The State as a superrational nanny

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 + √(∑_jc_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.

If players in this game were superrational they would opt for cooperation, and individual and global payoff would indeed be much higher than predicted by classical Game Theory. In a sense, the State (and its tax-enforcing machinery) can be seen then as an agent coercing society into superrationality, which is a Game Theory formulation of the classical social contract. The important question remains of how primitive societies can evolve to self-imposing such social contract, given that superrationality can hardly be attributed to most individuals (hence the existence of law-enforcing mechanisms, and even Law itself). A rather disturbing explanation is that a society can only be forced into entering the social contract by a benevolent dictator; once the social contract has been set in place, the system itself caters for its own preservation, so that formal dictatorship is no longer required.