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.