Monopolistic competition,aggregation of competitive information,and the amount of product differentiation

Consider an industry with a large number of homogeneous firms. Each firm's profits are a function of its own strategy and the strategies the other firms select. Suppose other firms' strategies enter into each firm's profit function only through one or more statistics. For example, average price in the market may parameterize every firm's profit function. We prove that, as a general rule, the industry's firms will in equilibrium follow at most M + 1 distinct strategies, where M is the number of statistics by which competitors' strategies affect each firm's profits.