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# Econ 805 Advanced Micro Theory 1 - PowerPoint PPT Presentation

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

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### 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

• 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)

• 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

• 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”)

• 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)

• 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)

• 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 functionp = 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 functionN = 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}

• 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

• 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