Subgame Perfect Equilibrium. Microeconomics II Extensive Form Games III. Sequential Rationality. Beliefs and Assessments. Definition (Belief System)

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1 icroeconomics II xtensive orm Games III event Koçkesen Koç University Subgame Perfect quilibrium,3 T B, 0,0 0, 0, Strategic form of the game,3,3 T, 0,0 B 0, 0, Set of Nash equilibria N(Γ) = {(T,),(,)} What is the set of SP? SP(Γ) = {(T,),(,)} (,) equilibrium is not plausible: is strictly dominated for player SP does not test for sequential rationality at every non-singleton information set event Koçkesen (Koç University) xtensive orm Games III / 57 event Koçkesen (Koç University) xtensive orm Games III / 57 Sequential ationality In this game sequential rationality of player requires playing How about in the following game? epends on player s beliefs,3 T, 0,0 0, 0, B We have to introduce beliefs in addition to strategies... and require sequential rationality at every information set given beliefs This immediately eliminates (,) equilibrium for the game in the previous slide Beliefs and ssessments Unless noted otherwise, from now on we restrict our analysis to extensive form games with perfect recall, and in which every information set is finite efinition (Belief System) belief system is a collection of probability distributions µ = (µ(. I))I I, where µ(h I) denotes the probability assigned to h I. We denote the set of all possible beliefs by. efinition (ssessment) n assessment (β,µ) B is a behavioral strategy profile combined with a belief system. event Koçkesen (Koç University) xtensive orm Games III 3 / 57 event Koçkesen (Koç University) xtensive orm Games III 4 / 57

2 utcomes and xpected Payoffs Take any assessment (β,µ) B Conditional probability assigned to history h given that history h has been reached: P β (h h ) If there is no sequence (a,a,...,ak) such that h = (h,a,a,...,ak), then P β (h h ) = 0 If there is a sequence (a,a,...,ak) such that h = (h,a,a,...,ak), then let h0 = h, hl = (h,a,a,...,al), for any l =,,...,k, and define P β (h h ) = k l=0 β ι(hl)(al+ hl) Perfect recall implies that each hl lies in a different information set and hence events {hl+ occurs conditional on hl occuring} are independent and multiplying their probabilities makes sense. utcomes and xpected Payoffs Conditional probability assigned to history h given that information set I has been reached: P β,µ (h I) P β,µ (h I) = h Iµ(h I)P β (h h ) Note that there is at most one h I such that P β (h h ) > 0. xpected payoff of the assessment (β,µ) conditional on reaching information set I to the player who is moving at that information set U ι(i)(β,µ I) = z ZP β,µ (z I)ui(z) event Koçkesen (Koç University) xtensive orm Games III 5 / 57 event Koçkesen (Koç University) xtensive orm Games III 6 / 57 Sequential ationality Is Sequential ationality Sufficient? Consider the following game, efinition (Sequential ationality) n assessment (β,µ) B is sequentially rational if for all I I ) β ι(i) argmax U ι(i) ((β ι(i),β ι(i)),µ I β ι(i) B ι(i) l r l r 0, 3,0,0 0, The following assessment is sequentially rational β( {a0}) =,β(r {,}) =,µ( {,}) = Yet, the strategy profile is not even a Nash equilibrium of this game There is something unsatisfactory about the beliefs We would like beliefs to be consistent with strategies event Koçkesen (Koç University) xtensive orm Games III 7 / 57 event Koçkesen (Koç University) xtensive orm Games III 8 / 57

3 Weak Sequential quilibrium or any β B and h H let I(h) be the information set that contains h, and P β (I(h)) = P β (h ) efinition (Weak Consistency) h I(h) n assessment (β,µ) B is weakly consistent if the belief system is derived from the strategies using Bayes rule whenever possible. In other words, for every h H such that P β (I(h)) 0 µ(h I) = Pβ (h) P β (I(h)) efinition (Weak Sequential quilibrium) n assessment (β,µ) B is a weak sequential equilibrium if it satisfies sequential rationality and weak consistency. event Koçkesen (Koç University) xtensive orm Games III 9 / 57 Is Weak Sequential quilibrium Good nough? Consider the following game 0,0 5,0 8,,8 0,0 The following assessment is a weak sequential equilibrium β( {a0}) =,β( {}) =,β( {(,),}) = µ((,) {(,),}) = Weak consistency is trivially satisfied since player s information set is off-the-equilibrium However, the strategy profile is not a SP event Koçkesen (Koç University) xtensive orm Games III 0 / 57 Consistency If a behavioral strategy profile β assigns a strictly positive probability to every action, we say β is completely mixed and write β 0. Clearly, if β 0, then P β (I(h)) > 0 for all h H or every β 0, there is a unique belief system µ such that (β,µ) is weakly consistent efinition (Consistency) n assessment (β,µ) B is consistent if there exists a sequence (β n,µ n ) (β,µ) such that β n 0 and (β n,µ n ) is weakly consistent for all n. Sequential quilibrium efinition (Sequential quilibrium) n assessment (β,µ) B is a sequential equilibrium if it satisfies sequential rationality and consistency. The intuition is that probability of an event conditional on zero probability events must approximate the probability that is derived from nearby completely mixed strategies. or on-the-path information sets, weak consistency uniquely determines beliefs and implies consistency. event Koçkesen (Koç University) xtensive orm Games III / 57 event Koçkesen (Koç University) xtensive orm Games III / 57

4 0,0 5,0 8,,8 0,0 Sequential quilibrium Consider again and the weak sequential equilibrium β( {a0}) =,β( {}) =,β( {(,),}) = µ((,) {(,),}) = or any weakly consistent (β n,µ n ) we must have µ n ((,) {(,),}) = β n( {}) The convergence condition requires that β n ( {}) 0 and hence µ n ((,) {(,),}) 0, But µ((,) {(,),}) = in the assessment event Koçkesen (Koç University) xtensive orm Games III 3 / 57 Sequential quilibrium What is the set of sequential equilibria for this game? 0,0 5,0 8,,8 0,0 There are three possibilities β( {(,),}) = sequential rationality implies β( {}) = consistency implies µ( {(,),}) = sequential rationality implies β( {(,),}) = 0, contradiction event Koçkesen (Koç University) xtensive orm Games III 4 / 57 Sequential quilibrium Sequential quilibrium 0,0 5,0 8,,8 0,0 0,0 β( {(,),}) = 0 sequential rationality implies β( {}) = consistency implies µ( {(,),}) = sequential rationality implies β( {(,),}) = 0 sequential rationality implies β( {a0}) = (β n,µ n ) below satisfies weak consistency and convergence criteria β n ( {a0}) = βn ( {}) = βn ( {(,),}) = n+ µ n ( {(,),}) = n+ event Koçkesen (Koç University) xtensive orm Games III 5 / 57 5,0 8,,8 0,0 3 β( {(,),}) (0,) sequential rationality of player implies β( {}) = consistency implies µ( {(,),}) = sequential rationality of player implies β( {(,),}) =, a contradiction event Koçkesen (Koç University) xtensive orm Games III 6 / 57

5 Selten s Horse Selten s Horse C c,, α d β 3 µ C c α d β 3 µ,, 3,3, 0,0,0 4,4,0 0,0, 3,3, 0,0,0 4,4,0 0,0, et µ = µ( {,(C,d)}),α = β( {a0}),β = β(d {C}),γ = β3( {,(C,d)}) There are three possible types of sequential equilibria: γ = γ = 0 3 γ (0,) et s investigate each. event Koçkesen (Koç University) xtensive orm Games III 7 / 57 γ = Sequential rationality (S) of player implies β = S of player implies α = 0 Bayes rule (B) implies µ = 0 But then S of player 3 implies γ = 0, a contradiction event Koçkesen (Koç University) xtensive orm Games III 8 / 57 Selten s Horse C c,, Selten s Horse or any µ (0,/3], let α n = µ n+,βn = µ n+,γn = n+,µn = µ µ n+ γ = 0 α µ 3 d β 3,3, 0,0,0 4,4,0 0,0, Sequential rationality (S) of player implies β = 0 S of player implies α = 0 B has no bite, but S of player 3 implies µ µ or µ /3 or µ = 0, let α n = (n+),βn = n+,γn = n+ n+,µn = n+(n+) Note that α n α,β n β,γ n γ,µ n µ and µ n α n = α n +( α n )β n so that consistency is satisfied. Therefore, the following is a class of S whose members differ only in off-the-equilibrium beliefs. {(α,β,γ,µ) : α = β = γ = 0,µ /3} event Koçkesen (Koç University) xtensive orm Games III 9 / 57 event Koçkesen (Koç University) xtensive orm Games III 0 / 57

6 Selten s Horse µ C α 3 d β 3,3, 0,0,0 4,4,0 0,0, γ (0,) Sequential rationality (S) of player 3 implies µ = /3 et s consider the three possibilities: β = : S of implies α = 0. But then B implies µ = 0, a contradiction. β (0,): S of implies γ = /4, which, by S of, implies α = 0. But then B implies µ = 0, a contradiction 3 β = 0: S of implies γ /4, which, by S of, implies α = 0. event Koçkesen (Koç University) xtensive orm Games III / 57 c,, Selten s Horse So let µ = /3,α = 0,β = 0,γ (0,/4] and α n = 3(n+),βn = 3(n+),γn = γ,µ n = 3 3(n+) Note that α n α,β n β,γ n γ,µ n /3 and µ n α n = α n +( α n )β n so that consistency is satisfied. Combining this with the class of S found previously, we have Proposition The set of S behavioral strategy profile is given by {(α,β,γ,µ) : α = β = 0,γ [0,/4]} ach such strategy profile can be combined with some µ /3 to constitute a S. event Koçkesen (Koç University) xtensive orm Games III / 57 Properties of Sequential quilibrium Properties of Sequential quilibrium Proposition (Kreps and Wilson (8)) very finite extensive form game with perfect recall has a sequential equilibrium. Proposition (Kreps and Wilson (8)) very sequential equilibrium is a subgame perfect equilibrium. Proposition (ne-eviation Property) consistent assessment (β,µ) is sequentially rational if, and only if, for all I I and all β B satisfying β ι(i) (. I ) = β ι(i)(. I ) for all I I, the following is true U ι(i)(β,µ I) U ι(i)(β,µ I) Proof. See Hendon et al (996), Games and conomic Behavior event Koçkesen (Koç University) xtensive orm Games III 3 / 57 event Koçkesen (Koç University) xtensive orm Games III 4 / 57

7 Perfect Bayesian quilibrium Consistency is difficult to work with which makes sequential equilibrium difficult to use in applications widely used alternative: perfect Bayesian equilibrium eveloped by udenberg and Tirole (99) for a particular class of extensive form games xtensive form games with incomplete information and observed actions Puts more restrictions on off-the-equilibrium beliefs than weak sequential equilibrium but less restrictions than sequential equilibrium Games with bserved ctions and Perfect Bayesian quilibrium In an extensive form game with incomplete information and observed actions ach player has a type that is revealed only to the player at the beginning of the game fter each history in the game, players simultaneously choose actions, which are revealed before players choose actions again Perfect Bayesian equilibrium requires that Posterior beliefs about a player s type is derived from prior beliefs using Bayes rule even at off-the-equilibrium information sets, as long as the player s action has positive probability conditional on the history Beliefs about a player s type depend only on that player s actions Beliefs are such that the players types are independently distributed event Koçkesen (Koç University) xtensive orm Games III 5 / 57 event Koçkesen (Koç University) xtensive orm Games III 6 / 57 Games with bserved ctions n extensive form game with incomplete information and observed actions is composed of the following elements: N: a finite set of players H: a set of histories 3 ι : H \Z N : the player function Players in ι(h) move simultaneously after history h There is a collection of sets {i(h)} i ι(h) such that (h) = {a : (h,a) H} = i ι(h)i(h) Games with bserved ctions 4 Θi: a finite set of types for player i Θ = i NΘi: set of type profiles 5 p: Nature s probability distribution over Θ Nature moves only once at the beginning of the game choosing a type profile θ according to p We assume types are independent: p(θ) = i N pi(θi) pi is the marginal distribution over Θi ssume pi(θi) > 0, θi Θi 6 n information partition I ach player observes only her own type and the history so far Player i s information set after history h is given by I(θi,h) = {((θi,θ i ),h) : θ i Θ i} Player i chooses from i(h) after history h 7 or each i N, a von Neumann-orgenstern payoff function ui : Θ Z event Koçkesen (Koç University) xtensive orm Games III 7 / 57 event Koçkesen (Koç University) xtensive orm Games III 8 / 57

8 Games with bserved ctions n assessment is given by a behavioral strategy profile and belief system (β,µ), where βi(ai θi,h) denotes the probability with which player i of type θi chooses action ai after history h µ(θi h) denotes the (common) probability assigned, after history h, to player i being of type θi efinition (Perfect Bayesian quilibrium) n assessment (β,µ) B is a perfect Bayesian equilibrium (PB) if it is sequentially rational and for all h H \Z Strong Bayes rule: or all i ι(h) and ai i(h), if there exists θ i Θi with µ(θ i h) > 0 and βi(ai θ i,h) > 0 (i.e., ai has positive probability conditional on h), then for any θi Θi Independence: or all θ Θ µ(θi h)βi(ai θi,h) µ(θi h,a) = µ( θi h)βi(ai θi,h) θi Θi µ(θ h) = i Nµ(θi h) 3 ction determined beliefs: or all θi Θi If i / ι(h) and a (h), then µ(θi h,a) = µ(θi h) If i ι(h) and a,a (h) and ai = a i, then µ(θi h,a) = µ(θi h,a ) event Koçkesen (Koç University) xtensive orm Games III 9 / 57 event Koçkesen (Koç University) xtensive orm Games III 30 / 57 PB Beliefs Strong Bayes rule: The denominator is the probability assigned to player i playing action ai conditional upon reaching history h, whereas the numerator is the probability assigned to player i playing action ai and having type θi conditional upon reaching history h. It applies even if h has zero probability under β, and h has positive probability but, under β, a is a zero probability event after h. This, of course, must be due to some other player deviating from a, as there is at least one type of player i who put positive probability on ai. It says that once beliefs are formed about player i at an off-the-path information set, they should, together with βi, form the basis for updating at subsequent information sets until player i deviates from βi again. In that sense it prevents belief reversals. ction determined beliefs: ther players actions cannot give information about a player s type. It fits our assumption that players strategies are independent. event Koçkesen (Koç University) xtensive orm Games III 3 / 57 The following figure illustrates the application of the Bayes rule µ Nature x y p p µ µ p µ p l r l r l r l r µ µ The standard Bayes rule leaves µ,µ,µ free. But the stronger rule defined above says that µ is free µ = µ µ is free event Koçkesen (Koç University) xtensive orm Games III 3 / 57

9 l m r Perfect Bayesian quilibrium Proposition very S is a PB. Proposition If each player has at most two types, or the game has at most two stages, then the sets of PB and S coincide. Proof. See udenberg and Tirole (99), Journal of conomic Theory PB vs. S Consider the following (portion of a) game Player has three types l,m,r, player has only one type the game has reached a history (h) where player gives probability to type r Strategies of the players are indicated by arrows, and beliefs in angled parentheses l m r Proposition (ne-eviation Property) n assessment (β,µ) that satisfies the strong Bayes rule is sequentially rational if, and only if, for all I I and all β B satisfying β ι(i) (a I) = β ι(i)(a I) whenever a / (I), the following is true U ι(i)(β,µ I) U ι(i)(β,µ I) event Koçkesen (Koç University) Proof. xtensive orm Games III 33 / 57 See Hendon et al (996), Games and conomic Behavior event Koçkesen (Koç University) xtensive orm Games III 34 / 57 l m r o these beliefs satisfy the conditions for PB? Since there is no θ such that µ(θ h) > 0 and β( θ,h) > 0 beliefs following are free. The same is true for beliefs that follow. Is the assessment consistent? oes there exist completely mixed (β n,µ n ) (β,µ) that satisfy Bayes rule for every n? event Koçkesen (Koç University) xtensive orm Games III 35 / 57 Bayes rule implies µ n µ n (m h)β n ( m,h) (m h,) = µ n (m h)β n ( m,h)+µ n (l h)β n ( l,h)+µ n (r h)β n ( r,h) ivide the numerator and denominator by µ n (m h) µ n (m h,) = β n ( m,h) β n ( m,h)+ µn (l h) µ n (m h) βn ( l,h)+ µn (r h) µ n (m h) βn ( r,h) Consistency and µ n (m h,),β n ( l,h) imply that µn (l h) µ n (m h) 0. Similarly, consistency and µ n (l h,),β n ( m,h) imply that µ n (m h) µ n (l h) 0, a contradiction. event Koçkesen (Koç University) xtensive orm Games III 36 / 57

10 PB vs. S This is an example proving that not every PB beliefs are consistent, and hence not every PB is a S. udenberg and Tirole (99) extend the definition to relative probabilities and establish equivalence between PB and S when beliefs about relative probabilities satisfy the three conditions given in the definition of PB. Kohlberg and eny (997) provide another characterization of consistency that does not use sequences. n pplication: eputation in Chain-Store Game Consider the following simple entry game with a > and 0 < b < Player is a potential entrant who may enter () or stay out () Player is the incumbent monopolist who may fight () or accommodate () The following is the game tree, where the first payoff number is that of the monopolist a,0,b 0,b Unique SP is event Koçkesen (Koç University) xtensive orm Games III 37 / 57 event Koçkesen (Koç University) xtensive orm Games III 38 / 57 n pplication: eputation in Chain-Store Game Now consider the K market version: the same game is played in markets,,...,k against different entrants in sequence irst, market K is played against entrant K, second against entrant K, and so on. onopolist s payoff is the sum of his payoffs in K markets. ntrant k observes the history of market interaction up to and including market k + What is the SP? In the unique SP every entrant enters and the monopolist accommodates in every market independent of the history Couldn t the monopolist build a reputation for being a tough competitor by fighting in earlier markets so as to deter entry later on? ormalization of this idea requires introducing incomplete information n pplication: eputation in Chain-Store Game There are two possible types of the monopolist: Tough (τ) and Weak (ω) Tough monopolist prefers to fight in any single market Weak has the preferences given before Nature determines the type of the monopolist: Tough with probability ε (0,), Weak with ε ssume that there is no integer k such that ε = b k onopolist knows his type but entrants don t Here is the K = version a,0 ε ω Nature τ a,0 ε event Koçkesen (Koç University) xtensive orm Games III 39 / 57,b 0,b 0,b,b event Koçkesen (Koç University) xtensive orm Games III 40 / 57

11 K = Nature ω τ Nature ε ε a,0 ε ω,b 0,b N = {,} H = {a0,,,(,),} ι(a0) =,ι() = Θ = {ω,τ},θ = {x} p(τ) = ε or all h : ι(h) =, I(x,h) = {((x,θ),h) : θ {ω,τ}} or all h : ι(h) =, I(θ,h) = {((θ,x),h)},θ {ω,τ} event Koçkesen (Koç University) xtensive orm Games III 4 / 57 τ a,0 ε 0,b,b a,0,b 0,b Perfect Bayesian quilibrium Sequential ationality (S) β( ω,) = 0,β( τ,) = ction etermined Beliefs (B) µ(τ a0) = ε B µ(τ ) = ε xpected payoff to entering is Therefore, S a,0 0,b,b ( ε)b+ε(b ) = b ε β( a0) = { 0, ε > b, ε < b event Koçkesen (Koç University) xtensive orm Games III 4 / 57 et s find PB for the two markets version: arket is played first and arket second. lso assume b < ε < b k= or any market history h, S β( ω,h,) = 0,β( τ,h,) = 0, µ(τ h) > b β( h) = [0,], µ(τ h) = b, µ(τ h) < b k= et s organize our search for PB according to the way tough monopolist plays. There are three possibilities: β( τ,) = 0 β( τ,) = 3 β( τ,) (0,) We will investigate each in turn. event Koçkesen (Koç University) xtensive orm Games III 43 / 57 event Koçkesen (Koç University) xtensive orm Games III 44 / 57

12 β( τ,) = 0 a. β( ω,) = SB and B µ(τ,) = 0,µ(τ,) = U((,β ),µ ω,) = +0 < 0+a = U((,β ),µ ω,), contradiction ( ) b. β( ω,) = 0 SB and B µ(τ,) [0,],µ(τ,) = ε U((,β ),µ τ,) = +0 < 0 U((,β ),µ τ,), c. β( ω,) (0,) SB and B µ(τ,) = 0 and β( τ,) = a. β( ω,) = SB and B µ(τ,) = ε,µ(τ,) [0,] U((,β ),µ ω,) = +0 < 0 U((,β ),µ ω,), b. β( ω,) = 0 SB and B µ(τ,) =,µ(τ,) = 0 U((,β ),µ ω,) = 0+0 < +a = U((,β ),µ ω,), µ(τ,) = ε ε +( ε) β( ω,) > ε U((,β ),µ ω,) = +0 < 0 U((,β ),µ ω,), event Koçkesen (Koç University) xtensive orm Games III 45 / 57 event Koçkesen (Koç University) xtensive orm Games III 46 / 57 β( τ,) = (cont d) c. β( ω,) (0,) SB and B µ(τ,) = 0 and µ(τ,) = ε ε +( ε) β( ω,) > ε S U((,β ),µ ω,) = 0+0 = +β(,)a β(,) = /a S U((,β ),µ τ,) = +0 < 0+( a) = a U((,β ),µ τ,), S and β(,) = /a b = µ(τ,) ntrant s expected payoff to ε(b ) +( ε) β( ω,) = ( b)ε b( ε) ( b ( b)ε ) b( ε) = b ε b Therefore, since we assumed ε > b, S β( a0) = 0 3 β( τ,) (0,) is left as an exercise. event Koçkesen (Koç University) xtensive orm Games III 47 / 57 Therefore, the following is a PB of the two-market game β( a0) =,β( τ,) =,β( ω,) = ( b)ε b( ε) β( ω,h,) = 0,β( τ,h,) = ε, h {a0,} µ(τ h) = 0, h = b, h = (,) { 0, h {,} β( h) = /a, h = (,) In this equilibrium, even the weak monopolist fights earlier with positive probability so as to deter entry later on. In this sense we may say that the weak monopolist builds reputation for being tough. event Koçkesen (Koç University) xtensive orm Games III 48 / 57

13 eputation in Chain-Store Game Now consider the K-market version. We will show that an equilibrium with the following features exists: Tough monopolist always fights Weak monopolist fights in the earlier markets and starts accommodating with positive probability towards the end. ntrants stay out in the earlier markets et k = K,K,...,, and hk {,,(,)} stand for the market k sequence of events. event Koçkesen (Koç University) xtensive orm Games III 49 / 57 et k(h) be the market number that corresponds to history h. or any h : ι(h) = β( τ,h) = 0, k(h) = β( ω,h) =, k(h) > and µ(τ h) b k(h) ( b k(h) )µ(τ h) ( µ(τ h))bk(h), k(h) > and µ(τ h) < bk(h) or any h : ι(h) = k,k = K,K,...,, µ(τ h) > b k βk( h) = a, µ(τ h) = bk 0, µ(τ h) < b k µ(τ a0) = ε and beliefs at market k as a function of beliefs at market k and hk are given by µ(τ h), hk = µ(τ h,hk) = max{b k,µ(τ h)}, hk = (,) and µ(τ h) > 0 0, hk = or µ(τ h) = 0 event Koçkesen (Koç University) xtensive orm Games III 50 / 57 et s verify that this is a PB. Beliefs ction determined beliefs (B) implies µ(τ a0) = ε and µ(τ h,) = µ(τ h). If, µ(τ h) = 0 we either have µ(τ h,hk) = 0 by strong Bayes rule (SB) or beliefs are free. Since β( τ,h) = 0, hk = implies µ(τ h,hk) = 0, by SB, or µ(τ h,hk) is free. If µ(τ h) > 0 and hk = (,), SB implies µ(τ h)β( τ,h,) µ(τ h,,) = µ(τ h)β( τ,h,) +( µ(τ h))β( ω,h,) If µ(τ h) b k, then β( ω,h,) =, and hence µ(τ h,,) = µ(τ h). If, µ(τ h) < b k, then β( ω,h,) = ( bk )µ(τ h) ( µ(τ h))b k and hence µ(τ h,,) = b k. Therefore, beliefs satisfy the three conditions for PB. event Koçkesen (Koç University) xtensive orm Games III 5 / 57 ntrant k If µ(τ h) b k, then β( θ,h) =,θ = τ,ω, so βk( h) = is optimal. If b k < µ(τ h) < b k, then probability of fight in market k is µ(τ h)+( µ(τ h)) ( bk )µ(τ h) µ(τ h) = ( µ(τ h))bk b k and the expected payoff to entering is b k µ(τ h) < 0, µ(τ h) > b k b k = 0, µ(τ h) = b k > 0, µ(τ h) < b k and hence the entrant s strategy is sequentially rational. event Koçkesen (Koç University) xtensive orm Games III 5 / 57

14 ,(,),(,),(,) (,) (,) (,) Tough monopolist By accommodating the tough monopolist gets today and 0 afterwards, whereas by fighting he gets 0 today and at worst 0 in the future. Therefore, always fighting is optimal. Weak monopolist If k(h) =, there is no tomorrow and is optimal. So, suppose k(h) > and note that playing brings a total payoff of zero. If µ(τ h) > b k(h), then µ(τ h,,) = µ(τ h) > b k(h) and therefore βk ( h,,) =, and the monopolist s payoff is at least +a > 0. In this case fighting is optimal. If µ(τ h) = b k(h), then µ(τ h,,) = b k(h) and therefore βk ( h,,) = /a. This implies that the payoff to is at least + aa = 0, and hence fighting is again optimal. Weak monopolist (cont d) ptimality in the case µ(τ h) < b k(h) can be proved by induction. In market, µ(τ h,,) = b and therefore βk ( h,,) = /a. The monopolist gets an expected payoff of + aa = 0 by fighting. Therefore, mixing is optimal. Now suppose mixing is optimal at market k when µ(τ h) < b k. This implies that, if µ(τ h) < b k, and bring the same expected total payoff (of zero) starting at k. Now suppose that µ(τ h) < b k in market k+. If the monopolist fights in this market, µ(τ h,,) = b k in market k. This implies that βk( h,,) = /a. urthermore, since µ(τ h,,) = b k < b k, if entry occurs the monopolist gets a total payoff of zero starting from market k, by the induction hypothesis. Therefore, the payoff to fighting is + a a+( a ) 0 = 0 in market k+. This proves that if µ(τ h) < b k, then mixing is optimal in market k +. event Koçkesen (Koç University) xtensive orm Games III 53 / 57 event Koçkesen (Koç University) xtensive orm Games III 54 / 57 et min{k : b k < ε} = 4. The following figure illustrates the evolution of beliefs and the equilibrium path. K The game starts in market K with belief µ K = ε. Since µ K > b K, entrant stays out and the next period belief µ K = ε. If the out-of-equilibrium outcome (,) () were to occur µ K = ε (µ K = 0). The game continues in this manner until market 4. 4 In market 4 we still have µ4 = ε > b 4, and therefore entrant stays out. However, since µ4 < b 3, weak monopolist would mix between and if entry were to occur. 3 In market 3 µ3 = ε < b 3, and entrant enters. Since µ3 < b, weak monopolist mixes between and. If market 3 outcome is, then µ = 0, and entrant enters in market. If the outcome in market 3 is (,), then µ = b and entrant mixes. Since µ < b, weak monopolist mixes as well. In market, weak monopolist accommodates if entry occurs. ntrant enters if the outcome of market is or, and mixes if it is (,). b ε K b k K K 4 3 event Koçkesen (Koç University) xtensive orm Games III 55 / 57 eputation in Chain-Store Game Note that the entrants stay out until market min{k : b k < ε}, which is independent of the number of markets. Therefore, in the above equilibrium U(β,µ) lim = a, for any ε > 0 N N n the other hand, if ε = 0, in the unique S, U(β,µ) = 0 N This shows that a small amount of uncertainty can lead to large changes in equilibrium outcomes. event Koçkesen (Koç University) xtensive orm Games III 56 / 57

15 eputation in Chain-Store Game There are other Nash equilibria of this game that are not sequential equilibria. or example, always fight for both types of the monopolist and always stay out for the entrant is such an equilibrium. or some parameters, there are also other S with different outcomes. But Kreps and Wilson (98, Journal of conomic Theory) show that if ε b k for k K and beliefs are such that µ(τ h,) µ(τ h,) for any history h, then the equilibrium we described is unique on the equilibrium path. event Koçkesen (Koç University) xtensive orm Games III 57 / 57