1 / 11

Econ 805 Advanced Micro Theory 1

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 A 1 , A 2 , …, A N

yakov
Télécharger la présentation

Econ 805 Advanced Micro Theory 1

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

  1. Econ 805Advanced Micro Theory 1 Dan Quint Fall 2008 Lecture 1 – Sept 2, 2007 A Quick Review of Game Theory and, in particular, Bayesian Games

  2. 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)

  3. 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

  4. 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?

  5. 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) u2 = Q2(100 - Q1 - Q2 - 30) u1 = Q1(100 - Q1 - Q2 - 25) u2 = Q2(100 - Q1 - Q2 - 20)

  6. 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)

  7. 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

  8. 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)

  9. 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}

  10. 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

  11. 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

More Related