M9302 Mathematical Models in Economics Instructor: Georgi Burlakov 5.1.Static Games of Incomplete Information Lecture 5 21.04.2011 OPVK_MU.tif Revision oWhen a combination of strategies ois a Nash equilibrium? nIf for any player i, is player i’s best response to the strategies of the n-1 other players oFollowing this definition we could easily find game that have no Nash equilibrium: nExample: Penny Game Example: Penny Game P1 Heads Tails Heads -1,1 1,-1 Tails 1,-1 -1,1 P2 No pair of strategies can satisfy N.E.: If match (H,H), (T,T) – P1 prefers to switch If no match (H,T), (T,H) – P2 prefers to switch Extended definition of Nash Equilibrium oIn the 2-player normal-form game G={S1,S2;u1,u2}, the MIXED strategies are a Nash equilibrium if each player’s mixed strategy is a best response to the other player’s MIXED strategy oHereafter, let’s refer to the strategies in Si as player i’s pure strategies oThen, a mixed strategy for player i is a probability distribution over the strategies in Si Example: Penny Game oIn Penny Game, Si consists of the two pure strategies H and T oA mixed strategy for player i is the probability distribution (q,1-q), where q is the probability of playing H, and 1-q is the probability of playing T, oNote that the mixed strategy (0,1) is simply the pure strategy T, likewise, the mixed strategy (1,0) is the pure strategy H o Example: Penny Game oComputing P1’s best response to a mixed strategy by P2 represents P1’s uncertainty about what P2 will do. oLet (q,1-q) denote the mixed strategy in which P2 plays H with probability q. oLet (r, 1-r) denote the mixed strategy in which P1 plays H with probability r. o Example: Penny Game oP1’s expected payoff from playing (r,1-r) when P2 plays (q,1-q) is: o o owhich is increasing in r for q<1/2 (i.e. P1’s best response is r=1) and decreasing in r for q>1/2 (i.e. P1’s best response is r=0). oP1 is indifferent among all mixed strategies (r,1-r) when q=1/2. Example: Penny Game (Tails) (Tails) (Heads) (Heads) 1/2 1 q r 1 r*(q) Because there is a value of q such that r*(q) has more than one value, r*(q) is called P1’s best-response correspondence. Example: Penny Game (Tails) (Tails) (Heads) (Heads) 1/2 1 q r 1 q*(r) The intersection of the best-response correspondences r*(q) and q*(r)yields the mixed-strategy N.E. in Penny Game. r*(q) General Definition of Mixed Strategy oSuppose that player i has K pure strategies, Si={si1,…, siK} oThen, a mixed strategy for player i is a oprobability distribution (pi1,…, piK), where pik is othe probability that player i will play strategy sik, ok=1,…,K oRespectively, for k=1,…,K oand oDenote an arbitrary mixed strategy by pi o General Definition of Nash Equilibrium oConsider 2-player case where strategy sets of the two players are S1={s11,…, s1J} and S1={s11,…, s1K}, respectively oP1’s expected payoff from playing the mixed strategies p1 = (p11,…,p1J) is: o oP2’s expected payoff from playing the mixed strategies p2 = (p21,…,p2K) is: General Definition of Nash Equilibrium oFor the pair of mixed strategies to be a Nash equilibrium, must satisfy: o o ofor every probability distribution p1 over S1, and must satisfy: o o o ofor every probability distribution p2 over S2. Existence of Nash Equilibrium oTheorem (Nash 1950): In the n-player normal-form game G={S1,…,Sn;u1,…,un), if n is finite and Si is finite for every i then there exists at least one Nash equilibrium, possibly involving mixed strategies. oProof consists of 2 steps: nStep1: Show that any fixed point of a certain correspondence is a N.E. nStep 2: Use an appropriate fixed-point theorem to show that the correspondence must have a fixed point. Revision oWhat is a strictly dominated strategy? nIf a strategy si is strictly dominated then there is no belief that player i could hold such that it would be optimal to play si. oThe converse is also true when mixed strategies are introduced nIf there is no belief that player i could hold such that it would be optimal to play si, then there exists another strategy that strictly dominates si. o Example /mixed strategy dominance/: P1 B1 B2 A1 3,— 0,— A2 0,— 3,— A3 1,— 1,— P2 For any belief of P1, A3 is not a best response even though it is not strictly dominated by any pure strategy. A3 is strictly dominated by a mixed strategy (½ , ½, 0) Example /mixed strategy best response/: P1 B1 B2 A1 3,— 0,— A2 0,— 3,— A3 2,— 2,— P2 For any belief of P1, A3 is not a best response to any pure strategy but it is the best response to mixed strategy (q,1-q) for 1/3 ui(a1,…,an; ti) o oA player is uncertain about o o{other player’s payoff function} = {other player’s type t-i} o owhere o Normal-form Representation oADD probability measure of types to account for uncertainty: o o ‑ player i‘s belief about the other oplayers’ types (t-i) given player i‘s knowledge of her own type, ti. o oBayesian Theorem o o o o oPLAYERS oACTIONS – A1, … ,An; Ai = {ai1,…, ain} oTYPES – Ti = {ti1,…, tin} oSystem of BELIEFS ‑ oPAYOFFS ‑ owhich is briefly denoted as o Normal-form Representation Timing of the Bayesian Games (Harsanyi, 1967) oStage 1: Nature draws a type vector ot = (t1,…,tn), where ti is drawn from the set of possible types Ti. oStage 2: Nature reveals ti to player i but not necessarily to the other players. oStage 3: Players simultaneously choose actions i.e. player i chooses ai from the feasible set Ai. oStage 4: Payoffs ui(a1,…,an; ti) are received. Strategy in a Bayesian Game oIn a static Bayesian game, a strategy for player i is a function , where for each type ti in Ti, si(ti) specifies the action from the feasible set Ai that type ti would choose if drawn by nature. oIn a separating strategy, each type ti in Ti chooses a different action ai from Ai. oIn a pooling strategy, in contrast, all types choose the same action. How to solve a Bayesian game? oBayesian Nash Equilibrium: o oIn the static Bayesian game o othe strategies are a (pure-strategy) Bayesian Nash equilibrium if for each player i and for each of i’s types ti in Ti, solves: o o oThat is, no player wants to change his or her strategy, even if the change involves only one action by one type. o Existence of a Bayesian Nash Equilibrium o oIn a finite static Bayesian game o(i.e., where n is finite and (A1,…,An) and (T1,…,Tn) oare all finite sets), there exists a Bayesian Nash equilibrium, perhaps in mixed strategies. o oMixed-strategy in a Bayesian game: o oPlayer i is uncertain about player j’s choice not because it is random but rather because of incomplete information about j’s payoffs. o oExamples: Battle of Sexes; Cournot Competition with Asymmetric Information o Summary oGame Theory distinguishes between pure and mixed strategy oMixed strategy is a probability distribution over the strategy set oTo be efficient in solving games including uncertainty, N.E. concept needs to be extended and defined for mixed strategies oGames with uncertainty are called Bayesian games and their solution concept – Bayesian N.E.