Game Development Reference

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(ii) If G is solved by IEWDS, then the resulting joint strategy is a Nash

equilibrium of G.

Exercise 1.2

Provide the proof.

Example 1.8
A nice example of a game that is solved by IEWDS is

the
Beauty Contest game
due to Moulin [1986]. In this game there are

n>
2 players, each with the set of strategies equal

. Each player

submits a number and the payoff to each player is obtained by splitting 1

equally between the players whose submitted number is closest to

{

1
,...,
100

}

2

3
of the

average. For example, if the submissions are 29
,
32
,
29, then the payoffs are

respectively

2
,
0
,
2
.

One can check that this game is solved by IEWDS and results in the joint

strategy (1
,...,
1). Hence, by the IEWDS Theorem 1.7 this joint strategy

is a (not necessarily unique; we shall return to this question in Section 1.5)

Nash equilibrium.

1

Exercise 1.3

Show that the Beauty Contest game is indeed solved by

IEWDS.

Note that in contrast to the IESDS Theorem 1.3 we do not claim in part

(
ii
) of the IEWDS Theorem 1.7 that the resulting joint strategy is a
unique

Nash equilibrium. In fact, such a stronger claim does not hold. Further, in

contrast to strict dominance, an iterated elimination of weakly dominated

strategies can yield several outcomes.

The following example reveals even more peculiarities of this procedure.

Example 1.9

Consider the following game:

LMR

T
0
,
1 1
,
0 0
,
0

B
0
,
0 0
,
0 1
,
0

It has three Nash equilibria, (
T,L
), (
B, L
) and (
B, R
). This game can be

solved by IEWDS but only if in the first round we do not eliminate all weakly

dominated strategies, which are
M
and
R
. If we eliminate only
R
, then we

reach the game

LM

T
0
,
1 1
,
0

B
0
,
0 0
,
0

that is solved by IEWDS by eliminating
B
and
M
. This yields

L

T

0
,
1

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