Finiteness and Inefficiency of Nash Equilibria,
YALE UNIV NEW HAVEN CONN COWLES FOUNDATION FOR RESEARCH IN ECONOMICS
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Our aim is to explore efficiency, as well as strongness, of the Nash Equilibria N.E. of finite-person noncooperative games in strategic form. We show that--with smooth strategy sets and payoff functions--it is almost always the case that the N.E. are finite in number efficient N.E. are extremal points and strong N.E. are inactive points. Extremal inactive points are points in the Cartesian product of the players strategy sets at which at least one at most one of the players is is not at a vertex of herhis strategy set. Both sets of points are therefore thin they are nonexistent if the strategy-sets are vertex-free. This result is not very surprising because efficiency or strongness is generally an outcome of cooperation. And, indeed, it has been part of the folklore of Game Theory witness the Prisoners Dilemma.
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