The Least-core and Nucleolus of Path Cooperative Games

Cooperative games provide an appropriate framework for fair and stable profit distribution in multiagent systems. In this paper, we study the algorithmic issues on path cooperative games that arise from the situations where some commodity flows through a network. In these games, a coalition of edges or vertices is successful if it enables a path from the source to the sink in the network, and lose otherwise. Based on dual theory of linear programming and the relationship with flow games, we provide the characterizations on the CS-core, least-core and nucleolus of path cooperative games. Furthermore, we show that the least-core and nucleolus are polynomially solvable for path cooperative games defined on both directed and undirected network

Learning Cooperative Games

This paper explores a PAC (probably approximately correct) learning model in cooperative games. Specifically, we are given $m$ random samples of coalitions and their values, taken from some unknown cooperative game; can we predict the values of unseen coalitions? We study the PAC learnability of several well-known classes of cooperative games, such as network flow games, threshold task games, and induced subgraph games. We also establish a novel connection between PAC learnability and core stability: for games that are efficiently learnable, it is possible to find payoff divisions that are likely to be stable using a polynomial number of samples.Comment: accepted to IJCAI 201

Quantum Cooperative Games

We study two forms of a symmetric cooperative game played by three players, one classical and other quantum. In its classical form making a coalition gives advantage to players and they are motivated to do so. However in its quantum form the advantage is lost and players are left with no motivation to make a coalition.Comment: Revised in the light of referee's comments. Submitted to Physics Letters A. LaTex, 9 pages, 1 figure. Parts of this paper are rewritte

Robust Dynamic Cooperative Games

Classical cooperative game theory is no longer a suitable tool for those situations where the values of coalitions are not known with certainty. Recent works address situations where the values of coalitions are modelled by random variables. In this work we still consider the values of coalitions as uncertain, but model them as unknown but bounded disturbances. We do not focus on solving a specific game, but rather consider a family of games described by a polyhedron: each point in the polyhedron is a vector of coalitions’ values and corresponds to a specific game. We consider a dynamic context where while we know with certainty the average value of each coalition on the long run, at each time such a value is unknown and fluctuates within the bounded polyhedron. Then, it makes sense to define “robust” allocation rules, i.e., allocation rules that bound, within a pre- defined threshold, a so-called complaint vector while guaranteeing a certain average (over time) allocation vector. We also present as motivating example a joint replenishment application

Non-cooperative Games

Non-cooperative games are mathematical models of interactive strategic decision situations.In contrast to cooperative models, they build on the assumption that all possibilities for commitment and contract have been incorporated in the rules of the game.This contribution describes the main models (games in normal form, and games in extensive form), as well as the main concepts that have been proposed to solve these games.Solution concepts predict the outcomes that might arise when the game is played by "rational" individuals, or after learning processes have converged.Most of these solution concepts are variations of the equilibrium concept that was proposed by John Nash in the 1950s, a Nash equilibrium being a combination of strategies such that no player can improve his payoff by deviating unilaterally.The paper also discusses the justifications of these concepts and concludes with remarks about the applicability of game theory in contexts where players are less than fully rational.noncooperative games

A note on the monotonicity and superadditivity of TU cooperative games

In this note we make a comparison between the class of monotonic TU cooperative games and the class of superadditive TU cooperative games. We first provide the equivalence between a weakening of the class of su- peradditive TU games and zero-monotonic TU games. Then, we show that zero-monotonic TU games and monotonic TU games are different classes. Finally, we show under which restrictions the classes of superadditive and monotonic TU games can be related.TU cooperative games; superadditivity; monotonicity

A Nucleolus for Stochastic Cooperative Games

This paper extends the definition of the nucleolus to stochastic cooperative games, that is, to cooperative games with random payoffs to the coalitions. It is shown that the nucleolus is nonempty and that it belongs to the core whenever the core is nonempty. Furthermore, it is shown for a particular class of stochastic cooperative games that the nucleolus can be determined by calculating the traditional nucleolus introduced by Schmeidler (1969) of a specific deterministic cooperative game.Nucleolus;cooperative game theory;random variables;preferences

A value for bi-cooperative games

Bi-cooperative games were introduced by Bilbao et al. as a generalization of TU cooperative games, in which each player can participate positively, negatively, or not at all. In this paper, we propose a definition of a share of the worth obtained by some players after they decided on their participation in the game. It turns out that the cost allocation rule does not look for a given player to her contribution at the opposite participation option to the one she chooses. The relevance of the value is discussed on several examples.Bi-cooperative games ;Value ;Efficiency