Cooperative games and multiagent systems

Stéphane Airiau
2013 Knowledge engineering review (Print)  
Forming coalitions is a generic means for cooperation: people, robots, web services, resources, firms, they can all improve their performance by joining forces. The use of coalitions has been successful in domains such as task allocations, sensor networks, and electronic marketplaces. Forming efficient coalitions requires the identification of matching synergies between different entities (finding complementary partners, or similar partners, or partners who add diversity). In addition, the
more » ... rent parties must negotiate a fair repartition of the worth created by the coalition. The first part of this paper is a tutorial on cooperative game theory (also called coalitional games). We then survey the different scenarios and the key issues addressed by the multiagent systems community. 2 S. AIRIAU computation of some solution concepts. Some other issues are related with dynamic environments: agents can enter and leave the system at any time, new tasks may appear in the environment, the environment may be uncertain (uncertainty about the value of the coalitions, about the competence of other agents, etc.). Safety and robustness issues should also be taken into account to guarantee a stable agent society. In addition, researchers must design protocols that are secure to prevent the possibility of manipulation or infiltration by agents or external forces. Another scenario is to consider that the goal of the agent is to maximise utilitarian social welfare. This scenario is not interesting for game theory as sharing the value between the members is no longer an issue. However, finding the optimal organisation is still a hard problem which can be addressed by AI techniques. The first part, section 2, consists of a tutorial on cooperative game theory. We first survey the case in which utility can be transferred between agents (i.e., agents are allowed to make side payments between them): the transferable utility games (TU games). This is the most important case treated by the game theory literature. We introduce the stability concepts for TU games and provide some results about their complexity. We will also study one special type of TU game that models voting situation, and some extensions of TU games. We then briefly introduce the case where no transfer or comparison of utility are possible between agents: the non-transferable utility games (NTU games) and provide some definitions of stability concepts. The rest of the paper introduces research from the multiagent systems literature. We first present some applications that have been used to study the formation of coalitions in Section 3. In particular, we discuss the task allocation domain, the electronic marketplace domain, and some variants. We also list some additional domains where coalitions of agents have been used. In Section 4 we survey the cooperative case where the agents' goal is to maximise utilitarian social welfare, i.e., the case where the utility of an agent is the total utility of the population. We survey some central algorithms that efficiently search for the optimal partition of agents into coalitions. Finally in Section 5, we survey some issues raised by the multiagent systems community for which game theory has little (or no) answer so far. Transferable Utility Games (TU games) In the following, we use a utility-based approach and we assume that "everything has a price": each agent has a utility function that is expressed in currency units. The use of a common currency enables the agents to directly compare alternative outcomes, and it also enables side payments. A TU game involves a set of players N and a characteristic function v : 2 N → R that provides a value for each possible coalition or subset of agents. The characteristic function is common knowledge for the entire population, and the value of a coalition depends only on the players present in its coalition. In a TU games, two questions are asked simultaneously: what coalitions should form (i.e., how to partition the set N into coalitions), and how to share the value of a coalition to each of its members. In general, it is not always possible to satisfy the interests of all players at the same time. Unfortunately, there is no single criterion for characterising an acceptable solution. After defining the TU games with more details, we will present some desirable criterion for a solution, and then, we will present the main solution concepts. Notations and types of TU games We consider a set N of n agents. A coalition is a non-empty subset of N . The set N is also known as the grand coalition. The set of all coalitions is C and its cardinality is 2 n . A coalition structure (CS) S = {C 1 , · · · , C m } is a partition of N : each set C i is a coalition with ∪ m i=1 C i = N and i = j ⇒ C i ∩ C j = ∅. The set of all CSs is S and its size is of the order O(n n ) and ω(n n 2 ) . The characteristic function (or valuation function) v : 2 N → R provides the worth or utility of a coalition. For TU games, it is assumed that the valuation of a coalition C does not depend on the other coalitions present in the population. Definition 2.
doi:10.1017/s0269888913000106 fatcat:df6iqtesb5ehdauxctolr64zbq