Game Development Reference
InDepth Information
Let us return now to our analysis of an arbitrary strategic game (
S
1
,...,S
n
,
p
1
,...,p
n
). Let
s
i
,s
i
be strategies of player
i
. We say that
s
i
strictly dom
inates
s
i
(or equivalently, that
s
i
is
strictly dominated by
s
i
)if
∀s
−i
∈ S
−i
p
i
(
s
i
,s
−i
)
>p
i
(
s
i
,s
−i
)
,
that
s
i
weakly dominates
s
i
(or equivalently, that
s
i
is
weakly dominated
by
s
i
)if
S
−i
p
i
(
s
i
,s
−i
)
>p
i
(
s
i
,s
−i
)
,
and that
s
i
dominates
s
i
(or equivalently, that
s
i
is
dominated by
s
i
)if
∀
p
i
(
s
i
,s
−i
) and
∀
s
−i
∈
S
−i
p
i
(
s
i
,s
−i
)
≥
∃
s
−i
∈
p
i
(
s
i
,s
−i
)
.
Further, we say that
s
i
is
strictly dominant
if it strictly dominates all
other strategies of player
i
and define analogously a
weakly dominant
and
a
dominant
strategy.
Clearly, a rational player will not choose a strictly dominated strategy. As
an illustration let us return to the Prisoner's Dilemma. In this game for each
player,
C
(cooperate) is a strictly dominated strategy. So the assumption of
players' rationality implies that each player will choose strategy
D
(defect).
That is, we can predict that rational players will end up choosing the joint
strategy (
D, D
) in spite of the fact that the Pareto e
cient outcome (
C, C
)
yields for each of them a strictly higher payoff.
The Prisoner's Dilemma game can be easily generalised to
n
players as
follows. Assume that each player has two strategies,
C
and
D
. Denote by
C
n
the joint strategy in which each strategy equals
C
and similarly with
D
n
.
Further, given a joint strategy
s
−i
of the opponents of player
i
denote by

s
−i
∈
S
−i
p
i
(
s
i
,s
−i
)
≥
s
−i
(
C
)
the number of
C
strategies in
s
−i
.
Assume now that
k
i
and
l
i
, where
i

∈{
1
,...,n
}
, are real numbers such
that for all
i
∈{
1
,...,n
}
we have
k
i
(
n
−
1)
>l
i
>
0. We put
p
i
(
s
):=
k
i
+
l
i
if
s
i
=
D
k
i
s
−i
(
C
)

if
s
i
=
C.
Note that for
n
=2
,k
i
= 2 and
l
i
= 1 we get the original Prisoner's Dilemma
game.
Then for all players
i
we have
p
i
(
C
n
)=
k
i
(
n −
1)
>l
i
=
p
i
(
D
n
), so for all
players the strategy profile
C
n
yields a strictly higher payoff than
D
n
.Yet
for all players
i
strategy
C
is strictly dominated by strategy
D
, since for all
s
−i
∈

s
−i
(
C
)

p
i
(
C, s
−i
)=
l
i
>
0.
Whether a rational player will never choose a weakly dominated strategy
is a more subtle issue that we shall not pursue here.
S
−i
we have
p
i
(
D, s
−i
)
−
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