Review of Bayesian Games
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Transcript Review of Bayesian Games
Econ 805
Advanced 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
Action spaces
Payoff functions
1, 2, …, N
A1, A2, …, AN
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)
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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
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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?
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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
“Nature”
make 2 weak
make 2 strong
Firm 2
Firm 2
Q2
Q2
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”
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)
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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=½
p=½
Q2W
Firm 2
Q2S
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)
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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
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Solving the Cournot example, with
p = ½ 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 ½q(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)
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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
t kj T j
p kj ui si (ti ), s j (t kj ), ti , t kj t T p kj ui ai , s j (t kj ), ti , t kj
k
j
j
for each potential deviation ai Ai
for every type ti Ti
for each player i {1,2}
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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 ui si (ti ), s j (t j ), ti , t j Et j T j ui ai , s j (t j ), ti , t j
The mapping si : Ti Ai maximizes “ex-ante expected payoffs”
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
ex-ante expected payoffs
Ex-post optimality is almost always simpler to verify
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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
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