Game Development Reference
In-Depth Information
This shows that the strategies s 1 (
·
) and s 2 (
·
) such that
s 1 ( U ):= F, s 1 ( D ):= B , s 2 ( L )= F, s 2 ( R )= B
form here an ex-post equilibrium.
However, there is a crucial difference between strategic games and pre-
Bayesian games.
Example 1.33
Consider the following pre-Bayesian game:
Θ 1 =
{
U, B
}
2 =
{
L, R
}
,
A 1 = A 2 =
{
C, D
}
.
L
CD
R
CD
C
2 , 2
0 , 0
C
2 , 1
0 , 0
U
D
3 , 0
1 , 1
D
3 , 0
1 , 2
CD
CD
C
1 , 2
3 , 0
C
1 , 1
3 , 0
B
D
0 , 0
2 , 1
D
0 , 0
2 , 2
Even though each θ -game has a Nash equilibrium, they are so 'positioned'
that the pre-Bayesian game has no ex-post equilibrium. Even more, if we
consider a mixed extension of this game, then the situation does not change.
The reason is that no new Nash equilibria are then added to the 'constituent'
θ -games. (Indeed, each of them is solved by IESDS and hence by the IESDMS
Theorem 1.16( ii ) has a unique Nash equilibrium.) This shows that a mixed
extension of a finite pre-Bayesian game does not need to have an ex-post
equilibrium, which contrasts with the existence of Nash equilibria in mixed
extensions of finite strategic games.
To relate pre-Bayesian games to mechanism design we need one more
notion. We say that a pre-Bayesian game is of a revelation-type if A i i
for all i
. So in a revelation-type pre-Bayesian game the strategies
of a player are the functions on his set of types. A strategy for player i is
called then truth-telling if it is the identity function π i (
∈{
1 ,...,n
}
)onΘ i .
Now, as explained in Ashlagi et al. [2006] mechanism design can be viewed
as an instance of the revelation-type pre-Bayesian games. Indeed, we have
the following immediate, yet revealing observation.
·
Theorem 1.34
Given a direct mechanism
n , Θ 1 ,..., Θ n ,u 1 ,...,u n , ( f, t ))
( D
× R
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