On the Existence of Pure Strategy Nash Equilibria in Large Games

Over the years, several formalizations of games with a continuum of players have been given. These include those of Schmeidler (1973), Mas-Colell (1984) and Khan and Sun (1999). Unlike the others, Khan and Sun (1999) also addressed the equilibrium problem of large ¯- nite games, establishing the existence of a pure strategy approximate equilibrium in su±ciently large games. This ability for their formal- ization to yield asymptotic results led them to argue for it as the right approach to games with a continuum of players. We challenge this view by establishing an equivalent asymptotic theorem based only on Mas-Colell's formalization. Furthermore, we show that it is equivalent to Mas-Colell's existence theorem. Thus, in contrast to Khan and Sun (1999), we conclude that Mas-Colell's for- malization is as good as theirs for the development of the equilibrium theory of large ¯nite games.