Game Development Reference
In-Depth 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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