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.