profile. Replacing such

profile by its expected output in each period maintains the buyer’s incentive (holding fixed

the payments). Under the seller-optimal equilibrium, each buyer type maintains its selection.

By Jensen’s inequality and convexity of C(·), such operation increases the seller’s surplus.

Lemma A.4. If the seller has full information, the equilibrium continues with first-best

quantity for the remaining periods.

51

Proof. Suppose the seller knows the buyer’s type. Given the continuation of any previous

agreement, the seller can improve the total surplus by renegotiating towards a first-best

(efficient allocation) continuation. In such renegotiation, she can increase her profit by

making the buyer type indifferent between the outside option (continuation of previous

agreement) and the new (first-best) contract.

Let us state a result that will be used multiple times, which relies on the induction

hypothesis of an interval partition. It gives sufficient conditions that if the future-self of the

seller prefers a certain mechanism, then the current-self also prefers having such continuation.

Lemma A.5. Consider horizon T . For two continuation (a) and (b) on h1B given the same

quantity allocation in period 0 and are implementable in period 1, if (a) yields higher a profit

than (b) in the period-1 perspective, and (a) has a lower utility for the highest type in h1B,

then (a) yields a higher profit in the period-0 perspective.

Proof. Denote the quantities of (a) and (b) by q and q. The seller optimality of the seller’s

future-self implies for h1B ≡ [v, v] that∫ v

v

(θq+1 (θ)− c+

1 (qs(θ)s≥1))dF (θ)−∫ v

v

q+1 (θ)(F (v)− F (θ))dθ

≥∫ v

v

(θq+1 (θ)− c+

1 (qs(θ)s≥1))dF (θ)−∫ v

v

q+1 (θ)(F (v)− F (θ))dθ.

Our goal is to show that the above inequality still holds if we replace F (v) by F (θ) > F (v).

In doing so, the l.h.s of the inequality decreases by

(F (θ)− F (v))

∫ v

v

q+1 (θ)dθ = (F (θ)− F (v))δ−1

[U+

0 (v)−∫ v

θ

q0(θ)dθ

],

and the r.h.s of the inequality decreases by

(F (θ)− F (v))

∫ v

v

q+1 (θ)dθ = (F (θ)− F (v))δ−1

[U+

0 (v)−∫ v

θ

q0(θ)dθ

].

As we assumed U+0 (v) ≤ U+

0 (v), the inequality holds when replacing F (v) by F (θ).

Lemma A.6. Denote q0 as the period-0 quantity allocation if one assumed (M) of period 0

is slack, and q0 as the actual period-0 allocation. Then∫ θθq0(v)dv ≥

∫ θθq0(v)dv.

Proof. The q0 and q0 are constant on the pooling masses h1. When (M) binds for some

boundary type in period 0, relative to quantities defined via virtual value, the shadow cost

52

further distorts the period-0 pooling quantities for lower types and alleviates the distortion

for higher types: this is equivalent to perform a mean-preserving spread of the virtual values.

By concavity of the inverse of C ′ (Claim 3), Appendix B.3 implies that such operation further

reduces the average period-0 quantity:∫ θθq0(v)dv ≥

∫ θθq0(v)dv. One obtains the result by

taking θ = θ.

Lemma A.7. For horizon T , consider (a) a seller-optimal equilibrium, and (b) a seller-

optimal equilibrium of horizon T − 1 for the first T periods, and first-best quantity for period

T . For each type, (a) yields a lower utility than (b). Moreover, (a) yields a higher profit.

Proof. We denote the quantities of (a) and (b) by q and q, and the utilities by U+0 and U+

0 .

Our goal is to show∫ θ

θ

q+0 (v)dv = U+

0 (v) < U+0 (v) =

∫ θ

θ

q+0 (v)dv ∀v ∈ [θ, θ].

In this inductive proof only, we assume interval partition.42

Clearly, (b) satisfies (M) as it is a convex combination of two monotone allocation rules.

The induction hypothesis of the above inequality implies the conditions of Lemma C.7:

replacing the period-1 continuation of (b) by the seller-optimal one yields higher continuation

profit and lower continuation utilities. By such lemma, one concludes the induction step:

(a) yields a higher profit than (b) and a lower continuation utility for each type.

Repeatedly applying the above lemma to the last periods to make them non-strategic,

or equivalently, applying the results for horizon T , T − 1, · · · , 1, yields:

Proposition A.1. The equilibrium utility of the highest type in the support is lower bounded

by that in the instant screening mechanism.

Lemma A.8. The seller derives the same surplus from a constrained-optimal full-pooling

mechanism and from an efficient mechanism.

Proof. Pooling gives θ q fb(θ)− C(q fb(θ)), while the efficient allocation gives

∫ θ

θ

dF (θ)[θ q fb(θ)− C(q fb(θ))]−∫ θ

θ

(1− F (θ)) q fb(θ)dθ.

Using the following envelope formula

θ q fb(θ)− C(q fb(θ)) = θ q fb(θ)− C(q fb(θ)) +

∫ θ

θ

dθ q fb(θ),

42This does not invalidate our proofs of other results. The proof of the separation and the pooling resultsuses this Lemma A.7 in the continuation game in period 1 only.

53

one can rewrite the principal’s surplus under efficient screening as

θ q fb(θ)− C(q fb(θ)) +

∫ θ

θ

dF (θ)

∫ θ

θ

dθ q fb(θ)−∫ θ

θ

(1− F (θ)) q fb(θ)dθ

= θ q fb(θ)− C(q fb(θ)) +

∫ θ

θ

(1− F (θ)) q fb(θ)dθ −∫ θ

θ

(1− F (θ)) q fb(θ)dθ

= θ q fb(θ)− C(q fb(θ)).

Therefore, we show that in a static scenario, the best pooling contract gives the same profit

as the efficient contract.

Proof of Proposition 1. We first prove period 0 does not have full pooling. If (a’) the seller

fully pools in period 0, then the continuation is identical to the equilibrium of a T -period

model. Such continuation is identical to the equilibrium play of the first T periods in (b)

a (T+1)-period model if the seller had to offer the first-best contracts in period T (i.e., the

play in period T has nothing to do with earlier plays). By δ < 1 and the profit-equivalence

of the first-best and full pooling, (b) yields a higher profit than (a’). Now, by Lemma A.7,

any seller-optimal equilibrium gives a higher ex-ante profit than (b), and therefore, (a’) is

not a seller-optimal equilibrium.

By Lemma A.9 and (M), (PC) does not bind for higher types, therefore, the above

argument applies to any history: At any non-degenerate history, the continuation equilibrium

involves some separation in the current period. The argument for variation in forward

quantities also applies.

Claim 3. C ′′′ ≥ 0 implies that the inverse function of C ′ is concave.

Proof. Denote C ′ = g and the inverse function by h. By definition h(g(v)) = v, and chain

rule gives h′(g(v))g′(v) = 1, and further differentiating (or equivalently implicit function

theorem) gives

h′′(g(v)) · g′(v)2 + h′(g(v)) · g′′(v) = 0.

By the above equation, we have h′′(g(v)) = −g′′(v)/[g′(v)]3. The second derivative h′′ is

negative, as g is an increasing function: C ′′′ ≥ 0 implies that g′′(v) > 0, and C ′′ > 0 implies

that g′(v) > 0. Therefore, the inverse function of C ′ is concave.

Let’s adopt the loss minimization program introduced in Section 9. The seller has a

differential loss function: l(v(θ|θ), w), where θ is the belief cutoff, v(θ|θ) is the conditional

virtual value and w is the virtual wedge. For a quantity assignment q for type θ, the

virtual wedge w := v(θ|θ) − C ′(q). Further suppose that l1 > 0, wl2 ≥ 0, l2,2 ≥ 0, and

54

ω|w|β ≤ |l2| ≤ Ω|w|β for the relevant domain for β = 1 and Ω > ω > 0. See Lemma A.12

for the foundation for such loss function and a proof that the above properties hold when C

is moderately concave.

Proof of Proposition 2. We first observe that the above formulation ensures the order estimates

for the thought experiment in the main text is correct: pooling ∆ of the lowest types in period

0 and screens them in period 1 yields a loss in period-0 allocation efficiency by at most Ω∆3

and a gain in period-1 rent extraction by at least δω inf(1− F (θ))∆2.

We then prove by an induction that if the seller pools an interval of lowest type within

the support and screens all higher types, then a seller-optimal continuation equilibrium (of

tomorrow) outperforms the above screening continuation in today’s perspective. We use the

induction hypothesis that for horizon less than T − 1, the optimal mechanism has interval

partition, and that the above statement holds for horizon less than T − 2. Therefore, it

suffices to prove the above result for period 0 of horizon T . Denote the the pooling interval

as [v, v], and the posterior upper bound as θ > v. Denote the continuation equilibrium trade

profile as qs for s ≥ 1, and the screening trade proposal as qscs which is q sb(θ|v) for period 1

and q fb(θ) for periods afterwards. By Lemma A.5, it suffices to show∫ v

v

qsc,+1 (θ)dθ ≥

∫ v

v

q+1 (θ)dθ.

Proposition A.1 gives the above inequality. Now, the hypothetical equilibrium demon-

strated in the main text, which is to pool a tail interval in period 0 and screen it in period

1, yields a higher profit than the instant screening mechanism. The actual equilibrium

following such period-0 tail-pooling, by the above analysis, yields higher ex ante profit than

the hypothetical continuation. Therefore, tail-pooling (with equilibrium continuation) gives

a profitable deviation to the seller, compared to instant screening.

Proof of Lemma 1. By induction, it suffices to show the suboptimality of the inside screening

for period 0 when buyer plays pure strategy. So let’s say the “original equilibrium” involves

some inside screening of some interval [θl, θr] nested in a pooling with convex hull [α, α]

in period 0. Analogous to Proposition 2, the proof consists of (i) constructing a profitable

deviation (implementable but hypothetical continuation) for the seller, and (ii) showing the

actual continuation equilibrium following period-0 deviation has a lower utility for type α.

The deviation pools the inside interval with the mass choosing the pooling contract of

interest, i.e., it pools the convex hull [α, α]. Consider the induction step.

For (i), the hypothetical equilibrium has a continuation that screens the inside interval

[θl, θr] in the next period, while the period-1 type separation and quantity assignment of the

55

other parts of the interval are the same as the continuation of the original equilibrium. Under

the original equilibrium, qt(θ−l ) < q fb(θl) for t ≥ 0 (c.f. Appendix B), while qt(θ

+l ) = q fb(θl)

for t ≥ 1 due to screening. As q+0 (θ) = q+

0 (θl) for θ ∈ [θl, θr], the convexity of C implies that

the hypothetical equilibrium reduces the cost for serving the inside interval: assigning them

with q0(θ−l ) in period 0 smooths the costs across time, especially for relatively higher types

in the inside interval. It is identical to the original equilibrium in other welfare aspects.

To prove (ii), first observe that the hypothetical continuation is implementable in period

1, hence the actual continuation has higher expected profit in period 1. Then, we use the

induction hypothesis that the continuation exhibits countable interval partition. As α is

the highest type in its period-1 pooling, the continuation equilibrium of the deviation has

qt(α) = q fb(α) for t ≥ 1, same as the original equilibrium. Therefore, there is no concern

of violating (M) at α. By not screening the inside interval, one relaxes the self-imposed

binding (M) over the interval, and obtains a mean-preserving spread of forward quantity

(over types). By Appendix B.3, this further decreases U+0 (α) and completes the proof of

induction step.

The base-case T = 1 uses a slightly different but simpler proof: the actual continuation

screens [α, α]. Let’s check the deviation is implementable. At α, for both the original

equilibrium and the deviation, q1 coincides with first-best quantity. Therefore, the deviation

satisfies (M) at the upper boundary. The internal (M) is trivially satisfied. Applying Lemma

A.6 as above completes the proof.

The remaining task is to prove Lemma 2. Let us follow the procedure laid out in step 1

to obtain the required properties for step 2, and then establish step 2.

Proof of Lemma 2. Suppose there are no other contracts having non-trivial overlap in support

with z and z. Consider two subcases: (1.1) contracts z and z specify the same period-0

quantity; (1.2) they specify different period-0 quantities.

For case (1.1), as the forward quantity has to be the same across the two contracts, one

has the total price for any fixed quantity to be the same under F1 and F1. When merging

the two contracts (in the sense that all types that choose either contract, all coordinate on

one of the two), the posterior becomes some Fm1 (a truncated prior). For the continuation

equilibrium under Fm1 , continuation utilities are weakly lower than those under F1. Lemma

C.2 and C.5 establish the base case and induction step. Under Fm1 , the period-1 continuation

of the contract z and z are still incentive compatible, hence feasible. By revealed preference,

one obtains that the continuation equilibrium yields a higher expected profit than offering

the continuation of z and z when the expectation is evaluated under belief Fm1 . Now, Fm

1 has

lower forward quantities and∫q+

0 (θ)dθ and higher continuation profit, hence the period-0

seller also prefers the continuation equilibrium under Fm1 .

56

For case (1.2), one either merges the two contracts or devises two new incentive contracts

(with the same period-0 quantity as the original contracts z and z respective) such that there

is a threshold v for which the mass (in F1 and F1) consisting of types below it selects one

of the contracts, and the mass above selects the other. The total mass (of F1 and F1) is a

truncated prior, and for such belief, the new contract(s) give rise to a higher expected profit

than the original contracts. Moreover, the continuation utilities are lower than those under

F1 and F1, which again implies that the constructed deviation using the two new incentive

contracts is profitable for the period-0 seller. Lemma C.3 and C.6 establish the base case

and induction step respectively.

When (M) is satisfied at the lower boundary following the above operation, then we

have found a profitable deviation for the seller. For step 2, consider when such (M) is

violated: one increases the cutoff type such that without changing period-0 quantities, the

new continuations match the forward quantity at the new cutoff. The change in cutoffs may

propagate in [θ, v] due to Lemma C.4. Such process restores profit by enhancing distortion

of the lower types (Lemma C.4) and having weakly higher interval boundaries (Lemma C.7).

Now consider when some other contracts share overlap in support with z and z. If all

these contracts have the same period-0 quantity, then the analysis is identical to case (1.1):

merging these contracts gives a profitable deviation. Otherwise, the analysis is similar to

(1.2) but with more distinct threshold types: it requires no more than (n− 1) thresholds for

n distinct period-0 quantities.

Proof of Theorem 1. By Lemma 1 and Lemma 2, a seller-optimal equilibrium exhibits an

interval partition structure. By Proposition 3, we further establish such partition involves

countable intervals.

A.3 Proofs of Section 7

Proof of Lemma 3. Consistent with the dependence diagram, constraints (SC) today only

affect today’s quantity and do not influence constraints in the future: these constraints

are set with respect to types outside the new pooling mass. Therefore, the only relevant

quantity adjustment is that of today, which has a positive shadow cost. This observation

further implies that the profit of any implementable mechanism is upper bounded by ignoring

the shadow cost (violating the constraints), which is further upper bounded by the actual

solution. The last claim follows from the observation that the seller’s solution identified in

the main text is the best among the candidate allocations for seller’s problem in Section 4

which treats the shadow cost of (SC) as 0.

57

Lemma A.9. A new agreement has a weakly higher forward quantity than the continuation

of the previous agreement.

Proof. In period-t with history ht, types in ht+1B reach the same agreement with the seller,

denoted by q+t . As period-(t+ 1) arrives, these types further separate by selecting different

contracts. Types in htB share the same old agreement in period-(t− 1), denoted q+t−1. Such

old agreement specifies a forward quantity for period-t, denoted q+t−1,t. By definition, q∗t (θ) =

qtt. I will call the period-t and period-(t − 1) agreements as current (or new) and previous

agreements respectively.

Suppose to the contrary that for θ ∈ ht+1B , q+

t−1,t > q+t . Consider the following seller

deviation: replace the period-(t− 1) trade proposal of htB, qt−1s s≥t−1, by qt−1

t−1 and qtss≥t(of ht+1

B ), and decrease the total payment by δv(q+t − q+

t−1,t), where v denotes the minimum

of htB. Maintain the quantities of the trade proposals for other subsets of ht−1B , i.e. ht−1

B \htB,

and trace out the relevant (decreased) incentive payment using the envelope formula.

From such deviation, the seller does not change the equilibrium trade profile while she

lowers the rent given to all types in ht+1. Therefore, the deviation maintains the level of

allocation efficiency and improves rent extraction. To see this, under the original menu,

types θ with q+,∗t (θ) < q+

t−1,t should weakly prefer the equilibrium contract to the period-t

agreement. To incentivize, the equilibrium U(θ) ≥ U(v) + (θ−v)q+t−1,t, where the r.h.s is the

utility by sticking with the period-t agreement. The deviation lowers the r.h.s envelope.

Lemma A.10. An agreement has a forward quantity weakly higher than the equilibrium

forward quantity of the highest type of the lower adjacent pooling.

Proof. An argument similar to the proof of Lemma A.9 suffices.

A.4 Proof of Section 8

Similar to the previous proofs, again we use an inductive argument, assuming the results

hold for horizons 0, · · · , T − 1.

Lemma A.11. Suppose there is a unique seller-optimal PBE. If in period-0, the seller offers

a menu consists of the no trade option, and q∗ss of the lowest type in h1B for each on-path

posterior support h1B, then there is a unique continuation equilibrium.

Proof. The base case T = 0 is trivial. To prove the induction step for horizon T , it suffices

to establish the result for period 0.

Let us consider the case where the seller uses the natural implementation, i.e., he offers

a menu consisting of the trade profiles of the lowest type of each equilibrium h1B in period-0.

58

Since these h1B are ordered, we can adopt the same order on the contracts of the menu. With

such order, we will show by induction that each contract is selected by the same interval of

types in any continuation equilibrium given that lower contracts are selected by the respective

same intervals. The following analysis deals with such induction step. Denote the utility

under the seller-optimal equilibrium by U and that under the alternative continuation by U .

We first show that there is no other equilibrium partition in pure strategy. Let us

inspect a contract, for which the seller-optimal equilibrium assigns types [v, v]. Given that

lower contracts are chosen by the same intervals of types, the contract (for the induction

step) cannot have a higher upper boundary of the designated type interval. Otherwise, a

higher upper bound lowers the lower types’ continuation utility in period 1 (Lemma C.4),

and type v has an incentive to deviate to choose the higher contract and stick with the

respective outside option. The contract cannot have a lower upper boundary, v, of type

interval. Otherwise, if continuation has upward or no renegotiation for v+, then its (PC)

binds: U(v+) = U(v) − (v − v)q+0 (v) < U(v) < U(v−), where the last inequality follows

from Lemma C.4. If continuation has downward renegotiation for v+, then it is assigned

with a q+1 ≥ q fb(v). Inside screening is suboptimal (for the seller) regardless of the outside

option, therefore, v+ is either locally pooled or screened (as inside and cross pooling are

not feasible). The “insight” following Lemma 3 says that outside option higher than trade

profile (already) gives high rents for high types, implying that a renegotiation to recover

distortion of low types has no effect in reducing the info rent. Therefore, q+1 (v+) should

be renegotiated either to some q+1 > q fb(v) if locally pooled, or to q+

1 = q fb(v) if it is

screened. Denote the lowest type that is not renegotiated downward as v′ < v. Denote the

renegotiated forward quantity to be q+0 (θ) for θ ≤ v′. Then, with downward renegotiation

of v+, U(v+) = U(v′)−∫ v+v′

q+0 (θ)dθ, where U(v′) = U(v)− (v − v′)q+

0 (v). For q+0 (θ), it has

q0(θ) = q0(v) > q0(θ) and qt(θ) ≥ q fb(θ) for t ≥ 1, therefore, q+0 (θ) > q+

0 (θ) for θ ∈ [v, v′],

which implies that once again U(v+) < U(v+). Hence, v+ is better off by pretending to v−.

Now consider a candidate equilibrium for which some buyer types mix between contract

z and the adjacent higher contract z in period 0. First, the set of types choosing either

contract (convex by Lemma 1) must contain the boundary v. Otherwise, Lemma C.4 and

the analysis in the previous paragraph will identify a type having profitable deviation. In

particular, type v should obtain the same forward quantities following the two contracts.

Following z, v obtains a lower forward quantity than the seller-optimal equilibrium identified

in Section 6: it is no longer the highest type choosing z, so it experiences downward distortion.

Similarly for types in (v−ε, v]. This particularly implies that v renegotiates downward when

choosing z (to match quantity), and so does types lower than v that choose z. From the

previous paragraph, these types, which includes (v − ε, v), are worse off compared to the

59

seller-optimal equilibrium. A lower utility of type v = v − ε, and lower forward quantities

for types in (v, v) implies that v obtains a lower utility under contract z compared to the

seller-optimal equilibrium. Then v strictly prefers z, by which it secures the same utility as

in the seller-optimal equilibrium.

A.5 Proof of Section 9

The following lemma ensures that the proof of Proposition 2 works with the correct conditions

on the reduced-form l, and thereby validates the claim that a PBE involves bootstrapped

recursion.

Lemma A.12. The loss function l defined as above satisfies l1 > 0, w · l2 ≥ 0, l2,2 ≥ 0 and

ω|w|β ≤ |l2| ≤ Ω|w|β for ω = 1/R, Ω = 1/r and β = 1.

Proof. If qC ′′/C ′ ∈ (r, R), then C ′/(qC ′′) ∈ (1/R, 1/r). Now, using the principal’s optimal-

ity in quantity – v = C ′(q) and dv = C ′′(q)dq – one can derive d ln q/d ln v ∈ (1/R, 1/r).

Therefore, for two pairs of (modified) virtual cost and the associated optimal quantity

(v1, q1), (v2, q2) and v1 < v2, a Taylor expansion gives an approximation of (q2 − q1)/q2

between 1/r · (v2−v1)/v2 and 1/R · (v2−v1)/v2. Plugging in v1 = v(θ|θ) and v2 = v(θ|θ)+w,

one can check that l(v(θ|θ), w) has an order of 2 in w, hence l2 has an order of 1 in w:

l(v(θ|θ), w) =

∫ v(θ|θ)+w

v(θ|θ)

(q fb(v(θ|θ) + w)− q fb(γ)

)dγ.

The integrand has order 1 in w by a Taylor expansion, and the integral over a range of w

gives a total order of 2, i.e., lww > 0.

Lemma A.13. For an equilibrium (with interval partition), for a pooling interval [v, v],

pooling it with [v, v + ε) for ε→ 0 will reduce the future loss of [v, v].

Proof. Denote today’s support upper bound by θ > v. Denote the trade profile under pooling

[v, v] as q and that under pooling [v, v + ε] as q. The optimality of q of pooling [v, v] gives∫ v

v

(θq+t+1(θ)− c+

t+1(qs(θ)s≥t+1))dF (θ)−∫ v

v

q+t+1(θ)(F (v)− F (θ))dθ

≥∫ v

v

(θq+t+1(θ)− c+

t+1(qs(θ)s≥t+1))dF (θ)−∫ v

v

q+t+1(θ)(F (v)− F (θ))dθ.

The optimality of q of pooling [v, v + ε] implies that the above inequality is reversed if one

replaces F (v) by F (v + ε), and hence the reversed inequality also holds if one replaces F (v)

by F (θ) > F (v). This also implies that U+t (v) is lower under the new pooling.

60

To prove Theorem 2, we use an induction on T (and the recursive structure) to reduce

the proof to that of the period-0 boundaries and (M). We use a transfinite induction to

reduce the analysis to a boundary θi, assuming properties are shown for the equilibrium

θ1, · · · , θi−1. Therefore, as in the main text, we focus on choosing θi within interval [θi−1, θ].

Lemma A.14. Assume monotone hazard rates and assign quantities based on pooling-

adjusted virtual values. When boundary θi increases, (M) at θi gains slackness.

Proof. Monotone hazard rates ensure the variation in (conditional) virtual values is greater

than the underlying variation in value. When boundary θi increases by dv, the period-1

forward quantities of θ−i increases by the same amount (up to normalization), and the period-

1 forward quantities of θ+i increases by a greater amount. The case of T = 1 is simple: by

monotone hazard rates, the conditional virtual value has a derivative (in the value type)

greater than 1; with positive distortion, the concavity of the inverse function of C ′ implies

that the same variation in (virtual) value induces a greater change in quantity when the value

is smaller. Results for T > 1 are by induction and the pooling quantity formula in Appendix

B. Specifically, the period-1 forward quantities of θ+i has positive distortion by induction

hypothesis (on the validity of the relaxed approach) while that of θ−i is at first-best. Putting

these together gives the result.

Proof of Theorem 2. Lemma A.14 proves step 1 in Section 9.3. Now, we prove the remaining.

When (M) binds, decreasing θi increases the differential loss. The dominating marginal

change in the differential loss is due to a further compression of period-0 pooling quantity

of the lower interval and an inflation of period-0 pooling quantity of the higher interval.

Moreover, lowering θi increases future loss. Hence, (M) does not bind in the equilibrium.

B. Derivations: Second-Best

B.1 Second-Best Allocation

With belief F , write down the profit and monotonicity constraint:

maxq∈Q

∫(q(θ)θ − C(q(θ)))dF (θ)−

∫(1− F (θ))q(θ)dθ

whereQ = q : Θ→ R+, increasing. First, one needs to establish the existence of a solution.

See supplemental material. Next, verify that the optimal cutoff is characterized by

θ(q) ≡ inf

θ

∣∣∣∣∣∫ θ

θ

(v − C ′(q))dF (v)−∫ θ

θ

(1− F (v))dv ≥ 0,∀θ ∈ [θ, θ]

.

61

Suppose to the contrary that the optimal cutoff θ is different from θ(q) at some quantity q.

First, suppose θ < θ(q). By definition of θ(q), there exists some q′ ∈ (θ, θ(q)) such that∫ θ′

θ

(v − C ′(q))dF (v)−∫ θ′

θ

(1− F (v))dv < 0.

Then R(q) increases when changing the cutoff for quantity q from θ to θ′; moreover, this

provides slackness to the (potential) constraints for lower q’s. Therefore, such change in

cutoff gives a profitable deviation. Contradiction.

Now suppose θ > θ(q). By the definition of θ(q),

∫ θ

θ(q)

(v − C ′(q))dF (v)−∫ θ

θ(q)

(1− F (v))dv > 0.

Then R(q) decreases when changing the cutoff for quantity q from θ(q) to θ. Therefore, θ

cannot be the optimal cutoff. Contradiction.

B.2 Virtual Values for Pooling

The main goal is to characterize the equilibrium quantity given the equilibrium separation in

a relaxed program that ignores (M). Therefore, we only consider pooling of adjacent types in

a posterior support, i.e., each pooling is convex in the sense that there are no other types in

the support that is in the convex hull of the pooling region but not included in the pooling.

For the following derivation, F denotes the posterior, θ denotes the highest type in the

support, and qp denotes the pooling quantity.

For pooling an interval [θl, θr], the following gives the marginal benefit and cost:

(F (θ)− F (θr))(θr − θl) + (F (θr)− F (θl))C ′(qp) = (F (θr)− F (θl))θl.

For the L.H.S, an increase in the pooling quantity increases higher type’s info rent by a factor

of θr − θl. The relevant increase in total surplus on the pooling region itself is linked to θl:

by pooling, the principal needs to satisfy the lowest pooling type’s incentive compatibility

and forgo the chance to discriminate the higher pooling types. The pooling virtual value is

defined by C ′(qp) = vp, which gives

vp([θl, θr]) = θl −(θr − θl

) F (θ)− F (θr)

F (θr)− F (θl)

62

Alternatively, one can take the average virtual value of the pooling types:

[F (θr)− F (θl)]−1

∫ θr

θl

(θf(θ)− [F (θ)− F (θ)]

)dθ

= θl + [F (θr)− F (θl)]−1

(∫ θr

θl(θ − θl)f(θ)dθ −

∫ θr

θl[F (θ)− F (θ)]dθ

)= θl + [F (θr)− F (θl)]−1

([(θ − θl)(F (θ)− F (θ))

]∣∣θrθl

)= θl −

(θr − θl

) F (θ)− F (θr)

F (θr)− F (θl),

where the second equality uses an integration by parts.

Under monotone hazard rates, the virtual value is increasing, and therefore, vp([θl, θr])

increases in θl and θr.

B.3 Quantities When (M) Binds

Here, we establish properties of virtual values and quantities on the two sides of a boundary

θi of the optimal partition, where (M) binds for such boundary. The mean-preserving spread

argument goes as follows: when (M) binds at θi in period 0, q0(θ−i ) is distorted further

downward while q0(θ+i ) is less distorted relatively to the pooling-adjusted virtual value pinned

down in the previous subsection so that

(F (θi)− F (θi−1))[vp([θi−1, θi])− C ′(q0(θ−i ))] = (F (θi+1)− F (θi))[C′(q0(θ+

i ))− vp([θi, θi+1])]. (1)

The above equations demonstrate a balanced weighted increase in net wedge, which gives

a mean-preserving spread of the marginal costs of quantity provision relative to those of

the second-best pooling quantities. By concavity of the inverse of C ′, for q0 denoting

the allocation ignoring (M), one has∫ θθq0(v)dF (v) ≥

∫ θθq0(v)dF (v). Clearly, one has∫ θ

θq0(v)dv ≥

∫ θθq0(v)dv when f is increasing. Now, let’s show for general F and the equi-

librium thresholds θi’s, one still has∫ θθq0(v)dv ≥

∫ θθq0(v)dv.

Suppose to the contrary that∫ θθq0(v)dv <

∫ θθq0(v)dv, then there is some irregularity (in

virtual value) in the higher interval such that having a higher boundary recovers efficiency

in period 0 and rent extraction in future periods, which contradicts the optimality of θi = θi.

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C. Derivations: Inequalities

C.1 Two-period Case

We first state an auxiliary property of θ(q|F1), i.e., comparative statics in marginal price is

useful in providing the base-case (T = 1) analysis of Section 6.

The marginal price in period-1 (under posterior F1) at quantity q is

θ(q|F1) ≡ inf

θ

∣∣∣∣∣∫ θ

θ

(v − C ′(q))dF1(v)−∫ θ

θ

(1− F1(v))dv ≥ 0,∀θ ∈ [θ, θ]

.

Denote θ(q|[v, v]) as the marginal price when posterior is the prior truncated on interval

[v, v]. By definition of θ(q|·):

Lemma C.1. θ(q|[v, v]) increases in v for a fixed v.

Proof. When F1 is the truncated prior, the sign of

∫ θ

θ

(v − C ′(q))dF1(v)−∫ θ

θ

(1− F1(v))dv

is the same as that of ∫ θ

θ

(v − C ′(q))dF (v)−∫ θ

θ

(F (v)− F (v))dv

For a higher v, the above expression strictly decreases, therefore the set of types assigned a

fixed quantity q is higher when v increases. Hence, θ(q|[v, v]) increases in v.

Use the same notation z and z as in the main text (Lemma 2), and F1 and F1 the

respective posterior belief. Then the set of equilibrium forward quantities q+1 under F1 and

F1 are either weakly ordered or one set nests the other. This is because inside pooling

and cross pooling are infeasible. Without loss of generality, the quantity range of F1 covers

lower quantities that F1 does not. Denote the posterior whose quantity range covers higher

quantities (that the other does not) by F1 ∈ F1, F1. The union of their supports is [v, v].

First, consider z and z specify the same q0. The previous paragraph implies that F1 and

F1 have the same upper bound of the support.

Lemma C.2. If Fm = βF1 +(1−β)F1 and for overlapped quantity range θ(q|F1) = θ(q|F1),

then Fm has weakly lower marginal price: θ(q|Fm) ≤ θ(q|F1) .

64

Proof. For any q in the range of equilibrium period-1 quantity for both F1 and F1, denote

θ = θ(q|F1) = θ(q|F1), one has

∫ θ

θ

(v − C ′(q))dG(v)−∫ θ

θ

(1−G(v))dv ≥ 0 ∀θ ∈ [θ, θ], (2)

for both G = F1 and G = F1. A convex combination of the above inequality using weight β

on G = F1 and complementary weight on G = F1 gives

∫ θ

θ

(v − C ′(q))dFm(v)−∫ θ

θ

(1− Fm(v))dv ≥ 0 ∀θ ∈ [θ, θ],

therefore θ(q|Fm) ≤ θ. Conversely, one cannot have θ(q|Fm) equals to some θ < θ. As

θ(q|F1) > θ, one has∫ θ

θ

(v − C ′(q))dG(v)−∫ θ

θ

(1−G(v))dv < 0 ∀θ ≤ θ,

for G = F1, and similarly for F1. Therefore, it particularly implies that the last inequality

holds at θ = θ for Fm, and hence θ(q|Fm) 6= θ.

For q in the low-range covered only by F1: as overlapped types do not contribute more

to the l.h.s of the above inequality, yet merging introduces more distortion

∫ θ

θ

(v − C ′(q))d[βF1(v)]−∫ θ

θ

(1− βF1(v))dv ≥ 0 ∀θ ∈ [θ, θ],

which effectively increases the distortion factor 1 in inequality (2) to 1/β. Therefore, for

lower q, θ(q|Fm) ≤ θ(q|F1).

Now, consider when z and z specify different q0’s, w.l.o.g the latter specifies a higher

q0. Since the two supports have non-trivial overlap, and to match up the total quantity, a

necessary condition is that the highest type of F1, denoted v, is in the interior of convex hull

of F1: such type receives first-best q1 under F1, and has some distortion under F1.

For such case, we consider two types of merges: (i) merge the mass below v, with posterior

F l and F h following a contract zl for types below v and a contract zh for types above; (ii)

merging z and z altogether to get posterior Fm following a merged contract zm. Specify q0

of zl (zr) to be the same as z (z). I will specify q0 of zm in the proof. For ease of discussion,

I use superscript to denote the contract of the discussed quantities.

Lemma C.3. When z and z specify different period-0 quantities, a merge of either form

65

increases the total price.

Proof. First observe that under z, q1 increases near v, hence v is not involved in an ironing,

which implies that in either way of merging, types above v receives period-1 quantity same

as q1 under z. For each θ in the overlap, q1 and q1 differ by the same amount as δ−1(q0− q0).

As z and z have identical ironing regions in period 1, under zm, for each non-ironed type in

the overlap of support, C ′(qm1 ) = βC ′(q1)+(1−β)C ′(q1), and qm1 ∈ [q1, q1]. Similar to Lemma

C.1, for types below such overlap, qm1 < q1. By convexity of C, q1(v)−qm1 (v) < q1(v)−qm1 (v)

for any v in the overlap of the support. Therefore, using the fact that v receives the same

forward quantity under z and z, we have qm1 (v)−qm1 (v) < (1−δ)q0+δq1(v)−(1−δ)q0−δq1(v).

Therefore, there is extra freedom in specifying qm0 so that (M) is satisfied at both v and v.

Choose qm0 to have binding (M) at v. As all types receive weakly lower q+0 under zm, total

price increases.

Now, consider contracts zl and zr. Similar to the above argument, for types below

v, ql1 ≤ q1. Compared to z and z, forward quantities decrease and total price increases.

Compared to zm, period-0 allocation is more efficient (due to earlier separation) and period-

1 has higher quantities.

Now, we check that the above construction increases profit in period-0 perspective. The

deviation in Lemma C.2 has higher profit in period-1 perspective, and by Lemma A.5, the

lower quantities translate to higher ex-ante profit.

By revealed preference, the continuation of zm in Lemma C.3 has higher profit in period-

1 perspective (truncated prior). Moreover, the integral of period-1 utility is lower, and by

Lemma A.5, the continuation has higher profit in period-0 perspective. The period-0 profit

is lower. If such loss in period-0 profit is small, then use merge contract zm. Otherwise, the

loss in allocation efficiency in period 0 (due to the merge) outweighs the extra rent extracted

from types [v, v], in which case, the new contracts zl and zr perform the duties as they

trade-off future rents (on types [v, v] choosing z) for current allocation efficiency.

The following deal with a general finite horizon. See proofs in supplemental material.

C.2 More than Two Periods

Let’s consider a truncated prior on [v, v] in period t. We first extend Lemma C.1.

Lemma C.4. For θ ∈ [v, v], the promised utility∫ θvq+t+1(v)dv decreases in v for a fixed v.

For this lemma, it suffices to show for period t = 0. Results for future periods follow

from the induction hypothesis on shorter horizons.

Next, we extend Lemma C.2 and C.3:

66

Lemma C.5. If z and z specify the same period-0 quantity, then merging these two contracts

yields lower promised utilities for all buyer types.

Lemma C.6. If z and z specify different period-0 quantities, then merging the mass below

v yields lower promised utilities for all buyer types; merging F1 and F1 also yields lower

forward quantities for all buyer types.

Finally, one needs to deal with the case where the constructed deviation incurs violation

of monotonicity at the low boundary.

Lemma C.7. For an allocation that satisfies (M) and interval partition at any history but

violates (SRC) at t = 1, suppose the period-1 seller-optimal continuation yields higher ex-

pected profit and lower utilities. Then the seller-optimal equilibrium yields a higher profit

and lower utilities in period 0.

Let me provide a sketch of the construction. Consider (a’) separating the same way as

the initial allocation in period 0, followed by a seller-optimal continuation. The induction

hypothesis of countable intervals implies that (a’) assigns first-best q+1 to the left of the

boundary type regardless. On the other hand, by assumption, (a’) assigns a lower q+1 to

the right of a boundary type. Therefore, relative to the original allocation, (a’) has more

stringent (M) in period 0, and by Lemma A.6, (a’) has a lower∫ θθq0(v)dv. The remaining

step is to construct a deviation (a) from (a’) that has higher interval boundaries than (a’) to

ensure that (a) obtains a higher period-0 profit than the original allocation while benefiting

from having the lower continuation utilities.

D. Derivations for the Examples

D.1 Derivation of Example 1

Here we derive the results shown in Table 1. For 1-pooling, the seller pools the lowest types

[1, θ], and screens types [θ, 2]; she looks for the profit-maximizing θ ∈ (1, 2). For ∞-pooling,

by the linear recursion argument in Section 9, the optimal cutoffs (of pooling intervals) takes

the form: θi = 2− (1− h)i for i ≥ 0; she looks for the profit-maximizing h.

The quantities are, assuming monotonicity (which is satisfied in equilibrium) easily to

compute. Under 1-pooling: q0(θ) = θ− 1 and q1(θ) = 2θ− θ if θ ∈ [1, θ] and q0(θ) = 2θ− 2,

q1(θ) = θ otherwise. Under ∞-pooling: q0(θ) = qi0 ≡ (θi + θi+1)− 2 and q1(θ) = 2θ − θi+1 if

θ ∈ [θi, θi+1]. Under separating (instant screening) mechanism, q0(θ) = 2θ− 2 and q1(θ) = θ.

Under commitment, q0(θ) = q1(θ) = 2θ − 2.

67

Given the quantities, the profit and welfare can be expressed as∫ 2

1

dθ

[g(θ)(q0(θ) + δq1(θ))− q0(θ)2 + δq1(θ)2

2

]where g(θ) = v(θ) = 2θ− 2 for the case of profit, and g(θ) = θ for the case of welfare. Table

1 uses δ = 1 to simplify algebra. So up to this point, one is done with the case of separating

mechanism and commitment benchmark. The remaining task is to derive the equilibrium

θ for the optimal 1-pooling and h for the optimal ∞-pooling. In either case, the relevant

policy maximizes profit among the respective constrained set of mechanisms.

With some algebra, the profit under 1-pooling using cutoff θ can be written as 2−0.5θ3 +

2θ2 − 2.5θ + 1/6, which is maximized at θ = 5/3. The profit under ∞-pooling using linear

recursive parameter λ is (when δ = 1)

∑i≥0

∫ θi+1

θi

dθ(2(θ − 1)qi0 − (qi0)2/2

)+

∫ θi+1

θi

dθ 2(θ − 1)(2θ − θi+1 − (2θ − θi+1)2/2

). (3)

To simplify notation, denote vi = θi− 1. The sum over these first integrals (of different i) is∑i

qi0((vi+1)2 − (vi)2)−

∑i

(vi+1 − vi)(qi0)2/2

=∑i

(vi+1 + vi)((vi+1)2 − (vi)2)−

∑i

(vi+1 + vi)2(vi+1 − vi)/2

= 1/2[∑

i

((vi+1)3 − (vi)

3)

+∑i

vi+1vi(vi+1 − vi)]

= 1/2

[1 +

λ(1− λ)

1− (1− λ)3

].

In the last equality, we telescope the first sum and use vi = 1− (1− λ)i for the second sum.

One can use a similar trick to rewrite the sum of the second integrals in Equation 3:

∑i

2

3((vi+1)3 − (vi)

3)−∑i

1

2[vi+1 − vi](1− λ)2i+2 =

2

3− 1

2

λ(1− λ)2

1− (1− λ)3.

To maximize the sum of these two terms 5/6 + 1/2 · λ2(1− λ)/(1− (1− λ)3), the first-order

condition yields λ = 1.5 −√

3/2 ' 0.634. The evaluation of the total welfare under such λ

(associated with ∞-pooling) requires a similar method in telescoping and recursive sum.

68

D.2 Derivation of Loss Functions in Section 10

To this end, write L/2 as the sum over these integrals (up to relevant time discounting)

∫ θi+1d

θid

(θ − θi+1

d + θid2

+θd − θe

2

)2

dθ =I3

12+

(θd − θe)2

4I,

where I = θi+1d − θid. Denote the discounted sum over the first term I3/12 as L0 and that

over the second term (θd − θe)2I/4 as L2. Then L = L0 + L2.

For V , one needs to expand L at a slight deviation

∫ θi+1d

θid

(θ − θi+1

d + θid2

+ε+ θd − θe

2

)2

dθ =I3

12+

(θd − θe)2

4I +

ε2

4I +

ε

2(θd − θe)I.

The discounted sum of the new third term is ε2/4. Denote the discounted sum over the

new fourth term as εL1. Furthermore, one can check that L1 is homogeneous of degree 2 in

the support size. Then evaluating ε at 1 − θ, and support size θ, one obtains L([0, θ]|1) =

θ3L+ θ(1− θ)2/4 + (1− θ)θ2L1. Hence, the objective to minimize is

f(θ) = (L0 + L2)((1− θ)3 + δθ3) +1− δ

12θ3 +

δ

4θ(1− θ)2 +

δL1

2(1− θ)θ2.

Claim 4. For the above expressions,

L0 =1− δ

12

λ3

1− (1− λ)3 − δλ3

L1 =δ

2

1− λ2− λ− λδ

L2 =δ

4

λ(1− λ)2

1− (1− λ)3 − δλ3

2− λ+ δλ

2− λ− δλ.

See supplemental material for a proof by recursion.

69