A Bayesian interpretation of
mixed-strategy equilibria
E621 Recitation
Han Zhang∗
Spring 2014
• Harsanyi (1973): player i’s mixed strategy represents player i’s uncertainty about j’s
choice of a pure strategy, and that j’s choice in turn depends on the realization of a
small amount of private information.
• A mixed-strategy equilibrium in a game of complete information can (almost always)
be interpreted as a pure-strategy Bayesian equilibrium in a closely related game with
a little bit of incomplete information.
An example
• There are three equilibria in the game shown in Table 1: the two obvious purestrategy equilibria and a mixed strategy equilibrium where Chris plays Opera with
probability 2/3 and Pat plays Opera with probability 1/3.
∗
Department of Economics, Indiana University–Bloomington. Email: [email protected].
Table 1: A BoS game.
Chris
Opera
Fight
Pat
Opera Fight
2,1
0,0
0,0
1,2
1
Table 2: The Bayesian version.
Pat
Chris
Opera
Fight
Opera
Fight
2 + tc , 1
0,0
0,0
1, 2 + tp
• Now suppose the players are not quite sure of each other’s payoffs, as in Table 2. tc
is private to Chris, and tp is private to Pat. Both are i.i.d. U [0, x], where x is a small
number.
• In the notations of a Bayesian game: G = {Ac , Ap ; Tc , Tp ; pc , pp ; uc , up }, where
– Ac = Ap = {Opera, Fight} are the action sets;
– Tc = Tp = [0, x] are the sets of types;
– pc , pp are the beliefs such that pc (tp ) = pp (tc ) = 1/x, ∀tp ∈ Tp , tc ∈ Tc ;
– uc , up are the payoffs as represented in Table 2.
• We focus on a Bayesian equilibrium where Chris plays Opera iff tc > c, where c is a
critical value to be determined, and Pat plays Fight iff tp exceeds some p.
• Then Chris plays Opera with probability (x − c)/x, and Pat plays Fight with probability (x − p)/x.
• Claim. As the incomplete information disappears (i.e., as x ↓ 0), the players’ behavior in this pure-strategy Bayesian equilibrium approaches their behavior in the
mixed-strategy equilibrium in the original game of complete information.
Proof. Given Pat’s strategy, Chris’ expected payoffs from playing Opera and Fight
are respectively
h
pi
p
p
(2 + tc ) + 1 −
· 0 = (2 + tc )
(1)
x
xi
x
h
p
p
p
·0+ 1−
·1=1− .
(2)
x
x
x
Thus playing Opera is optimal if
tc ≥
x
− 3 = c.
p
2
(3)
Similarly, given Chris’ strategy, Pat’s expected payoffs from playing Fight and Opera
are respectively
h
c
c
ci
· 0 + (2 + tp ) = (2 + tp )
(4)
1−
x
x
hx
i
c
c
c
1−
·1+ ·0=1− .
(5)
x
x
x
Thus playing Fight is optimal if
tp ≥
x
− 3 = p.
c
(6)
Solving (3) and (6) simultaneously yields p = c and p2 + 3p − x = 0. Hence the
probability that Chris plays Opera and the probability that Pat plays Fight are both
equal to
√
x−p
−3 + 9 + 4x
x−c
=
=1−
(7)
x
x
2x
Using L’Hospital’s rule,
√
2(9 + 4x)−1/2
2
−3 + 9 + 4x
= 1 − lim
= 1 − lim(9 + 4x)−1/2 =
lim 1 −
x↓0
x↓0
x↓0
2x
2
3
(8)
which is exactly the probabilities in the mixed-strategy equilibrium of the original
game.
References
Gibbons, R. (1992): Game Theory for Applied Economists. Princeton University Press.
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