Game Development Reference
In-Depth Information
By the choice of s i the sets A and B are disjoint. Moreover, both sets are
convex subsets of
R |S −i | .
By the Separating Hyperplane Theorem 1.25 for some non-zero c
R |S −i |
and d
R
c
·
x
d for all x
A ,
(1.1)
c
·
y
d for all y
B .
(1.2)
But 0
B , so by (1.2) d
0. Hence by (1.1) and the definition of A for
all s −i
0. Again by (1.1) and the definition of A this
excludes the contingency that d> 0, i.e., d = 0. Hence by (1.2)
s −i ∈S −i c s −i p i ( m i ,s −i ) s −i ∈S −i c s −i p i ( s i ,s −i ) for all m i Δ S i . (1.3)
Let c := s −i ∈S −i c s −i . By the assumption c
S −i we have c s −i
= 0. Take
m −i :=
s −i ∈S −i
c s −i
c
s −i .
Then (1.3) can be rewritten as
p i ( m i ,m −i )
p i ( s i ,m −i ) for all m i
Δ S i ,
i.e., s i is a best response to m −i .
1.6 Variations on the definition of strategic games
The notion of a strategic game is quantitative in the sense that it refers
through payoffs to real numbers. A natural question to ask is: do the payoff
values matter? The answer depends on which concepts we want to study. We
mention here three qualitative variants of the definition of a strategic game
in which the payoffs are replaced by preferences. By a preference relation
on a set A we mean here a linear order on A .
In Osborne and Rubinstein [1994] a strategic game is defined as a sequence
( S 1 ,...,S n ,
1 ,...,
n ) ,
where each i is player's i preference relation defined on the set S 1 ×
···×
S n of joint strategies.
In Apt et al. [2008] another modification of strategic games is consid-
ered, called a strategic game with parametrised preferences . In this
approach each player i has a non-empty set of strategies S i and a preference
relation s −i
on S i parametrised by a joint strategy s −i of his opponents.