Game Development Reference

In-Depth Information

By definition, no player achieves a higher payoff by switching from a

dominant strategy to another strategy. This explains the following obvious

observation.

Note 1.1
(Dominant Strategy)
Consider a strategic game G. Suppose that

s is a joint strategy such that each s
i
is a dominant strategy. Then it is a

Nash equilibrium of G.

In particular, the conclusion of the lemma holds if each
s
i
is a strictly

or a weakly dominant strategy. In the former case, when the game is finite,

we can additionally assert (see the IESDS Theorem 1.3 below) that
s
is a

unique Nash equilibrium of
G
. This stronger claim does not hold if each
s
i
is

a weakly dominant strategy. Indeed, consider the game

LR

T
1
,
1 1
,
1

B
1
,
1 0
,
0

Here
T
is a weakly dominant strategy for the player 1,
L
is a weakly dominant

strategy for player 2 and, as prescribed by the above Note, (
T,L
), is a

Nash equilibrium. However, this game has two other Nash equilibria, (
T,R
)

and (
B, L
).

The converse of the above Note of course is not true. Indeed, there are

games in which no strategy is dominant, and yet they have a Nash equilibrium.

An example is the Battle of the Sexes game that has two Nash equilibria,

but no dominant strategy.

So to find a Nash equilibrium (or Nash equilibria) of a game it does not

su
ce to check whether a dominant strategy exists. In what follows we

investigate whether iterated elimination of strategies can be of help.

1.3 Iterated elimination of strategies I

1.3.1 Elimination of strictly dominated strategies

We assumed that each player is rational. So when searching for an outcome

that is optimal for all players we can safely remove strategies that are strictly

dominated by some other strategy. This can be done in a number of ways.

For example, we could remove all or some strictly dominated strategies

simultaneously, or start removing them in a round robin fashion starting

with, say, player 1. To discuss this matter more rigorously we introduce the

notion of a restriction of a game.

Given a game
G
:= (
S
1
,...,S
n
,p
1
,...,p
n
) and (possibly empty) sets

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