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The existence of a second sub-game-perfect equilibrium strategy vector, where the potential entrant becomes the monopolist and the monopolist exits the market may be disturbing to some, however, this equilibrium strategy would disappear if one were to introduce an exit cost, so that the monopolist's payoff would be negative in the event of exit.
These arguments are summarized by the following proposition (a formal proof of this proposition is available from the authors per request): the two stage market game has two sub-game-perfect equilibrium strategy vectors, {Pm = C([Y.sub.o])/ [Y.sub.o], Not Enter} and {[P.sub.m] >C([Y.sub.o])/[Y.sub.o], Enter}.
The key observation (Theorem 1 below) is that, for such a game, a mixed strategy vector that is optimal, in the sense of maximizing the common utility function, is necessarily a Nash equilibrium.
If the common interest game is derived from a voting environment as above, and there is any strategy vector that is acceptable for the aggregation rule f, then ([Mathematical Expression Omitted], ..., [Mathematical Expression Omitted]) must also be acceptable for f.
Applying multilinearity and symmetry, we find that ([Mu], ..., [Mu]) is inferior to the symmetric strategy vector obtained by having each agent combine a high proportion of [Mu] with a small bit of [Upsilon], contrary to hypothesis.
Therefore, to produce an equilibrium symmetric (mixed) strategy vector that is acceptable for f, it suffices to prove the existence of a (possibly nonequilibrium) symmetric mixed strategy vector with this property.
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