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4 Chapter 2. Nash Equilibrium
she receives a payoff of 2. If she is prohibited from using her second action in this second
game then she obtains an equilibrium payoff of 3, however.
27.2 (BoS with imperfect information) The Bayesian game is as follows. There are two players,
say N = {1, 2}, and four states, say Ω = {(B, B), (B, S), (S, B), (S, S)}, where the state
(X, Y ) is interpreted as a situation in which player 1’s preferred composer is X and player 2’s
is Y . The set A
i
of actions of each player i is {B, S}, the set of signals that player i may
receive is {B, S}, and player i’s signal function τ
i
is defined by τ
i
(ω) = ω
i
. A belief of each
player i is a probability distribution p
i
over Ω. Player 1’s preferences are those represented by
the payoff function defined as follows. If ω
1
= B then u
1
((B, B), ω) = 2, u
1
((S, S), ω) = 1,
and u
1
((B, S), ω) = u
1
((S, B), ω) = 0; if ω
1
= S then u
1
is defined analogously. Player 2’s
preferences are defined similarly.
For any beliefs the game has Nash equilibria ((B, B), (B, B)) (i.e. each type of each
player chooses B) and ((S, S), (S, S)). If one player’s equilibrium action is independent
of his type then the other player’s is also. Thus in any other equilibrium the two types
of each player choose different actions. Whether such a profile is an equilibrium depends
on the beliefs. Let q
X
= p
2
(X, X)/[p
2
(B, X) + p
2
(S, X)] (the probability that player 2
assigns to the event that player 1 prefers X conditional on player 2 preferring X) and let
p
X
= p
1
(X, X)/[p
1
(X, B) + p
1
(X, S)] (the probability that player 1 assigns to the event
that player 2 prefers X conditional on player 1 preferring X). If, for example, p
X
≥
1
3
and
q
X
≥
1
3
for X = B, S, then ((B, S), (B, S)) is an equilibrium.
28.1 (Exchange game) In the Bayesian game there are two players, say N = {1, 2}, the set of
states is Ω = S × S, the set of actions of each player is {Exchange, Don’t exchange}, the
signal function of each player i is defined by τ
i
(s
1
, s
2
) = s
i
, and each player’s belief on Ω is
that generated by two independent copies of F . Each player’s preferences are represented
by the payoff function u
i
((X, Y ), ω) = ω
j
if X = Y = Exchange and u
i
((X, Y ), ω) = ω
i
otherwise.
Let x be the smallest possible prize and let M
i
be the highest type of player i that chooses
Exchange. If M
i
> x then it is optimal for type x of player j to choose Exchange. Thus if
M
i
≥ M
j
and M
i
> x then it is optimal for type M
i
of player i to choose Don’t exchange,
since the expected value of the prizes of the types of player j that choose Exchange is less
than M
i
. Thus in any possible Nash equilibrium M
i
= M
j
= x: the only prizes that may
be exchanged are the smallest.
28.2 (More information may hurt) Consider the Bayesian game in which N = {1, 2}, Ω =
{ω
1
, ω
2
}, the set of actions of player 1 is {U, D}, the set of actions of player 2 is {L, M, R},
player 1’s signal function is defined by τ
1
(ω
1
) = 1 and τ
1
(ω
2
) = 2, player 2’s signal function
is defined by τ
2
(ω
1
) = τ
2
(ω
2
) = 0, the belief of each player is (
1
2
,
1
2
), and the preferences of
each player are represented by the expected value of the payoff function shown in Figure 5.1
(where 0 < <
1
2
).
This game has a unique Nash equilibrium ((D, D), L) (that is, both types of player 1
choose D and player 2 chooses L). The expected payoffs at the equilibrium are (2, 2).
In the game in which player 2, as well as player 1, is informed of the state, the unique
Nash equilibrium when the state is ω
1
is (U, R); the unique Nash equilibrium when the state
is ω
2
is (U, M ). In both cases the payoff is (1, 3), so that player 2 is worse off than he is
when he is ill-informed.