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We then focus on the iterated elimination of never best responses, in short
IENBR , obtained by using the
N relation. The following counterpart of
the IESDS Theorem 1.3 then holds.
Suppose that G is an outcome of IENBR from
Theorem 1.10 (IENBR)
a strategic game G.
(i) If s is a Nash equilibrium of G, then it is a Nash equilibrium of G .
(ii) If G is finite and s is a Nash equilibrium of G , then it is a Nash
equilibrium of G.
(iii) If G is finite and solved by IENBR, then the resulting joint strategy is
a unique Nash equilibrium.
Exercise 1.4
Provide the proof.
Further, as shown by Apt [2005], we have the following analogue of the
Order Independence I Theorem 1.5.
Theorem 1.11 (Order Independence II) Given a finite strategic game all
iterated eliminations of never best responses yield the same outcome.
In the case of infinite games we encounter the same problems as in the case
of IESDS as Example 1.6 readily applies to IENBR, as well. In particular,
if we solve an infinite game by IENBR we cannot claim that we obtained a
Nash equilibrium. Still, IENBR can be useful in such cases.
Example 1.12 Consider the following infinite variant of the location game
considered in Example 1.4. We assume that the players choose their strategies
from the open interval (0 , 100) and that at each real in (0 , 100) there resides
one customer. We have then the following payoffs that correspond to the
intuition that the customers choose the closest vendor:
s i + s 3 i
if s i <s 3 −i
s i + s 3 i
p i ( s i ,s 3 −i ):=
if s i >s 3 −i
if s i = s 3 −i .
It is easy to check that in this game no strategy strictly or weakly dominates
another one. On the other hand each strategy 50 is a best response to some
strategy, namely to 50, and no other strategies are best responses. So this
game is solved by IENBR, in one step. We cannot claim automatically
that the resulting joint strategy (50 , 50) is a Nash equilibrium, but it is
straightforward to check that this is the case. Moreover, by the IENBR
Theorem 1.10( i ) we know that this is a unique Nash equilibrium.
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