Econ 805 Advanced Micro Theory 1

Econ 805Advanced Micro Theory 1 Dan Quint Fall 2009 Lecture 1 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 functions ui : 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)

Solution concepts for games of complete information • Pure-strategy Nash equilibrium: sÎA1 x … x AN s.t. ui(si,s-i) ³ ui(s’i,s-i) for all s’iÎAi for all iÎ{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

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?

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 weak make 2 strong Firm 2 Firm 2 Q2 Q2 Firm 1 Q1 Q1 u1 = Q1(100 - Q1 - Q2 - 25) u2 = Q2(100 - Q1 - Q2 - 30) u1 = Q1(100 - Q1 - Q2 - 25) u2 = Q2(100 - Q1 - Q2 - 20)

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 make 2 strong Firm 2 Firm 2 p = ½ p = ½ Q2 Q2W Q2 Q2S Firm 1 Q1 Q1 u1 = Q1(100 - Q1 - Q2 - 25) u2 = Q2(100 - Q1 - Q2 - 30) u1 = Q1(100 - Q1 - Q2 - 25) u2 = Q2(100 - Q1 - Q2 - 20)

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

Solving the Cournot example, with p = ½ that firm 2 is strong… • Strong firm 2 best-responds by choosing Q2S = arg maxqq(100-Q1-q-20) Maximization gives Q2S = (80-Q1)/2 • Weak firm 2 sets Q2W = arg maxqq(100-Q1-q-30) giving Q2W = (70-Q1)/2 • Firm 1 maximizes expected profits: Q1 = arg maxq½q(100-q-Q2S-25) + ½q(100-q-Q2W-25) giving Q1 = (75 – Q2W/2 – Q2S/2)/2 • Solving these simultaneously gives equilibrium strategies: Q1 = 25, (Q2W, Q2S) = (22½ , 27½)

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 {p11, …, p1K, p21, …, p2K’} • 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 for each potential deviation aiÎAi for every type tiÎ Ti for each player i Î {1,2}

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 • The mapping si : Ti  Ai 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

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

Econ 805 Advanced Micro Theory 1