Adaptive Coalition Structure Generation in Cooperative Multi-agent Systems Giovanni Rossi and Gabriele D’Angelo Department of Computer Science, University of Bologna Mura Anteo Zamboni 7, 40126 Bologna, Italy Email: {giorossi, gdangelo}@cs.unibo.it Abstract: In multiagent systems a coalition structure is a col- lection of pair-wise disjoint subsets of agents whose union yields the entire population. Given a characteristic function quantifying the worth of agent subsets, searching for optimal coalition structures (i.e. where the sum of subsets’ worth is maximal) is a well-known NP-hard combinatorial optimiza- tion problem. While existing algorithms (either deterministic or stochastic) deal with time-invariant goal functions, the fo- cus here is on dynamic settings, where the worth of agent sub- sets possibly varies over time in an unknown and unpredictable fashion. The aim is to design an adaptive dynamic process gen- erating coalition structures with high worth most of the times. To this end, detecting variations in the worth of agent sub- sets becomes crucial. The proposed method takes into account such (possible) changes by intensifying the exploration activity whenever they are detected. The performance with respect to the worth of optimal coalition structures is evaluated through simulations. Keywords: Adaptive Coalition Structure Generation, Coali- tional Game, Simulation, Dynamic and Non-superadditive En- vironment, Cooperative Multiagent System. 1. Introduction MAS (multiagent systems) are said to be cooperative when agents are assumed to collaborate in order to achieve some optimal outcome of the overall system [15] [20] [21]. In this setting, a great deal of attention has been paid to coalition structure generation, where outcomes are partitions of agents, that is, collections of disjoint coalitions or subsets of agents, called blocks, whose union yields the entire population. Given a characteristic function CF or coalitional game, assigning a worth to each coalition, the worth of coalition structures obtains as the sum of their blocks’ worth, and optimality attains where such a global worth is maximal. Searching for optimal coalition structures is a NP-hard combinatorial optimization problem [16] [17], whose generic instance consists of the 2 m - dimensional real-valued vector specifying the worth of (non- empty) coalitions, where m ∈ N is the (finite) number of agents (while N is the set of naturals). A main aim of this paper is to formally organize and mathe- matically approach coalition structure generation in dynamic settings, where the worth of coalitions varies over time in an unknown and unpredictable fashion. In the static scenario, searching amounts to (efficiently) explore the space of candi- date solutions (namely, the lattice of partitions of agents), iden- tifying the optimal ones. Conversely, in the dynamic scenario the set of optimal coalition structures changes over time. In this case, any solver can only aim at generating as often as possible near-optimal coalition structures. In fact, a main as- sumption shall be that the initial goal function is not known and that no information is available concerning if, when and how such a goal function shall vary, the only available infor- mation at each time being simply the worth of (the blocks of) the coalition structure generated at that time. Hence, any solver crucially has to detect changes of the goal function and allow for re-generating coalition structures previously found to be poor when such changes do (or seem to) occur. While in the static formulation any stochastic search method can be compared with some deterministic one [11] [16] [18], the dynamic formulation lacks benchmarks for comparisons. There is no univocally defined solution, and different solvers can only be compared, through simulation results, in terms of some performance index. The proposed mechanism identifies coalitions as decisional units, and thus it is distributed, that is, without a central authority. Although it is adjustable for coali- tion structure generation in the non-CF form [18], this paper focuses on the CF form. In particular, the time-varying worth of coalitions is chosen in a way such that at each time the set of optimal coalition structures and the associated maximal worth are easily determined. This is used for comparing, at each time, the worth of generated coalition structures with the worth of optimal coalition structures. The contribution is methodolog- ical, providing a solver for a novel dynamic setting, which is therefore dealt with in essentially abstract terms. Such a solver is tested through simulations in the challenging scenario where the time-varying CF is double-peaked (i.e. displaying two maxima, a global and a local one) and bi-symmetric (i.e. with the population partitioned into two types and the worth of coalitions depending only on members’ type). Although this paper addresses dynamic environments, it seems worth recalling that searching for optimal partitions of a (finite) set with given CF is a problem arising in a variety of applications. Mainly, in combinatorial auctions (where agents are to be interpreted as goods to sell and the CF gets deter- mined by the available bids), maximizing the revenue amounts to optimally partition the goods and sell the blocks. Similarly, in task allocation mechanism design, if the system has to per- form a set of basic tasks each of which may be performed more or less efficiently by different coalitions, then global task performance is maximized when both agents and tasks to be performed are optimally partitioned with a bijection between these two partitions such that each block of agents performs exactly one block of tasks [7] [19]. In fact, this issue is em- siwn.org.uk ISSN 1757-4439 print / ISSN 1757-4447 cd-rom © 2008 The Systemics and Informatics World Network. All rights reserved. sai: cosiwn.2008.06.181 Communications of SIWN June 2008 Volume 4 pp. 116-122