Game Development Reference
In-Depth Information
Theorem 1.19 (IEWDMS)
Suppose that G is a finite strategic game.
(i) If G is an outcome of IEWDMS from G and m is a Nash equilibrium
of G , then m is a Nash equilibrium of G.
(ii) If G is solved by IEWDMS, then the resulting joint strategy is a Nash
equilibrium of G.
Here is a simple application of this theorem.
Corollary 1.20 Every mixed extension of a finite strategic game has a
Nash equilibrium such that no strategy used in it is weakly dominated by a
mixed strategy.
Proof It su ces to apply Nash's Theorem 1.14 to an outcome of IEWDMS
and use item ( i ) of the above theorem.
Finally, observe that the outcome of IEWMDS does not need to be unique.
In fact, Example 1.9 applies here, as well.
1.5.3 Rationalizability
Finally, we consider iterated elimination of strategies that are never best
responses to a joint mixed strategy of the opponents. Following Bernheim
[1984] and Pearce [1984], strategies that survive such an elimination process
are called rationalizable strategies. 3
Formally, we define rationalizable strategies as follows. Consider a restric-
tion R of a finite strategic game G . Let
RAT ( R ):=( S 1 ,...,S n ) ,
where for all i
}
S i := {s i ∈ R i |∃m −i ∈× j = i Δ R j s i is a best response to m −i in G}.
Note the use of G instead of R in the definition of S i . We shall comment on
it below.
Consider now the outcome G RAT
∈{
1 ,...,n
starting with G .We
call then the strategies present in the restriction G RAT rationalizable .
We have the following counterpart of the IESDMS Theorem 1.16, due to
Bernheim [1984].
of iterating
RAT
Theorem 1.21
Assume a finite strategic game G.
3 More precisely, in each of these papers a different definition is used; see Apt [2007] for an
analysis of the conditions for which these definitions coincide.