Games of Incomplete Information EC202 Lectures V & VI Francesco Nava London School of Economics January 2011 Nava (LSE) EC202 Lectures V & VI Jan 2011 1 / 22
Summary Games of Incomplete Information: Definitions: Incomplete Information Game Information Structure and Beliefs Strategies Best Reply Map Solution Concepts in Pure Strategies: Dominant Strategy Equilibrium Bayes Nash Equilibrium Examples EXTRA: Mixed Strategies & Bayes Nash Equilibria Nava (LSE) EC202 Lectures V & VI Jan 2011 2 / 22
Incomplete Information (Strategic Form) An incomplete information game consists of: N the set of players in the game A i X i player i s action set player i s set of possible signals A profile of signals x = (x 1,..., x n ) is an element X = j N X j f a distribution over the possible signals u i : A X R player i s utility function, u i (a x) Nava (LSE) EC202 Lectures V & VI Jan 2011 3 / 22
Bayesian Game Example Consider the following Bayesian game: Player 1 observes only one possible signal: X 1 = {C } Player 2 s signal takes one of two values: X 2 = {L, R} Probabilities are such that: f (C, L) = 0.6 Payoffs and action sets are as described in the matrix: 1\2.L y 2 z 2 1\2.R y 2 z 2 y 1 1,2 0,1 y 1 1,3 0,4 z 1 0,4 1,3 z 1 0,1 1,2 Nava (LSE) EC202 Lectures V & VI Jan 2011 4 / 22
Information Structure Information structure: X i denotes the signal as a random variable belongs to the set of possible signals X i x i denotes the realization of the random variable X i X i = (X 1,..., X i 1, X i+1,..., X n ) denotes a profile of signals for all players other than i Player i observes only X i Player i ignores X i, but knows f With such information player i forms beliefs regarding the realization of the signals of the other players x i Nava (LSE) EC202 Lectures V & VI Jan 2011 5 / 22
Beliefs about other Players Signals [Take 1] In this course we consider models in which signals are independent: f (x) = j N f j (x j ) This implies that the signal x i of player i is independent of X i Beliefs are a probability distribution over the signals of the other players Any player forms beliefs about the signals received by the other players by using Bayes Rule Independence implies that conditional observing X i = x i the beliefs of player i are: f i (x i x i ) = j N \i f j (x j ) = f i (x i ) Nava (LSE) EC202 Lectures V & VI Jan 2011 6 / 22
Extra: Beliefs about other Players Signals [Take 2] Also in the general case with interdependence players form beliefs about the signals received by the others by using Bayes Rule Conditional observing X i = x i the beliefs of player i are: f i (x i x i ) = Pr(X i = x i X i = x i ) = = Pr(X i = x i X i = x i ) Pr(X i = x i ) = = = Pr(X i = x i X i = x i ) y i X i Pr(X i = y i X i = x i ) = f (x i, x i ) y i X i f (y i, x i ) Beliefs are a probability distribution over the signals of the other players Nava (LSE) EC202 Lectures V & VI Jan 2011 7 / 22
Strategies Strategy Profiles: A strategy consists of a map from available information to actions: α i : X i A i A strategy profile consists of a strategy for every player: α(x ) = (α 1 (X 1 ),..., α N (X N )) We adopt the usual convention: α i (X i ) = (α 1 (X 1 ),..., α i 1 (X i 1 ), α i+1 (X i+1 ),..., α N (X N )) Nava (LSE) EC202 Lectures V & VI Jan 2011 8 / 22
Bayesian Game Example Continued Consider the following game: Player 1 observes only one possible signal: X 1 = {C } Player 2 s signal takes one of two values: X 2 = {L, R} Probabilities are such that: f (C, L) = 0.6 Payoffs and action sets are as described in the matrix: 1\2.L y 2 z 2 1\2.R y 2 z 2 y 1 1,2 0,1 y 1 1,3 0,4 z 1 0,4 1,3 z 1 0,1 1,2 A strategy for player 1 is an element of the set α 1 {y 1, z 1 } A strategy for player 2 is a map α 2 : {L, R} {y 2, z 2 } Player 1 cannot act upon 2 s private information Nava (LSE) EC202 Lectures V & VI Jan 2011 9 / 22
Dominant Strategy Equilibrium Strategy α i weakly dominates α i if for any a i and x X: u i (α i (x i ), a i x) u i (α i (x i ), a i x) [strict somewhere] Strategy α i is dominant if it dominates any other strategy α i Strategy α i is undominated if no strategy dominates it Definitions (Dominant Strategy Equilibrium DSE) A Dominant Strategy equilibrium of an incomplete information game is a strategy profile α that for any i N, x X and a i A i satisfies: u i (α i (x i ), a i x) u i (α i (x i ), a i x) for any α i : X i A i I.e. α i is optimal independently of what others know and do Nava (LSE) EC202 Lectures V & VI Jan 2011 10 / 22
Interim Expected Utility and Best Reply Maps The interim stage occurs just after a player knows his signal X i = x i It is when strategies are chosen in a Bayesian game The interim expected utility of a (pure) strategy profile α is defined by: U i (α x i ) = X i u i (α(x) x)f (x i x i ) : X i R With such notation in mind notice that: U i (a i, α i x i ) = X i u i (a i, α i (x i ) x)f (x i x i ) The best reply correspondence of player i is defined by: b i (α i x i ) = arg max ai A i U i (a i, α i x i ) BR maps identify which actions are optimal given the signal and the strategies followed by others Nava (LSE) EC202 Lectures V & VI Jan 2011 11 / 22
Pure Strategy Bayes Nash Equilibrium Definitions (Bayes Nash Equilibrium BNE) A pure strategy Bayes Nash equilibrium of an incomplete information game is a strategy profile α such that for any i N and x i X i satisfies: U i (α x i ) U i (a i, α i x i ) for any a i A i BNE requires interim optimality (i.e. do your best given what you know) BNE requires α i (x i ) b i (α i x i ) for any i N and x i X i Nava (LSE) EC202 Lectures V & VI Jan 2011 12 / 22
Bayesian Game Example Continued Consider the following Bayesian game with f (C, L) = 0.6: 1\2.L y 2 z 2 1\2.R y 2 z 2 y 1 1,2 0,1 y 1 1,3 0,4 z 1 0,4 1,3 z 1 0,1 1,2 The best reply maps for both player are characterized by: b 2 (α 1 x 2 ) = { y2 if x 2 = L z 2 if x 2 = R { y1 if α b 1 (α 2 ) = 2 (L) = y 2 z 1 if α 2 (L) = z 2 The game has a unique (pure strategy) BNE in which: α 1 = y 1, α 2 (L) = y 2, α 2 (R) = z 2 DO NOT ANALYZE MATRICES SEPARATELY!!! Nava (LSE) EC202 Lectures V & VI Jan 2011 13 / 22
Relationships between Equilibrium Concepts If α is a DSE then it is a BNE. In fact for any action a i and signal x i : u i (α i (x i ), a i x) u i (a i, a i x) a i, x i u i (α i (x i ), α i (x i ) x) u i (a i, α i (x i ) x) α i, x i X i u i (α(x) x)f i (x i x i ) X i u i (a i, α i (x i ) x)f i (x i x i ) α i U i (α x i ) U i (a i, α i x i ) α i Nava (LSE) EC202 Lectures V & VI Jan 2011 14 / 22
BNE Example I: Exchange A buyer and a seller want to trade an object: Buyer s value for the object is 3$ Seller s value is either 0$ or 2$ based on the signal, X S = {L, H} Buyer can offer either 1$ or 3$ to purchase the object Seller choose whether or not to sell B\S.L sale no sale B\S.H sale no sale 3$ 0,3 0,0 3$ 0,3 0,2 1$ 2,1 0,0 1$ 2,1 0,2 This game for any prior f has a BNE in which: α S (L) = sale, α S (H) = no sale, α B = 1$ Selling is strictly dominant for S.L Offering 1$ is weakly dominant for the buyer Nava (LSE) EC202 Lectures V & VI Jan 2011 15 / 22
BNE Example II: Entry Game Consider the following market game: Firm I (the incumbent) is a monopolist in a market Firm E (the entrant) is considering whether to enter in the market If E stays out of the market, E runs a profit of 1$ and I gets 8$ If E enters, E incurs a cost of 1$ and profits of both I and E are 3$ I can deter entry by investing at cost {0, 2} depending on type {L, H} If I invests: I s profit increases by 1 if he is alone, decreases by 1 if he competes and E s profit decreases to 0 if he elects to enter E \I.L Invest Not Invest E \I.H Invest Not Invest In 0,2 3,3 In 0,0 3,3 Out 1,9 1,8 Out 1,7 1,8 Nava (LSE) EC202 Lectures V & VI Jan 2011 16 / 22
BNE Example II: Entry Game Let π denote the probability that firm I is of type L and notice: α I (H) = Not Invest is a strictly dominant strategy for I.H For any value of π, α I (L) = Not Invest and α E = In is BNE: u I (Not, In L) = 3 > 2 = u I (Invest, In L) U E (In, α I (X I )) = 3 > 1 = U E (Out, α I (X I )) For π high enough, α I (L) = Invest and α E = Out is also BNE: u I (Invest, Out L) = 9 > 8 = u I (Not, Out L) U E (Out, α I (X I )) = 1 > 3(1 π) = U E (In, α I (X I )) E \I.L Invest Not Invest E \I.H Invest Not Invest In 0,2 3,3 In 0,0 3,3 Out 1,9 1,8 Out 1,7 1,8 Nava (LSE) EC202 Lectures V & VI Jan 2011 17 / 22
Extra: Mixed Strategies in Bayesian Games Strategy Profiles: A mixed strategy is a map from information to a probability distribution over actions In particular σ i (a i x i ) denotes the probability that i chooses a i if his signal is x i A mixed strategy profile consists of a strategy for every player: σ(x ) = (σ 1 (X 1 ),..., σ N (X N )) As usual σ i (X i ) denotes the profile of strategies of all players, but i Mixed strategies are independent (i.e. σ i cannot depend on σ j ) Nava (LSE) EC202 Lectures V & VI Jan 2011 18 / 22
Extra: Interim Payoff & Bayes Nash Equilibrium The interim expected payoff of mixed strategy profiles σ and (a i, σ i ) are: U i (σ x i ) = u i (a x) σ j (a j x j )f (x i x i ) X i a A j N U i (a i, σ i x i ) = u i (a x) σ j (a j x j )f (x i x i ) X i a i A i j =i Definitions (Bayes Nash Equilibrium BNE) A Bayes Nash equilibrium of a game Γ is a strategy profile σ such that for any i N and x i X i satisfies: U i (σ x i ) U i (a i, σ i x i ) for any a i A i BNE requires interim optimality (i.e. do your best given what you know) Nava (LSE) EC202 Lectures V & VI Jan 2011 19 / 22
Extra: Computing Bayes Nash Equilibria Testing for BNE behavior: σ is BNE if only if: U i (σ x i ) = U i (a i, σ i x i ) for any a i s.t. σ i (a i x i ) > 0 U i (σ x i ) U i (a i, σ i x i ) for any a i s.t. σ i (a i x i ) = 0 Strictly dominated strategies are never chosen in a BNE Weakly dominated strategies are chosen only if they are dominated with probability zero in equilibrium This conditions can be used to compute BNE (see examples) Nava (LSE) EC202 Lectures V & VI Jan 2011 20 / 22
Extra: Example I Consider the following example for f (1, L) = 1/2: All BNEs for this game satisfy: 1\2.L X Y 1\2.R W Z T 1,0 0,1 T 0,0 1,1 D 0,1 1,0 D 1,1 0,0 σ 1 (T ) = 1/2 and σ 2 (X L) = σ 2 (W R) Such games satisfy all BNE conditions since: U 1 (T, σ 2 ) = (1/2)σ 2 (X L) + (1/2)(1 σ 2 (W R)) = = (1/2)(1 σ 2 (X L)) + (1/2)σ 2 (W R) = U 1 (D, σ 2 ) u 2 (X, σ 1 L) = σ 1 (T ) = 1 σ 1 (T ) = u 2 (Y, σ 1 L) u 2 (W, σ 1 R) = (1 σ 1 (T )) = σ 1 (T ) = u 2 (Z, σ 1 R) Nava (LSE) EC202 Lectures V & VI Jan 2011 21 / 22
Extra: Example II Consider the following example for f (1, L) = q 2/3: 1\2.L X Y 1\2.R W Z T 0,0 0,2 T 2,2 0,1 D 2,0 1,1 D 0,0 3,2 All BNEs for this game satisfy σ 1 (T ) = 2/3 and: σ 2 (X L) = 0 (dominance) and σ 2 (W R) = 3 2q 5 5q Such games satisfy all BNE conditions since: U 1 (T, σ 2 ) = 2(1 q)σ 2 (W R) = = q + 3(1 q)(1 σ 2 (W R)) = U 1 (D, σ 2 ) u 2 (X, σ 1 L) = 0 < 2σ 1 (T ) + (1 σ 1 (T )) = u 2 (Y, σ 1 L) u 2 (W, σ 1 R) = 2σ 1 (T ) = σ 1 (T ) + 2(1 σ 1 (T )) = u 2 (Z, σ 1 R) Nava (LSE) EC202 Lectures V & VI Jan 2011 22 / 22