Game Development Reference

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strategy
m
of
G
, when we say that
m
is a Nash equilibrium of
G
we implicitly

stipulate that each strategy used (with positive probability) in
m
is a strategy

in
G
.

Theorem 1.16
(IESDMS)
Suppose that G is a finite strategic game.

(i) If G
is an outcome of IESDMS from G, then m is a Nash equilibrium

of G iff it is a Nash equilibrium of G
.

(ii) If G is solved by IESDMS, then the resulting joint strategy is a unique

Nash equilibrium of G (in, possibly, mixed strategies).

Exercise 1.6

Provide the proof.

To illustrate the use of this result let us return to the Beauty Contest

game discussed in Example 1.8. We explained there why (1
,...,
1) is
a
Nash

equilibrium. Now we can draw a stronger conclusion.

Example 1.17
One can show that the Beauty Contest game is solved by

IESDMS in 99 rounds. In each round the highest strategy of each player is

removed and eventually each player is left with the strategy 1. On account

of the above theorem we now conclude that (1
,...,
1) is a
unique
Nash

equilibrium.

Exercise 1.7
Show that the Beauty Contest game is indeed solved by

IESDMS in 99 rounds.

As in the case of strict dominance by a pure strategy we now address the

question of whether the outcome of IESDMS is unique. The answer, as before,

is positive. The following result was established by Osborne and Rubinstein

[1994].

Theorem 1.18
(Order independence III)
All iterated eliminations of strat-

egies strictly dominated by a mixed strategy yield the same outcome.

1.5.2 Elimination of weakly dominated strategies

Next, we consider iterated elimination of pure strategies that are weakly

dominated by a mixed strategy.

As already noticed in Subsection 1.3.2 an elimination by means of weakly

dominated strategies can result in a loss of Nash equilibria. Clearly, the

same observation applies here. We also have the following counterpart of the

IEWDS Theorem 1.7, where we refer to Nash equilibria in mixed strategies.

Instead of 'the iterated elimination of strategies weakly dominated by a

mixed strategy' we write
IEWDMS
.

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