Game Development Reference
In-Depth Information
this context we talk about a pure Nash equilibrium , when each of the
constituent strategies is pure, and refer to an arbitrary Nash equilibrium of
the mixed extension as a Nash equilibrium in mixed strategies of the
initial finite game. In what follows, when we use the letter m we implicitly
refer to the latter Nash equilibrium.
Lemma 1.13 (Characterisation) Consider a finite strategic game
( S 1 ,...,S n ,p 1 ,...,p n ) . The following statements are equivalent:
(i) m is a Nash equilibrium in mixed strategies, i.e.,
p i ( m ) ≥ p i ( m i ,m −i )
and all m i
for all i
∈{
1 ,...,n
}
Δ S i ,
(ii) for all i
∈{
1 ,...,n
}
and all s i
S i
p i ( m )
p i ( s i ,m −i ) ,
(iii) for all i
∈{
1 ,...,n
}
and all s i
support ( m i )
p i ( m )= p i ( s i ,m −i )
and for all i ∈{ 1 ,...,n} and all s i ∈ support ( m i )
p i ( m )
p i ( s i ,m −i ) .
Exercise 1.5
Provide the proof.
Note that the equivalence between ( i ) and ( ii ) implies that each Nash
equilibrium of the initial game is a pure Nash equilibrium of the mixed
extension. In turn, the equivalence between ( i ) and ( iii ) provides us with
a straightforward way of testing whether a joint mixed strategy is a Nash
equilibrium.
We now illustrate the use of the above theorem by finding in the Battle of
the Sexes game a Nash equilibrium in mixed strategies, in addition to the
two pure ones exhibited in Section 1.3. Take
m 1 := r 1
·
F +(1
r 1 )
·
B,
m 2 := r 2
·
F +(1
r 2 )
·
B,
where 0 <r 1 ,r 2 < 1. By definition
p 1 ( m 1 ,m 2 )=2
·
r 1 ·
r 2 +(1
r 1 )
·
(1
r 2 ) ,
p 2 ( m 1 ,m 2 )= r 1 ·
r 2 +2
·
(1
r 1 )
·
(1
r 2 ) .
Suppose now that ( m 1 ,m 2 ) is a Nash equilibrium in mixed strategies. By
the equivalence between ( i ) and ( iii ) of the Characterisation Lemma 1.13
p 1 ( F, m 2 )= p 1 ( B, m 2 ), i.e., (using r 1 = 1 and r 1 = 0 in the above formula