Game Theory
Section 7: Bayes-Nash Equilibrium
Agenda
•
•
•
•
•
What is Incomplete Information?
What is Bayes-Nash Equilibrium?
A Quick Example: Simple Game
Strategies for finding solutions
Bayes Rule: Prep for tomorrow’s lecture
Main Ideas
What is Incomplete Information?
• When some players don’t know others’ payoffs
– This is often true in the real world, of course.
• The notion of “type” is very helpful in this setting
– A player’s “type” , typically private information, is
typically associated with his cost.
– More generally, the “type” of a player embodies any
private information relevant to the player’s decision
making
• By “private information”: info that is not common knowledge
• In addition to the player’s payoff function this may include his
beliefs about other players’ payoff functions, his beliefs about
what other players believe his beliefs are, and so on.
Main Ideas
How do we handle games of incomplete information?
• Harsanyi: introduce a prior move by nature
that determines the player(s)’s type(s).
• In the transformed game, player 2’s
incomplete information about player 1’s
cost becomes imperfect information about
nature’s moves.
• This transformed game can be analyzed
with standard techniques.
Main Ideas
What is a Bayesian Nash Equilibrium?
• The Nash Equilibrium of the imperfectinformation game
– A Bayesian Equilibrium is a set of strategies such that
each player is playing a best response, given a
particular set of beliefs about the move by nature.
– All players have the same prior beliefs about the
probability distribution on nature’s moves.
– So for example, all players think the odds of player 1 being of
a particular type is p, and the probability of her being the other
type is 1-p
Example: Incomplete Information
P
L
A
Y
E
R
1
Build
Don’t
Build
PLAYER 2
PLAYER 2
Enter Don’t
Enter
0, -1
2, 0
Enter Don’t
Enter
3, -1
5, 0
2, 1
3, 0
Payoffs if 1’s building costs are high
State of the World SH
P
L
A
Y
E
R
1
Build
Don’t
Build
2, 1
3, 0
Payoffs if 1’s building costs are low
State of the World SL
When does player 2 choose to enter?
In SH, Pl 1 doesn’t build (dominant strat) and player 2 enters: payoffs are (2, 1)
In SL, Pl 1 builds (dominant strat) and player 2 doesn’t enter: payoffs are (5, 0)
Let p1 denote prior probability player 2 assigns to p(SH); implies p(SL) = 1-p1
Decision is based on p1 *1 + (1- p1 )*(-1) > p1 *0 + (1- p1 )*(0)
Example, Modified in SL
PLAYER 2
P
L
A
Y
E
R
1
Build
Don’t
Build
PLAYER 2
Enter Don’t
Enter
0, -1
2, 0
2, 1
3, 0
Payoffs if 1’s building costs are high
State of the World SH
P
L
A
Y
E
R
1
Enter Don’t
Enter
Build 1.5, -1 3.5, 0
Don’t
Build
2, 1
3, 0
Payoffs if 1’s building costs are low
State of the World SL
What is Player 1’s optimal strategy?
In SH, Pl 1 doesn’t build (dominant strat), player 2 enters: payoffs are (2, 1)
In SL, Pl 1 does not build if Player 2 is likely to enter; let y be p(player 2 enters)
Player 1 must have prior probability about pl 2’s behavior to choose own action.
Let y = player 2’s probability of entry
Strategy: SH don’t build; If SL, build if 1.5*y +3.5*(1-y) > 2y+3*(1-y)
Let’s Transform the Game
PLAYER 2
P
L
A
Y
E
R
1
Build
Don’t
Build
PLAYER 2
Enter Don’t
Enter
0, -1
2, 0
2, 1
P
L
A
Y
E
R
1
3, 0
Payoffs if 1’s building costs are high
State of the World SH
High
Build
Enter
Don’t
Enter Don’t
Enter
Build 1.5, -1 3.5, 0
Don’t
Build
3, 0
Payoffs if 1’s building costs are low
N
State of the World SL
Low
Don’t Build
Enter
2, 1
Don’t
Build
Enter
Don’t
Don’t Build
Enter
Don’t
Transformed into a Game of Imperfect Information
The Modified Example
PLAYER 2
P
L
A
Y
E
R
1
Build
Don’t
Build
PLAYER 2
Enter Don’t
Enter
0, -1
2, 0
2, 1
3, 0
Payoffs if 1’s building costs are high
State of the World SH
P
L
A
Y
E
R
1
Enter Don’t
Enter
Build 1.5, -1 3.5, 0
Don’t
Build
2, 1
3, 0
Payoffs if 1’s building costs are low
State of the World SL
What is the Bayesian Nash Equilibrium?
Let p1 denote prior probability player 2 assigns to p(SH); implies p(SL) = 1-p1
Let x = player 1’s probability of building when her cost is low;
Let y = player 2’s probability of entry
What is Player 2’s optimal strategy? Enter (y =1) if p1(1) +(1-p1)[-x+(1-x)] > 0
This is equivalent to enter if x < 1/(2(1-p )
Modifying the Example
PLAYER 2
P
L
A
Y
E
R
1
Build
Don’t
Build
PLAYER 2
Enter Don’t
Enter
0, -1
2, 0
2, 1
3, 0
Payoffs if 1’s building costs are high
State of the World SH
P
L
A
Y
E
R
1
Enter Don’t
Enter
Build 1.5, -1 3.5, 0
Don’t
Build
2, 1
3, 0
Payoffs if 1’s building costs are low
State of the World SL
What is the Bayesian Nash Equilibrium?
Let p1 denote prior probability player 2 assigns to p(SH); implies p(SL) = 1-p1
Let x = player 1’s probability of building when her cost is low;
Let y = player 2’s probability of entry
Already shown that the best response for Low-Cost Player 1 is identified by:
Build (x = 1) if 1.5*y +3.5*(1-y) > 2y+3*(1-y), so when y < ½
Modifying the Example
PLAYER 2
P
L
A
Y
E
R
1
Build
Don’t
Build
PLAYER 2
Enter Don’t
Enter
0, -1
2, 0
2, 1
3, 0
Payoffs if 1’s building costs are high
State of the World SH
P
L
A
Y
E
R
1
Enter Don’t
Enter
Build 1.5, -1 3.5, 0
Don’t
Build
2, 1
3, 0
Payoffs if 1’s building costs are low
State of the World SL
3 Bayesian Nash Equilibrium
Recall Player 2 indifferent when x = 1/(2(1-p1), Player 1 indifferent when SL, y = ½
Mixed-strategy Bayesian Eqm: x and y as above, for all p1 in (0, ½), pl 1 in SL never builds
Pure strategy Bayesian Nash Equilibria:
(x = 0, y = 1 for any p1), (x = 1, y = 0 for for all p1 in (0, ½)
Main Ideas
What is Bayes’ Rule?
• A mathematical rule of logic explaining how
you should change your beliefs in light of
new information (updating)
• Bayes’ rule is essential knowledge for all
rigorous scientific methodologies
• Bayes’ Rule: P(A|B) = P(B|A)*P(A)/P(B)
– Typically, P(A|B) is your scientific inquiry, with A as
your quantity of interest—maybe A is your regression
coefficient, and you want to know if it equals zero.*
“What’s the probability parameter is A, given data B ?”
*See Chapter 2 in Gary King’s “Unifying Political Methodology”
Main Ideas
What is Bayes’ Rule?
• A mathematical rule of logic explaining how
you should change your beliefs in light of
new information (updating)
• Bayes’ rule is essential knowledge for all
rigorous scientific methodologies
• Bayes’ Rule: P(A|B) = P(B|A)*P(A)/P(B)
– To use Bayes’ Rule, you need to know a few things:
– You need to know P(B|A)
– You also need to know the probabilities of A and B
A Bayes’ Rule Exercise
• Situation: There are 2 States of the World
• We have “priors” on the causal mechanisms
• Priors about the true data-generating process
• State 1: Unstable political order: occurs with probability r
• State 2: Stable political order: occurs with probability 1 - r
• Many citizens get independent signals of which state it is
• If State 1
• Probability(stable) = v, probability(unstable) = (1-v)
• If State 2
• Probability(stable) = q, probability(unstable) = (1-q)
• If a “stable” signal is received, what is the probability that it is State 1?
• P(State1|stable) = P(stable|State1)*P(State1)/P(stable)
= v*r/P(stable)
= v*r/[P(stable & State1)+ P(stable & State2)]
= v*r/[rv+ (1-r)q]
A Bayes’ Rule Exercise
• Situation: There are 2 States of the World
• State 1: Unstable political order: occurs with probability r
• State 2: Stable political order: occurs with probability 1 - r
• Many citizens get independent signals of which state it is
• If State 1
• Probability(stable) = v, probability(unstable) = (1-v)
• If State 2
• Probability(stable) = q, probability(unstable) = (1-q)
• If 3 stable signals and 2 unstable signals are received, what is
P(State1)? Notation: Call 3 stable signals and 2 unstable signals
<3,2>
• P(State1|<3,2>) = P(<3,2>|State1) *P(State1)/P<3,2>
= [v3*(1-v)2 ] *(r)/ P<3,2>
= [v3*(1-v)2 ] *(r)/[(P<3,2>& State1) +(P<3,2>& State2)]
3
2
3
2
3
2