Arbitration and Stability in Cooperative Games in Overlapping Coalitions

Consider the following scenario: a group of agents is tasked with completing some projects; the agents divide into groups, and using the resources available to each group, agents generate profits, which must in turn be divided among group members. Cooperative game theory [Peleg and Sudhölter, 2007] studies such scenarios; formally, Given a set of agents N = {1, . . . , n}, the value of each subset S of N is given by a function v : 2 → R. Agents first form a coalition structure CS by partitioning into disjoint sets; then, the value of each subset S ∈ CS is divided among the members of S. Such payoff divisions are also called imputations. Given a game G = 〈N, v〉, a solution concept for G is a set of imputations that share some desirable properties; for example, the core of a game G is the set of all payoff divisions such that for all S ⊆ N , the total payoff to S is at least v(S). That is, the core is the set of all stable payoff divisions, from which no subset of agents would want to deviate. Classic cooperative game theory assumes that when agents form coalition structures, each agent is a member of only one coalition. In Overlapping Coalition Formation (OCF) games [Chalkiadakis et al., 2010], agents can participate in several coalitions. Each agent i ∈ N controls some finite resource such as time, computational power, or money. The key feature of OCF games is that unlike classic cooperative games, agents are allowed to concurrently commit resources to several coalitions. Thus, a coalition is no longer a subset of N , but rather a vector c in [0, 1], where the i-th coordinate of c, c, denotes how much of i’s resource is devoted to c. The valuation function v is now from [0, 1] to R, rather than from 2 to R. Under this setting, a coalition structure CS is a list of vectors in [0, 1], (c1, . . . , ck), and its value is simply ∑k j=1 v(cj). Having formed CS , agents must divide the payoffs from CS in some manner; such a payoff division, x = (x1, . . . ,xk), consists of vectors xj , such that ∑n i=1 x i j = v(cj). Similarly to the non-overlapping setting, if cj = 0, i.e. agent i does not contribute to cj , then imay not receive any payoff from cj . Those agents for which cj > 0 are called the support of cj . The pair (CS ,x) is called an outcome of G. As noted in [Chalkiadakis et al., 2010], stability in OCF games is a complicated matter; when deviating from (CS ,x), a set S may abandon some, but not all of the coalitions it is involved in. The main issue is whether S gets to keep its payoffs under (CS ,x) from coalitions that are unaffected by the deviation. [Chalkiadakis et al., 2010] introduce three possible reactions to deviation: the conservative, refined, and optimistic. Under the conservative reaction, S may expect no payoffs from any coalition; like in the non-overlapping case, it assumes that it is “on its own” if it deviates; under the refined reaction, S may expect payoff from all coalitions that were not changed by the deviation; finally, under the optimistic, S may still receive payoff from a coalition cj , if it can reduce its contribution to cj while still paying all agents in N \ S the same amount they got from cj under (CS ,x).