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economics literature (see, for example, [Dekel et al., 1998, Halpern, 2001,
Halpern and Rego, 2008, Heifetz et al., 2006a, Modica and Rustichini, 1994,
8.5 Iterated regret minimisation
Consider the well-known traveller's dilemma [Basu, 1994, 2007]. Suppose
that two travellers have identical luggage, for which they both paid the same
price. Their luggage is damaged (in an identical way) by an airline. The
airline offers to recompense them for their luggage. They may ask for any
dollar amount between $2 and $100. There is only one catch. If they ask for
the same amount, then that is what they will both receive. However, if they
ask for different amounts - say one asks for $ m and the other for $ m , with
m<m - then whoever asks for $ m (the lower amount) will get $( m + p ),
while the other traveller will get $( m
p ), where p can be viewed as a reward
for the person who asked for the lower amount, and a penalty for the person
who asked for the higher amount.
It seems at first blush that both travellers should ask for $100, the maximum
amount, for then they will both get that. However, as long as p> 1, one
of them might then realise that he is actually better off asking for $99 if
the other traveller asks for $100, since he then gets $(99+p). In fact, $99
weakly dominates $100, in that no matter what Traveller 1 asks for, Traveller
2 is always at least as well off asking for $99 than $100, and in one case
(if Traveller 2 asks for $100) Traveller 1 is strictly better off asking for $99.
Thus, it seems we can eliminate 100 as an amount to ask for. However, once
we eliminate 100, a similar argument shows that 98 weakly dominates 99.
And once we eliminate 99, then 97 weakly dominates 98. Continuing this
argument, both travellers end up asking for $2! In fact, it is easy to see that
(2,2) is the only Nash equilibrium. Indeed, with any other pair of requests,
at least one of the travellers would want to change his request if he knew
what the other traveller was asking. Since (2,2) is the only Nash equilibrium,
it is also the only sequential and perfect equilibrium. Moreover, it is the
only rationalizable strategy profile; and, once we allow mixed strategies, (2,2)
is the only strategy that survives iterated deletion of strongly dominated
strategies. (It is not necessary to understand these solution concepts in detail;
the only point that I am trying make here is that all standard solution
concepts lead to (2,2).)
This seems like a strange result. It seems that no reasonable person - even a
game theorist! - would ever play 2. Indeed, when the traveller's dilemma was
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