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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 805 advanced micro theory 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


Games of complete information
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
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
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
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)


Bayesian nash equilibrium
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
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 that firm 2 is strong
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 n 2 and finite independent types
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}


Ex post versus ex ante formulations
Ex-post versus ex-ante formulations function

  • 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
Auctions are typically modeled as Bayesian games function

  • 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


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