Binary Games with Many Players

• Published in: Economic theory Vol. 28; no. 1; pp. 125 - 143
• Main Authors: Menezes, Flavio M., Pitchford, Rohan
• Format: Journal Article
• Language: English
• Published: Heidelberg Springer-Verlag 01.05.2006; Springer Nature B.V
• Subjects: Competition; Distribution; Economic competition; Economic theory; Equilibrium; Game theory; Games; Games of strategy; Market entry; Mathematical methods; Mixed strategy; Nash equilibrium; Probability; Public goods; Regulatory theory; Studies; Sufficient conditions; Transaction costs
• DOI: 10.1007/s00199-005-0611-z
• ISSN: 0938-2259

We examine a problem with n players each facing the same binary choice. One choice is superior to the other. The simple assumption of competition - that an individual's payoff falls with a rise in the number of players making the same choice, guarantees the existence of a unique symmetric equilibrium (involving mixed strategies). As n increases, there are two opposing effects. First, events in the middle of the distribution - where a player finds itself having made the same choice as many others - become more likely, but the payoffs in these events fall. In opposition, events in the tails of the distribution - where a player finds itself having made the same choice as few others - become less likely, but the payoffs in these events remain high. We provide a sufficient condition (strong competition) under which an increase in the number of players leads to a reduction in the equilibrium probability that the superior choice is made.