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 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 ) i )
p i ( a i ,s −i ( θ −i ) 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 i ) ≥ p i ( a i ,a −i 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 −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
}
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