Cores of combined games

This paper studies the core of combined games, obtained by summing two coalitional games. It is shown that the set of balanced transferable utility games can be partitioned into equivalence classes of component games whose core is equal to the core of the combined game. On the other hand, for non balanced games, the binary relation associating two component games whose combination has an empty core is not transitive. However, we identify a class of non balanced games which, combined with any other non balanced game, has an empty core.Cooperative games, Core, Additivity, Issue Linkage, Multi Issue Bargaining

Sex differences in the structure and stability of children’s playground social networks and their overlap with friendship relations

Gender segregated peer networks during middle childhood have been highlighted as important for explaining later sex differences in behaviour, yet few studies have examined the structural composition of these networks and their implications. This short-term longitudinal study of 119 children (7-8 years) examined the size and internal structure of boys' and girls' social networks, their overlap with friendship relations, and their stability over time. Data collection at the start and end of the year involved systematic playground observations of pupils' play networks during team and non-team activities and measures of friendship from peer nomination interviews. Social networks were identified by aggregating play network data at each time point. Findings showed that the size of boy's play networks on the playground, but not their social networks, varied according to activity type. Social network cores consisted mainly of friends. Girl's social networks were more likely to be composed of friends and boys' networks contained friends and non-friends. Girls had more friends outside of the social network than boys. Stability of social network membership and internal network relations were higher for boys than girls. These patterns have implications for the nature of social experiences within these network contexts

Coalitional Games for Transmitter Cooperation in MIMO Multiple Access Channels

Cooperation between nodes sharing a wireless channel is becoming increasingly necessary to achieve performance goals in a wireless network. The problem of determining the feasibility and stability of cooperation between rational nodes in a wireless network is of great importance in understanding cooperative behavior. This paper addresses the stability of the grand coalition of transmitters signaling over a multiple access channel using the framework of cooperative game theory. The external interference experienced by each TX is represented accurately by modeling the cooperation game between the TXs in \emph{partition form}. Single user decoding and successive interference cancelling strategies are examined at the receiver. In the absence of coordination costs, the grand coalition is shown to be \emph{sum-rate optimal} for both strategies. Transmitter cooperation is \emph{stable}, if and only if the core of the game (the set of all divisions of grand coalition utility such that no coalition deviates) is nonempty. Determining the stability of cooperation is a co-NP-complete problem in general. For a single user decoding receiver, transmitter cooperation is shown to be \emph{stable} at both high and low SNRs, while for an interference cancelling receiver with a fixed decoding order, cooperation is stable only at low SNRs and unstable at high SNR. When time sharing is allowed between decoding orders, it is shown using an approximate lower bound to the utility function that TX cooperation is also stable at high SNRs. Thus, this paper demonstrates that ideal zero cost TX cooperation over a MAC is stable and improves achievable rates for each individual user.Comment: in review for publication in IEEE Transactions on Signal Processin

Cores of Cooperative Games in Information Theory

Cores of cooperative games are ubiquitous in information theory, and arise most frequently in the characterization of fundamental limits in various scenarios involving multiple users. Examples include classical settings in network information theory such as Slepian-Wolf source coding and multiple access channels, classical settings in statistics such as robust hypothesis testing, and new settings at the intersection of networking and statistics such as distributed estimation problems for sensor networks. Cooperative game theory allows one to understand aspects of all of these problems from a fresh and unifying perspective that treats users as players in a game, sometimes leading to new insights. At the heart of these analyses are fundamental dualities that have been long studied in the context of cooperative games; for information theoretic purposes, these are dualities between information inequalities on the one hand and properties of rate, capacity or other resource allocation regions on the other.Comment: 12 pages, published at http://www.hindawi.com/GetArticle.aspx?doi=10.1155/2008/318704 in EURASIP Journal on Wireless Communications and Networking, Special Issue on "Theory and Applications in Multiuser/Multiterminal Communications", April 200

Complexity of Determining Nonemptiness of the Core

Coalition formation is a key problem in automated negotiation among self-interested agents, and other multiagent applications. A coalition of agents can sometimes accomplish things that the individual agents cannot, or can do things more efficiently. However, motivating the agents to abide to a solution requires careful analysis: only some of the solutions are stable in the sense that no group of agents is motivated to break off and form a new coalition. This constraint has been studied extensively in cooperative game theory. However, the computational questions around this constraint have received less attention. When it comes to coalition formation among software agents (that represent real-world parties), these questions become increasingly explicit. In this paper we define a concise general representation for games in characteristic form that relies on superadditivity, and show that it allows for efficient checking of whether a given outcome is in the core. We then show that determining whether the core is nonempty is $\mathcal{NP}$-complete both with and without transferable utility. We demonstrate that what makes the problem hard in both cases is determining the collaborative possibilities (the set of outcomes possible for the grand coalition), by showing that if these are given, the problem becomes tractable in both cases. However, we then demonstrate that for a hybrid version of the problem, where utility transfer is possible only within the grand coalition, the problem remains $\mathcal{NP}$-complete even when the collaborative possibilities are given