On the fair division of multiple stochastic pies to multiple agents within the Nash bargaining solution.

The fair division of a surplus is one of the most widely examined problems. This paper focuses on bargaining problems with fixed disagreement payoffs where risk-neutral agents have reached an agreement that is the Nash-bargaining solution (NBS). We consider a stochastic environment, in which the overall return consists of multiple pies with uncertain sizes and we examine how these pies can be allocated with fairness among agents. Specifically, fairness is based on the Aristotle's maxim: "equals should be treated equally and unequals unequally, in proportion to the relevant inequality". In this context, fairness is achieved when all the individual stochastic surplus shares which are allocated to agents are distributed in proportion to the NBS. We introduce a novel algorithm, which can be used to compute the ratio of each pie that should be allocated to each agent, in order to ensure fairness within a symmetric or asymmetric NBS.