Consider the following N-player symmetric game where each player is given the option to cooperate or defect: we define c_{i} as 1 if the i-th player cooperates, and 0 if she defects, and her payoff is given by:

u_{i} = −c_{i} + √(∑_{j}c_{j}).

Let M be the number of players who cooperate; the payoff of a cooperative player is then √M − 1, and that of a defecting player is √M. Overall utility is

∑u_{i} = M(√M − 1) + (N−M)√M = N√M − M,

which is maximized when N = M, i.e. if everybody cooperates. On the other hand, each player is better off defecting, as the individual marginal payoff for contributing is never greater than 1 (this follows from the inequality √(M+1) − √M ≤ 1); but everybody defecting will yield and individual and global payoff of zero, which is the lowest possible outcome, both individually and overall.

This game is an example of the Free Rider Problem. As I see it, it can also be regarded as a model for the economic role of the State as a tax-funded public goods producer: public services (infrastructure, health care, etc.) have a utility which conforms to the Law of Diminishing Returns, but this ineficiency (as compared with the situation where every tax-payer keeps their money) is compensated by the fact that the service is enjoyed by every citizen in a non-exclusive manner.

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