Equilibrium concepts so far.

Equilibrium concepts so far.
4.3 Dynamic Games of Incomplete Information: Perfect Bayesian (Nash) Equilibrium 4.3.1 Introduction Equilibrium concepts so far. Static/SMG games under symmetric info : NE + Mixed Strategy NE Dynamic/Seq.MG games under symmetric info : Subgame Perfect NE Static/SMG games under asymmetric info : Bayesian NE. For the Seq.MG games of asymmetric info, we will introduce ‘Perfect Bayesian NE’ Big picture so far, NEXT.

Equilibrium Concepts this semester
4.3 Dynamic Games of Incomplete Information: Perfect Bayesian (Nash) Equilibrium 4.3.1 Introduction Equilibrium Concepts this semester Complete/Symmetric Info Incomplete/Asymmetric Info Static(SMG) (1) NE MSNE (3) Bayesian NE (prob assigned to NE; conditional Eq.) Dynamic(Seq.MG) (2) Subgame perfect NE (NE in every subgame) Rollback Eq. (4) Perfect Bayesian NE (Bayesian NE in every continuation game) 지금까지 배운 균형들의 포함관계

4.3 Dynamic Games of Incomplete Information: Perfect Bayesian (Nash) Equilibrium
4.3.1 Introduction 일반화 (조건완화) 조건 강화 As we consider progressively richer games [(1) to (2), (2) to (4), (3) to (4)], we progressively strengthen the equilibrium concepts, in order to rule out implausible equilibria in richer games that would survive if we applied equilibrium concepts suitable for simpler games. Ex) NE 에 조건 추가하여 타당하지 않은 균형 제거  균형개념 강화 (NE  SPNE) In each case, the stronger equilibrium concept differs from the weaker concept only for the richer games, not for the simpler game (강균형은 복잡한 게임에서는 약균형과 다르나, 간단한 게임에서는 다르지 않음 강균형 ⊂약균형) In particular, PBNE is equivalent to BNE in static games of incomplete info, equivalent to SPNE in dynamic games of complete and perfect info (and many dynamic games of complete but imperfect info), and equivalent to NE in static games of complete info.

4.3.2 Sub-game vs. Continuation Game
4.3 Dynamic Games of Incomplete Information: Perfect Bayesian (Nash) Equilibrium 4.3.2 Sub-game vs. Continuation Game SPNE = NE for the entire game and NE for every subgame(하위게임). PBNE = BNE for the entire game and for every continuation(연속) game; replacing the idea of subgame with the more general idea of continuation game.(연속게임의 더 일반적인/많은 조건을 만족시켜야 함  더 어려움  균형이 더 축소됨/강해짐) Subgame can’t begin at a non-singleton node; can begin only at singleton node. Continuation game can begin at any complete info set (whether singleton or not): An extension of subgame with prob. assigned. Subgame ⊂ Continuation game Continuation game subgame

4.3 Dynamic Games of Incomplete Information: Perfect Bayesian (Nash) Equilibrium 4.3.2 Sub-game vs. Continuation Game PBNE: BNE 중에서 연속게임에서도 BNE인 BNE PBNE⊂ SPNE Sub-game을 통과하는 조건 Sub-game Perfect NE 통과하면 연속게임과 하위게임 관계 (더 적은 조건을 통과하여야 하므로 덜 엄격) 큰 것 작은 것 모두 통과 Continuation Game Sub-game Continuation game을 통과하는 조건 Perfect Bayesian NE 통과하면 작은 것만 통과 (더 많은 조건을 통과하여야 하므로 더 엄격) 따라서, PBNE⊂ SPNE

(=2 subgames with prob.=1 on right node for 2)
4.3 Dynamic Games of Incomplete Information: Perfect Bayesian (Nash) Equilibrium 4.3.2 Sub-game vs. Continuation Game How many subgames? (click) 2, excluding the entire game. How many continuation games? (click) 2, excluding the entire game. (why? click) *아래 경우, subgame=continuation game  subgame 2개면, continuation game도 자동으로 2개 1 L R 2 2 L’ R’ L’ R’ 3, 1 1, 2 2, 1 0, 0 Continuation game 2 sub-games 2 continuation games (=2 subgames with prob.=1 on right node for 2) subgame

={left node, right node}
4.3 Dynamic Games of Incomplete Information: Perfect Bayesian (Nash) Equilibrium 4.3.2 Sub-game vs. Continuation Game In the below game, there is no subgame, excluding the entire game. But continuation game? 1, excluding the entire game. It can begin at any complete information set (singleton or not) 1 a complete info. set ={left node, right node} L R 2 2 p=0 (why?) 1-p=1 L’ R’ L’ R’ 3, 1 1, 2 2, 1 0, 0 No subgame, 1 continuation game

4.3.2 Sub-game vs. Continuation Game Perfect Bayesian NE strengthens the requirements of SPNE by explicitly analyzing the players’ beliefs(확률할당), as in BNE. Thus, PBNE⊂ SPNE. cf.) All human beings are accepted  Only men are accepted (strengthened requirement)  men ⊂ Human We describe a game of incomplete info as though it were a game of imperfect info; nature reveals player i’s type to i but not to j, so player j does not know the complete history of the game. Thus, an eq. concept (PBNE) designed to strengthen BNE in static games of incomplete info also can strengthen SPNE in dynamic games of complete but imperfect info.

4.3 Dynamic Games of Incomplete Information: Perfect Bayesian (Nash) Equilibrium 4.3.2 Sub-game vs. Continuation Game Remember this. Asymmetric info: some info is NOT known to others. Incomplete Info: players lack some info (frequently on payoffs) Imperfect info: players don’t know where they are. Imperfect-complete-symmetric info game: Rock-Paper-Scissors game. In many cases Incomplete Information Asymmetric Information Always Always Imperfect 하지만 complete 할 수 있음 Not always Not always Imperfect 하지만 symmetric 할 수 있음 Imperfect Information

4.3 Dynamic Games of Incomplete Information: Perfect Bayesian (Nash) Equilibrium 4.3.2 Sub-game vs. Continuation Game Because, PBNE⊂ SPNE, we have the following Venn-diagram. 균형 정보 완벽 완전-대칭 BNE x MSNE o NE x/o SPNE RE PBNE *NE를 기점으로, 동시선택게임이 되어 정보의 불완벽성이 증가하거나, 정보 비대칭성/불완전성이 증가하면 여러 가능성에 대응하여 균형은 더 커지게 됨 *반면, NE를 기점으로, 순차선택게임이 되어 정보의 불완벽성이 제거되거나, 보다 강화된 균형조건 (부분게임연속게임)이 적용되면서 균형은 점차 작아지게 됨 [End of Chapter]

4.3.3 Introduction to PBNE: motivation
4.3 Dynamic Games of Incomplete Information: Perfect Bayesian (Nash) Equilibrium 4.3.3 Introduction to PBNE: motivation Consider the following dynamic game of complete but imperfect information. NE? (L, L’) and (R, R’). SPNE? The same. There is no subgame other than the entire game. NE=SPNE for games w/o subgame. 1 R Not a subgame b/c it begins at the initial node. Chapter03-2, slide 54 1, 3 NE Player 2 L’ R’ Player 1 L 2, 1 0, 0 M 0, 2 0, 1 R 1, 3 L M 2 L’ R’ L’ R’ 2, 1 0, 0 0, 2 0, 1

4.3.3 Introduction to PBNE: motivation We introduced SPNE to exclude non-credible threat. But in this game, the SPNE includes a non-credible threat, (R, R’), which is an implausible equilibrium. Why? If 2 gets to move, then playing L’ dominates playing R’ (1>0; 2>1), so player 1 should not be induced to play R by 2’s threat to play R’ if given the move. SPNE may fail to exclude a non-credible threat for complete and imperfect info game, even though SPNE excludes a non-credible threat for complete and perfect info game. Thus, we need additional requirements to strengthen SPNE.

4.3.4 PBNE: the requirements Requirement 1: At each info set, the player with the move must have a belief about which node in the info set has been reached. For a singleton info set, the player’s belief puts prob 1 on the single decision node. For a non-singleton info set, a belief is a prob distribution over the nodes in the info set. Implication: If the play of the game reaches player 2’s nonsingleton info set, then player 2 must have belief about which node has been reached (or, equivalently, about whether player 1 has played L or M). This belief is represented by the prob p and (1- p) attached to the relevant nodes.

4.3.4 PBNE: the requirements Implication of Requirement 1: probabilities assigned. 1 R 1, 3 L M [ p ] [ 1- p ] 2 L’ R’ L’ R’ 2, 1 0, 0 0, 2 0, 1

4.3.4 PBNE: the requirements
4.3 Dynamic Games of Incomplete Information: Perfect Bayesian (Nash) Equilibrium 4.3.4 PBNE: the requirements Requirement 2: Given their beliefs, the players’ strategies must be sequentially rational. That is, at each info set the action taken by the player with the move (and the player’s subsequent strategy) must be optimal given the player’s belief at that info set and the other players’ subsequent strategies (where a “subsequent strategy” is a complete plan of action covering every contingency that might arise after the given info set has been reached.)  Req.2 rejects any strategy profiles which specify at any info set an action which is dominated at that info set. A strategy profile (sometimes called a strategy combination) is a set of strategies for all players which fully specifies all actions in a game. A strategy profile must include one and only one strategy for every player.

4.3 Dynamic Games of Incomplete Information: Perfect Bayesian (Nash) Equilibrium 4.3.4 PBNE: the requirements Implication of Requirement 2: sequentially rational. 2’s expected payoff playing R’=0*p+1*(1-p)= 1-p 2’s expected payoff playing L’=1*p+2*(1-p)= 2-p Since 1-p<2-p (R’ is dominated by L’), Requirement 2 implies that it is sequentially rational for 2 to choose L’. 1 R 1, 3 L M The continuation game [ p ] [ 1- p ] 2 L’ R’ 2p, 2-p 0, 1-p L’ R’ L’ R’ 2, 1 0, 0 0, 2 0, 1

4.3 Dynamic Games of Incomplete Information: Perfect Bayesian (Nash) Equilibrium 4.3.4 PBNE: the requirements Thus, Requirement 1&2 removes the implausible equilibrium (R, R’). However, Req.1&2 does not insist that these beliefs be reasonable. Consider the following example. 2 NE, (L,L’) and (M,R’) 1 2 L’ R’ 1 L 2, 1 0, 0 M 0, 1 L M [ p ] [ 1- p ] 2 L’ R’ L’ R’ 2, 1 0, 0 0, 0 0, 1

4.3 Dynamic Games of Incomplete Information: Perfect Bayesian (Nash) Equilibrium 4.3.4 PBNE: the requirements Consider (L,R’; and believes p=0). It satisfies Req.1(prob distribution assigned) and Req.2(The expected payoff of playing L’ is 1*p+0*(1-p)=p. The expected payoff of playing R’ is 0*p+1*(1-p)=1-p. Thus, if p=0, R’ dominates L’ (E(L’)=0 <E(R’)=1) sequentially rational.)  Req.1&2 satisfied. However, (L,R’) is not even a NE. Why did this happen? Because 2’s belief on p=0 was not reasonable; if 2 believes 1 takes L, p should be 1, not 0, p≠0. Req.3은 이러한 비합리적인 경우를 배제시킴.

4.3.4 PBNE: the requirements In order to impose further requirements on the players’ beliefs, we define the following. An info set is on the equilibrium path if it will be reached with positive probability if the game is played according to the equilibrium strategies. An info set is off the equilibrium path if it will NOT be reached at all if the game is played according to the equilibrium strategies. Equilibrium can mean Nash, subgame perfect, Bayesian, or perfect Bayesian equilibrium.

4.3 Dynamic Games of Incomplete Information: Perfect Bayesian (Nash) Equilibrium 4.3.4 PBNE: the requirements Implication of definition: (L, L’), is on the equilibrium path. (R, R’) is off the equilibrium path. 1 R 1, 3 (L,L’) is on the equilibrium Path (R,R’) is off the equilibrium Path L M [ p ] [ 1- p ] 2 L’ R’ L’ R’ 2, 1 0, 0 0, 2 0, 1 2’s info set is reached. 2’s info set is not reached.

4.3 Dynamic Games of Incomplete Information: Perfect Bayesian (Nash) Equilibrium 4.3.4 PBNE: the requirements Requirement 3: At info sets on the equilibrium path, beliefs are determined by Bayes’ rule and the players’ equilibrium strategies. Implication of Requirement 3: Suppose there were a mixed strategy eq. in which player 1 plays L with probability q1, M with prob q2, and R with prob 1- q1 - q2. Then Req.3 would force player 2’s belief to be p= q1 /(q1 + q2). Bayes’ rule 1- q1 - q2 q1 q2 Why not p= q1 /(q1 + q2+(1-q1-q2))? R is excluded by req.2 1-q1-q2=0

4.3.4 PBNE: the requirements Requirements 1-3  An equilibrium no longer consists of just a strategy for each player but now also includes a belief for each player at each info set at which the player has the move. This crucial feature of this equilibrium concept is due to Kreps and Wilson (1982): beliefs are elevated to the level of importance of strategies in the definition of equilibrium.

4.3.4 PBNE: the requirements The advantage of making the players’ belief explicit in this way is that, just as in earlier chapters we insist that the players choose credible strategies, we can now also insist that they hold reasonable beliefs, both on the equilibrium path (in req. 3) and off the equilibrium path (in req. 4, which follows next). Req. 1-3 most capture the idea of PBNE. But more requirements need to be imposed to eliminate implausible equilibria.

4.3 Dynamic Games of Incomplete Information: Perfect Bayesian (Nash) Equilibrium 4.3.4 PBNE: the requirements Requirements 4: At info sets off the equilibrium path, beliefs are determined by Bayes’ rule and the players’ equilibrium strategies where possible. Motivation: Example of 3 persons game below. 1 A 2,0,0 D 2 L R [ p ] 3 [ 1- p ] L’ R’ L’ R’ 1,2,1 3,3,3 0,1,2 0,1,1

4.3 Dynamic Games of Incomplete Information: Perfect Bayesian (Nash) Equilibrium 4.3.4 PBNE: the requirements Implication: One subgame (click). NE of the subgame? (L, R’) (click). The unique SPNE of the entire game? (D,L,R’). The strategy profile ‘(D,L,R’) and p=1’ satisfies req. 1&2 [E(L’)=1*p+ 2*(1-p)=2-p=1<E(R’)=3*p+ 1*(1-p)=1+2*p=3] and req.3 [p=q1/(q1+q2) =1/(1+0) for player 3]. 1 2,0,0 Subgame Player 3 L’ R’ Player 2 L 2, 1 3, 3 R 1, 2 1, 1 A D subgame 2 L R q1 q2 [ p ] 3 [ 1- p ] L’ R’ L’ R’ 1,2,1 3,3,3 0,1,2 0,1,1

4.3 Dynamic Games of Incomplete Information: Perfect Bayesian (Nash) Equilibrium 4.3.4 PBNE: the requirements Also, ‘(D,L,R’) and p=1’ for player 3 satisfy req. 4, b/c there is no info set off the equilibrium path. Thus, “(D,L,R’) and p=1 for player 3” is a perfect Bayesian NE. ‘(D,L,R’) and p=1’ is on the equilibrium path. Info set(점선)이 Eq. path에 연결되어 있음

4.3 Dynamic Games of Incomplete Information: Perfect Bayesian (Nash) Equilibrium 4.3.4 PBNE: the requirements Now consider the strategies (A,L,L’) together with p=0 (b/c 1 chooses A). The strategy profile, (A,L,L’; p=0) is off the equilibrium path. 1 A 2,0,0 D 2 L R q1 q2 [ p=0 ] 3 [ 1- p=? ] L’ R’ L’ R’ 1,2,1 3,3,3 0,1,2 0,1,1

4.3 Dynamic Games of Incomplete Information: Perfect Bayesian (Nash) Equilibrium 4.3.4 PBNE: the requirements “(A,L,L’) with p=0” is a NE. These also satisfy req. 1-3 (player 3 has a belief and acts optimally given it, and players 1 and 2 act optimally given the subsequent strategies of the other players, next slide). But this is not a SPNE b/c the unique NE of the game’s only subgame is (L, R’). 1 plays A Player 3 L’ R’ Player 2 L 2, 0, 0 R 1 plays D Player 3 L’ R’ Player 2 L 1, 2, 1 3, 3, 3 R 0, 1, 2 0, 1, 1

4.3.4 PBNE: the requirements The strategy profile (A,L,L’; p=0) satisfies Req.1-3? Req.1: prob distribution assigned? YES. Req.2: Sequentially rational? YES. Check: E(L’)=1*p+ 2*(1-p)=2-p=2>E(R’)=3*p+ 1*(1-p)=1+2*p=1] Req.3: p reasonable on the eq. path (follows Bayes’ rule)? YES. B/c Req.3, which puts restriction on p that is on the equilibrium path, puts no restriction on p b/c it is off the equilibrium path  Req.3 is trivially satisfied. But the problem is that player 3’s belief (p=0) is inconsistent with player 2’s strategy (L), b/c if 2 choose L, p=1. But Req.1-3 impose no restriction b/c it is off the path.

4.3.4 PBNE: the requirements Therefore, Req.4, which puts a restriction on p off the path, is needed to rule out (A,L,L’; p=0). Req.4 forces player 3’s belief to be determined by player 2’s strategy: if 2’s strategy is L then 3’s belief must be p=1; if 2’s strategy is R then 3’s belief must be p=0. But if 3’s belief is p=1 then Req.2 forces 3’s strategy to be R’, so the strategies (A,L,L’) and the belief p=0 do not satisfy Req.1-4. Definition: A perfect Bayesian (Nash) equilibrium consists of strategies and beliefs satisfying Requirements 1 through 4.

Next Steps Cooperative games Bargaining Theory in Agriculture (Fall semesters) More details on Bayesian games with continuous strategies.(Skipped) Evolutionary games: Equilibrium Selection. And other topics such as Mechanism Design(Reverse Game Theory).

A Simple Poker Game (skip)
Nature Prob. of High=p Prob. of Low=1-p AIS1 AIS2 AISi: Alice’s ith Information Set Alice Fold Fold BIS: Bob’s Info. Set Raise Raise -1, 1 -1, 1 Bob BIS Meet Pass Meet Pass 2, -2 1, -1 -2, 2 1, -1 Stage 0: Alice and Bob place bet: $1 each. Stage 1: Nature moves. High or Low card. Each prob.=1/2. Stage 2: Alice moves. Raise or Fold. If Alice folds, Bob wins and takes the Alice’s bet of $1. Stage 3: If Alice Raises at Stage 2, If High card AND Alice Raises (+$1) AND Bob Meets  Alice wins. Takes the bet of $2. If High card AND Alice Raises (+$1) AND Bob Pass  Alice wins. Takes the bet of $1. If Low card AND Alice Raises (+$1) AND Bob Meets  Bob wins. Takes the bet of $2. If Low card AND Alice Raises (+$1) AND Bob Pass  Alice wins. Takes the bet of $1.

Strategies of Alice and Bob
(skip) Strategies of Alice and Bob Alice : Does know where she is. Alice’s strategies are: If High card, Raise OR If Low card, Raise If High card, Raise OR If Low card, Fold If High card, Fold OR If Low card, Raise If High card, Fold OR If Low card, Fold Bob : Doesn’t know where he is. Bob’s strategies are: Meet Pass

Transformation of Extensive Form into Normal Forma or Strategic Form
(skip) Transformation of Extensive Form into Normal Forma or Strategic Form 4 strategies for Alice, 2 strategies for Bob Payoffs?  Expected payoffs.

The Expected Payoffs. (skip)
Multiply ½ to the payoff for the pure strategies Nash eq.? None. Perfect Bayesian Equilibrium? (Next)

For Alice (skip) At AIS1(High card),
1. If Alice Raises, her Expected Payoff is E(Raise)=2*q+1*(1-q)=q+1 (q; the prob that Bob plays Meet) 2. If Alice Folds, the game ends and her Payoff is -1 (=E(Fold)). Then, 0<q<1.  E(Raise)=q+1>-1=E(Fold). That is, at AIS1, Alice always gets better payoff when she plays Raise. Thus, at AIS1, the prob that Alice plays Raise is 1. The prob that she plays Fold is 0.

For Alice (cont’d) (skip) At AIS2(Low card),
1. If Alice Raises, her Expected Payoff is -2*q+1*(1-q)=-3q+1 2. If Alice Folds, the game ends and her Payoff is -1. Thus, If -3q+1>-1 (or q<2/3), Alice Raises. If -3q+1<-1 (or q>2/3), Alice Folds. If -3q+1=-1, Alice either Raises or Folds (same payoffs).

(skip) For Bob. Alice will NOT choose Fold at AIS1. Why? Alice will play Raise if she gets High card. Thus, Fold-Raise, Fold-Fold will not be used by Alice. Thus, Bob only needs to consider Alice’s strategies of Raise-Raise and Raise-Fold. In this case, Bob’s mixed strategies? 0*q+(1)*(1-q) Alice’s Payoff 0.5*q+(0)*(1-q) That is, Bob’s best strategy is To meet with prob of 2/3. 2/3 1 q

(skip) For Alice. Bob’s strategy is playing q=2/3. Thus, Alice should react to mix Raise or Fold. Alice’s Payoff 0*p+(0.5)*(1-p) 1*p+(0)*(1-p) 1/3 1 p That is, at AIS2(low card), Alice should play Raise with the prob of 1/3 and play Fold with prob of 2/3.

Perfect Bayesian Equilibrium for the poker game.
(skip) Perfect Bayesian Equilibrium for the poker game. Alice plays Raise if q<2/3(Bob Meets with prob lower than 2/3). Alice plays Fold if q>2/3(Bob Meets with prob higher than 2/3). If q=2/3, Alice plays Raise with prob of 1/3 and plays Fold with prob of 2/3 Bob Meets with prob of 2/3.

Bayes’ Rule (Law): An Example
P(A) is the prior probability or marginal probability of A. It is "prior" in the sense that it does not take into account any information about B. P(A|B) is the conditional probability of A, given B. It is also called the posterior probability because it is derived from or depends upon the specified value of B. P(B|A) is the conditional probability of B given A. P(B) is the prior or marginal probability of B, and acts as a normalizing constant. P(G) is the prob the student is a girl, 0.4. P(B) is the prob that the student is a boy, 0.6. P(T|B)=1 (Boys wear trousers) P(T|G)=0.5 (Girls wear trousers half the time) P(T) is the prob that the student is wearing trousers. P(T) = P(T|G)P(G) + P(T|B)P(B)= 0.5× ×0.6 = 0.8. P(G|T)=P(T|G)*P(G)/P(T)=0.5*0.4/0.8=0.25; the prob that a student wearing trouser is a girl. 20/80=0.25 = 0.2/( )

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