Econ 805 Advanced Micro Theory 1 Dan Quint Fall 2008 Lecture 1 Sept 2 2007 A Quick Review of Game Theory and in particular Bayesian Games Games of complete information A static simultaneous move game is defined by Players 1 2 N Action spaces A1 A2 AN Payoff functionsui A1 x x AN 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 A1 x x AN s t ui si s i ui s i s i for all s i Ai 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 Q1 Q2 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 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 weak make 2 strong Firm 2 Firm 2 Q2 Q2 Firm 1 Q1 Q1 u1 Q1 100 Q1 Q2 25 u1 Q1 100 Q1 Q2 25 u2 Q2 100 Q1 Q2 30 u2 Q2 100 Q1 Q2 20 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 A2 R Firm 1 maximizes expected payoff in expectation over firm 2 s types given firm 2 s equilibrium strategy Nature make 2 weak Firm 2 make 2 strong p p Q2 W Firm 2 Q2 S Firm 1 Q1 Q1 u1 Q1 100 Q1 Q2 25 u1 Q1 100 Q1 Q2 25 u2 Q2 100 Q1 Q2 30 u2 Q2 100 Q1 Q2 20 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 that firm 2 is strong Strong firm 2 best responds by choosing Q2S arg maxq q 100 Q1 q 20 Maximization gives Q2S 80 Q1 2 Weak firm 2 sets Q2W arg maxq q 100 Q1 q 30 giving Q2W 70 Q1 2 Firm 1 maximizes expected profits Q1 arg maxq q 100 q Q2S 25 q 100 q Q2W 25 giving Q1 75 Q2W 2 Q2S 2 2 Solving these simultaneously gives equilibrium strategies Q1 25 Q2W Q2S 22 27 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 T1 t11 t12 t1K and T2 t21 t22 t2K for each player and a probability for each type 11 1K 21 2K A set of possible actions Ai for each player A payoff function mapping actions and types to payoffs for each player ui A1 x A2 x T1 x T2 R A pure strategy Bayesian Nash Equilibrium is a mapping si Ti Ai for each player such that k k k k k k u s t s t t t u a s t t t k potentialj deviation t T j i i j j i j i i i for each aj j A i j t kj T j i i j j for every type ti Ti 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 si ti maximizes ex post expected payoffs for each type ti Et j T j uissi i Tti i s j A ti j maximizes ti t j Et j ex ante T j ui ai s expected j t j ti t j payoffs The mapping among all such mappings Eti Ti t j T j ui si ti s j t j ti t j Eti Ti t j T j ui si ti s j t j ti t j 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 exante 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 Thursday some common auction formats and the baseline model 11