Performance and Turnover in a StochasticPartnership

David McAdams, Duke University

NBER Conference on Relational Contracts, November2009

Motivation: endogenous stability

A long-lasting (“stable”) relationship is essential for effectiveinformal incentives.

Stability is often treated as exogenous, but empirical research hasfound that real-world partnerships display endogenous stability.

▶ Stylized fact: older partnerships tend to be more stable andmore productive.

This paper develops a theory of endogenous stability anddynamic performance in partnerships, when payoffs evolve overtime according to a controlled stochastic process.

As an application, I derive the value and steady-state distributionof a partnership economy with anonymous re-matching.

Shanghai GM: cooperation breakdown?

“Kevin Wale, the president of GM’s China operations,said that the company remained ‘very comfortable’ withthe partnership, and that the Shanghai company’s recentintroduction of its own sedans in competition had shown‘no significant impact’ on GM’s own sales.”

“General Motors announced ... that it would build a[solely-operated] advanced research center in Shanghai todevelop hybrid technology and other designs, in the latestresearch investment in China by a foreign automakerdespite chronic problems with purloined car designs.”– Int Herald Tribune, October 29, 2007.

Should we expect the GM-Shanghai Automotive partnership todissolve soon, given their apparent recent failure to cooperate fully?

Outline of talk

▶ Model & overview of results

▶ Simple example: dynamic Prisoners’ Dilemma

▶ Partnership economy with anonymous rematching

▶ Related literature

▶ Directions for future work

Model

Each period t = 0, 1, 2, ... in an active partnership proceeds as:

v

Uncertainty realized

?v

Observable efforts

6

v

Both stay?

@@R��

�����Yv

PPPPPPPN v Exit

Retention bonuses

v

?-

▶ Each player seeks to maximize expected present value ofstream of stage-game payoffs.

▶ Players share common discount factor � ∈ (0, 1).

Model

Each period t = 0, 1, 2, ... in an active partnership proceeds as:

v

Uncertainty realized

Uncertainty realized

?v

Observable efforts

6

v

Both stay?

@@R��

�����Yv

PPPPPPPN v Exit

Retention bonuses

v

?-

▶ State xt ∈ Xt publicly observed; (Xt ,ર) partially ordered.

▶ Distribution of Xt depends only on (t, xt−1, et−1).

▶ “Persistence”: x ′t ર xt implies Xt+1∣(x ′t , et) FOSDXt+1∣(xt , et) for all et .

Model

Each period t = 0, 1, 2, ... in an active partnership proceeds as:

v

Uncertainty realized

?v

Observable efforts

Observable efforts

6

v

Both stay?

@@R��

�����Yv

PPPPPPPN v Exit

Retention bonuses

v

?-

▶ Effort eit ∈ ℰt simultaneous then observed; (ℰt ,ર) partiallyordered with minimal element “0”.

▶ Stage-game payoff �it(et ; xt) from efforts et = (eit , ejt) (i)weakly decreasing in eit , (ii) weakly increasing in ejt , (iii)satisfies weakly increasing differences in (et ; xt), and (iv)�it(0; xt) = 0 for all xt .

Model

Each period t = 0, 1, 2, ... in an active partnership proceeds as:

v

Uncertainty realized

?v

Observable efforts

6

v

Both stay?

Both stay?

@@R��

�����Yv

PPPPPPPN v Exit

Retention bonuses

v

?-

▶ The partnership ends if either player chooses to quit, or withexogenous probability � ∈ [0, 1] if both stay.

▶ Upon exit, player i enjoys exogenous outside option vi andzero payoffs thereafter.

Overview of Results

Welfare-maximizing equilibrium construction. Focus onsymmetric pure-strategy subgame-perfect equilibrium (SPSPE). Inspirit of APS, but additional structure yields comparative statics.

Comparative statics in this eqm. Joint welfare is increasing inthe state. If Xt depends only on (t, xt−1), the following hold truein higher states:

▶ joint stage-game payoff is higher

▶ joint continuation payoff is higher

▶ hazard of exit is lower; indeed conditional stopping time of thepartnership is FOSD-higher

Partnership economy. Steady state of a partnership economy isderived, assuming “anonymous re-matching”, thereby endogenizingthe outside option.

Outline of talk

▶ Model & overview of results

▶ Simple example: dynamic Prisoners’ Dilemma

▶ Partnership economy with anonymous rematching

▶ Related literature

▶ Directions for future work

Example: dynamic Prisoners’ Dilemma

Symmetric outside option v ≥ 0.

Stage-game payoffs from effort take the form:

Work Shirk

Work wt ,wt −wt − dt ,wt + dt

Shirk wt + dt ,−wt − dt 0,0

xt = (wt , dt) ∈ R2+ follows an exogenous stochastic process.

Example 1: wt iid, dt = d > 0 [Ramey Watson 97]

Example 2: log(wt) random walk, d = 0 [Levinthal 91]

Example 3: wt =∑

s≤t yst , ys ∼ N(w , �) iid, d = 0 [Jovanovic 79a]

I will focus here on another special case in which � = 12 and

▶ wt = 1 for all t

▶ log(dt) symmetric random walk

Summary of findings

Efficient benchmark: work forever regardless of the state.

-

incentiveto shirkdt

6

outside option v10

WORK EXIT

Summary of findings

Unchanging benchmark: either work forever or quit immediately.

-

1���

incentiveto shirkdt

6

outside option v10@@

@@@

@@@

@@@

@@@

WORK

EXIT

Summary of findings

Dynamic game: log(dt) evolves according to a random walk.

-

exitthreshold

12

workthreshold

incentiveto shirkdt

6

outside option v1vv0

EXIT

SHIRK

WORK

������

“DOOMED”

Structure of the “optimal” SPE.

Corollary. Suppose that v ∈ [0, 1). There exist dW ≤ dout suchthat, in a SPE that maximizes joint (discounted) payoffs:

1. [good times] both work and stay when dt ≤ dW

2. [hard times] both shirk and stay when dW < dt ≤ dout

3. [exit] both shirk and quit when dt > dout

4. [after any deviation] both shirk and quit

STATE

dW dout

work shirk exit

Cooperation is harder in the changing game

Players cooperate in equilibrium in an even smaller set of states,compared to the unchanging case, for two main reasons:

1. Exit: fewer future periods in which to cooperate.

2. Hard times: losses in periods without cooperation.

Stability

100%0

Eqmgains

1

v

vv

Cooperation is harder in the changing game

Players cooperate in equilibrium in an even smaller set of states,compared to the unchanging case, for two main reasons:

1. Exit: fewer future periods in which to cooperate.

2. Hard times: losses in periods without cooperation.

Stability

100%0

Eqmgains

1

v

v

v

Cooperation is harder in the changing game

Players cooperate in equilibrium in an even smaller set of states,compared to the unchanging case, for two main reasons:

1. Exit: fewer future periods in which to cooperate.

2. Hard times: losses in periods without cooperation.

Stability

100%0

Eqmgains

1

vv

v

Graphical intuition why dW = 12 when v = 0

STATEd0 1

2

work shirk

If d0 ≤ 12 , cooperation can be supported both now and

frequently in the future.

Stability 10

Eqmgains

0%

50%

100%

0%

50%

100%

0%

50%

100%

10

0

d0

Graphical intuition why dW = 12 when v = 0

STATEd01

2

work shirk

If d0 >12 , cooperation can be supported neither now nor

frequently in the future.

Stability 10

Eqmgains

0%

50%

100%

0%

50%

100%

0%

50%

100%

10

0

d0

Graphical intuition why dW = 12 when v = 0

STATE12

work shirk

If d0 = 12 , future gains equal current incentive to shirk.

Stability 10

Eqmgains

0%

50%

100%

0%

50%

100%

0%

50%

100%

10

0

d0

Summary of behavior in the optimal SPE

Example: log(dt) follows random walk

-

exitthreshold

workthreshold

incentiveto shirkdt

6

outside payoff v1vv0

EXIT

SHIRK

WORK

������

“DOOMED”

Dynamics of partnership performance

STATE

dW

both work both shirk

Regime persistence: partners who have enjoyed several periodsof cooperation are more likely to remain cooperative.

Period 2 3 4 5 10 25

% work resumes 25% 16.7% 12.8% 9.8% 5.0% 2.0%

Table: Probability that cooperation first breaks down in period t,conditional on d0 = dW and cooperation in periods 1, ..., t − 1.[Assuming v = 0 and log(dt+1) = log(dt) + �t where �t ∼ U[−1, 1] iid.]

Partnership persistence: when v > 0, similarly, hazard of exitmonotonically decreasing in age once partnership is old enough.

Outline of talk

▶ Model & overview of results

▶ Simple example: dynamic Prisoners’ Dilemma

▶ Partnership economy with anonymous rematching

▶ Related literature

▶ Directions for future work

Model: partnership economy

Unit mass of players, with flow (1− �) of births and deaths. Eachplayer dies with iid probability (1− �) each period, seeks tomaximize expected lifetime earnings.

Each period that a player is not already matched, he isautomatically and costlessly matched with a new partner.

Each new match is a “fresh start” in two senses:

1. The stochastic state process is iid across partnerships (andcontrolled only by efforts in the current partnership).

2. Neither player knows anything about his partner’s priorhistory, including age, number of past partnerships, etc.

Thus, any welfare-maximizing equilibrium of the overall economyspecifies welfare-maximizing play in each partnership, taking intoaccount the endogenous outside options v∗ ≥ 0 from re-matching.

Fit within stochastic partnership model

Each individual partnership can be interpreted as an instance ofour stochastic partnership game, under the following parameterrestrictions:

▶ Discount factor �: Given death-rate (1− �), each player actsas if maximizing discounted earnings.

▶ Exogenous separation probability � = (1− �): Conditional onown survival, each player’s partner dies with prob. (1− �).

▶ Outside options v∗ = (v∗i , v∗j ): determined endogenously by

players’ expected continuation payoff from being re-matched.

Maximal economy-wide equilibrium welfare

Theorem 2 (Maximal social welfare). Suppose thatE [Π

eqmΣ0 (X0; vΣ)] is continuous in vΣ Sufficient conditions . In all

social-welfare maximizing partnership-economy equilibria, players’expected joint payoff when first matched is v∗Σ/�, where v∗Σ isunique solution to

v∗Σ = E [�ΠeqmΣ0 (X0; v∗Σ)]. (1)

Further, players in each partnership play a joint-welfare maximizingSPE given their endogenous outside options.

Uniqueness arises as LHS of (1) rises more quickly than RHS.Intuition:

▶ Holding equilibrium efforts fixed, marginal joint benefit fromhigher outside options is at most one-for-one.

▶ Set of IC efforts shrinks as vΣ increases ⇒ performance insidethe partnership deteriorates with higher outside options.

Steady-state distribution of partnership histories

Through the process of death and re-birth, all histories reached onthe eqm path communicate and are positively recurrent. Details

⇒ the Markov chain of partnership histories is ergodic.

Corollary (Steady-state distribution). In a partnership economywith welfare-maximizing SPSPE play within each partnership, thereexists a unique steady-state distribution over partnership histories.

Aside: Multiple welfare-maximizing SPSPE may exist with differentwage transfers. (As defined above, “history” does not includewages.)

A “typical” life in the partnership economy

Play tends to pass through a few (mostly) distinct phases:

▶ Dating.

▶ Honeymoon.

▶ Good times.

▶ Hard times.

▶ Golden years.

with typical transitions

▶ Dating → Honeymoon

▶ Honeymoon → Good times or Hard times

▶ Hard times → Dating or Good times

▶ Good times → Golden Years or Hard times

▶ Golden years absorbing (until death do they part)

Dating

When searching, players will churn through a sequence ofone-period, zero-effort partnerships.

Discussion: The assumption that X0 is atomless is crucial for thisresult. Given a non-stochastic repeated partnership,welfare-maximizing equilibria exhibit an “incubation period” withthe first person that you meet (Kranton (1996), MailathSamuelson (2006)).

“Dating equilibria” also arise as welfare-maximizing if such playershave access to a public randomization device.

Once there is non-trivial payoff-relevant variation in “initial fit”,dating equilibria strictly outperform incubation equilibria.

Good times / Hard times

STATE

dW

both work both shirk

The partnership transitions, in equilibrium, between “good times”when both players work (dt ≤ dW ) and “hard times” when bothshirk (dt > dW ).

Given v > 0, exit is typically preceded by hard times.

Players remain in a rocky relationship because of the (endogenous)option value created by the possibility of future cooperation.

Honeymoon

STATEdW dout

work shirk exit

Players only form an enduring partnership in good times, ifbeing unmatched yields a positive payoff flow. (Otherwise, they arestill “likely” to form a partnership in good times.)

Consistent with findings on the survival of organizations but not onthe survival of marriages. How to reconcile this?

Survivorship bias

STATEdW dout

work shirk exit

Partnerships that have survived a long time are likely to lasteven longer.

Consistent with findings on the survival of organizations (Levinthal1991), employment relationships (Topel Ward 1992), andmarriages (Stevenson Wolfers 2007).

Golden years

After high enough histories, the partnership becomespermanent (until death).

Golden years can arise for two distinct reasons:

1. Absorbing increasing subset of the state-space.

2. High efforts in high states, combined with overwhelmingpositive feedback from effort.

Outline of talk

▶ Model & overview of results

▶ Simple example: dynamic Prisoners’ Dilemma

▶ Partnership economy with anonymous rematching

▶ Related literature

▶ Directions for future work

Dynamics in relationships: some classic models

Green Porter (1984) — imperfect monitoring

▶ More precise prediction: during non-cooperative regime, therate at which cooperation resumes falls over time.

Ghosh Ray (1996), Watson (2002) — building trust (signaling)

▶ Different implication: trust not monotonically increasing induration of partnership, though survivors tend to be thosewho have built up more trust.

Jovanovic (1979a,b), Pissarides (1994) — learning about matchquality, match-specific investment, and on-the-job search

▶ Key difference: this paper adds two-sided moral hazard, aswell as rich investment [but no on-the-job search].

Outline of talk

▶ Model & overview of results

▶ Simple example: dynamic Prisoners’ Dilemma

▶ Partnership economy with anonymous rematching

▶ Related literature

▶ Directions for future work

Promising future directions

The tractability and flexibility of the model invite variousextensions and suggest various explorations as future work:

▶ Macroeconomic volatility.

▶ Poaching.

▶ Changing individuals.

▶ Networking.

▶ Endogenous learning.

Macroeconomic volatility

Consider a partnership economy with costly re-matching andcommon productivity shocks, e.g. with stage-game payoffs

Work Shirk

Work t , t − t(1 + dt), t(1 + dt)

Shirk t(1 + dt),− t(1 + dt) 0,0

where t is a common to all partnerships and dt is idiosyncratic.

Conjecture: In the welfare-maximizing equilibrium, the players’dt-thresholds to exit and to work are decreasing in t .

Implication #1: In a deep recession, players’ outside option isunemployment. This makes them work harder in an existingpartnership, dampening the pain of a deep recession.

Implication #2: In boom times, players will dissolve relationshipsmore freely, and realize lower effort for any given dt .

Poaching

The current partnership economy analysis does not exploit thepossibility of on-the-job search, e.g.

▶ “search intensity” = the number of potential mates that youmeet, in either a directed context (“bar”) or undirectedcontext (“street”)

▶ “outside option” = payoff from leaving your current partnerand matching with the most desirable of these potentialmates, among those who are willing to match with you.

Conjecture: Welfare-maximizing search cost is non-zero as long as“initial fit” is sufficiently unimportant.

Changing individuals / Networking

Assumptions on the stochastic process rule out interestingapplications. Relaxing these assumptions will be an importantdirection for future work, e.g.

▶ Stochastic partnerships are iid across states. This rules outthe possibility that individuals have stochastic characteristics(such as age, beauty, skills) that they carry with them.

▶ Investments today have no impact on future outside options.This rules out the possibility that players may develop anetwork of contacts.

Endogenous learning

This paper’s model can capture various sorts of exogenous publiclearning (a la Jovanovic (1979a)): let xt = summary statistic ofshared belief.

Further, players’ “effort” could impact on the learning process.

Speculation: should players be able to influence the precision oftheir signal each period about the underlying (changing) state,

▶ Players will pay to observe more precise signals at first

▶ ... but seek to learn less about the match once in good times

▶ ... until their relationship is “on the rocks”, when they willonce again seek to learn more.

THANK YOU

More on the “honeymoon effect”

Stinchcombe (1965) argued that partnerships can be especiallyunstable when they are young if, among other reasons, players areuncertain about each other’s type and quickly learn whether theyare a good match.

This is consistent with the dynamics of my model, when onecounts “dates” as relationships. Indeed, the model generates alarge mass of quickly-dissolving partnerships, followed by a lull inbreak-ups. This pattern is found in data on American marriages.

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Sufficient conditions for continuity of E [ΠeqmΣ0 (X0; vΣ)]

Definition: The initial state x0 includes an “independenteffort-incentivizing type” s0 if (i) xt = (s0, yt) for all t ≥ 0, (ii)S0 ∈ R is atomless and independent of (Yt : t ≥ 0), and (iii)�it(e ′t ; s0, yt)− �it(et ; s0, yt) is strictly increasing in s0 for alli , t, yt , e

′t ≻ et .

Claim 2: Suppose that the initial state includes an independenteffort-incentivizing partnership type, and that ℰt is finite for all t.Then E [Π

eqmΣ0 (X0; vΣ)] is continuous in vΣ.

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Steady-state distribution of partnership histories

Some notation:

▶ ht = (xt , et−1): payoff-relevant history at start of period t.

▶ et(ht): symmetric effort chosen at history ht .

▶ pexitt (ht): probability at least one player exits.

▶ Xt+1(ht) ∼ Xt+1∣(ht , et(ht)): random next-period state.

Transition probabilities among histories are fully described by:

▶ With probability 1− �2, each partnership will end by death,replaced by a new one having random initial history H0 = X0.

▶ With probability �2pexitt (ht), it will end due to some partner’s

endogenous departure, and also be replaced by a new one.

▶ Otherwise, the partnership will continue to time t + 1, withaugmented random history Ht+1 = (ht ; et(ht); Xt+1(ht)).

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