Game Development Reference
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(i) Then m is a Nash equilibrium of G iff it is a Nash equilibrium of G RAT .
(ii) If each player has in G RAT exactly one strategy, then the resulting joint
strategy is a unique Nash equilibrium of G.
Exercise 1.8
Provide the proof.
In the context of rationalizability a joint mixed strategy of the opponents
is referred to as a belief . The definition of rationalizability is generic in
the class of beliefs w.r.t. which best responses are collected. For example,
we could use here joint pure strategies of the opponents, or probability
distributions over the Cartesian product of the opponents' strategy sets,
so the elements of the set Δ S −i (extending in an expected way the payoff
functions). In the first case we talk about point beliefs and in the second
case about correlated beliefs .
In the case of point beliefs we can apply the elimination procedure entailed
by
to arbitrary games. To avoid discussion of the outcomes reached in
the case of infinite iterations we focus on a result for a limited case. We refer
here to Nash equilibria in pure strategies.
RAT
Theorem 1.22 Assume a strategic game G. Consider the definition of the
RAT operator for the case of point beliefs and suppose that the outcome
G RAT
is reached in finitely many steps.
(i) Then s is a Nash equilibrium of G iff it is a Nash equilibrium of G RAT .
(ii) If each player is left in G RAT with exactly one strategy, then the
resulting joint strategy is a unique Nash equilibrium of G.
Exercise 1.9
Provide the proof.
A subtle point is that when G is infinite, the restriction G RAT
may have
empty strategy sets (and hence no joint strategy).
Example 1.23 Bertrand competition , originally proposed by Bertrand
[1883], is a game concerned with a simultaneous selection of prices for the
same product by two firms. The product is then sold by the firm that chose
a lower price. In the case of a tie the product is sold by both firms and the
profits are split.
Consider a version in which the range of possible prices is the left-open
real interval (0 , 100] and the demand equals 100
p , where p is the lower
price. So in this game G there are two players, each with the set (0 , 100] of
strategies and the payoff functions are defined by:
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