An algebraic framework for the greedy algorithm with applications to the core and Weber set of cooperative games

An algebraic model generalizing submodular polytopes is presented, where modular functions on partially ordered sets take over the role of vectors in ${\mathbb R}^n$. This model unifies various generalizations of combinatorial models in which the greedy algorithm and the Monge algorithm are successful and generalizations of the notions of core and Weber set in cooperative game theory. As a further application, we show that an earlier model of ours as well as the algorithmic model of Queyranne, Spieksma and Tardella for the Monge algorithm can be treated within the framework of usual matroid theory (on unordered ground-sets), which permits also the efficient algorithmic solution of the intersection problem within this model. \u

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

Monge extensions of cooperation and communication structures

Cooperation structures without any {\it a priori} assumptions on the combinatorial structure of feasible coalitions are studied and a general theory for mar\-ginal values, cores and convexity is established. The theory is based on the notion of a Monge extension of a general characteristic function, which is equivalent to the Lovász extension in the special situation of a classical cooperative game. It is shown that convexity of a cooperation structure is tantamount to the equality of the associated core and Weber set. Extending Myerson's graph model for game theoretic communication, general communication structures are introduced and it is shown that a notion of supermodularity exists for this class that characterizes convexity and properly extends Shapley's convexity model for classical cooperative games.

Cooperative Games with Lattice Structure

A general model for cooperative games with possibly restricted and hierarchically ordered coalitions is introduced and shown to have lattice structure under quite general assumptions. Moreover, the core of games with lattice structure is investigated. Within a general framework that includes the model of classical cooperative games as a special case, it is proved algorithmically that monotone convex games have a non-empty core. Finally, the solution concept of the Shapley value is extended to the general class of cooperative games with restricted cooperation. It is shown that several generalizations of the Shapley value that have been proposed in the literature are subsumed in this model

A system-theoretic approach to multi-agent models

A system-theoretic model for cooperative settings is presented that unifies and ex- tends the models of classical cooperative games and coalition formation processes and their generalizations. The model is based on the notions of system, state and transi- tion graph. The latter describes changes of a system over time in terms of actions governed by individuals or groups of individuals. Contrary to classic models, the pre- sented model is not restricted to acyclic settings and allows the transition graph to have cycles. Time-dependent solutions to allocation problems are proposed and discussed. In par- ticular, Weber’s theory of randomized values is generalized as well as the notion of semi-values. Convergence assertions are made in some cases, and the concept of the Cesàro value of an allocation mechanism is introduced in order to achieve convergence for a wide range of allocation mechanisms. Quantum allocation mechanisms are de- fined, which are induced by quantum random walks on the transition graph and it is shown that they satisfy certain fairness criteria. A concept for Weber sets and two dif- ferent concepts of cores are proposed in the acyclic case, and it is shown under some mild assumptions that both cores are subsets of the Weber set. Moreover, the model of non-cooperative games in extensive form is generalized such that the presented model achieves a mutual framework for cooperative and non-co- operative games. A coherency to welfare economics is made and to each allocation mechanism a social welfare function is proposed

Allocation of fixed costs and the weighted Shapley value

The weighted value was introduced by Shapley in 1953 as an asymmetric version of his value. Since then several approximations have been proposed including one by Shapley in 1981 specifically addressed to cost allocation, a context in which weights appear naturally. It was at the occasion of a comment in which he only stated the axioms. The present paper offers a proof of Shapley's statement as well as an alternative set of axioms. It is shown that the value is the unique rule that allocates additional fixed costs fairly: only the players who are concerned contribute to the fixed cost and they contribute in proportion to their weights. A particular attention is given to the case where some players are assigned a zero weight.cost allocation, Shapley value, fixed cost

Allocation of fixed costs: characterization of the (dual) weighted Shapley value.

The weighted value was introduced by Shapley in 1953 as an asymmetric version of his value. Since then several axiomatizations have been proposed including one by Shapley in 1981 specifically addressed to cost allocation, a context in which weights appear naturally. It was at the occasion of a comment in which he only stated the axioms. The present paper offers a proof of Shapley's statement as well as an alternative set of axioms. It is shown that the value is the unique rule that allocates additional fixed costs fairly: only the players who are concerned contribute to the fixed cost and they contribute in proportion to their weights. A particular attention is given to the case where some players are assigned a zero weight.cost allocation, Shapley value, fixed cost.

Bilateral and Community Enforcement in a Networked Market with Simple Strategies

We present a model of repeated games in large buyer-seller networks in the presence of reputation networks via which buyers share information about past transactions. The model allows us to characterize cooperation networks - networks in which each seller cooperates (by providing high quality goods) with every buyer that is connected to her. To this end, we provide conditions under which: [1] the incentives of a seller s to cooperate depend only on her beliefs with respect to her local neighborhood - a subnetwork that includes seller s and is of a size that is independent of the size of the entire network; and [2] the incentives of a seller s to cooperate can be calculated as if the network was a random tree with seller s at its root. Our characterization sheds light on the welfare costs of relying only on repeated interactions for sustaining cooperation, and on how to mitigate such costs.Networks, moral hazard, graph theory, repeated games

Note on Representations of Ordered Semirings

The article studies ordered semigroups and semirings with respect to their representations in lattices. Such structures are essentially the pseudolattices of Dietrich and Hoffman. It is shown that a subadditive representation implies the semigroup to be a lattice in its own right. In particular, distributive lattices can be characterized as semirings admitting subadditive supermodular representations. The cover problem asks for a minimal cover of a ground set by representing sets with respect to a semiring. A greedy algorithm is exhibited to solve the cover problem for the class of lattices with weakly subadditive and supermodular representation