Game Development Reference
In-Depth Information
(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