Game Development Reference

In-Depth Information

Battle of the Sexes

FB

F

2
,
1

0
,
0

B

0
,
0

1
,
2

Matching Pennies

H

T

H

1
,−
1

−
1
,

1

T

−

1
,

1

1
,

−

1

We introduce now some basic notions that will allow us to discuss and

analyse strategic games in a meaningful way. Fix a strategic game

(
S
1
,...,S
n
,p
1
,...,p
n
)
.

We denote
S
1

S
a
joint strategy
,or

a
strategy profile
, denote the
i
th element of
s
by
s
i
, and abbreviate the

sequence (
s
j
)
j
=
i
to
s
−i
. Occasionally we write (
s
i
,s
−i
) instead of
s
. Finally,

we abbreviate
×
j
=
i
S
j
to
S
−i
and use the '
−i
' notation for other sequences

and Cartesian products.

We call a strategy
s
i
of player
i
a
best response
to a joint strategy
s
−i

of his opponents if

×

...

×

S
n
by
S
, call each element
s

∈

s
i
∈

p
i
(
s
i
,s
−i
)
.

∀

S
i
p
i
(
s
i
,s
−i
)

≥

Next, we call a joint strategy
s
a
Nash equilibrium
if each
s
i
is a best

response to
s
−i
, that is, if

∀i ∈{
1
,...,n}∀s
i
∈ S
i
p
i
(
s
i
,s
−i
)
≥ p
i
(
s
i
,s
−i
)
.

So a joint strategy is a Nash equilibrium if no player can achieve a higher

payoff by
unilaterally
switching to another strategy.

Finally, we call a joint strategy
s
Pareto e
cient
if for no joint strategy
s

p
i
(
s
)

p
i
(
s
)
>p
i
(
s
).

∀

i

∈{

1
,...,n

}

≥

p
i
(
s
) and

∃

i

∈{

1
,...,n

}

That is, a joint strategy is Pareto e
cient if no joint strategy is both a weakly

better outcome for all players and a strictly better outcome for some player.

Some games, like the Prisoner's Dilemma, have a unique Nash equilibrium,

namely (
D, D
), while some other ones, like the Matching Pennies, have no

Nash equilibrium. Yet other games, like the Battle of the Sexes, have multiple

Nash equilibria, namely (
F, F
)and(
B, B
). One of the peculiarities of the

Prisoner's Dilemma game is that its Nash equilibrium is the only outcome

that is not Pareto e
cient.

Search Nedrilad ::

Custom Search