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

}

,Θ
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
=Θ
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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