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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