11Recap{ Games in normal formz NBS vs. ABC, Prisoner’s dilemma, Tragedy of the commonsz Dominant or dominated actions, outcomesz Best response, Nash equilibrium (pure vs. mixed strategy){ Duopoly modelsz Cournot, Stackelberg, Bertrand; comparison with monopoly{ Multi-stage games with observed actionsz Stackelberg, Strategic investmentz Extensive form of a game, information setsz Subgame-perfect Nash equilibrium{ Repeated games (finitely or infinitely)z Prisoner’s dilemma, Cournot competitionz Trigger strategies, Friedman’s theorem{ Supply Chain Applications of Theory2Example: One-card poker{ A player is dealt an ace or king with equal probability23Example: One-card poker{ Two players{ One deck of cards, half aces, half kings{ Pay $a to play{ Each player is dealt a card face down{ After seeing his/her card, each player (simultaneously)z Action B: bets b, orz Action P: passes{ Payoffsz (B,P) or (P,B) → betting player gets the potz (B,B) or (P,P) → higher card gets the pot; in case of a tie, the pot is split4Record the resultsPlayer 1 Player 2 Payoffs---------------------------------------------------------------Game 1 Ace-Bet King-Bet (a+b, –(a+b))Game 2 King-Pass King-Bet (-a, a)Game 3 King-Pass King-Pass (0, 0)etc35Example: One-card pokerBetPassBP(K,K)(K,A)(A,K)(A,A)BetBetBetPassPassPassBBBBBBBPPPPPPP(0,0)(0,0)(0,0)(0,0)(a,-a)(a,-a)(a,-a)(a,-a)(a,-a)(-a,a)(-a,a)(-a,a)(-a,a)(-a,a)(a+b,-a-b)(-a-b,a+b)Player 1 OptionsPlayer 2 Options0.250.250.250.256Example: One-card poker (cont.){ The payoffs depend on players’actions andon card combinations{ Need to compute expected payoffs for each outcome47Example: One-card poker (cont.){ Payoffs for outcome (B,B) under different card combinations(0,0)0.25(K,K)(-a-b,a+b)0.25(K,A)(a+b, -a-b)0.25(A,K)(0,0)0.25(A,A)PayoffsProbabilityCard combination{ Expected payoff for player 1:(0.25)(0)+(0.25)(a+b)+(0.25)(-a-b)+(0.25)(0)=0{ Similarly, expected payoff for player 2 is 0{ Expected payoffs for (B,B) = (0,0)8One-card poker in normal form0, 0-a, aPassa, -a0, 0BetPassBetPlayer 1Player 2{ The unique NE is (B,B)59ISYE6230 one-card poker strategies10Cournot competition with incomplete information{ The market price, P is determined by (inverse) market demand: z P=a-Q if a>Q, P=0 otherwise. { Each firm decides on the quantity to sell (market share): q1and q2 {Q= q1+q2 total market demand{ Both firms seek to maximize profits{ The marginal cost of producing each unit of the good: c1and c2¾c1is common knowledge, however c2is known only by firm 2¾ Firm 1 believes that c2is “high” cHwith probability p and “low” cLwith probability (1-p) ¾ Firm 1’s belief about firm 2’s cost is common knowledge611Cournot Competition with Incomplete Information: Best response of Firm 2{ Suppose firm 1 produces q1{ Firm 2’s profits, if it produces q2are:π2= (P-c2)q2= [a-(q1+q2)]q2 – c2q2 = (Residual) revenue – Cost{ First order conditions:d π2/dq2= a - 2q2 – q1– c2= = RMR – MC = 0 →q2=(a-c2-q1)/2If Firm 2’s type is “high”: qH2=(a-cH-q1)/2If Firm 2’s type is “low”: qL2=(a-cL-q1)/212Cournot Competition with incomplete information: Best response of Firm 1{ Firm 1’s expected profits, if it produces q1are:π1= (P-c1)q1= p[a-(q1+qH2)]q1 +(1-p)[a-(q1+qL2)]q1 -c1q1 { First order conditions (FOC):d π1/dq1=p(a - 2q1 -qH2) – (1-p)( a - 2q1 -qL2)- c1= 0 →q1= p(a- c1 -qH2)/2 + (1-p)(a - c1-qL2)/2Cournot with complete information: q1=(a - c1– q2)/2713Cournot equilibrium under incomplete informationIf Firm 2’s type is “high”: qH2=(a-cH-q1)/2If Firm 2’s type is “low”: qL2=(a-cL-q1)/2q1= p(a- c1 -qH2)/2 + (1-p)(a - c1-qL2)/2Simultaneously solve to get:q1= (a - 2c1+ p cH+(1-p) cL)/3qH2= (a-2cH+c1)/3 + (1-p)(cH-cL)/6qL2= (a-2cL+c1)/3 - p(cH-cL)/614Comparison with the Cournot equilibrium under complete information{ Complete informationqC1=(a-2c1+c2)/3 qC2=(a-2c2+c1)/3 q’H= (a-2cH+c1)/3q’L= (a-2cL+c1)/3{ Incomplete informationq1= (a - 2c1+ p cH+(1-p) cL]/3qH2= (a-2cH+c1)/3 + (1-p)(cH-cL)/6qL2= (a-2cL+c1)/3 -p(cH-cL)/6qH2> q’Hand qL2< q’L.815Games of Incomplete Information (Bayesian Games){ In a game of complete information the players’ payoff functions are common knowledge{ In a game of incomplete information, at least one player is uncertain about another player’s payoff function16Normal Form Representation of Static Bayesian Games{ The normal form representation of an n-player static Bayesian game specifies the players’z action spaces A1,…,An, z type spaces T1,…,Tn, z beliefs p1,…,pn, andz payoff functions π1,…, πn.{ Player i’s type tiis privately known by player i, determines player i’s payoff function πi(a1,…,an; ti) and is a member of the set of possible types.{ Player i’s belief pi(t-i| ti) describes i’s uncertainty about the n-1 other players’ possible types t-i, given i’s own type ti. We denote this game by G=(A1,…,An; T1,…,Tn; p1,…,pn; π1,…, πn).917Example: Cournot Competition with incomplete information{ Actionsz Quantity choices q1 and q2 .{ Type spacesz Firm 1: T1={c1}z Firm 2: T2={cH, cL}{ Payoffsz Firm 1: π1(q1,qL2, qH2; c1) = p[a-(q1+qH2)]q1 +(1-p)[a-(q1+qL2)]q1 -c1q1.z Firm 2: π2(q1,qH2; cH) = (a- q1-qH2-cH) q2.π2(q1,qL2; cL) = (a- q1- qL2-cL) q2.{ Beliefsz Firm 1: p1(cH|c1) = p p1(cL|c1) = 1-pz Firm 2: p2(c1|t2) = 1 t2= cH, cL18Timing of a Static Bayesian Game{ Nature draws the type vector t=(t1,…,tn) according to probability distribution p(t){ Nature reveals tito player i but not to any other playerz Player i can compute his/her belief pi(t-i|ti) using Bayes’ rule (P(A|B)=P(A,B)/P(B)):pi(t-i| ti) = pi(t-i, ti) / pi(ti) = pi(t-i, ti) / ∑t-i∈T-ipi(t-i, ti) { The players simultaneously choose actions{ Payoffs are received1019Strategies in a Static Bayesian Game{ In the static Bayesian game G=(A1,…,An; T1,…,Tn; p1,…,pn; π1,…, πn)a strategyfor player i is a function si(ti), where for each type ti∈Ti, si(ti) specifies the action from the feasible set Aithat type i would choose if drawn by nature.{ Set of possible (pure) strategies for player i: Si: Ti→ Ai.{ In a separating strategy, each type ti∈Tichooses a different action from ai∈Ai.{ In a pooling strategy, for player i, each type ti∈Tichooses the same actions. 20Strategies in the Cournot GameFirm 1’s strategy: q1such thatq1= (a - 2c1+ p cH+(1-p) cL]/3Firm 2’s strategy: (q2(cH), q2(cL))q2(cH) = (a-2cH+c1)/3 + (1-p)(cH-cL)/6 = qH2q2(cL) =
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