Econ 805 Advanced Micro Theory 1 Dan Quint Fall 2009 Lecture 1 A Quick Review of Game Theory and, in particular, Bayesian Games

1 Games of complete information  A static (simultaneous-move) game is defined by:  Players1, 2, …, N  Action spacesA 1, A 2, …, A N  Payoff functionsu i : A 1 x … x A N  R all of which are assumed to be common knowledge  In dynamic games, we talk about specifying “timing,” but what we mean is information  What each player knows at the time he moves  Typically represented in “extensive form” (game tree)

2 Solution concepts for games of complete information  Pure-strategy Nash equilibrium: s  A 1 x … x A N s.t. u i (s i,s -i )  u i (s’ i,s -i ) for all s’ i  A i for all i  {1, 2, …, N}  In dynamic games, we typically focus on Subgame Perfect equilibria  Profiles where Nash equilibria are also played within each branch of the game tree  Often solvable by backward induction

3 Games of incomplete information  Example: Cournot competition between two firms, inverse demand is P = 100 – Q 1 – Q 2  Firm 1 has a cost per unit of 25, but doesn’t know whether firm 2’s cost per unit is 20 or 30  What to do when a player’s payoff function is not common knowledge?

4 John Harsanyi’s big idea “Games with Incomplete Information Played By Bayesian Players” (1967)  Transform a game of incomplete information into a game of imperfect information – parameters of game are common knowledge, but not all players’ moves are observed  Introduce a new player, “nature,” who determines firm 2’s marginal cost  Nature randomizes; firm 2 observes nature’s move  Firm 1 doesn’t observe nature’s move, so doesn’t know firm 2’s “type” “Nature” make 2 weakmake 2 strong Firm 2 Q2Q2 Q2Q2 Firm 1 Q1Q1 Q1Q1 u 1 = Q 1 (100 - Q 1 - Q ) u 2 = Q 2 (100 - Q 1 - Q ) u 1 = Q 1 (100 - Q 1 - Q ) u 2 = Q 2 (100 - Q 1 - Q )

5 Bayesian Nash Equilibrium  Assign probabilities to nature’s moves (common knowledge)  Firm 2’s pure strategies are maps from his “type space” {Weak, Strong} to A 2 = R +  Firm 1 maximizes expected payoff  in expectation over firm 2’s types  given firm 2’s equilibrium strategy “Nature” make 2 weakmake 2 strong Firm 2 Q2Q2 Q2Q2 Firm 1 Q1Q1 Q1Q1 u 1 = Q 1 (100 - Q 1 - Q ) u 2 = Q 2 (100 - Q 1 - Q ) u 1 = Q 1 (100 - Q 1 - Q ) u 2 = Q 2 (100 - Q 1 - Q ) p = ½ Q2WQ2W Q2SQ2S

6 Other players’ types can enter into a player’s payoff function  In the Cournot example, firm 1 only cares about firm 2’s type because it affects his action  In some games, one player’s type can directly enter into another player’s payoff function  Poker: you don’t know what cards your opponent has, but they affect both how he’ll plays the hand and whether you’ll win at showdown  Either way, in BNE, simply maximize expected payoff given opponent’s strategy and type distribution

7 Solving the Cournot example, with p = ½ that firm 2 is strong…  Strong firm 2 best-responds by choosing Q 2 S = arg max q q(100-Q 1 -q-20) Maximization gives Q 2 S = (80-Q 1 )/2  Weak firm 2 sets Q 2 W = arg max q q(100-Q 1 -q-30) giving Q 2 W = (70-Q 1 )/2  Firm 1 maximizes expected profits: Q 1 = arg max q ½q(100-q-Q 2 S -25) + ½q(100-q-Q 2 W -25) giving Q 1 = (75 – Q 2 W /2 – Q 2 S /2)/2  Solving these simultaneously gives equilibrium strategies: Q 1 = 25, (Q 2 W, Q 2 S ) = (22½, 27½)

8 Formally, for N = 2 and finite, independent types…  A static Bayesian game is  A set of players 1, 2  A set of possible types T 1 = {t 1 1, t 1 2, …, t 1 K } and T 2 = {t 2 1, t 2 2, …, t 2 K’ } for each player, and a probability for each type {  1 1, …,  1 K,  2 1, …,  2 K’ }  A set of possible actions A i for each player  A payoff function mapping actions and types to payoffs for each player u i : A 1 x A 2 x T 1 x T 2  R  A pure-strategy Bayesian Nash Equilibrium is a mapping s i : T i  A i for each player, such that for each potential deviation a i  A i for every type t i  T i for each player i  {1,2}

9 Ex-post versus ex-ante formulations  With a finite number of types, the following are equivalent:  The action s i (t i ) maximizes “ex-post expected payoffs” for each type t i  The mapping s i : T i  A i maximizes “ex-ante expected payoffs” among all such mappings  I prefer the ex-post formulation for two reasons  With a continuum of types, the equivalence breaks down, since deviating to a worse action at a particular type would not change ex-ante expected payoffs  Ex-post optimality is almost always simpler to verify

10 Auctions are typically modeled as Bayesian games  Players don’t know how badly the other bidders want the object  Assume nature gives each bidder a valuation for the object (or information about it) from some ex-ante probability distribution that is common knowledge  In BNE, each bidder maximizes his expected payoffs, given  the type distributions of his opponents  the equilibrium bidding strategies of his opponents  Next week: some common auction formats and the baseline model