Game Development Reference
InDepth 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 nonzero
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 nonempty set of strategies
S
i
and a
preference
relation
s
−i
on
S
i
parametrised
by a joint strategy
s
−i
of his opponents.
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