A note on purifying mixed strategy equilibria in the search-theoretic model of fiat money

The simple search-theoretic model of fiat money has three symmetric Nash equilibria: all agents accept money with probability 1; all agents accept money with probability 0; and all agents accept money with probability y between 0 and 1. Here the author constructs a nonsymmetric pure strategy equilibrium, payoff-equivalent to the symmetric mixed strategy equilibrium, where a fraction N between 0 and 1 of agents always accepts money and 1-N never accepts money. Counter to what has been conjectured previously, the author finds N>y. The author also studies evolutionary dynamics and shows that the economy converges to monetary exchange if and only if the initial proportion of agents accepting money exceeds N.