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to their experimental observations.) As we show in [Halpern and Pass, 2009],
iterated regret minimisation captures experimental behaviour in a number
of other game as well, including the Centipede Game [Rosenthal, 1982],
Nash bargaining [Nash, 1950b], and Bertrand competition [Dufwenberg and
Gneezy, 2000].
I conclude this discussion by making precise the sense in which iterated re-
gret minimisation does not require knowledge of the other players' strategies
(or the fact that they are rational). Traditional solution concepts typically
assume common knowledge of rationality, or at least a high degree of mutual
knowledge of rationality. For example, it is well known that rationalizability
can be characterised in terms of common knowledge of rationality [Tan and
Werlang, 1988], where a player is rational if he has some beliefs according
to which what he does is a best response in terms of maximizing expected
utility; Aumann and Brandenburger [1995] show that Nash equilibrium re-
quires (among other things) mutual knowledge of rationality (where, again,
rationality means playing a utility-maximizing best response); and Branden-
burger, Friedenberg, and Keisler [2008] show that iterated deletion of weakly
dominated strategies requires su ciently high mutual assumption of rational-
ity, where 'assumption' is a variant of 'knowledge', and 'rationality' means
'does not play a weakly dominated strategy'. This knowledge of rationality
essentially also implies knowledge of the strategy used by other players.
But if we make this assumption (and identify rationality with minimising
regret), we seem to run into a serious problem with iterated regret minimi-
sation, which is well illustrated by the traveller's dilemma. As we observed
earlier, the strategy profile (97 , 97) is the only one that survives iterated
regret minimisation when p = 2. However, if agent 1 knows that player 2 is
playing 97, then he should play 96, not 97! That is, among all strategies, 97
is certainly not the strategy that minimises regret with respect to
.
Some of these di culties also arise when dealing with iterated deletion of
weakly dominated strategies. The justification for deleting a weakly dom-
inated strategy is the existence of other strategies. But this justification
may disappear in later deletions. As Mas-Colell, Whinston, and Green [1995,
p. 240] put in their textbook when discussing iterated deletion of weakly
dominated strategies:
{
97
}
[T]he argument for deletion of a weakly dominated strategy for player i is that he
contemplates the possibility that every strategy combination of his rivals occurs
with positive probability. However, this hypothesis clashes with the logic of iterated
deletion, which assumes, precisely, that eliminated strategies are not expected to
occur.
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