Attaining Fairness in Communication for Omniscience

This paper studies how to attain fairness in communication for omniscience that models the multi-terminal compress sensing problem and the coded cooperative data exchange problem where a set of users exchange their observations of a discrete multiple random source to attain omniscience—the state that all users recover the entire source. The optimal rate region containing all source coding rate vectors that achieve omniscience with the minimum sum rate is shown to coincide with the core (the solution set) of a coalitional game. Two game-theoretic fairness solutions are studied: the Shapley value and the egalitarian solution. It is shown that the Shapley value assigns each user the source coding rate measured by their remaining information of the multiple source given the common randomness that is shared by all users, while the egalitarian solution simply distributes the rates as evenly as possible in the core. To avoid the exponentially growing complexity of obtaining the Shapley value, a polynomial-time approximation method is proposed which utilizes the fact that the Shapley value is the mean value over all extreme points in the core. In addition, a steepest descent algorithm is proposed that converges in polynomial time on the fractional egalitarian solution in the core, which can be implemented by network coding schemes. Finally, it is shown that the game can be decomposed into subgames so that both the Shapley value and the egalitarian solution can be obtained within each subgame in a distributed manner with reduced complexity.

Playing Fair

Norms requiring individuals to treat their partners fairly can evolve even in populations that lack well-defined identity groups. The emergence of fairness equilibria in the bargaining problem and in Augmented Stag Hunt is analyzed with inductive and evolutionary learning models applied to populations that are not subdivided into preexisting groups. Inductive learning models applied to the bargaining problem yield distributions of equilibrium solutions centered around the egalitarian solution that corresponds to a norm of equal division of benefits. Inductive and evolutionary learning models applied to the Augmented Stag Hunt yield distributions of equilibrium solutions where each side contributes to a commonly desired good that are supported by costly punishments for noncontributors. These results support the strong reciprocity hypothesis in the social sciences without employing the controversial idea of group selection.

LEARNING BARGAINING CONVENTIONS

I examine from a conventionalist perspective the Nash bargaining problem that philosophers use as a tool for analyzing fair division. From this perspective, the solutions to bargaining problems are conventions that can emerge from inductive learning and focal point effects. I contrast the conventionalist approach to analyzing the bargaining problem with the better-known rational choice approach, which I criticize for having overly demanding epistemic presuppositions and for producing disappointing results. I apply a simple model of inductive learning to specific bargaining problems to show that agents can learn from repeated experience to follow a variety of bargaining conventions in a given problem. I conclude that such agents can come to regard two such conventions as focal for the bargaining problem, one that assigns claimants equal shares of a good and another egalitarian solution of equal payoff gains, and that the egalitarian solution tends to prevail when these two solutions differ. I conclude further that the above analysis lends support for admitting interpersonal utility comparisons into the analysis of fair division problems, and also suggests a focal point explanation of the wide acceptance of the Aristotelian proportionality principle of distributive justice.