Micro III Dirk Engelmann Overview A Few Remarks on Subgame - - PowerPoint PPT Presentation

Micro III Dirk Engelmann

Overview

• A Few Remarks on Subgame Perfection
• Perfect Bayesian and Sequential Equilibria
• Static Oligopoly
• Repeated Games, Folk Theorems
• Auctions
• Theories of Other-Regarding Preferences

Subgame Perfection and Backward Induction Definition (Subgame) A (proper) subgame of an ex- tensive form game ΓE consists of a single node and (exactly) all its successors in ΓE, such that if x is in the subgame and x′ ∈ H(x) then x′ is in the subgame (i.e. information sets are not “broken”) Note: “proper” in this definition does not refer to “proper subset”, i.e. the game is a subgame of itself, but to a proper subgame as a opposed to continuation games in games of incomplete information starting at non-singleton information sets.

Definition (MWG 9.B.2, SPNE) A profile of strate- gies σ = (σ1, . . . , σI) in an I-player extensive form game ΓE is a subgame perfect Nash equilibrium (SPNE) if it induces a Nash equilibrium in every subgame of ΓE. Note: The term “perfect equilibrium” is used by some authors as a synonym for SPNE and by others as a synonym for trembling hand perfect equilibrium.

Backward induction in finite games of perfect in- formation: determine optimal play at the final decision

• nodes. Then move to the next-to-last decision nodes

assuming that the actions at the final decision nodes will be correctly anticipated and so on. In this class

• f games backward induction and subgame perfection
• coincide. In more general games, not all SPNE can be

found by backward induction. Extension of backward induction: In multi-stage games (i.e. where players move simultaneously in each stage but are informed about previous stages) if the last stage can be solved by iterated strict dominance, replace last stage by non-dominated strategies and move up.

Examples:

• 1. Trust Game: {pure Strategy Nash Equ}={SPNE}
• 2. (Mini-) Ultimatum Game: {Nash Equ}={SPNE}

Nash Equilibria can include non-credible (i.e. non-subgame perfect) threats (because they don’t have to be ex- ecuted in equilibrium), but not non-subgame perfect promises (because the equilibrium path would reach the subgame and hence require best-reply behavior).

Critiques of backward induction:

• 1. In case of many players common knowledge as-

sumptions become very demanding.

• 2. Example: centipede game: if player 1 deviated once,

why should player 2 assume that 1 will stick to the backward induction solution in the future? Resolutions: (a) payoff uncertainty ⇒ give up backward induction, (b) trembles ⇒ stick to backward induction Experimental evidence: McKelvey and Palfrey (Econo- metrica 1992)

Critiques of subgame perfection: In an SPNE all players have to have the same expec- tations concerning the equilibrium to be played in a

• subgame. But in some situations it appears reasonable

that they might expect different equilibria to be played, e.g. FT Figure 3.20

Extensive Form Games of Imperfect Information An extensive form game of imperfect information con- tains continuation games starting with an information set that is not a singleton. These are hence no proper subgames and subgame perfection has no bite (see MWG, Figure 9.C.1). The idea of Perfect Bayesian Equilibria is to extend the notion of sequential rational- ity to such games. Definition (MWG 9.C.1, System of Beliefs): A sys- tem of beliefs µ in extensive form game ΓE is a spec- ification of a probability µ(x) ∈ [0, 1] for each decision node x in ΓE such that

x∈H µ(x) = 1 for all informa-

tion sets H.

Definition (MWG 9.C.2, Sequential Rationality): A strategy profile σ = (σ1, . . . , σI) in extensive form game ΓE is sequentially rational at information set H given a system of beliefs µ if, denoting by ι(H) the player who moves at H, we have E[uι(H)|H, µ, σι(H), σ−ι(H)] ≥ E[uι(H)|H, µ,

• σι(H), σ−ι(H)]

for all

• σι(H) ∈ ∆(Sι(H)). If strategy profile σ satisfies

this condition for all information sets H, σ is sequentially rational given belief system µ.

Definition (MWG 9.C.3, weak Perfect Bayesian Equilibrium): A profile of strategies and system of beliefs (σ, µ) is a wPBE in extensive form game ΓE if it has the following properties (i) σ is sequentially rational given µ (ii) µ is derived from σ through Bayes’ rule whenever

• possible. That is for any H with Pr(H|σ) > 0

µ(x) = Pr(x|σ) Pr(H|σ)∀x ∈ H Note: This is called a weak PBE because it does not ensure subgame perfection (see MWG Figure 9.C.5).

Hence for applications additional consistency require- ments are added to obtain a Perfect Bayesian Equilib- rium. These, however, differ with the application (e.g. FT’s consistency requirement B(ii) (p.332) that states that Bayes’ rule is also applied, if possible, to derive beliefs following probability 0 information sets, ensures consis- tency of a player’s belief across information sets but does not help in the example above) leading to a non- consensus in definitions even concerning questions such as whether different players can have different beliefs about the type of a third player or more extremely, whether different types of one player can have different beliefs (“type dependent beliefs”).

This non-consensus of the definition of a PBE makes a sequential equilibrium, though at a first glance more complicated, more appealing. In a sequential equi- librium consistency of beliefs across information sets, players and types is guaranteed.

Definition (MWG 9.C.4, Sequential Equilibrium): A profile of strategies and system of beliefs (σ, µ) is a sequential equilibrium in extensive form game ΓE if it has the following properties (i) σ is sequentially rational given µ (ii) There exists a sequence of completely mixed strate- gies {σk}∞

k=1, with limk→∞ σk = σ, such that µ = limk→∞ µk,

where µk denotes the beliefs derived from σk using Bayes’ rule. Trembling hand perfect equilibrium is similar, but also requires that σk form an equilibrium of the per- turbed game where all players make minimal trembles.

Refinements Because sequential equilibria are often not unique, and sometimes implausible, a large number of refinements has been suggested. We will only discuss here the most famous one (the Intuitive Criterion).

The Intuitive Criterion Consider a game where there is uncertainty only about the type of player 1. Let Θ be the set of types of player 1 and let u∗

1(θ) denote the equilibrium payoff of

type θ and let S∗( Θ, a) be the set of possible equilibrium responses for some beliefs with support Θ ⊂ Θ following an action a by player 1. Definition (Equilibrium Dominance): An action a is equilibrium dominated for type θ if u∗

1(θ) >

max

s∈S∗(Θ,a) u1(a, s, θ)

Let Θ∗∗(a) denote the set of types for which a is not equilibrium dominated.

Definition (Intuitive Criterion): A Sequential Equi- librium violates the Intuitive Criterion if there exists a type θ of player 1 and an action a such that min

s∈S∗(Θ∗∗(a),a) u1(a, s, θ) > u∗ 1(θ).

That is, an equilibrium is eliminated if there is a type θ who can by deviating to an action a assure himself a payoff above the equilibrium payoff as long as the other players do not assign positive probability to the devia- tion being made by a type for whom it is equilibrium dominated.

Example (The Beer-Quiche-Game, FT Figure 11.6) Player 1 can be of weak type θw (p = 0.1) or strong type θs (p = 0.9). First player 1 chooses his breakfast. θw prefers quiche and θs prefers beer (utility difference 1), but both prefer avoiding a fight over their preferred breakfast (avoiding fight yields additional utility of 2). Player 2 chooses whether to fight player 1 after observ- ing 1’s breakfast. 2 prefers to fight θw and not to fight θs (additional utility 1). There are two pooling equilibria, one where both have quiche, one where both have beer and player 2 doesn’t fight in this case but would fight with sufficient proba- bility if he observed the other breakfast.

This requires that player 2’s belief about the type after a deviation assigns at least probability 1/2 to θw. The Quiche equilibrium is eliminated by the IC: the equilibrium outcome having quiche and no fight is the best of all worlds for θw, hence deviating to beer is equilibrium dominated for θw. Thus Θ∗∗(beer) = {θs}. But player 2 won’t fight θs, so S∗(Θ∗∗(beer), beer) = {NoFight}, but for type θs it then pays to deviate to beer: min

s∈S∗(Θ∗∗(beer),beer) u1(beer, s, θs) = u1(beer, NoFight, θs) = 3

> 2 = u1(quiche, NoFight, θs) = u∗

1(θs)

Generally, requiring that beliefs are restricted to Θ∗∗, i.e. no player ever assigns positive probability to another type playing an equilibrium dominated action is even stricter than the IC. In case of only two types of player 1, however, they coincide. MWG, Example 13.AA.1: Let there be three types

• f player 1, {θ1, θ2, θ3}. Consider a PBE where out-of

equilibrium action ˆ a is equilibrium dominated for θ1, so Θ∗∗(ˆ a) = {θ2, θ3}. Assume θ2 would wish to deviate exactly for beliefs µ(θ2|ˆ a) ≥ 1

4 while θ3 would wish to

deviate exactly for beliefs µ(θ2|ˆ a) ≤ 3

4.

The IC does not eliminate the PBE, because for both 2 and 3, the “worst” belief in Θ∗∗(ˆ a) does not justify deviating. But for any particular belief, at least one type prefers to deviate, so requiring beliefs to be in Θ∗∗(ˆ a) eliminates the PBE.

Industrial Organization Applications Monopoly Let x(p) be the demand function, p(q) = Min{p : x(p) = q} be the inverse demand function facing a monopolist and let ¯ p be the minimal price such that demand is

• zero. Let c(q) be the monopolist’s costs and p and c be

continuous and twice differentiable. The monopolist’s decision problem is Maxq≥0p(q)q − c(q) Assume p(0) > c′(0) and there is a unique qo such that p(qo) = c′(qo). Then a solution to the max problem exists and the optimal quantity qm has to satisfy FOC p′(qm)qm + p(qm) = c′(qm)

That is, marginal revenue d(pq)/dq = p′q + p equals marginal costs. If p′(q) < 0 for all q, p(qm) > c′(qm), hence deadweight loss. Example: Linear inverse demand p(q) = a − bq, con- stant marginal cost c. Then qm = (a − c)/2b and pm = (a+c)/2, while welfare maximizing is qo = (a−c)/b and p0 = c.

Bertrand Duopoly: Assumptions:

• Firms compete through price choice
• Goods are homogeneous, firms have identical constant

marginal costs c.

• Prices are chosen simultaneously
• The firm with the lowest price captures the whole

demand

• In case of a tie, demand is split equally
• There is ¯

p such that x(¯ p) = 0

Proposition: The unique Nash-equilibrium of the Bertrand Game is that both firms choose p = c. Profits are zero. Remarks: - Two firms are sufficient to achieve out- come of perfect competition

• For n > 2 the crucial result continues to hold: all sales

are at marginal costs.

• If there is a smallest money unit ǫ then there are two

equilibria: p1 = p2 = c and p1 = p2 = c + ǫ.

• If there are capacity constraints, the equilibrium is no

longer to price at marginal costs.

Cournot Duopoly Assumptions:

• Firms simultaneously choose quantities q1, q2
• Costs c1(q1), c2(q2).
• Price is determined according to inverse demand func-

tion p(q1 + q2)

• technical assumptions as for monopoly.

Then duopolist i’s decision problem for given value of ¯ qj is Maxqi≥0p(qi + ¯ qj)qi − ci(qi)

The optimal quantity qi has to satisfy FOC p′(qi + ¯ qj)qi + p(qi + ¯ qj) ≤ c′(qi) (equality for qi > 0.) As for a monopolist, for the optimal quantity marginal revenue d(pqi)/dqi = p′(qi + ¯ qj)qi +p(qi + ¯ qj) equals mar- ginal costs c′(qi). Solution: bi(¯ qj), best response function bi(.). Cournot equilibrium is the Nash-equilibrium of the simultaneous move quantity setting game, hence pair q∗

1, q∗ 2 such that q∗ 1 = b1(q∗ 2) and q∗ 2 = b2(q∗ 1).

Proposition: Consider symmetric case c′

1(q1) = c′ 2(q2) = c.

Then q∗

1 = q∗ 2, q∗ 1 + q∗ 2 > qm and p > c.

proof: Due to c′

1 = c′ 2 = c, both qi are positive and

FOC’s hold with equality. Hence q∗

1 = q∗

• 2. p > c because

in FOC p′ < 0. Assume q∗

1 + q∗ 2 < qm. Then increasing

qi would increase joint profit but decrease j’s, hence for sure increases i’s. Assume q∗

1 + q∗ 2 = qm. This implies

p′(qm)qm

2 + p(qm) = c, violating monopoly FOC.

Asymmetric case: (qm

i :

i’s monopoly quantity). If costs are sufficiently asymmetric, such that b1(qm

2 ) = 0

• r b2(qm

1 ) = 0, in equilibrium one firm leaves the market

and the other acts as monopolist. The same can result even in the symmetric case if there are sufficiently high fixed costs F. example: Linear inverse demand p(q) = a−bq, constant marginal cost c. Then q1 = q2 = (a−c)/3b and p = (a+ 2c)/3. So Cournot is closer to the welfare maximizing qo = (a − c)/b, p0 = c than monopoly. Indeed, as the number of firms increases, the equilibrium converges towards perfectly competitive equilibrium.

Stackelberg equilibrium Like Cournot, but one firm (’leader’) moves first, other firm (’follower’) second, after observing leader’s quan- tity. The Stackelberg-equilibrium is the SPNE of the sequen- tial quantity setting game. Follower chooses bf(ql). Leader can hence choose his preferred point on the follower’s best response function and will be better off then in Cournot. Follower is worse

• ff because leader will choose larger quantity.

Supergames Bertrand game (price competition, lower price firm cap- tures the whole market, in case of equal prices the de- mand is split) with 2 firms, repeated T + 1 times. This is a “repeated game” or “supergame”. Πi(pit,pjt) : i’s profit at t = 0, . . . , T; marginal cost c. Πm monopoly profit, pm monopoly price. ⇒ discounted value of profits:

T

• t=0

δtΠi(pit, pjt)

no physical link between periods i.e. past prices do not affect profits, but the price strategy depends on the history Ht = (p10, p20, . . . , p1t−1, p2t−1) Finite horizon T < ∞ : Subgame Perfect Nash Equilibrium by backward induc- tion p1T = p2T = c ⇒ . . . (b.i) ⇒ p10 = p20 = c

⇒ finite repetition does not matter. But the usual critique of backward induction could be applied here, i.e. if firm i deviates now, is it still reasonable for firm k to believe that firm i will stick to the SPNE in the future

• r should this not rather be interpreted as a signal that

maybe firm i’s objective function is different than firm k believed up to now?

Infinite horizon T = ∞ Infinite repetition of Bertrand equilibrium is a SPNE (if the other always plays the stage game equilibrium it is a best reply in each period and hence in the whole game) But there are new equilibria: Set p = pm if pm has been chosen up to now by both and choose p = c forever otherwise (“trigger” strategy,

• r “grim trigger”, because it is not forgiving)

pit = pm if Ht = (pm, pm, . . . , pm, pm) c

• therwise

Deviating to pm − ε yields (almost) Πm now and 0 in the future. Sticking to the equilibrium strategy yields

Πm 2

forever. Πm ≤ Πm 2 (1 + δ + δ2 + . . .) = Πm 2 · 1 1 − δ ⇔ 2 ≤ 1 1 − δ ⇔ δ ≥ 1 2 ⇒ trigger strategies form SP Nash-equilibrium for δ ≥ 1

2

(since the infinite repetition of the Bertrand equilibrium is a SPNE, the punishment phase is a SPNE)

Note that collusion is achieved through a non-cooperative mechanism (i.e. collusion is not “nice” behavior or en- forced through external threats, but is an equilibrium strategy) Problems: 1. there are infinitely many equilibria. All prices p ∈ [c, pm] can be supported as a SPNE through such Nash- reversion strategies if δ ≥ 1

• 2. (“Folk Theorem”)

⇒ selection problem One might argue, that pm serves as a focal point, be- cause the outcome is Pareto-optimal and symmetric.

In case of cost asymmetries there are different monopoly prices and it is hence unclear which of these to coordi- nate on. For practical purposes, the selection problem persists even in case of symmetric costs, because the monopoly price is usually not perfectly known (firms might differ not with respect to costs, but with respect to their expectations of demand)

• 2. The equilibrium is not renegotiation proof: in pun-

ishment phase both firms would prefer to get back to collusive phase

The Chain Store Paradox (Selten, 1978)

• An Incumbent I firm faces a series of N (large, e.g.

20) potential entrants Ei.

• In each market i, if Ei enters, I decides whether to

fight (”prey”) or to accommodate

• I prefers to accommodate if Ei enters, but he prefers

that Ei does not enter

• Each Ei prefers to stay out if I fights, but prefers

to enter if I accommodates

• I could try to deter entry by fighting in early peri-
• ds, but
• The only SPNE is obtained by backward induction:

– In market N, I will accommodate if EN enters ⇒ EN will enter ⇒ in market N − 1, there is no reason for I to fight (because EN will enter in any case) ⇒ I will accommodate if EN−1 enters ⇒ EN−1 will enter ⇒ all Ei will enter and I will never fight

But Selten states: ”I would follow the deterrence theory. I would be very surprised if it failed to work.. . . My ex- perience suggests that mathematically trained persons recognize the logical validity of the induction argument, but they refuse to accept it as a guide to practical behavior.. . . The fact that the logical inescapability of the induction theory fails to destroy the plausibility of the deterrence theory is a serious phenomenon which merits the name of a paradox.” Resolutions to the Chain Store Paradox Incomplete information about I’s payoffs: with proba- bility p the incumbent is strong and prefers to fight if E enters (but still prefers no entry over entry and fight), with 1 − p he is weak

Payoffs for entrant: 0 if Out, b with 0 < b < 1 if (In, Accommodate), b − 1 if (In, Fight) Payoffs for weak incumbent Iw: a > 0 if Out, −1 if (In, Fight), 0 if (In, Accommodate) Payoffs for strong incumbent Is: a > 0 if Out, 0 if (In, Fight), −1 if (In, Accommodate)

Number periods backwards, i.e. last market is market 1, first is market N. Denote by pn the belief of En that I is strong if I has always fought so far Let N = 1, then p1 = p

• Clearly Is fights entry, Iw does not
• So E1 enters if

p(b − 1) + (1 − p)b > ⇐ ⇒ b − p > 0 ⇐ ⇒ p < b

Assume in the following that p < b (otherwise no en- trant without knowledge would ever enter) Let N = 2, then p2 = p Again in period 1 Is fights entry, Iw does not In period 1 E1 enters if p1(b − 1) + (1 − p1)b > ⇐ ⇒ b − p1 > 0 ⇐ ⇒ p1 < b Denote by wi the probability that Iw fights in period i and by si the probability that Is fights in period i. p1 = p2s2 p2s2 + (1 − p2)w2 = ps2 ps2 + (1 − p)w2

Assume Is fights in 2 and Iw does not, Then s2 = 1 and w2 = 0 and p1 = 1 and if I does not fight, E1 knows that I = Iw. Hence E1 will stay out if I fights in 2 and will enter if I does not fight in 2. Case a < 1 : Iw would not find fighting in 2 profitable, hence E2 enters and E1 enters if I does not fight in 2

Case a > 1 : Iw would find it profitable to fight in 2 ⇒ Contradiction to assumption that Iw does not fight Assume both Is and Iw fight in 2, Then p1 = p < b, so E1 will enter But then Iw will prefer to accommodate in 2 ⇒ con- tradiction So we need E1 to be indifferent after fight in 2, which requires p1 = b and hence b = p p + (1 − p)w2 ⇐ ⇒ w2 = p(1 − b) (1 − p)b < 1

Iw mixing in 2 requires that Iw is indifferent, thus that E1 enters after fight in 2 with probability e1 such that −1 + (1 − e1)a = 0 ⇐ ⇒ e1 = a − 1 a E2 would enter if (p + (1 − p)w2) (b − 1) + (1 − p)(1 − w2)b > 0 ⇐ ⇒

• p + p(1 − b)

b

• (b − 1) + (1 − p)b − p(1 − b) > 0

⇐ ⇒ p − p b + b − p > 0 ⇐ ⇒ p < b2

So if p < b2, E2 will enter, Iw will mix in 2 and E1 will mix if I fought in 2 and enter for sure if I accommodated in 2 If b2 < p < b, E2 will not enter, but E1 will enter If p = b2, E2 will mix

• If entry results, Iw will mix and E1 will mix if I

fought in 2

• If no entry results in 2, E1 will enter

We can continue with this logic for N periods. If p > bN, initially no entrant will enter and entry would be fought. Note that since b < 1, bN → 0 for N → ∞.

Auctions Introduction Definition: An Auction is a selling institution that elic- its information from potential buyers in the form of bids and where the outcome (i.e. who obtains the objects and who pays how much) is determined solely by this information. This implies that auctions are universal (i.e. any object can be sold by means of an auction) and anonymous (i.e. the identity of the bidders does not matter, hence if the bids of two bidders are ex- changed, the allocation and payments are exchanged accordingly and no other bidder is affected).

Procurement auctions used to buy a good from po- tential sellers work exactly correspondingly. Hence we can restrict the discussion to auctions employed to sell goods. Auctions are useful when the seller is unsure about the valuations (i.e. the maximal willingness to pay) of the

• buyers. Otherwise he could just offer the good to the

buyer with the highest valuation at a price just below this valuation. Auctions have been used for a long time, e.g. govern- ment bonds, drilling rights. Recent important applica- tions: privatization, spectrum auctions, internet auc- tion platforms.

Different auction formats are evaluated on the basis

• f revenue (expected selling price) and efficiency (allo-

cation to the bidder with the highest valuation). For practical purposes simplicity and the susceptibility to collusion are further (and possibly more) important cri- teria.

Common (Single Unit) Auction Forms

• 1. First-Price auction: All bidders submit a sealed bid,

the highest bidder wins and pays his bid

• 2. Second-Price auction: All bidders submit a sealed

bid, the highest bidder wins and pays the second highest bid

• 3. English Ascending Price auction (Japanese auction):

The auctioneer continuously raises the price until

• nly one bidder remains active, who obtains the
• bject at the price where the auction ended (i.e.

where the second to last bidder dropped out).

• 4. Dutch Descending Price auction:

The auctioneer starts with a high price (presumed to be higher than the maximal valuation) and continuously lowers the price until one bidder signals to buy the object at the current price. A number of other auctions formats is possible. Some might appear as unusual when thought of as a classi- cal auction, but less so when seen in another context (e.g. an all-pay auction, where all bidders pay their bids seems unusual for selling a painting, but a patent-race,

• r lobbying are essentially all-pay auctions). Some auc-

tions may at a first glance not conform to a straight- forward idea of an auction.

Single Unit Private Value Auctions Definition: Bidders are said to have private values if each bidder knows the value of the object to himself (and only to himself) for sure at the time of bidding. Otherwise, if the value of the object to a bidder may depend on information that other bidders have, values are said to be interdependent. An extreme case is that

• f a pure common value, where the value of the object

is the same for all bidders, but unknown by the time of bidding. Private values do not have to be statistically indepen- dent and in the case of interdependent values the sig- nals of the bidders can still be statistically independent. Independent private values are, however, the standard case.

Equivalences For the single unit case, the First-Price and the Dutch auction are equivalent in a strong sense, they are strate- gically equivalent (the games have the same normal form). The available strategies to a bidder consist of plainly choosing one number, the bid in the FPA or the price where the bidder would agree to buy the object in the DA in case it has not been sold yet. Also the

• utcomes are derived from the strategies in the same

way: the bidder choosing the highest number wins and pays this number. A bidder does not learn anything in DA, because when he does learn something the auction is over (and the fact that no other bidder has bought the object yet is not informative, because the strategies condition on this event in the first place).

Between the Second-Price and the English auction there is a weaker form of equivalence, because in the Eng- lish auction (with more then two bidders) a bidder can learn something by the drop-out prices of other bid- ders and could in principle condition his strategy on his observations (hence the games are not strategically equivalent). But in the private value case, the informa- tion gathered from the drop-out prices of other bidders is not informative and hence the equilibria in both auc- tions are identical.

Equilibria in the Symmetric Model There is a single object for sale to N bidders. Bidder i assigns a value Xi (this is a random variable) and the Xi are independently and identically distributed in some interval [0, ω] with distribution function F which has a continuous density with full support f = F ′. While ω = ∞ is allowed, E[Xi] < ∞ is assumed.

Bidder i knows the realization xi of Xi but only that the

• ther bidders’ values are distributed according to F.

Except for the realizations of the values, all aspects of the model, in particular F and N, are common knowl- edge. Bidders are risk neutral, they try to maximize expected profits. Bidders do not face liquidity constraints, i.e. bidder i is willing and able to pay up to xi. An auction determines a game with the strategies being bid functions βi : [0, ω] → R+. The focus will be on symmetric equilibria, because bidders are symmetric.

Second-Price and English auction: Proposition: In SPA and EA it is a weakly dominant strategy to bid one’s own valuation: βII(x) = x. Proof: Assume to the contrary that bidder i bids zi >

• xi. This changes the outcome only if for p, the highest
• f the other bidders’ bids, xi < p < zi.

In this case bidder i now wins the object and makes a loss. Hence

• verbidding is dominated. Bidding zi < xi only changes

the outcome if for p, the highest of the other bidders’ bids, zi < p < xi. In this case bidder i now misses a profitable deal that he could have made by bidding xi. Hence also underbidding is dominated. QED

The argument is even more obvious in EA: it cannot pay to drop out before the price p reaches xi and it cannot pay to stay in once the price exceeds xi, so it is (weakly) dominant to drop out at p = xi. Note that this result depends neither on risk neutrality nor on the symmetry of the bidders (not even on the independence of the distributions, only on values being private).

Some Notes on Order Statistics: Consider bidder 1. Let Y1 ≡ Y (N−1)

1

denote the highest value of the re- maining N − 1 bidders, i.e. Y1 is the first order statistic

• f X2, . . . , XN. The distribution function of Y1 is given

by G(y) = F(y)N−1, with density g. First-Price auction: In FPA, each bidder submits a sealed bid bi and payoffs are Πi =

• xi − bi if bi > maxj=i bj

0 if bi < maxj=i bj A bidder will clearly not submit a bid equal to his val- uation, because this guarantees a profit of 0. Raising the bid implies a trade-off. The chance of winning the

• bject are increased, but so is the expected price.

Assume that there is a symmetric equilibrium with in- creasing, differentiable strategy β. Obviously, bidding b > β(ω) is dominated, and β(0) = 0. Bidder 1 wins if his bid b > max

i=1 β(Xi) = β(max i=1 Xi) = β(Y1),

hence if Y1 < β−1(b). His expected payoff is G(β−1(b)) · (x − b). Maximizing w.r.t b yields g(β−1(b)) β′(β−1(b))(x − b) − G(β−1(b)) = 0 In a symmetric equilibrium b = β(x) implying G(x)β′(x) + g(x)β(x) = xg(x) d dx(G(x)β(x)) = xg(x)

Since β(0) = 0 βI(x) = 1 G(x)

x

0 yg(y)dy = E[Y1|Y1 < x]

The equilibrium bidding strategy can be rewritten as βI(x) = E[Y1|Y1 < x] = x −

x

G(y) G(x)dy = x −

x

• F(y)

F(x)

N−1

dy hence the degree of bid shading decreases in the num- ber of bidders. Example: Values are uniformly distributed over [0, 1]. Then F(x) = x, G(x) = xN−1 and hence βI(x) = N − 1 N x.

Revenue Comparison The expected payment of a bidder with value x is in SPA mII(x) = Pr(Win)·E[2nd highest bid|x is the highest bid] = Pr(Win) · E[2nd highest value|x is the highest value] = G(x) · E[Y1|Y1 < x] In FPA it is mI(x) = Pr(Win) · Amount Bid = G(x) · E[Y1|Y1 < x] = mII(x) Hence the expected payment for a bidder with value x is identical in FPA and SPA.

Thus the ex-ante expected payment of each bidder is the same and therefore so is the expected revenue of the seller (it is E

• Y (N)

2

• ).

Note, however, that for given values, the revenue is usually different. Furthermore, the revenues in SPA vary more than in FPA (e.g. in the case of 2 bidders with uniformly distributed values in [0, 1] the maximal revenue in FPA is 1

2, but in SPA it

is 1.) Precisely, the distribution of equilibrium prices in SPA is a mean preserving spread of the distribution of prices in FPA. Hence a risk-averse seller would prefer FPA (given that the bidders are risk-neutral).

The Revenue Equivalence Principle The observed identity of expected revenues between SPA and FPA holds in much more generality. Definition: An auction is standard if the highest bidder

• btains the object.

Proposition: Suppose values are iid and bidders are risk-neutral. Then any symmetric and increasing equi- librium of any standard auction, such that the expected payment of a bidder with value 0 is 0, yields the same expected revenue to the seller.

Proof: Let mA(x) be the expected payment in the symmetric equilibrium β of a standard auction A and let mA(0) = 0. Assume all bidders except for bidder 1 follow β and that bidder 1 with value x bids β(z) instead

• f β(x). Bidder 1 wins when β(z) > β(Y1) or z > Y1. His

expected payoff is ΠA(z, x) = G(z)x − mA(z) Note that mA(z) depends on β and z but not on x. Maximization yields ∂ ∂zΠA(z, x) = g(z)x − d dzmA(z) = 0 In equilibrium z = x is optimal, hence for all y d dymA(y) = g(y)y

Thus mA(x) = mA(0) +

x

0 yg(y)dy

=

x

0 yg(y)dy = G(x) · E[Y1|Y1 < x]

since by assumption mA(0) = 0. The right hand side does not depend on A and hence the expected payment

• f each bidder for a particular value does not depend
• n the particular auction, and therefore, the ex-ante

expected payment and thus the expected revenue of the seller do not either. QED

Example: Values are uniformly distributed on [0, 1] F(x) = x ⇒ G(x) = xN−1, thus for any standard auction with mA(0) = 0 mA(x) = N − 1 N xN E[mA(x)] = N − 1 N(N + 1) E[RA] = N · E[mA(x)] = N − 1 N + 1

Application: All-Pay Auction The revenue equivalence principle can be used to de- rive equilibria in other auctions, e.g. the all-pay auction where all bidders pay their bid. Consider an all-pay auction with symmetric, independent private values. Suppose there is a symmetric, increasing equilibrium such that mAP(0) = 0. The expected payment in an all-pay auction equals the bid and hence due to the revenue equivalence principle the equilibrium bid is βAP(x) = mA(x) =

x

0 yg(y)dy

Qualifications

• 1. Risk Aversion

Suppose bidders are risk-averse, all with the same utility function. In the case of iid private values, FPA yields a higher expected revenue than SPA. Intuition: In SPA risk-aversion does not change that bidding one’s value is a dominant strategy (by rais- ing the bid, one can increase the probability of win- ning, but only in cases where one does not want to win). In FPA at the equilibrium bid there is a perfect trade-off between the probability of winning and the amount won.

A risk-averse bidder will prefer, compared to a risk- neutral bidder, a smaller gain with a higher proba- bility and will hence choose a higher bid. By bidding higher he insures against ending up with 0.

• 2. Budget constraints

Suppose that bidder i has an absolute budget Wi. In SPA a bidder is more likely to be constrained than in FPA and hence FPA leads to higher expected revenues.

• 3. Symmetry

Consider the case where bidders are ex-ante asym- metric, i.e. their valuations are drawn from different distributions. In SPA, bidding one’s own value again remains a dominant strategy and hence SPA allocates effi- ciently even in case of asymmetries. In FPA, the weak bidder will bid more aggressively, because he faces a stochastically higher distribution

• f bids.

This more aggressive bidding of the weak bidder can lead to inefficiencies (if the weak bidder’s value is slightly smaller than the strong bidder’s), hence the revenue equivalence principle fails. Furthermore, there is no definite ranking of revenues between FPA and SPA.

• 4. Resale and Efficiency

One might argue that, since the auction reveals the valuations of the bidders, the auctioneer does not have to worry about efficient allocation because post-auction transaction will lead to an efficient al- location. This, however, is not correct even ab- sent transaction costs. If there is the opportunity for post-auction trade, bidders have an incentive not to completely reveal their valuation and hence post-auction bargaining takes place under incom- plete information which can lead to profitable deals being missed.

Single Unit Auctions with Interdependent Values As opposed to the case of private values, with interde- pendent values, the value of the object to bidder i, Vi is assumed to depend also on the information that other bidders have. Each bidder i has some private informa- tion Xi ∈ [0, ωi], called i’s signal. Vi is assumed to be given by a function Vi = vi(X1,X2, . . . , XN) that is non- decreasing in all the arguments, strictly increasing in Xi and twice continuously differentiable. Hence the value is completely determined by the signals. More general cases where there is some remaining uncertainty can be accommodated by considering the expected value given all signals (bidders are assumed to be risk neutral) vi(x1,x2, . . . , xN) = E[Vi|X1 = x1, . . . XN = xN]

Further assumption are vi(0, . . . , 0) = 0 and E[Vi < ∞]. Extreme cases: private values: vi(X1, . . . , XN) = Xi pure common value: Vi = V = v(X1, . . . , XN) for all i. Special case (mineral rights model): conditional on V = v, the signals Xi of the bidders are iid with E[Xi|V = v] = v.

The Winner’s Curse If values are not private, the estimate of the value to

• ne-self has to take information into account that is ob-

tained during or even after the auctions. In particular, it brings bad news if one wins the auction: bidder 1’s initial estimate of the value upon receiving the signal x is E[V |X1 = x]. Now if bidders are symmetric and fol- low the same strategy then winning the auction means that bidder 1 has the highest signal. The estimate of the value is then E[V |X1 = x, Y1 < x] < E[V |X1 = x]. Failure to take this into account, i.e. bidding plainly ac- cording to the initial signal instead of shading the bid below the initial estimate, leads to the winner’s curse, paying more than the value.

The Winner’s Curse magnitude increases with the num- ber of bidders. Note that the winner’s curse results from the failure to take the interdependence into account, it does not

• ccur in equilibrium.

Experimental Evidence: Kagel, Levin: Common Value Auctions and the Winner’s Curse, 2002, Princeton Uni- versity Press)

Equivalences Dutch and FPA are still strategically equivalent, be- cause that their normal forms are identical does not depend at all on the distribution of values and signals. But English and SPA are not equivalent with interde- pendent values, since information gathered in the Eng- lish auction (the drop-out prices of the other bidders) conveys valuable information about their signals and hence about one’s own value. In the case of only two bidders, however, the auction is over as soon as one ob- tains information and hence the two auctions are equiv- alent.

Consider again symmetric bidders, i.e. signals are drawn from the same distribution, and for bidder i the valua- tion does not change if we swap the signals of bidders j and k. Let v(x, y) = E[V1|X1 = x, Y1 = y] denote the expected value to bidder 1 if his signal is x and the highest of the other signals is y. Second-Price Auctions Proposition: Symmetric equilibrium strategies in the second price auction are given by βII(x) = v(x, x)

English Auctions In an English auction the bidders learn the prices where

• ther bidders drop out (in the symmetric model, the

identities of the bidders who dropped out are irrelevant) and as this information becomes available, this can (and should) influence when the remaining bidders plan to drop out. Symmetric equilibrium strategies are as follows:

• As long as no bidder has dropped out, drop out if

the price reaches the expected value assuming that all bidders have the same signal as yourself.

• If a bidder (say bidder k) drops out before you do,

infer his signal xk from the price where he dropped

• ut.
• Then drop out if the price reaches the expected

value given xk and assuming that all remaining bidders have the same signal as you.

• Continue in this fashion, i.e. infer the signals of the

bidders who drop out and recalculate the expected value.

An interesting implication is that the price at which you intend to drop out decreases over the course of the auction. Experimental evidence: CERGE-EI prep semester stu- dents learn this (while they do not learn to play the equilibrium, they learn to lower their drop-out prices when other bidders drop out).

Applications of Auction Theory See Paul Klemperer’s papers in the syllabus There are situations that at a first glance may not look like auctions, but can be interpreted as auctions, and where applying the revenue equivalence theorem yields interesting insights

Suggestion to reform US legal system: Losing party should pay winning party an amount equal to the losing party’s expenditures. Intuition: if spending $1 extra costs me $2, I will spend less. But this is wrong: assume that each party has a pri- vately known value from winning rather than losing the law-suit, that they independently decide how much money to spend and that the party that spends more money wins.

Both the existing system (pay your expenses) and the suggested new system (pay twice your expenses if you lose) lead to the party with the higher value winning (since in equilibrium the expenses increase with the value) and a party who does not value winning at all has zero surplus (because it will not spend anything and will hence lose) and thus RET applies. Hence the expected total expenses (i.e. the “expected revenue”) are the same and the new system would not reduce the legal expenses.

The suggested reform would also not reduce the in- centives for (and hence number of) lawsuits: while the legal expenditures for specific values change (they de- crease for low values and increase for high values), RET also implies that for any value the expected payment and hence the incentive to bring a lawsuit are the same (because the probabilities to win are also the same). Alternative system: the loser pays a part of the win- ner’s expenses (common in Europe). RET does not apply, because a party with value zero of winning the lawsuit does not have an expected payment of 0. In fact, expected payments are higher, but incentives to bring suits are lower.

Queueing: Assume that values from obtaining a ticket are distributed as in RET. Bids correspond to the show- up time. Obviously, the buyers with the highest values win and a potential buyer with 0 value will not show up (or show up exactly at the time when the tickets are sold and hence have 0 waiting time) and hence his expected payment is 0. Thus the RET applies, implying that any change to the system (e.g. making the waiting time more or less comfortable) will not affect the expected cost from waiting for each potential buyer and hence not the total social cost (it will affect the waiting time, but not the costs of waiting). Other types of such all-pay auctions, where auction theory can yield useful insights, are lobbying, political campaigns, patent races etc.

Bertrand paradox: it is often claimed that the unique equilibrium in a Bertrand game with inelastic demand is to price at marginal costs (assuming that prices are infinitely divisible). This, however, is wrong: the Bertrand game corre- sponds to a sealed-bid auction. But uniqueness of an equilibrium in an auction requires that the set of admissable bids is bounded, usually by constraining bids to be above the minimal possible

• value. If negative bids are allowed and if it is common

knowledge that both bidders have a value of 0, then it is a mixed-strategy equilibrium to bid below a price −p with probability k

p for any fixed non-negative k.

Similarly, in the Bertrand-game with two sellers, it is a mixed-strategy equilibrium to set prices larger than any price p with probability k

p for any fixed k (and of course,

with probability 1 above any price < k). To see this, observe that the expected profit from charg- ing any p ≥ k is then equal to pk

p = k and thus each

seller is indifferent between all prices p ≥ k. Hence if de- mand is perfectly inelastic, there is an equilibrium with arbitrarily large profits (because k is arbitrary).

Experimental Evidence on Single-Unit Auctions Some well-established results

• Common Value Auctions: Winner’s curse frequent,

but subjects learn to avoid it.

• Private Value Auctions:

– EA: players learn dominant strategy relatively quickly. – SPA: players also learn dominant strategy, but there is more overbidding than in EA.

– FPA: most experiments find substantial and per- sistent overbidding of the risk-neutral equilibrium We will be concerned with the last issue. See Goeree et al. Two prominent explanations:

• Risk aversion
• “Joy of winning”

First proposed explanation: risk aversion Harrison’s “flat maximum” critique: around the opti- mum, the profit function is very flat. Hence the incen- tives are small and deviations not very informative. Problem: the loss function is almost symmetric, so it is unclear why the flatness leads to overbidding.

Goeree et al.:

• incorporate asymmetry of loss functions.
• In one treatment losses are higher for overbidding,

in the other for underbidding

• Flatness explanation should lead to overbidding in
• ne treatment
• overbidding in both treatments indicates risk aver-

sion or joy of winning

• First-price auction, discrete valuations, bids inte-

gers.

• Treatment 1: $0, $2, $4, $6, $8, $11, all with same

probability

• Equilibrium: bid half the valuation (b = $5 for v =

$11).

• overbidding the equilibrium by $1 costs 0.25, but

underbidding only 0.08 (except for v = $11)

• Treatment 2: $0, $3, $5, $7, $9, $12, all with same

probability

• Equilibrium: bids as above
• now underbidding the equilibrium by $1 costs 0.25,

but overbidding only 0.08 (except for v = $0, $12)

• So if deviations from equilibrium are just noise, ex-

pect underbidding in treatment 1, but overbidding in treatment 2.

Results

• Overbidding in both treatments, consistent with

risk aversion and joy of winning

• Overbidding proportional to value, consistent with

risk aversion, but not with joy of winning

• More overbidding in treatment 2, consistent with

intuition for noisy behavior.

Model of noisy behavior: Quantal Response Equilib- rium

• probabilistic choice rule: let Ue(i) denote expected

payoff from action i. Then i is chosen with P(i), P(i) = exp(Ue(i)/µ)

n

k=1 exp(Ue(i)/µ)

• Alternatively, use Ue(i)1/µ instead of exp(Ue(i)/µ)
• for µ → 0 behavior is noise free, for µ → ∞ it is

random.

• this captures differences between treatments: e.g.

in treatment 2 overbidding is relatively cheaper, so

• ccurs with higher probability than in treatment 1.
• in Quantal Response Equilibrium beliefs and ac-

tions are consistent. E.g. for µ = 0 is just Nash- equilibrium.

• players play noisy best response to noisy behavior
• f others

• assume bidders are risk averse, expected utility func-

tion, b: bid, v: value, P w(b): Probability to win, r coefficient of relative risk aversion Ue(b|v) = (v − b)(1−r) 1 − r P w(b)

• can find QRE for each r, µ combination
• maximum-likelihood estimation used to find best

fitting noise and risk-aversion parameters.

• fits data well, much better than when imposing risk-

neutrality.

• QRE with joy of winning also performs reasonably

well, but not as good as QRE with risk aversion.