ECO290E: Game Theory Lecture 12 Static Games of Incomplete Information Review of Lecture 11 In the repeated Bertrand games, the following trigger strategies achieve collusion if 1/ 2. Each firm charges a monopoly price until so meone undercuts the price, and after such de viation she will set a price equal to the margi nal cost c, i.e., get into a price war. t t+1
t+2 collusion deviation 2
0 0 Finite Repetitions Q: If the Bertrand games are played only finitely, i.e., ends in period T, then collusion can be sustained? A: NO! In the last period (t=T), no firm has an incentive to collude since there is no future play. The only possible outcome is a stage game NE. In the second to the last period (t=T-1), no firm has an incentive to collude since the future play will be a price war no matter how each firm plays in period T1. By backward induction, firms end up getting into price wars in every period.
No collusion is possible! Finitely Repeated Games If the stage game G has a unique NE, then for any T, the finitely repeated game G(T) has a unique SPNE: the NE of G is played in every stage irrespective of the history. If the stage game G has multiple NE, then for any T, any sequence of those equilibrium profiles can be supported as the outcome of a SPNE. Moreover, non-NE strategy profiles can be sustained as a SPNE in this case. Games of Incomplete Information In a game of incomplete information, at least one player is uncertain about what other play ers know, i.e., some of the players possess p
rivate information, at the beginning of the ga me. For example, a firm may not know the cost of the rival firm, a bidder does not usually know her competitors valuations in an auction. Following Harsanyi (1967), we can translate a ny game of incomplete information into a Bay esian game in which a NE is naturally extend ed as a Bayesian Nash equilibrium. Cournot Game with Unknow n Cost Firm 1s marginal cost is constant (c), while fi rm 2s marginal cost takes either high (h) wit h probability or low (l) with probability 1-. Firm 1s strategy is a quantity choice, but fir m 2s strategy is a complete action plan, i.e., she must specify her quantity choice in each
possible marginal cost. Assume each firm tries to maximize an expec ted profit. Calculation It is important to consider the types of player 2 as separate players. Equilibrium strategies can be derived by the following maximization problems: max q1 1 {a b(q1 q2 (h)) c}q1 (1 ){a b(q1 q2 (l )) c}q1 max q2 ( h ) 2 (h) {a b(q1 q2 (h)) h}q2 (h) max q2 (l ) 2 (l ) {a b(q1 q2 (l )) l}q2 (l ) Solution a 2c h (1 )l 3b
a 2h c 1 q2* (h) (h l ) 3b 6b a 2l c q2* (l ) (h l ) 3b 6b q1* Notice that firm 2 will produce more (/less) than she would in the complete information case with high (/low) cost, since firm 1 does not take the best response to firm 2s actual quantity but maximizes his expected profit.
Static Bayesian Games 1. Nature draws a type vector t, accordin g to a prior probability distribution p (t). 2. Nature reveals is type to player i, but not to any other player. 3. The players simultaneously choose ac tions. 4. Payoffs are received. Remarks A belief about other players types is a condit ional probability distribution of other players types given the players knowledge of her ow n type. A (pure) strategy for a player is a complete a ction plan, which specifies her action for eac
h of her possible type. Bayes rule: Pr( A, B) Pr( A | B) Pr( B) Bayesian Nash Equilibrium A strategy profile s* is a Bayesian NE if isi ( ) * * * u ( s (
t ), s ( t ); t ) u ( s ( t ), s i i i i i i i i i i (t ); ti ) t
t which is equivalent to isi ( )ti * * u ( s ( t ), s i i i i (t i ); ti ) Pr(t i | ti ) t i ui ( si (ti ), s*i (t i ); ti ) Pr(t i | ti )
t i Calculation E[ui ] ui ( s (t ); ti ) Pr(t ) t ui ( s (t ); ti ) Pr(t i | ti ) Pr(ti ) t [ ui ( s (t ); ti ) Pr(t i | ti )] Pr(ti ) ti t i Maximizing RHS is identical to maximizing i nside the brackets for all possible is type.