Howtoplay?
We will now describe the most common “solution concepts” for normal-form games. We
will first describe the concept of “dominant strategy equilibrium,” which is implied by
the rationality of the players. We then discuss “rationalizability” which corresponds
to the common knowledge of rationality, and finally we discuss the Nash Equilibrium,
which is related to the mutual knowledge of players’ conjectures about the other players’
actions.
9
2.1 Dominant-strategy equilibrium
Let us use the notation s
−i
to mean the list of strategies s
j
played by all the players j
other than i, i.e.,
s
−i
= (s
1
, .
..s
i−1
, s
i+1
, .
..s
n
).
Definition 7 A strategy s
∗
i
strictly dominates s
i
if and only if
u
i
(s
∗
i
, s
−i
) > u
i
(s
i
, s
−i
),
∀
s
−i
∈
S
−i
.
That is, no matter what the other players play, playing s
∗
i
is strictly better than
playing s
i
for player i. In that case, if i is rational, he would never play the strictly
dominated strategy s
i
.
3
A mixed strategy σ
i
dominates a strategy s
i
in a similar way: σ
i
strictly dominates
s
i
if and only if
σ
i
(s
i1
)u
i
(s
i1
, s
−i
) + σ
i
(s
i2
)u
i
(s
i2
, s
−i
) + · · · σ
i
(s
ik
)u
i
(s
ik
, s
−i
) > u
i
(s
i
, s
−i
),
∀
s
−i
∈
S
−i
.
A rational player i will never play a strategy s
i
iff s
i
is dominated by a (mixed or pure)
strategy.
Similarly, we can define weak dominance.
Definition 8 A strategy s
∗
i
weakly dominates s
i
if and only if
u
i
(s
∗
i
, s
−i
) ≥ u
i
(s
i
, s
−i
),
∀
s
−i
∈
S
−i
and
u
i
(s
∗
i
, s
−i
) > u
i
(s
i
, s
−i
)
for some s
−i
∈
S
−i
.
That is, no matter what the other players play, playing s
∗
i
is at least as good as
playing s
i
, and there are some contingencies in which playing s
∗
i
is strictly better than
s
i
. In that case, if rational, i would play s
i
only if he believes that these contingencies
will never occur. If he is cautious in the sense that he assigns some positive probability
for each contingency, he will not play s
i
.
3
That is, there is no belief under which he would play s
i
. Can you prove this?
10
Definition 9 A strategy s
di
is a (weakly) dominant strategy for player i if and only if s
di
weakly dominates all the other strategies of player i. A strategy s
di
is a strictly dominant
strategy for player i if and only if s
di
strictly dominates all the other strategies of player