Game Development Reference

In-Depth Information

all players selected their actions, each player knows his payoff but does not

know the payoffs of the other players. Note that given a pre-Bayesian game,

every joint type
θ

Θ uniquely determines a strategic game, to which we

refer below as a
θ
-game.

A
strategy
for player
i
in a pre-Bayesian game is a function
s
i
:Θ
i
→

∈

A
i
.

The previously introduced notions can be naturally adjusted to pre-Bayesian

games. In particular, a joint strategy
s
(

·

):=(
s
1
(

·

)
,...,s
n
(

·

)) is called an

ex-post equilibrium
if

∀

θ

∈

Θ

∀

i

∈{

1
,...,n

}∀

a
i

∈

A
i
p
i
(
s
i
(
θ
i
)
,s
−i
(
θ
−i
)
,θ
i
)

≥

p
i
(
a
i
,s
−i
(
θ
−i
)
,θ
i
)
,

where
s
−i
(
θ
−i
) is an abbreviation for the sequence of actions (
s
j
(
θ
j
))
=
i
.

In turn, a strategy
s
i
(

·

) for player
i
is called
dominant
if

∀θ
i
∈
Θ
i
∀a ∈ Ap
i
(
s
i
(
θ
i
)
,a
−i
,θ
i
)
≥ p
i
(
a
i
,a
−i
,θ
i
)
.

So
s
(
·
) is an ex-post equilibrium iff for every joint type
θ ∈
Θ the sequence

of actions (
s
1
(
θ
1
)
,...,s
n
(
θ
n
)) is a Nash equilibrium in the corresponding

θ
-game. Further,
s
i
(

·

) is a dominant strategy of player
i
iff for every type

θ
i
∈

Θ
i
,
s
i
(
θ
i
) is a dominant strategy of player
i
in every (
θ
i
,θ
−i
)-game.

We also have the following immediate counterpart of the Dominant Strategy

Note 1.1.

Note 1.31
(Dominant Strategy)
Consider a pre-Bayesian game G. Suppose

that s
(
·
)
is a joint strategy such that each s
i
(
·
)
is a dominant strategy. Then

it is an ex-post equilibrium of G.

Example 1.32

As an example of a pre-Bayesian game, suppose that

•

Θ
1
=

{

U, D

}

,Θ
2
=

{

L, R

}

,

•

A
1
=
A
2
=

{

F, B

}

,

and consider the pre-Bayesian game uniquely determined by the following

four
θ
-games. Here and below we marked the payoffs in Nash equilibria in

these
θ
-games in bold.

L

FB

R

FB

F

2
,
1

2
,
0

F

2
,
0

2
,
1

U

B

0
,
1

2
,
1

B

0
,
0

2
,
1

FB

FB

F

3
,
1

2
,
0

F

3
,
0

2
,
1

D

B

5
,
1

4
,
1

B

5
,
0

4
,
1

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