Internet Trading Mechanisms and Rational

Expectations

Michael Peters and Sergei SeverinovUniversity of Toronto and Duke University

First Version -Feb 03

This Version - December 11, 2003

Abstract

This paper studies an internet trading mechanism similar to theone described in (Peters and Severinov 2001) in a market where tradersvalues are interdependent. Conditions are given for which this mech-anism has a perfect Bayesian equilibrium which supports allocationsthat are the same as the allocations supported by a rational expecta-tions equilibrium. In particular, this equilibrium supports allocationsthat are ex post efficient. We show how to construct the rationalexpectations equilibrium belief function from posterior beliefs at ter-minal information sets of the internet trading game. The mechanismis also compared to a double auction.

This paper models competition in an auction market where buyers’ andsellers’ valuations are interdependent. Rather than having sellers competein reserve prices, as has been commonly assumed in the previous literature,1

we assume that the competition occurs in an internet auction market similarto the one studied by (Peters and Severinov 2001). Sellers cannot committo public reserve prices, but they are free to bid in any auction (includingtheir own) at any time during the bidding process. We give conditions thatresemble those found in a very large market, such that there is an equilibriumfor the internet trading mechanism for which trade is post efficient and occursat a uniform price.

1For example (McAfee 1993, Peters and Severinov 1997).

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Internet auctions provide a rich source of readily accessible data on bidderbehavior in auctions. One way to exploit this is to use the outcomes in thedifferent auctions as separate data points that can be used to test particularpredictions of auction theory.2 One of the main implications of (Peters andSeverinov 2001) is that a market based approach to the data may be moreeffective than an auction theoretic approach. So for example, the price thatprevails in some sellers’ internet auction will vary with aggregate demand andsupply, but will otherwise not be sensitive to the number of actual biddersthat participate in the seller’s auction.

We show that a similar conclusion can be drawn when values are interde-pendent. The equilibria that we derive for the internet auction market can beassociated with a rational expectations equilibrium for the market in whichall traders’ beliefs conditional on any particular price and trade are just theexpectation of traders’ posterior beliefs when the internet auction marketsleads to the same outcome and price. This again suggests that it may beadvantageous to consider data from internet auction markets by applyingmarket based rational expectations reasoning to the whole auction marketinstead of treating each auction as an independent observation.3

From the opposite perspective, the internet trading mechanism also pro-vides some insight into rational expectations equilibrium. Like all competi-tive solutions, rational expectations equilibrium is best interpreted as an ’asif’ solution concept. Traders are actually engaged in some complicated non-cooperative game, but under the right conditions they behave as if they wereengaged in a competitive market. This kind of interpretation is difficult withrational expectations because it presumes that agents view prices, revise be-liefs, then take appropriate actions. Non-cooperative mechanisms work theother way around with traders actions determining price. On the surface itis difficult to see how this ’as if’ interpretation might work.

We deal with this with the following approach - we first show, as men-tioned above, that the perfect Bayesian equilibrium that we study supportsan outcome where all trades occur at a common price. Nonetheless, thisprice is the outcome of agents actions, not something that agents respond to

2For example, (Bajari and Hortacsu 2002) measure the winners’ curse in ebay auctionsby measuring the impact that the number of bidders has on the trading price in eBayauctions.

3One implication of both (Peters and Severinov 2001) and this paper is that it wouldbe an error to treat data from different eBay auctions, for example, as independent drawsfrom some common distribution.

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when they make decisions. Once the mechanism plays itself out it generatesan outcome consisting of a price and a set of trades (or decision not to trade)for all agents. Looking at this terminal outcome, agents can figure out, fromtheir knowledge of the strategies the others are using, what types for theother agents are consistent with the part of the outcome that they can see -that is, the price and their own trading decision. This calculation is the basisfor a posterior belief conditional on price (and the agents trading outcome).We then ask whether the trading outcome is optimal for the agent given theterminal price and this posterior belief. If it is, we say that the mechanismhas the rational expectations property. If an outcome function has the ra-tional expectations property it will clearly support a rational expectationsequilibrium.4 A corollary to our main theorem shows that the internet auc-tion mechanism along with the perfect Bayesian equilibium we describe hasthe rational expectations property.

So in the end, our results provide two important insights. The first simplyshows what it means for rational expecations to approximate the behaviorof a trading mechanism. The second more applied contribution is to showthat the inherantly complex and chaoitic behavior in an interenet auctionmarket can acutally be understood using simpler rational expectations stylereasoning.

Our results are clearly related to the papers by (Reny and Perry 2002) andearlier (?) who effectively give conditions under which the function relatingprice in the double auction to the array of types possessed by the playersconverges to the full information price function as the number of bidders inthe auction or double auction becomes large. In a sense this means that thedouble auction will have the rational expecations property in the limit. Thereare a number of differences between their results and ours. For example, wedo not explore limit results, but give an exact theorem instead. However,the primary difference is in the nature of the mechanism that is explored.

The essence of rational expectations is that traders need informationabout what others are doing in order to make the right decision. Tradersparticipating in a trading mechanism cannot directly condition on price, butthey can typically condition on messages sent by the other players. Thisconditioning and belief revision is precisely what the rational expectationsargument is supposed to approximate. At first glance, the double auction

4For example, the full information price function has the rational expectations propertyif it is one to one.

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does not appear to allow traders to condition on other messages at all, sincebids are submitted ex ante. However, this is misleading since a bid or askrepresents a type of contingent plan. If there are m more individuals thanobjects, for example, a bid is a commitment to buy when the mth lowest bidor ask of the others is at or below the bid. This is nonetheless a very crudeplan since it restricts the set of messages that the trader can condition on,and imposes tight restrictions on trading decisions across information sets -for example a bid p commits the buyer to buy when the double auction priceis p, but also when it is at any level lower than p. This constraint couldbe very costly in an interdependent value auction if lower prices caused thebuyer to want to revise his decision about trading. For this reason the doubleauction will have the rational expecations property only in a very narrow setof environments - in some sense the other traders information can’t be tooimportant. The internet auction relaxes these restrictions considerably. Aswill be seen, the trading decision in the internet auction can be based onmore messages from the other traders - and because bidders make sequen-tial decisions, trading decisions relevant for the price p no longer constraindecisions at other prices. Both these properties allow us to verify the ra-tional expectations property in a different and informationally richer set ofenvironments. We illustrate this in an example below.

We do not invoke large numbers directly in our results, though our as-sumptions have a large number flavor to them. In this sense our results arerelated to the existence and characterization results for double auctions withlarge numbers. In the private value case, for example, (Rustichini, Satterth-waite, and Williams 1994) prove a theorem about the rate of convergenceto efficiency with independently distributed valuations. Cripps and Swinkels(02) prove a convergence theorem for the case where valuations are corre-lated. We have already mentioned the connection of the model in this paperto (Peters and Severinov 2001) who prove a convergence result for the pri-vate (but correlated) value case that uses a internet auction framework thatclosely resembles the approach that we use here.

All of the convergence results are based on some variant of the followingargument - a seller who asks strictly more than his true value will sometimesraise the price he recieves for the object he is selling, and sometimes lose anopportunity to trade at that higher price that he would have had anyway.Convergence theorems impose restrictions on the joint distribution of typesso that the latter event is much more likely than the former (neither event isvery likely).

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In an internet auction large numbers cannot be used to get results of thiskind. The reason is that when sellers are allowed to bid they will encounterinformation sets (albeit with very small probability) in which bidding abovetheir value is a weakly dominant strategy. Such a situation occurs when aseller has a high bidder at his own auction at a standing bid that makes theseller just indifferent about whether or not he trades, while at the same time,the seller is the only active trader who continues to bid at other auctions.In this case, if the seller bids at his own auction, he will become high bidderthere. If no one else bids then he fails to trade, but he is just indifferentabout trading at the current price anyway. Alternatively, if the bidder hedisplaces continues to bid, the price at his own auction may be driven up. Todeal with this, we impose a restriction on the joint distribution of types, thatappears reasonable in large auction markets, that convinces the seller thatbidding will not be profitable. This restriction has other advantages since itallows us to deal with exact equilibria rather than dealing with conceptuallycomplicated limit arguments.

Finally, we mention a couple of alternative papers that use models thatare much different from ours, but which have a similar flavor. Wolinsky(Wolinsky 1988). studies a pairwise matching environment in which tradersattempt to bilaterally ’negotiate’ a trading price. Traders who meet a seriesof tough bargaining partners infer that the value of the commodity beingtraded is higher. His results are negative - as the discount factor goes to oneso that is becomes easier for traders to collect information by meeting morepartners, the quality of the information conveyed by each meeting deterio-rates, so trading prices do not converge to full information prices. Gottardiand Serrano (?) consider a model in which a set of informed sellers competefor uninformed buyers over a large number of periods. They consider a va-riety of different trading rules and illustrate conditions under which sellers’price announcements will reveal the common value of a commodity. Thoughthe models discussed in these papers are different from ours, they share thefeature that traders are given the opportunity to see the messages of othertraders before they make their consumption and trade decisions.

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1 The Model

1.0.1 Fundamentals

There are n sellers and m buyers trading in a market. Let I denote the set oftraders (both buyers and sellers). Each seller has one unit of a homogeneousgood, while each buyer has an inelastic demand for one unit of this good.

Buyers’ and sellers’ values are determined by the types of all traders.We assume that types and values come from finite sets. The grid of valuesis V ⊂ R+ which contains η real numbers. The distance between any twopoints on the grid of values is δv. The minimum and maximum grid pointsare u and u respectively. The grid of types is also a subset X of R withthe distance between any two grid points equal to δx, with minimum andmaximum points x and x. We discuss order statistic extensively below. Onenotation we will use frequently is the following: for any vector y ∈ R

n andany i ≤ n, let y(i) be the ith lowest element in the vector. So by the ith orderstatistic in some vector, we always mean the ith lowest element of the vector.

We assume that a trader i with type xi ∈ X has utility u (xi, x−i) wherex−i ∈ R

n+m−1 is the array of types of all the other traders. The function u

is assumed to generate values that lie in the grid V, to be strictly increasingin the agent’s own type,and to be non-decreasing in the types of the otheragents.

We assume that traders share a common prior belief about the jointdistribution of types. This common prior is assumed to have the propertythat two traders who share the same type have the same belief about thedistribution of the types of the other traders.

The fact that values lie in a grid does not mean that lotteries over valueswill also lie in the grid. For this reason, we assume

Assumption 1 If F and G are any two lotteries on x−i, F is strictly pre-ferred by i to lottery G iff

max

{

u′ ∈ V : u′ ≤

∫

u (xi, x−i) dF (x−i)

}

≥

max

{

u′ ∈ V : u′ ≤

∫

u (xi, x−i) dG (x−i)

}

.

An internet auction market consists of n sellers, m buyers, and a jointdistribution Fmn over the types of the traders.

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1.0.2 The bidding Process

Buyers and sellers are indexed in such a way that i < j means that i bidsbefore j. At the beginning of the game, each seller is considered to be highbidder at his own auction. The standing bid at each auction is the lowestgrid point v. The traders then take ’turns’ bidding. The first buyer in thelist of traders has the first turn to bid.

At the beginning of any player i’s turn to bid, each of the traders whoare not high bidders at some auction simultaneously announce whether theywish to remain active or to pass on the chance to bid during the current turn.The game ends if all these traders announce that they want to pass or if allstanding bids have risen to the highest grid point v.

If trader i (whose turn it is to bid) announces that he is active then heis given the opportunity to bid. If i says that he wants to pass, then theopportunity to bid is given to the ’next’ active trader j who is not yet a highbidder.5 Traders who pass on some turn can become active again wheneverthey like. After some active player j submits a bid, it becomes player j + 1’sturn to bid (or player 1’s turn to bid if j = m + n).

If a trader is given the opportunity to bid, then he must submit a bidwith some seller that lies in the grid V, and that strictly exceeds the currentstanding bid at the auction where it is submitted.

When a new bid is received it is compared with the current high bid inthe auction. If the new bid is higher than the current high bid at an auction,the bidder is made the new high bidder. The only exception occurs if a sellerbids in his own auction - in that case it is assumed that he becomes highbidder if he matches the existing high bid. Whatever bid is submitted, thestanding bid at the auction is adjusted so that it equals the second highestbid that has been submitted to the auction. The identity of the high bidderand the value of the current standing bid are then made public.

When the game ends the high bidder at each auction pays the sellerrunning the auction the standing bid or second highest bid that has beensubmitted to that auction. Of course, any seller who is high bidder at hisown auction does not sell.

5The term ’next’ means the player with the smallest index j ≥ i who is active, or if nosuch traders exist, it means the player with the smallest index who is active.

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1.0.3 Assumptions

We are searching for conditions under which the internet auction mechanismwill have an ex post efficient equilibrium. For this to be true, preferencesmust satisfy four conditions. Two of them are substantive. Two of them aretechnical conditions that we use to deal with the fact that we are workingon a grid. We begin with the substantive assumptions.

Assumption 2 (Single Crossing Condition) If xi > xj where xj is the jth

component of x−i. Then u (xi, x′, {x−ij}) ≥ u (xj, xi, {x−ij})∀x′ < xi.

This strengthens the usual idea that every trader’s own signal is moreimportant than any other trader’s signal (which in turn ensures that thetraders with the highest types also have the highest values).6 It ensures thattraders with the highest types have the highest values (simply replace x′

by xj in the definition). It goes further by requiring that if trader i has ahigher type than trader j, then trader i will have a higher value than j evenif he underestimates j’s type. This assumption is an attempt to capture theidea that traders’ values do not depend on the types of particular players,though they may depend on, for example, the distribution of values of theother players.

The second substantive assumption is the following:

Assumption 3 For each i, xi and vector {x1 ≤ x2 · · · ≤ xm−1 ≤ x′}, con-sider the event

E ={

x̃−i : {x̃−i}(1) = x1, . . . {x̃−i}(m−1) = xm−1, {x̃−i}(m) ≥ x′}

.

Then if E occurs with positive probability conditional on xi then Pr(

{x̃−i}(m) = x′|E)

=

1.6Two trivial examples of utility functions satisfying this restriction are as follows:

u (xi, x−i) = xi +∑

j 6=i

max [xi, xj ]

If W : R → I is the function that gives for each real x the largest integer which is less thanor equal to x, then

u (xi, x−i) = xi + W

1

n + m − 1

∑

j 6=i

xj

satisfies Assumption 2 for all xi if the maximum grid point in X divided by n + m − 1 isless δ.

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The assumption says that if a trader thinks that a certain type is ’pivotal’then he must be sure that there is some other trader with the same type.7

The following assumptions are technically convenient and restrict theamount by which expected values can change when other traders types varyby a single grid point.

Assumption 4 (Local Invariance) For any vector of types (xi, x−i) let x+−i

be the vector of types formed by replacing any xj ∈ x−i such that xj = xi

with xi + δx. Similarly, let x−−i be the vector of types formed by replacing each

xj ∈ x−i such that xj = xi with xi − δ. Then

u (xi, x−i) = u(

xi, x+−i

)

= u(

xi, x−−i

)

Assumption 5 For any xi and vector y ∈ Rm

maxp′

{

p′ ≤ E

[

u (xi, x̃−i) | {x̃−i}(1) = y(1), . . . {x̃−i}(m) = y(m)

]}

=

maxp′

{

p′ ≤ E

[

u (xi, x̃−i) | . . . , {x̃−i}(m) = y(m), {x̃−i}(m+1) ≥ y(m) + δx]}

1.0.4 Values

To state the theorem, we begin by defining the ’values’ of the traders. It isimportant to note that these values depend on the vector of types of all thetraders, so these aren’t things that traders know at the time they bid.

Definition 6 For any vector (xi, x−i) of types for the traders, the implicitvalue vi for bidder i with type xi is the largest grid point in V that is lessthan or equal to

E

[

u (xi, x̃−i) | {x̃−i}(1) = min[

xi, {x−i}(1)

]

, . . . {x̃−i}(m) = min[

xi, {x−i}(m)

]]

To simplify the notation slightly, we leave out any explicit conditioningon trader i’s own type xi. This condition should be assumed in each of theformulas that follow. The implicit value assumes that trader i knows thetypes either of all the traders whose types are lower than his (if his type isless than {x−i}(m)) or knows the types of the traders with the m lowest types.

7This assumption resembles the ’overlapping’ property assumed by (Cole, Malaith, andPostlewaite 2001)

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The rules of the internet auction allow traders to see when other traders quitbidding. This allows them to make inferences about these quantities, whichis why this knowledge appears in the definition of the implicit value.

Let v = {v1, . . . vm+n} be the entire array of ’values’ computed this way.Then v(m) is the mth (lowest) order statistic among the vector of values forall the traders. Observe that by our single crossing condition, the traderwho has the mth lowest implicit value is also the trader who has the mth

lowest type. The most important of these values is the one for the traderwho has the mth lowest type. His implicit value is conditional on the m − 1types below his, and on the assumption that the mth lowest type of the othertraders (which must be at least as high as his, since he has the mth lowesttype) is exactly equal to his.

Our main theorem can now be stated.

Theorem 7 If the auction market satisfies 2,3, 5 and 4, then there exists aweak perfect Bayesian equilibrium for the internet trading mechanism suchthat for each vector of types, all trades occur at the same price v(m). Traderswhose type are strictly above x(m) will win some auction, traders who types arestrictly below x(m) will not win an auction. Other traders will win auctionswith positive probability at prices that make them indifferent about whetheror not to trade.

The proof is constructive. Traders use a strategy rule that causes standingbids at all auctions to rise together, with traders gradually dropping out ofthe bidding until all traders with high values are high bidders. We deferdiscussion of the details of this rule and begin by illustrating how to usethis theorem (and more generally how to think about bidding in the internetauction) by providing an example which fits all of our assumptions.

In this first example there are 6 traders, one of them is a seller, the othersare buyers. There are only two possible arrays of types for the traders givenby

{9, 8, 8, 8, 8, 8}

and{9, 10, 10, 3, 3, 3}

Ex ante, each of these arrays of types is equally like and every permutationof the identities of the traders is equally likely, so each trader can have anyone of four possible types. Beliefs are symmetric. In this formulation, most

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traders actually learn the true state from knowledge of their own type, exceptin the case where they have type 9.

Payoffs are given by

u (xi, x̃−i) = xi + W

(

∑

j 6=i xj

5

)

where W (x) is the largest integer that is less than or equal to x. Roughlythe payoff is the sum of a traders own type and the average of the types ofthe other traders.

A trader whose type is 9 can infer the state from knowledge of lowertypes. So in this example, the full information values of the traders in eachstate coincide with their implicit values, and are given by

{17, 16, 16, 16, 16}

and{14, 15, 15, 8, 8, 8}

respectively.It isn’t hard to verify (brute force) that if the grid of types is restricted

to {3, . . . , 10}, then the single crossing condition holds. The very specialprobability distribution ensures that assumption 3 holds. The technical as-sumptions follow from the properties of the function W (·). The assumptionsof Theorem 7 all hold. Then since implicit value of the trader with the 5th

highest type is 16 in the first state and 15 in the second, those are the pricesat which trade will occur under the internet trading mechanism in each ofthese states.

The example has the following plausible but unusual property - the seller,whose type is 9 wants to sell at ’price’ 15, but wants to keep the output atthe higher price 16 - in another terminology, his supply curve is downwardsloping. In the first state all the other traders have high valuations, whichraises the seller’s value because of interdependence. In the second state, onlythe traders who ultimately determine the trading price have high valuations,while the other traders have quite low valuations. Knowing this, the seller’s’supply’ of the good is inversely related to price.

There doesn’t seem to be anything particularly perverse about this exam-ple. It doesn’t create any problems for the internet trading mechanism. Yetit prevents the double auction from supporting an ex post efficient outcome.

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Recall that in a double auction all traders simultaneously submit bids orasks, the market maker orders them in a single vector of values and executestrades that maximize apparent gains from trade by allocating the goods tothe traders with the n highest bids and asks. A whole class of double auc-tions can be defined depending on the way in which the final price is setbetween the m-th and the m + 1-st lowest bids.8 Yet, since all the auctionsare outcome-equivalent, we can focus on the seller’s offer double auctionwhere the price is set equal to the m-th lowest bid or ask (from the bottom)

Let us show that the seller’s offer double auction has no ex-post efficientequilibrium. If the seller, for example, has type 9 then she will submit thesame bid in both states. All of the other traders know their own values atthis point. If the bid is above 16 then the seller will surely win in both states,if it is below 14 then she will surely lose in both states. Neither outcome isex post efficient. So if there is an ex post efficient outcome, the seller mustsubmit the bid either 14, 15 or 16. Whichever of these bids she submits, shewill win in the wrong state.

It might pay to point out here how it is that the internet auction over-comes this problem. In the first state, the other bidders in the internetauction continue to bid aggressively until the price reaches 16. The sellerobserves this, and conditions his bid behavior on it by continuing to bid toensure that he wins his auction. In the other state, the low type bidders allstop bidding when the price reaches 8, which is just the market signal thatthe seller needs to know that he should abandon bidding himself when theprice reaches 15.

2 Beliefs and Bidding

The proof of the main theorem is constructive. We specify a common be-lief function and bidding rule then show that these beliefs and bidding ruleconstitute a weak perfect Bayesian equilibrium. The full proof is complexbecause we need to specify beliefs and check deviations in arbitrary infor-mation sets, not just those that occur along the equilibrium path. We defermuch of this argument to the appendix, and specify here only those parts ofthe argument that are needed to discuss rational expectations.

8These mechanisms have been studied by (Rustichini, Satterthwaite, and Williams1994) in the private value case.

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Let ι (i) be any information set for trader i. Trader i shares public infor-mation on standing bids, the identities of the high bidders and other publiclyavailable information about the bidding history. He has private informationabout his own type and the unobservable parts of his own bidding history.To decide whether or not to bid, the bidder has to form beliefs about thetypes of the other bidders and the high bids that they have submitted. Werefer to the unknown types and high bids as the state of the market. Aswe have formulated it, the state is a random element in R

2n+m. We wishto describe a bidding rule in which traders behavior in each information setdepends only on the current array of standing bids, and a summary statisticfor their beliefs.9 Since beliefs are a summary statistic for the bid history, itis convenient to express the bidding rule itself as something that depends onthese beliefs. We compress all the relevant information contained in traders’beliefs into something called the trader’s willingness to pay, which dependson beliefs. We then specify the common bidding strategy as if it depends onthis willingness to pay.

Say that a bid is successful if the bidder who makes the bid becomeshigh bidder. A bidder i′ is inactive in some information set, if he is not highbidder at any auction in that information set, and if he chose to pass on thelast k chances to bid, where k ≥ 1. All other bidders are said to be active.For each inactive bidder i′, let qi′ be the lowest standing bid that prevailedon his kth to last chance to bid. In this case, we say that bidder i′ dropped outat price qi′ . Suppose that r bidders are inactive in the information set. Letqr+1 be the lowest standing bid in the information set. Let q = {q1, . . . qr+1}be the vector consisting of the low standing bids that prevailed when each ofthe inactive bidders dropped out, along with the lowest standing bid in theinformation set.

Among the active bidders, only some of them will be high bidders. Ahigh bidder is vulnerable in an information set if he is the seller runningan auction where no bids have been submitted, or if unsuccessful bids havebeen submitted against him at the auction where he is high bidder in theinformation set. A high bidder is secure otherwise.

Beliefs are assumed to have the following properties:

9In the private value case this is even more striking. Traders don’t care about othertraders valuations so their beliefs don’t matter and bidding strategies don’t depend on thebidding history at all. See (Peters and Severinov 2001).

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Condition 8 (B.1) In any information set, bidder i believes that the highbid at each auction where the high bidder is vulnerable is equal to the standingbid in that auction, while the high bid at each auction where the high bidderis secure is exactly one grid point above the standing bid in the auction.

We now describe the beliefs that traders hold about the types of otherbidders. For inactive bidders, the definition is inductive. In any informationset where all traders are active, if trader i chooses to pass, then all tradersother than i should infer that i’s type is

x̂i = max{

x : E

[

u (xi, x̃−i) | {x̃−i}(1) = · · · = {x̃−i}(m) = x]

≤ qr+1

}

In the special case where qr+1 = u the inference should be that i’s types isless than or equal to x̂i. Then in any information set where there are inactivebidders, suppose that each of the inactive bidders has been assigned a type x̂i

for i = 1, . . . r. Then if trader i drops out in the information set, all tradersother than i should infer that i’s type is

x̂i = max {x :

E

[

u (xi, x̃−i) | {x̃−i}(1) = x̂1; {x̃−i}(r) = x̂r; {x̃−i}(r+1) = · · · = {x̃−i}(m) = x]

≤ qr+1}

where again, the inference is that i’s type is less than or equal to x̂i in thespecial case where qr+1 = u. Let x̂ be the vector of such types.

Condition 9 (B.2) In any information set, the type of the i′th inactive bid-der is exactly x̂i′ unless i′ dropped out at u in which case i′’s type is less thanor equal to x̂i′ .

Observe that there must always be n high bidders. There are exactlym − r bidders who are not high bidders but who are still active. For anyq ≥ qr+1, any vector x−i of types such that x−i and x̂ have the same first r

order statistics, and any k = r, . . .m define

x̂p = max {x :

E

[

u (xi, x̃−i) | {x̃−i}(1) = x̂1; . . . {x̃−i}(r) = x̂(r); {x̃−i}(r+1) = · · · = {x̃−i}(m) = x]

14

≤ qr+1} (1)

In any information set, the active bidders are of four possible types -they are either secure high bidders at an auction with the lowest standingbid, vulnerable high bidders at an auction with the lowest standing bid, highbidders at an auction where the standing bid is strictly higher than the loweststanding bid, or not high bidders at all. Let x̂a = x̂p + δx.

Condition 10 (B.3) Traders who are secure high bidders have types at leastas high as x̂p + δx = x̂a.

Condition 11 (B.4) Traders who are vulnerable high bidders have types atleast x̂p.

Condition 12 (B.5) Active traders who are not high bidders are assumedto have types at least x̂a

r+1 if they have declared themselves to be active at thecurrent low standing bid, and to have types at least x̂

pr+1 otherwise.

In any information set ι, let Si (ι) be the set of states that bidder i thinksare possible if his beliefs obey (8) through (12). Observe that bidding endsonce there are m inactive bidders.

Definition 13 Define the willingness to pay for bidder i in information setι to be

E[

u(

xi, x̃−i ∨{

x̃(r) = x̃(m) = x̂pr+1

})

|Si (ι)]

if i is inactive and

E[

u(

xi, x̃−i ∨{

x̃(r+1) = x̃(m) = x̂pr+1

})

|Si (ι)]

if i is active. In this formulation x̃−i ∨{

x̃(k) = x̃(m) = x̂pr+1

}

is the vectorformed by replacing the kth through mth order statistic in x̃−i with x̂

pr+1.

We now describe the strategy rule that all traders will follow in the equi-librium we are interested in.

Definition 14 The symmetric strategy σ∗ is defined as follows: in any in-formation set ι (i)

(a) if i has to decide whether to pass or continue in ι (i), then if i’s willing-ness to pay is less than or equal to the lowest standing bid, he shouldpass. Otherwise he should remain active;

15

(b) if i has to bid in ι (i) and there is a unique lowest standing bid, then i

should submit a bid with the auction where the standing bid is lowest.The bid should be equal to the lowest value on the grid that exceeds thislow standing bid;

(c) otherwise if i has to bid in ι (i) and there are many auctions where thestanding bid is lowest, then i should submit the same bid as in (b)randomly choosing among auctions where the high bidder is vulnerable.If all high bidders are secure at all auctions where the standing bid islowest, the trader should choose randomly among them and bid as in(b).

This strategy rule has the following properties (proofs are all relegated tothe appendix):

Lemma 15 Along the path generated when all traders use σ∗ beliefs satisfy-ing Conditions (B.1) through (B.5) satisfy Bayes rule in every informationset that occurs with positive probability.

Lemma 16 The strategy rule σ∗ for all players, along with common belieffunction satisfying (B.1) though (B.5) constitutes a weak perfect Bayesianequilibrium for the internet trading process.

Lemma 17 In each terminal information set that occurs with positive prob-ability along the path generated by σ∗, each trader who wins an auction haswillingness to pay at least as high as the price at which he wins the auction.Each trader who does not win an auction has willingness to pay that is nohigher than the lowest standing bid in the terminal information set.

3 Rational Expectations

An outcome for any individual is a price and a decision about whether ornot to trade at that price. An outcome function describes a probabilitydistribution over outcomes for all individuals for each array of signals thatthe traders’ possess. Given an outcome function and a particular price andrecommendation about whether or not to trade, each trader could calculate aposterior belief about the state conditional on any price and trading outcome

16

that occurs with positive probability. If the trading outcome is optimal atthat price given this posterior belief for every trader and for every outcomethat occurs with positive probability, then we say that the outcome functionhas the rational expectations property.10

It is well known that the full information price function will have thisproperty if it is fully revealing. That is one possibility in the interdependentvalue case.11 The full information price function is not typically revealing inthe environment studied in this paper.

To show that the outcome function associated with the internet tradingmechanism has the rational expectations property, fix the assumptions re-quired for Theorem 7 to hold. Let (p, k) be an outcome for trader i withk = 0, 1 meaning that i does not win, or does win respectively some auctionat the end of the bidding process where the standing bid is p. Let Hi bethe (finite) collection of terminal information sets for trader i, and γi (·) thefunction that assigns an outcome (p, k) for i to each information set in Hi.For any vector of types x−i for the other traders, the strategy σ∗ determinesa probability distribution over terminal information sets for i. The composi-tion of the strategy rule and the function γi constitutes the outcome functionfor the internet trading mechanism.

The prior distribution over Xm+n and the strategy rule together gen-erate a random element in Hi for trader i conditional on xi in the sensethat there is some probability space Pi and mapping πi : Pi → Hi suchthat the set {ρ ∈ Pi : πi (ρ) = h′} is Pi measurable for h′ ∈ Hi. The setHi (p, k) = {h′ ∈ Hi|γi (h

′) = (p, k)} is the union of a finite number of ele-ments in Hi, so {ρ ∈ Pi : πi (ρ) ∈ Hi (p, k)} is Pi measurable, and a condi-tional probability distribution exists on x−i for each possible outcome (p, k).Let E {·| (p, k)} mean taking expectations with respect to this conditionalprobability distribution. The outcome function associated with the internettrading mechanism will have the rational expectations property if for eachtrader and for each outcome (p, k) ,the expected value a unit of output condi-tional on (p, 1) is at least p (the expected value of a unit of output conditionalon (p, 0) is no larger than p).

10This condition is different from ex post individual rationality, but similar to it in spirit.The primary difference is that the direct mechanism might map many different arrays oftypes for the other traders into the same outcome for an individual trader, so this tradercould not discover the true state simply by observing his own outcome.

11Though in the private value version of our model, it is not hard to see that the fullinformation price function is never revealing.

17

Theorem 18 If the assumptions required by Theorem 7 hold, then the out-come function associated with the internet trading mechanism has the rationalexpectations property.

Proof. Let (x1, . . . xm+n) be a vector of types that has positive proba-bility conditional on (p, 1). Then by Theorem 7 v(m) = p, and xi ≥ x(m),so vi ≥ p. For each vector x(·) ≡

{

x(1), . . . x(m)

}

with i ≤ j ≤ m ⇒ x(i) ≤x(j) ≤ xi, let Pr

{

x(·)| (p, 1)}

be the probability of x(·) conditional on (p, 1).Let H

(

p, 1, x(·)

)

be the set of terminal information sets that occur with pos-itive probability conditional on

(

p, 1, x(·)

)

. For each h ∈ H(

p, 1, x(·)

)

letPr

{

h′|(

p, 1, x(·)

)}

be the conditional probability of h. Then i’s expectedvalue conditional on (p, 1) can be written

E {u (xi, x̃−i) | (p, 1)}

as∑

x(·)

Pr{

x(·)| (p, 1)}

· (2)

∑

h′∈H(p,1,x(·))

Pr{

h′|(

p, 1, x(·)

)}

E

{

u (xi, x̃−i) | {x̃−i}(1) = x(1), . . . {x̃−i}(m) = x(m)|h′}

(3)

Since beliefs evolve according to Bayes rule, the expectation in this formulais i’s willingness to pay in the terminal information set associated with h′.By Lemma 17, i’s willingness to pay in the terminal information set is atleast equal to p. So the entire expectation is no less than p and trade is atleast as good not trading. The argument is identical for the event (p, 0).

This theorem says that if traders play this equilibrium when they partic-ipate in the internet trading mechanism, they behave exactly as if they wereplaying a rational expectations equilibrium where belief functions are givenas described before Theorem 18. Furthermore, the formula (2) that appearsin the proof of Theorem 18 shows that the rational expectations belief func-tion associated with the internet trading mechanism is a weighted average ofthe posterior beliefs that traders hold in different terminal information sets.This is a nice way of interpreting the rational expectations belief function,which is typically described as the solution to a fixed point problem. Thebelief function is simply a weighted average of posterior beliefs in differentinformation sets, all derived using Bayes rule.

18

It is informative to consider and example where the full information pricefunction is not revealing. A slight modification of the example consideredabove generates a non-revealing full information function and illustrates theway the internet trading mechanism aggregates information in this case.

Recall that the example involved only two possible arrays of types. Onearray is given by {9, 8, 8, 8, 8, 8}. Suppose now that the second array of types

is { 9, 11, 11, 3, 3, 3}. Using the utility function u (xi, x̃−i) = xi +W(∑

j 6=i xj

5

)

from the previous example, the full information values are {17, 16, 16, 16, 16}and {15, 16, 16, 10, 10, 10}, and the full information price function no longerreveals the types of the others to the trader who has type 9. The implicitvalues for the trader with the 5th highest type is also 16 in both states.By Theorem 7, the trading price in the equilibrium for the internet tradingmechanism is also 16 in both states. The outcome for the trader of type 9is different in each of the two states. In the first state, the trader will seethat the other traders remain in the bidding after the price rises past 10and will infer that their types are then all equal to 8. In the second state,the trader will observe that other traders have stopped bidding an make theappropriate inference. The traders’ own bids depend on this, which is whyhis own trading outcome aggregates this information.

What of the double auction? In the seller’s offer double auction, it is wellknown that bidding true value is a weakly dominant strategy for buyers. Sosuppose that all buyers who know their values bid their values. Then a sellerwith type 9 cannot affect the price that will prevail in the double auction.His bid only determines whether or not he trades. Whatever he chooses todo in equilibrium, the equilibrium price in the double auction will be 16 inboth states. In particular, the equilibrium price in the double auction willnot reveal the true state. Furthermore, conditioning on the outcome of tradedoes not help either since the seller either wins of loses in both states. Theonly way for traders to deduce the true state from the outcome is for theseller to bid 16. If ties are broken by some exogenously imposed rule, thenthe outcome for the seller will still not reveal the array of types. If ties arebroken in way that depends on the array of bids, then the auctioneer candetermine the identities of the traders efficiently from the array of all bids.In this case, the seller may be able to infer the state from his own outcome.

19

4 Conclusion

The internet auction mechanism supports ex post efficient trade in a fairlyrich set of environments. However, the mechanism is complex. The proof iscomplicated because so many different kinds of information sets have to beconsidered, verifying in a sort of tedious brute force way both that deviationsfrom our equilibrium strategy are unprofitable, and that Bayes rule holds.The equilibrium strategy itself is very simply - almost a rule of thumb thatcan be easily implemented without any sophisticated calculation or reason-ing. The behavior that this strategy induces in the market is also simple.Prices rise is unison until enough traders drop out of the market to satisfyall demand at that price. This suggests that a simpler way to model thisenvironment might be to consider an ascending double auction with pricesrising smoothly and traders dropping out until demands can be met. Weleave this kind of model for future research.

At a more practical level, the theorm also suggests a method for analyzinginternet auction data. A complete game thoeretic treatment of this data isdifficult because of the complexity of the extensive form game that is drivingthe results. Traders behaviour in this game can be understood using a muchsimpler rational expecations argument.

5 Appendix - Proofs

This section consists of a number of parts. The first part shows that whentraders use σ∗, their bidding behavior can be predicted from knowledge oftheir implicit values. This makes it possible to calculate outcomes along thecontinuation path from knowledge of traders types alone. This allows tradersto compare the outcome associated with different current actions in each statethat they believe is possible. The first subsection of the appendix presentsthis lemma. The second subsection provides the state by state prediction ofoutcomes.

The third subsection shows that beliefs satisfying (B.1) through (B.5)satisfy Bayes rule along the path associated with σ∗. The fourth subsectionchecks that all deviations for buyers are unprofitable (this section makesheavy use of the results in subsection 2). The final subsection shows thatdeviations for sellers are unprofitable.

20

5.1 Predicting bidding behavior from implicit values

Lemma 19 Suppose that all trader i’s beliefs satisfy (B.1) through (B.5).Let ι0 be an information set in which trader i moves and let ι1 be any infor-mation set that i can induce with positive probability on the continuation pathby taking some action provided that all traders follow σ∗ in every successorto ι0. Then every trader’s willingness to pay in ι1 exceeds the lowest stand-ing bid in ι1 if and only if every traders’ implicit value exceeds the loweststanding bid in every state that i believes is possible in ι1 given his beliefs inι0.

Proof. Let x−i be an array of types that i thinks is possible in ι0. Let ι1be an information set that i believes occurs with positive probability followingsome action that she takes in ι0. Trader j’s willingness to pay in ι1 is givenby Definition 13 as

E[

u(

xj, x̃−j ∨{

x̃(r) = x̃(m) = x̂pr+1

})

|Sj (ι1)]

=

ESj(ι1)

[

u (xj, x̃−j) | . . . {x̃−j}(r−1) = {x̂−j}(r−1) , {x̃−j}(r) = · · · = {x̃−j}(m) = xpr+1

]

if j is inactive in ι1 and

E[

u(

xj, x̃−j ∨{

x̃(r+1) = x̃(m) = x̂pr+1

})

|Sj (ι1)]

=

ESj(ι1)

[

u (xj, x̃−j) | . . . {x̃−j}(r) = {x̂−j}(r) , {x̃−j}(r+1) = · · · = {x̃−j}(m) = xpr+1

]

if j is still active in the information set. In these formulas, r refers to the(observable) number of dropouts in ι1 while x̂

pr+1 is calculated with reference

to these r dropouts.Focusing on this same array of values, suppose first that i believes that

xj > xpr+1 in information set ι1. Then since i expects all traders to use σ∗ on

the continuation path after ι0, i expects j to be active in ι1. Then

ESj(ι1)

[

u (xj, x̃−j) | . . . {x̃−j}(r) = {x̂−j}(r) , {x̃−j}(r+1) = · · · = {x̃−j}(m) = xpr+1

]

≥ E

[

u (xj, x̃−j) | . . . {x̃−j}(r) = {x̂−j}(r) , {x̃−j}(r+1) = · · · = {x̃−j}(m) = xpr+1

]

≥ E

[

u(

xar+1, x̃−j

)

| . . . {x̃−j}(r) = {x̂−j}(r) , {x̃−j}(r+1) = · · · = {x̃−j}(m) = xpr+1

]

> qr+1 (4)

21

The first inequality follows by monotonicity, while the second follows fromthe fact that xj > x̂

pr+1 and the definition of xa

r+1. So j’s willingness to paystrictly exceeds qr+1.

We now show that j’s implicit value (when i has type xi and the othershave the types x−i that i thinks are possible) also strictly exceeds qr+1. Thisimplicit value for j is given by

E

[

u (xj, x̃−j) | . . . {x̃−j}(r) = {x−j}(r) . . .

. . . {x̃−j}(r+1) = min[

xj, {x−j}(r+1)

]

. . . , {x̃−j}(m) = min[

xj, {x−j}(m)

]]

≥

E

[

u(

xar+1, x̃−j

)

| . . . {x̃−j}(r) = {x−j}(r) , {x̃−j}(r+1) = · · · = {x̃−j}(m) = xpr+1

]

(5)because of the fact that xj ≥ xa

r+1. If trader i is active in ι1 then in anystate that i thinks is possible in ι1, xk = x̂k for each inactive bidder, so each{x−j}(k) in (5) can be replaced with {x̂−j}(k). The expression then strictly

exceeds qr+1 by (4).If trader i is inactive in ι1 then in this last expression xk = x̂k for every

trader except trader i. If xi ≥ x̂i then when the {x−j}(k) are replaced with

{x̂−j}(k) on the right hand side of (5), the expression on the right hand side

falls, so again the implicit value strictly exceeds qr+1 by (4). If xi < x̂i thenthe inequality follows from the single crossing condition and the fact thatx̂i ≤ x

pr+1 < xa

r+1.We conclude that if xj > x̂

pr+1, then both j’s willingness to pay, and j’s

implicit value strictly exceed the lowest standing bid qr+1 in the informationset ι1.

On the other hand, if trader j’s type is less than or equal to xpr+1 in ι1,

then j’s willingness to pay (using Assumption 5) is less than or equal to

E

[

u(

xpr+1, x̃−j

)

| . . . {x̃−j}(r) = {x̂−j}(r) , {x̃−j}(r+1) = · · · = {x̃−j}(m) = xpr+1

]

by monotonicity.12 So j’s willingness to pay is less than or equal to qr+1.Since j’s type is less than or equal to x

pr+1 trader j’s implicit value is less

than or equal to

E

[

u(

xpr+1, x̃−j

)

| . . . {x̃−j}(r) = {x−j}(r) , {x̃−i}(r+1) = · · · = {x̃−i}(m) = xpr+1

]

12There are two parts to this inequality. j’s type is less than or equal to xpr+1 by

hypothesis, and if j is inactive, then in the calculation his dropout xj is replaced by xpr+1.

22

Since the state used in this calculation must be one that i believes is possiblein ι1, this is less than or equal to

E

[

u(

xpr+1, x̃−j

)

| . . . {x̃−j}(r) = {x̂−j}(r) , {x̃−i}(r+1) = · · · = {x̃−i}(m) = xpr+1

]

if x̂i ≥ xi by monotonicity. If xi < x̂i the same inequality follows from thesingle crossing condition.

This argument establishes that both the willingness to pay and the im-plicit value strictly exceed qr+1 for any type xj > x

pr+1 and that both these

numbers are less than or equal to qr+1 otherwise.

5.2 The outcome associated with σ∗

In this section we fix a vector of types for the traders and consider howthe game will evolve from different information sets if traders follow σ∗ inthe continuation game. Define trader i’s augmented implicit value to be themaximum of his true implicit value and the highest winning bid that he hasin any auction. Let v̂ be augmented vector of implicit values.

We begin with some simple properties of the augmented vector of implicitvalues.

Claim 20 Let v be a vector in Rm+n. Then the number of components of

v which are at or below v(m) must be at least as large as the number ofcomponents that strictly exceed v(m).

Proof. The number of components of v at or below v(m) must be at leastm since v(m) is the mth lowest value in v. Let b1, b2, b3 and s1, s2, s3 be thenumbers of components that are below, at or above v(m) respectively. Then

b1 + b2 + s1 + s2 ≥ m = b1 + b2 + b3

Subtracting the common terms from both sides of the inequality gives s1 +s2 ≥ b3

Claim 21 Let v̂ be the vector of augmented implicit values associated withsome state that i believes is possible in information set ι (i). The number oftraders whose augmented implicit values strictly exceed v̂(m) and who are notalready high bidders at auctions where the standing bid exceeds v̂(m) can beno larger than the number of auctions where the standing bid is at or belowv̂(m).

23

Proof. Of the n auctions, let n1 be the number where the standing bidstrictly exceeds v̂(m). By definition, the augmented implicit values of the highbidders at each of these auctions must be strictly higher than v̂(m). So thereare n1 traders whose values exceed v̂(m) who are high bidders at standing bidsabove v̂(m). In the n − n1 other auctions, the standing bids are less than orequal to v̂(m). By the definition of v̂(m) there cannot be more than n traderswith augmented implicit values strictly above v̂(m), and so at most n − n1

of the traders who are not already high bidders where standing bids exceedv̂(m) can have implicit values that strictly exceed v̂(m).

Lemma 22 Consider any information set and state in Si (ι). Suppose thatall bidders use strategy σ∗in the continuation. Then there are two cases: (i)any trader who does not have a high bid above v̂(m) in ι will pay a price thatis no higher than v̂(m) in any auction he or she wins in the continuation; (ii)any trader who has a high bid above v̂(m) in ι will win each of the auctionswhere she has submitted those bids.

Proof. If either (i) or (ii) is false, then some trader using σ∗ must bewilling to submit a bid at price p > v̂(m). This requires three things to betrue. First this bidder must have an implicit value strictly larger than v̂(m).Second, this bidder cannot be a high bidder at the time that she submitsher bid. Third, no other auction can have a lower standing bid. All n ofthe auctions must have high bidders. In any state that i thinks is possiblein Si (ι), each of these high bidders, with the possible exception of i has animplicit value that strictly exceeds v̂(m). In case (i) no bidder will bid at aprice above v̂(m) unless his adjusted bid price exceeds v̂(m). Thus there mustthen be n + 1 bidders with valuations strictly above v̂(m). This contradictsthe definition of v̂(m). In case (ii), in the initial information set, the numberof traders whose implicit values strictly exceed v̂(m) and who are not highbidders at auctions where the standing bid exceeds v̂(m) can be no largerthan the number of auctions where the standing bid is at or below v̂(m) byClaim 21. Thus there can be no more than n− 1 traders other than i whoseimplicit values exceed v̂(m). Since n−1 such traders are already high biddersat standing bids above v̂(m) in the information set where the price p is bid,there is no trader to submit this bid.

Lemma 23 Beginning in any information set and state in Si (ι). Supposethat all buyers other than i use σ∗ in the continuation. Then trader i cannotwin an auction at a price strictly below v̂(m).

24

Proof. Suppose that i wins at price p < v̂(m). Then there is at least oneauction where the standing bid is strictly less than v̂(m) in the terminal in-formation set. Suppose there are r1 such auctions. The terminal informationset is regular for i since all traders other than i use σ∗ in the continuation. Ofthe n traders whose adjusted bid prices are at or above v̂(m), at most n − r1

of them can be high bidders at standing bids at or above v̂(m). So there mustbe at least r1 traders whose implicit values are at least v̂(m) who are eitherhigh bidders at standing bids below v̂(m) or who aren’t high bidders at all.If no bids are submitted at the price p, then because the other b bidders areusing σ∗, they must all have implicit values strictly below v̂(m). This contra-dicts the definition of v̂(m). We conclude that there are strictly more traderswhose implicit values exceed p and who either aren’t high bidders, or arehigh bidders at auctions where the standing bid is less than v̂(m),than thereare auctions with standing bids below v̂(m). If the price p is not bid up, thenat least one of these traders must pass when his implicit value exceeds p, ormust submit a bid at an auction where the standing bid exceeds p. Either ofthese events is inconsistent with σ∗.

Corollary 24 Consider some information set ι and state in Si (ι). Sup-pose that all buyers use σ∗ in the continuation. Then trader i will win someauction and pay v̂(m) if his implicit value is strictly larger than v̂(m). Alter-natively if i’s implicit value is strictly less than v̂(m) then trader i will notwin an auction.

Proof. If i wins an auction, then he will pay v̂(m) by Lemmas 23 and 22.Trader i will bid at v̂(m) if and only if his implicit value exceeds v̂(m).

5.3 Proof of Lemma 15

Lemma 25 Along the path generated when all traders use σ∗ beliefs satisfy-ing Conditions (B.1) through (B.5) satisfy Bayes rule in every informationset that occurs with positive probability.

Proof. Consider first, the information set in which trader i gets his firstopportunity to bid. If i is the first trader to bid, then i believes that allhigh bidders are vulnerable, and that no seller has a high bid above thelowest standing bid. So beliefs satisfying 8 through 12 coincide with i’soriginal beliefs. If i is not the first to bid, then one or more traders may have

25

submitted bids, while one or more traders might have passed. If some traderj passes on his first chance to bid, then by Lemma 19, it must be that j hasan implicit value that is no higher than the lowest grid point u, i.e.,

E

{

u (x, x̃−i) : {x̃−i}(1) = . . . {x̃−i}(m) = x}

≤ u

which coincides with i’s inference according to 8. Let x be the highest gridpoint that satisfies this condition. If no such grid point exists, then j hasdeviated from σ∗ and any inference is legitimate. Otherwise each trader whohas passed must have a type less than or equal to x.

In all other information sets, suppose that i has already correctly inferredthe types of bidders who dropped out in previous information sets. Let i

be any bidder who passes for the first time in the current information set(bidders using σ∗ will not pass then re-enter). Then from the first propertyof σ∗ and Lemma 19, the highest type that this player could have is givenby x̂

pr+1. On the other hand, by monotonicity, lowering i’s type by one grid

point, lowers i’s value by at least one grid point in every state. So if i’s typeis even one grid point lower, i would have passed at the previous price (if thelowest standing bid jumps by many grid points, traders are not using σ∗ ).

Observe that a trader with type xar+1 = x

pr+1 + δx must have an implicit

value that strictly exceeds qr+1. If this isn’t the case, then

E

[

u(

xpr+1 + δx, x̃−i

)

| {x̃−i}(1) = x̂1; . . . {x̃−i}(r) = x̂(r); {x̃−i}(r+1) = · · · = {x̃−i}(m) = x̂pr+1

]

=

E

[

u(

xpr+1 + δx, x̃−i

)

| {x̃−i}(1) = x̂1; . . . {x̃−i}(r) = x̂(r); {x̃−i}(r+1) = · · · = {x̃−i}(m) = x̂pr+1 + δx

]

≤ qr+1

by the local invariance property of preferences. This contradicts the factthat x

pr+1 is the largest type for which this inequality is supposed to hold.

Thus any secure high bidder must have a type at least xar+1 if he is using

σ∗,as must any bidder who has declared himself to be active at the currentprice. Bidders who become high bidders at auctions where the standing bidstrictly exceeds the lowest standing bid are not using σ∗ so any inference isconsistent with weak perfect Bayesian equilibrium.13

13The specification of beliefs given by 10 and 11 may not be consistent with ’perfectBayesian Equilibrium’ which, depending on the source, may involve restrictions on offequilibrium beliefs. The complication occurs when these beliefs are applied to high bidders

26

5.4 Outline of the Proof of Lemma 16

We can now give a heuristic account of the proof of the theorem. We needto show that it will always pay each trader to follow σ∗ if he thinks theother traders are following σ∗ in every conceivable information set includingthose that are never attained when traders use σ∗. The key argument isLemma 19, which has two implications. The first is that if other traders areusing σ∗ then every traders’ behavior in each future information set couldbe predicted from knowledge of his implicit value. For each state that atrader thinks is possible given his belief, the trader could readily computethese implicit values, and use them to predict the outcome if that state weretrue. Second, once a trader has decided that a state is possible, each traders’implicit value is the same in all future information sets. So when a tradercompares two alternative actions for the current information set, he can do astatewise comparison of the terminal information set that is associated witheach different action for every state that he thinks is possible. The outcomesin these terminal information sets are tightly limited by Lemmas 22 and 23.

The main idea behind Lemmas 22 and 23 is that traders using σ∗ behavein every state as if they had private values equal to their implicit values. Thestrategy σ∗ then ensures then all trades will occur at the lowest ’competitive’price consistent with the implicit values of the traders in that state (whenappropriately qualified by the conditions in the initial information set). Theargument for this is similar to the one in (Roth and Sotomayor 1990). Thecomplications that are resolved in Lemmas 22 and 23 are to show that thisargument can be applied starting in any information set, not just from theoutset of the game as is done in (Roth and Sotomayor 1990).

The lowest competitive trading is always the implicit value of a traderwho fails to win an auction. Thus no buyer who trades can lower his tradingprice. A buyer who deviates and raises his trading price above his implicitvalue, will then trade at a price that exceeds his willingness to pay by Lemma19. So buyers can’t do better than to follow σ∗.

The argument is different for sellers since they can affect the lowest com-

at auctions where the standing bid is strictly above the lowest standing bid. 10 and 11require that in this case traders make the same inference that they do for active traderswho are not high bidders, or for traders who are high bidders at the lowest standing bid.An implication of this is that beliefs about these traders will change as the lowest standingbid goes up - as if the types of the active bidders and these off equilibrium bidders werecorrelated.

27

x1

xm

xm+n

1 m m+ n

types

Implicit Values

(x1, . . . , xm+n)→ (v1, . . . vm+n)

Figure 1: Figure 2

petitive trading price when they are outbid in their own auction. Figure 2illustrates.

The Figure lists the types of trader participating in an internet auctionmechanism is ascending order. The types are converted into implicit valueswith the same ranking, so the trader with type xm in the picture also has thecritical implicit value v(m) which determines the trading price in the internettrading mechanism.

As the picture is drawn, his type (and therefore his implicit value) hap-pens to be strictly higher than the trader with the m− 1st highest type, andstrictly lower than the trader with the m + 1st highest type. Suppose thatthis pivotal trader is a seller. In the course of the bidding this seller willeventually arrive at an information set having two characteristics. First, oneof the other traders will be high bidder at his auction (and he won’t be a highbidder anywhere) and the standing bid at his auction will be exactly equalto his willingness to pay. Second, the bidders with types below his will allhave dropped out of the bidding, allowing him to correctly infer their types.

At this point, the seller has one of two options. He could pass, as σ∗

suggests that he do. If he does this, he should expect the bidding to end(since m− 1 of the others have already dropped out and are not expected toreturn). He will then sell at a price that is exactly equal to his willingnessto pay. Or, he could remain active and submit a bid above his willingness topay. For example, he could submit a bid at his own auction.

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This has no immediate benefit, since it will simply make him the newhigh bidder at the auction. However, it will displace the current high bidderand force him to bid again elsewhere if he wants to trade. If this displacedbidder is using σ∗ he won’t immediately bid again at the same auction, but ifthe valuations of the other traders are ’high enough’ someone will eventuallyoutbid the seller and he will sell at a higher price. The seller would like thisto happen, but if it doesn’t he won’t care, since he gets the same payoff ashe would have received by passing on the chance to bid in the first place.

For this reason, ascending bid auctions have different incentive propertiesfrom, say, double auction. The reason that this deviation is unprofitable hereis because of assumption 3 which says that whenever a trader thinks that hisown type is pivotal (as the seller is in this example), he must believe thatwith probability 1 there is some other trader who has the same valuation.When the seller bids in his own auction, other traders will simply bid in away that displaces this other trader, at which point the bidding will stop.The deviation will never be profitable.

5.5 σ∗ is a best reply for buyers

In the argument for buyers and sellers, we show that the one deviation prop-erty holds - i.e., it cannot pay any trader to deviate from σ∗ in one informationset then revert to σ∗ in every subsequent information set. Since the game isfinite, this is sufficient to verify that no deviation is profitable.14

Suppose that the buyer is supposed to pass in the current information set.Then his willingness to pay is no higher than the lowest standing bid qr+1

in the information set. By Lemma 19, his implicit value is no higher thanthe lowest standing bid. Since the lowest standing bid can only rise in thecontinuation, he will not bid again, and his payoff from passing as prescribedby σ∗ is zero. If he bids, then by Lemma 23, he cannot win an auction at aprice below v̂(m). In any state that the buyer thinks is possible, v̂(m) ≥ qr+1,so by Lemma 19, the buyers willingness to pay in the terminal informationset cannot exceed v̂(m) and the buyer cannot gain by bidding.

If the buyer is supposed to bid, then by 22, if the buyer wins an auction insome state he will pay a price that does not exceed v̂(m) if he bids accordingto σ∗. If he bids in a way that differs from σ∗ he cannot lower his augmented

14For example (Osborne and Rubinstein 1994), excercize 227.1.

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implicit value. Then by Lemma 23 he cannot trade at a price that is lowerthan v̂(m) in any state he thinks is possible.

5.6 σ∗ is a best reply for sellers

If the seller deviates from the action that he is supposed to take accordingthe σ∗ in the current information set, then reverts to σ∗ in each subsequentinformation set, the trading price and outcome in each state can be calculatedfrom traders’ implicit values by Lemma 19, 23 and 22.

If the seller is supposed to pass in the current information set, then hiswillingness to pay is no higher than the lowest standing bid qr+1. By Lemma19, his implicit value is no higher than the lowest standing bid. If he bids,he may change the vector of augmented implicit values associated with eachstate since he will have to submit a bid above his implicit value.

Let v̂ be the augmented vector of implicit values for some state that theseller thinks is possible in the current information set. Let v̂ ′ be the vectorof augmented implicit values for the same state that applies after the sellersubmits his bid. Since the seller’s bid is at least as high as his implicit value,v̂′

(m) ≥ v̂(m). If the seller wins an auction in the continuation, then by Lemma23, he will pay a price at least v̂′

(m). This price must be at least as high as thelowest standing bid qr+1 in the current information set. The lowest standingbid is no less than his implicit value by hypothesis, so by Lemma 19, thisprice can be no less than his willingness to pay in the terminal informationset. The implication is that the seller cannot gain by deviating in any stateif he wins an auction.

The same argument applies if the seller is supposed to submit a bid inthe current information set, but decides to deviate somehow from the oneprescribed by σ∗. If he wins an auction in the continuation, then by 23, theprice he pays can be no lower than v̂(m) where v̂(m) is computed from theadjusted vector of implicit bid prices that prevails after he submits his bid.

On the other hand, since v̂′(m) ≥ v(m), the seller might gain in the contin-

uation if he does not win an auction, and if the price in his own auction ispushed above v(m).

Whether the seller deviates by bidding when he is supposed to pass, or bybidding more than specified by σ∗, he assumes that all traders will follow σ∗

in the continuation. Then by Lemma 22, none of the other traders will paya price above v̂′

(m) in the continuation. Hence the selling price in his auctioncannot exceed v̂′

(m). If the seller takes the action specified by σ∗, then he will

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sell at price v̂(m).We show that v̂(m) = v̂′

(m) in any state the seller thinks is possible. Tochange the selling price at his auction, the seller must submit a bid thatstrictly exceeds v̂(m). In any state that the seller thinks is possible, no traderhas a high bid above his or her implicit value. Hence v̂(m) is simply the mth

highest implicit value among the traders. Furthermore, by construction, theseller’s posterior beliefs never assign positive probability to a state that haszero probability ex ante. Assumption 3 then implies that the mth and m+1st

highest types, and therefore the mth and m + 1st highest implicit values, inthis state are the same. If the seller’s bid strictly exceeds v(m) it must strictlyexceed v(m+1) in this state which gives v̂(m) = v̂′

(m).We conclude that the seller does not believe that it is possible to raise

the price at which he sells his output. Deviations are unprofitable.

5.7 Proof of Lemma 17

Any trader who does not win an auction has willingness to pay that doesnot exceed the lowest standing bid, otherwise they would have bid on theirlast opportunity instead of passing. When all traders are using σ∗, no tradercan win an auction at a price that is more than one grid point higher thanthe standing bid when he submits his bid. Since traders willingness to paymust exceed the standing bid before they will submit a bid, and since theirwillingness can’t fall in the course of the subsequent bidding, their willingnessto pay must be at least as high as the price that they pay in the auction.

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