Surplus sharing is a kind of a

fair division Fair division is the problem in game theory of dividing a set of resources among several people who have an entitlement to them so that each person receives their due share. That problem arises in various real-world settings such as division of i ...

problem where the goal is to share the financial benefits of cooperation (the "

economic surplus In mainstream economics, economic surplus, also known as total welfare or total social welfare or Marshallian surplus (after Alfred Marshall), is either of two related quantities: * Consumer surplus, or consumers' surplus, is the monetary gai ...

") among the cooperating agents. As an example, suppose there are several workers such that each worker ''i'', when working alone, can gain some amount ''u_i''. When they all cooperate in a joint venture, the total gain is ''u₁''+...+''u_n+s'', where ''s''>0. This ''s'' is called the ''surplus'' of cooperation, and the question is: what is a fair way to divide ''s'' among the ''n'' agents? When the only available information is the ''u_i'', there are two main solutions: * Equal sharing: each agent ''i'' gets ''u_i''+''s''/''n'', that is, each agent gets an equal share of the surplus. * Proportional sharing: each agent ''i'' gets ''u_i''+''(s*u_i''/Σ''u_i)'', that is, each agent gets a share of the surplus proportional to his external value (similar to the proportional rule in bankruptcy). In other words, ''u_i'' is considered a measure of the agent's contribution to the joint venture. Kolm calls the equal sharing "leftist" and the proportional sharing "rightist". Chun presents a characterization of the proportional rule. Moulin{{Cite journal, last=Moulin, first=H., date=1987-09-01, title=Equal or proportional division of a surplus, and other methods, url=https://doi.org/10.1007/BF01756289, journal=International Journal of Game Theory, language=en, volume=16, issue=3, pages=161–186, doi=10.1007/BF01756289, s2cid=154259938 , issn=1432-1270 presents a characterization of the equal and proportional rule together by four axioms (in fact, any three of these axioms are sufficient): # ''Separability'' - the division of surplus within any coalition ''T'' should depend only on the total amount allocated to ''T'', and on the opportunity costs of agents within ''T''. # ''No advantageous reallocation'' - no coalition can benefit from redistributing its ''u_i'' among its members (this is a kind of

strategyproofness In game theory, an asymmetric game where players have private information is said to be strategy-proof or strategyproof (SP) if it is a weakly-dominant strategy for every player to reveal his/her private information, i.e. given no information abou ...

axiom). # ''Additivity'' - for each agent ''i'', the allocation to ''i'' is a linear function of the total surplus ''s''. # ''Path independence'' - for each agent ''i'', the allocation to ''i'' from surplus ''s'' is the same as allocating a part of ''s'', updating the ''u_i'', and then allocating the remaining part of ''s''. Any pair of these axioms characterizes a different family of rules, which can be viewed as a compromise between equal and proportional sharing. When there is information about the possible gains of sub-coalitions (e.g., it is known how much agents 1,2 can gain when they collaborate in separation from the other agents), other solutions become available, for example, the

Shapley value The Shapley value is a solution concept in cooperative game theory. It was named in honor of Lloyd Shapley, who introduced it in 1951 and won the Nobel Memorial Prize in Economic Sciences for it in 2012. To each cooperative game it assigns a u ...

References

Fair division

See also

References