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 δ≥1/2. Each firm charges a monopoly price until someone undercuts the price, and after such deviation she will set a price equal to the marginal cost c, i.e., get into a price war. tt+1t+2… collusionπππ… deviation2π2π00…

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 T- 1. 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 players know, i.e., some of the players possess private information, at the beginning of the game. 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 any game of incomplete information into a Bayesian game in which a NE is naturally extended as a Bayesian Nash equilibrium.

Cournot Game with Unknown Cost Firm 1’s marginal cost is constant (c), while firm 2’s marginal cost takes either high (h) with probability θ or low (l) with probability 1- θ. Firm 1’s strategy is a quantity choice, but firm 2’s 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 expected 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:

Solution 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 2’s actual quantity but maximizes his expected profit.

Static Bayesian Games 1.Nature draws a type vector t, according to a prior probability distribution p(t). 2.Nature reveals i’s type to player i, but not to any other player. 3.The players simultaneously choose actions. 4.Payoffs are received.

Remarks A belief about other players’ types is a conditional probability distribution of other players’ types given the player’s knowledge of her own type. A (pure) strategy for a player is a complete action plan, which specifies her action for each of her possible type. Bayes’ rule:

Bayesian Nash Equilibrium A strategy profile s* is a Bayesian NE if which is equivalent to

Calculation Maximizing RHS is identical to maximizing inside the brackets for all possible i’s type.