Optimal Social Contract, Public Participation Game

This paper attempts to introduce a public player, whose utility function is just thesocial welfare function, into the traditional noncooperative games and propose thedefinition of social optimal agreement as the solution concept based on Nashequilibrium.In order to find an appropriate social welfare function, we explore many possibilitiesand impossibilities of aggregating individual preferences and utilities into socialpreference and utility. The preference function is defined, and the usual definitions andproperties of collective preference are redescribed. The indifference principle andpositive responsiveness together are sufficient and necessary conditions of the majorityrule. Collecitve rationality, indifference principle, and Pareto principle are inconsistent.The veto power theorem is proposed and proved in this paper, and the famousoligarchy theorem and Arrow's impossibility theorem can be considered as directcollories. New characterizations of Pareto-Extention rule, hierarchy, and Borda rule arealso given.The minimal justice is proposed as a property of social welfare functional. Underfomal welfarism, Nash social welfare function is the only social welfare functionsatisfying weak Pareto principle, anonymity, ratio-scale noncomparability, and minimaljustice; Rawlsian social welfare function is the only social welfare function satisfyingweak Pareto principle, anonymity, cardinal full comparability (or ordinal levelcomparability), and minimal justice.The concepts of public player and social optimal agreement are respectivelyexplored in strategic and extensive games. In strategic games with completeinformation, the definitions of social optimal agreements, including utilitarian,Rawlsian, and Nash, are based on the set of Nash equilibria. The existence of socialoptimal agreement is estabilished on the nonempty and compactness of Nashequilibrium and the continuity of the public player's untility function. In addition, it isnot hard to show that the utilitarian optimal agreements satisfy linearity and cardinalunit comparability, Rawlsian optimal agreements satisfy full comparability, and Nashoptimal agreements satisfy cardinal noncomparability.In strategic games with incomplete information, the definition of social optimalBayesian agreement is based on Bayesian (Nash) equilibrium. In Aumann's model of incomplete information, the knowledge function is generalized, and it can be shownthat if the solution concept of a game is common knowledge, then the game solution isBayesian Nash equilibrium if and only if it satisfies Bayesian rationality. It can also beshown that the social optimal Bayesian agreements always exist if properly defined.In extensive games with perfect information, the dynamic optimal agreements aredefined upon the set of subgame perfect equilibria, which is nonempty and compactand thus guarantees the existence of dynamic optimal agreements. In extensive gameswith inperfect information, sequential optimal agreement and trembling hand optimalagreement are defined respectively upon the set of sequential equilibiria and the set oftrembling hand perfect equilibria. The nonemptiness and compactness of the sets ofequential and trembling hand perfect equilibria are also shown, and thus the existenceof sequential and trembling hand optimal equilibria follows.Finally, correlated equilibria of strategic games are also considered, and it is easy toshow that the social optimal correlated agreements as defined always exist. The socialoptimal correlated agreement is similar in linearity and invariance to the social optimalagreement.