Conitzer & Sandholm

1. Introduction sometimes empty. Three other solution concepts which we consider are: Shapley value, the nucleolus, and equal excess theory. In an environment teeming with autonomous agents, it is only natural for agents to cooperate in order to achieve goals, even under the assumption that the agents are selfish. Cooperation is realized through the formation of coalitions: the members of each coalition share their resources, and ultimately divide their payoff. The trick is to divide the payoff in a way that keeps the coalition structure stable. All solution concepts to such cooperative games require that agents know the capabilities and resources of their peers. In environments where communication is restricted, communication complexity may prove an obstacle. Communication Complexity Each player i knows only a private input xi. The players wish to jointly compute a function f(x1,…,xn). At each stage, some player i broadcasts a bit of information; the bit depends on xi , as well as on the data communicated so far. Ultimately, all players must know the value of f(x1,…,xn). The communication complexity is the worst-case number of bits required to compute f (in the best protocol). I t is self-evident that in numerous Multiagent settings, selfish agents stand to benefit from cooperating by forming coalitions. Nevertheless, negotiating a stable distribution of the payoff among agents may prove challenging. The issue of coalition formation has been investigated extensively, but until recently little attention has been given to the complications that arise when the players are software agents. The bounded rationality of such agents has motivated researchers to study the computational complexity of the aforementioned problems. We examine the communication complexity of coalition formation, in an environment where each of the n agents knows only its own initial resources and utility function. Specifically, we give a tight (n) bound on the communication complexity of the following solution concepts in unrestricted games: Shapley value, the nucleolus and the modified nucleolus, equal excess theory, and the core. Moreover, we show that in some appealing restricted games the communication complexity is constant, suggesting that it is possible to achieve sublinear complexity by constraining the environment or choosing a suitable solution concept. Cooperative Games Input is 10; transmit 1 Input is 11 and received 1; transmit 1 A cooperative n-person game is a pair (N; v), where N = {1, 2, . . . , n} is a set of players, and v(S) is the value of S  N: the payoff the players in coalition S can obtain by cooperating. The collection of payoffs to the players is expressed as a payoff vector: x = x1, x2,…, xn . A coalition structure is a partition of N, of the form  ={S1, S2, . . . , Sr}, which specifies how the players in N divide themselves into coalitions. A payoff configuration is a pair (x;) = {x1, x2, . . . , xn; S1, S2, . . . , Sr}, where x is a payoff vector and  is a coalition structure, such that for all j, the sum of payoffs to players in Sj is v(Sj). Different solution concepts, which differ in their notion of stability, have been proposed. The core: Simply the set of all payoff configurations {(x;): SN, x(S)v(S)}. This is the strongest solution concept, and is, in fact, Fig 1. f is a xor of 4 bits; Alice and Bob hold two bits each. The communication complexity is 2. Results Theorem: The communication complexity of computing the payoff of an arbitrary player according to the above solution concepts is (n) (even to decide whether the payoff of the given player is greater than 0). Nevertheless, this result can be circumvented in restricted cooperative games. Theorem: In constant-sum apex games, the communication complexity of computing the payoff of an arbitrary agent in any solution concept is 1. This suggests that complexity can be lowered w.r.t. certain classes of games or solutions. Conitzer & Sandholm Complexity of Determining Nonemptiness of the Core. IJCAI, pp. 613-618, 2003. Kahan & Rapoport Theories of Coalition Formation. Lawrence Erlbaum Associates, 1984. Shehory & Kraus Coalition Formation Among Autonomous Agents: Strategies and Complexity. LNAI 957, pp. 57-72, 1995.