Class Of Cooperative Games Research Articles

Measure-Theoretic Analysis of Stochastic Competence Sets and Dynamic Shapley Values in Banach Spaces

We develop a measure-theoretic framework for dynamic Shapley values in Banach spaces, extending classical cooperative game theory to continuous-time, infinite-dimensional settings. We prove the existence and uniqueness of strong solutions to stochastic differential equations modeling competence evolution in Banach spaces, establishing sample path continuity and moment estimates. The dynamic Shapley value is rigorously defined as a càdlàg stochastic process with an axiomatic characterization. We derive a martingale representation for this process and establish its asymptotic properties, including a strong law of large numbers and a functional central limit theorem under α-mixing conditions. This framework provides a rigorous basis for analyzing dynamic value attribution in abstract spaces, with potential applications to economic and game-theoretic models.

Rough Shapley functions for games with a priori unions

A family of allocation rules for cooperative games with a priori unions is introduced in this paper. These allocation rules, which will be called rough Shapley values, are extensions of the well-known Shapley value for classical cooperative games. The family of rough Shapley values, which is constructed by using rough sets to describe different cooperative options, includes several of the extensions of the Shapley value found in the literature. We prove that the rough Shapley values are the allocation rules for games with a priori unions that satisfy some reasonable properties.

The Unique Characterization of the Shapley Value for Bi-cooperative Games

The aim of the present paper is to study the unique characterization for bicooperative games. Shapley value is the expected marginal contribution of the alliance. We will introduce some properties for bicooperative games. Our first characterization is based on the classical axioms determining the Shapley value with the symmetry axiom replaced excluded null axiom. In our second axiomatization we use structural axiom and a zero excluded axiom instead of effective axiom in classical cooperative games. Finally, We provide here linearity, anonymity, dummy and efficiency and structural axioms to study a one-point solution concept for Bi-cooperative games.

On the Structural Stability of Values for Cooperative Games

It is generally assumed that any set of players can form a feasible coalition for classical cooperative games. But, in fact, some players may withdraw from the current game and form a union, if this makes them better paid than proposed. Based on the principle of coalition split, this paper presents an endogenous procedure of coalition formation by levels and bargaining for payoffs simultaneously, where the unions formed in the previous step continue to negotiate with others in the next step as “individuals,” looking for maximum share of surplus by organizing themselves as a partition. The structural stability of the induced payoff configuration is discussed, using two stability criteria of core notion for cooperative games and strong equilibrium notion for noncooperative games.

The equal-surplus Shapley value for chance-constrained games on finite sample spaces

Many interactions from linear production problems, financial markets, or sequencing problems are modeled by cooperative games where payoffs to a coalition of players is a random variable. For this class of cooperative games, we introduce a two-stage value as an ex-ante agreement among players. Players are first promised their prior Shapley shares which are exactly their respective shares by the Shapley value of the expectation game. The final payoff vector is obtained by equally re-allocating the surplus when a realization of the random payoff of the grand coalition is observed. In support of the tractability of the newly introduced value called equal-surplus Shapley value, we provide a simple and compact formula. Depending on which probability distributions over the sample spaces are admissible, we present several characterization results of the equal-surplus Shapley value. This is achieved by using some classical axioms together with some other appealing axioms such as the independence of local duplication which simply requires that individual shares in a game remain unchanged when only certain events are duplicated in the sample space of a coalition without altering the probability of observing the others.

Uncertainty in cooperative interval games: how Hurwicz criterion compatibility leads to egalitarianism

We study cooperative interval games. These are cooperative games where the value of a coalition is given by a closed real interval specifying a lower bound and an upper bound of the possible outcome. For interval cooperative games, several (interval) solution concepts have been introduced in the literature. We assume that each player has a different attitude towards uncertainty by means of the so-called Hurwicz coefficients. These coefficients specify the degree of optimism that each player has so that an interval becomes a specific payoff. We show that a classical cooperative game arises when applying the Hurwicz criterion to each interval game. On the other hand, the same Hurwicz criterion can also be applied to any interval solution of the interval cooperative game. Given this, we say that a solution concept is Hurwicz compatible if the two procedures provide the same final payoff allocation. When such compatibility is possible, we characterize the class of compatible solutions, which reduces to the egalitarian solution when symmetry is required. The Shapley value and the core solution cases are also discussed.

Public resources allocation using an uncertain cooperative game among vulnerable groups

PurposeThis paper aims to solve the problem of public resource allocation among vulnerable groups by proposing a new method called uncertainα-coordination value based on uncertain cooperative game.Design/methodology/approachFirst, explicit forms of uncertain Shapley value with Chouqet integral form and uncertain centre-of-gravity of imputation-set (CIS) value are defined separately on the basis of uncertainty theory and cooperative game. Then, a convex combination of the two values above called the uncertainα-coordination value is used as the best solution. This study proves that the proposed methods meet the basic properties of cooperative game.FindingsThe uncertainα-coordination value is used to solve a public medical resource allocation problem in fuzzy coalitions and uncertain payoffs. Compared with other methods, theα-coordination value can solve such problem effectively because it balances the worries of vulnerable group’s further development and group fairness.Originality/valueIn this paper, an extension of classical cooperative game called uncertain cooperative game is proposed, in which players choose any level of participation in a game and relate uncertainty with the value of the game. A new function called uncertainα-Coordination value is proposed to allocate public resources amongst vulnerable groups in an uncertain environment, a topic that has not been explored yet. The definitions of uncertain Shapley value with Choquet integral form and uncertainCISvalue are proposed separately to establish uncertainα-Coordination value.

On stability of collaborative supplier selection

This note discusses the possibility of fair gain sharing in cooperative situations where players optimally partition themselves across a number of alternative channels. An example is group purchasing among a set of buyers facing with a range of suppliers. We introduce channel selection games as a new class of cooperative games and give a representation of their cores. With two channels (suppliers), the game has a non-empty core if the gain functions across every individual channel is supermodular.

Analyzing Power in Weighted Voting Games with Super-Increasing Weights

Weighted voting games (WVGs) are a class of cooperative games that capture settings of group decision making in various domains, such as parliaments or committees. Earlier work has revealed that the effective decision making power, or influence of agents in WVGs is not necessarily proportional to their weight. This gave rise to measures of influence for WVGs. However, recent work in the algorithmic game theory community have shown that computing agent voting power is computationally intractable. In an effort to characterize WVG instances for which polynomial-time computation of voting power is possible, several classes of WVGs have been proposed and analyzed in the literature. One of the most prominent of these are super increasing weight sequences. Recent papers show that when agent weights are super-increasing, it is possible to compute the agents’ voting power (as measured by the Shapley value) in polynomial-time. We provide the first set of explicit closed-form formulas for the Shapley value for super-increasing sequences. We bound the effects of changes to the quota, and relate the behavior of voting power to a novel function. This set of results constitutes a complete characterization of the Shapley value in weighted voting games, and answers a number of open questions presented in previous work.

The extended Shapley value for generalized cooperative games under precedence constraints

We introduce a new class of games where cooperation among players is restricted by precedence constraints vis a vis the worth of a coalition depends on the order in which the players enter into the coalition. The idea combines two existing classes of cooperative games, namely the cooperative games under precedence constraints due to Faigle and Kern (Int J Game Theory 21(3):249–266, 1992) and the games in generalized characteristic function due to Nowak and Radzik (Games Econ Behav 6(1):150–161, 1994). A Shapley value for this special class of games is proposed, we call it the extended Shapley value to distinguish it for the existing one. Two axiomatic characterizations of the extended Shapley value are given: one uses Efficiency, Null player, and Linearity; the other uses Efficiency, Marginality, and Null game. Some interesting properties of the extended Shapley value are studied. Furthermore, two extensions of the extended Shapley value, called the extended probabilistic value and the extended order value, are proposed and characterized. Our study shows that the results in cooperative games under precedence constraints cannot have a trivial extension to the generalized constrained games and conversely.

A further discussion on fuzzy interval cooperative games

The F-core and the F-balancedness have been introduced for fuzzy interval cooperative games by Mallozzi et al. (2011). It was pointed out that the F-balancedness was a necessary but not sufficient condition for the non-emptiness of the F-core. In this paper, a numerical example is given to show that the F-balancedness can not guarantee the non-emptiness of the F-core. Furthermore, the F-cores of trapezoidal fuzzy interval games are discussed in detail. Some properties of the F-core are obtained and three sufficient conditions for the non-emptiness of the F-core are given. These conclusions are generalizations of the corresponding results in both interval cooperative game theory and classical cooperative game theory. Lastly, an application example is also provided.

Cooperative grey games and an application on economic order quantity model

Purpose – The purpose of this paper is to extend the results of Meca et al. (2004) depending on the grey information revealed by the individual firms. Design/methodology/approach – The authors introduce cooperative grey games and focus on sharing ordering cost rule (SOC-rule) to distribute the joint cost. Findings – In this study, the authors introduce a model, where inventory costs are assumed as grey numbers instead of crisp or stochastic ones studied in literature. At first, grey numbers and classical cooperative inventory games are recalled. Then, cooperative grey games are introduced and related results are given. Finally, an application is performed for three shotgun companies in Turkey. Originality/value – It is an effective approach for theoretical analysis of systems with imprecise information and incomplete samples. Therefore, grey system theory, rather than the traditional probability theory and fuzzy set theory, is better suited to model the inventory problems by using cooperative game theory. To the best of the knowledge no study exists modeling inventory situations by using cooperative grey games. From this point of view this study is a pioneering work on a promising topic.

The Shapley–Shubik power index for dichotomous multi-type games

This work focuses on multi-type games in which there are a number of non-ordered types in the input, while the output consists of a single real value. When considering the dichotomous case, we extend the Shapley–Shubik power index and provide a full characterization of this extension. Our results generalize the literature on classical cooperative games.

Shapley's Axiomatics for Lexicographic Cooperative Games

In classical cooperative game theory one of the most important principle is defined by Shapley with three axioms common payoff fair distribution’s Shapley value (or Shapley vector). In the last decade the field of its usage has been spread widely. At this period of time Shapley value is used in network and social systems. Naturally, the question is if it is possible to use Shapley’s classical axiomatics for lexicographic cooperative games. Because of this in the article for m dimensional lexicographic cooperative T m v v v ) ,..., ( 1  game is introduced Shapley’s axiomatics, as the principle of a fair distribution in the case of m dimensional payoff functions, when the criteria are strictly ranking. It has been revealed that axioms discussed by Shapley for classical games are sufficient in lexicographic cooperative games corresponding with the payoffs of distribution. Besides we are having a very interesting case: according to the proved theorem, Shapley’s classical principle simultaneously transforms on the composed scalar m v v ,..., 1 games of a lexicographic cooperative game, nevertheless, m v v ,..., 2 games could not be superadditive.

Pillage games with multiple stable sets

We prove that pillage games (Jordan in J Econ Theory 131.1:26–44, 2006, “Pillage and property”, JET) can have multiple stable sets, constructing pillage games with up to $$2^{\tfrac{n-1}{3}}$$ stable sets, when the number of agents, $$n$$ , exceeds four. We do so by violating the anonymity axiom common to the existing literature to establish a power dichotomy: for all but a small exceptional set of endowments, powerful agents can overcome all the others; within the exceptional set, the lesser agents can defend their resources. Once the allocations giving powerful agents all resources are included in a candidate stable set, deriving the rest proceeds by considering dominance relations over the finite exceptional sets—reminiscent of stable sets’ derivation in classical cooperative game theory. We also construct a multi-good pillage game with only three agents that also has two stable sets.

Marginal Pricing of Transmission Services Using Min-Max Fairness Policy

We consider the problem of allocating the cost of a transmission system among load and generator entities. It is a known instance of the classical cooperative game theoretic problem. To solve this problem is a formidable task, as we are dealing with a combinatorial game with transferable utilities. This is an NP hard problem. Therefore, it suffers from the curse of dimensionality. Marginal pricing is a pragmatic alternative to solve this problem since both Kirchhoff current law and voltage law are strictly adhered to. However, the generic complexity of cooperative game theoretic problems cannot be just wished away. It now manifests as the difficulty in choosing an economic slack bus, which may even be dispersed. We propose the application of min-max fairness policy to solve this problem and give an algorithm which will run in polynomial time. During network cost allocation, min-max fairness policy minimizes the maximum regret among participating entities at each step. Maximum regret is measured in terms of price, and lexicographic application of this principle leads to a fair and unique equilibrium price vector. Results on a large network demonstrate fairness as well as tractability of the proposed approach.

Cooperation with Externalities and Uncertainty

We introduce a new solution concept to problems with externalities, which is the first in the literature to take into account economic, regulatory and physical stability aspects of network problems in the very same model. A new class of cooperative games is defined where the worth of a coalition depends on the behavior of other players and on the state of nature as well. We allow for coalitions to form both before and after the resolution of uncertainty, hence agreements must be stable against both types of deviations. The appropriate extension of the classical core concept, the Sustainable Core, is defined for this new setup to test the stability of allocations in such a complex environment. A prominent application, a game of consumers and generators on an electrical energy transmission network is examined in details, where the power in- and outlets of the nodes have to be determined in a way, that if any line instantaneously fails, none of the remaining lines may be overloaded. We show that fulfilling this safety requirement in a mutually acceptable way can be achieved by choosing an element in the Sustainable Core.

Cooperative grey games and the grey Shapley value

This contribution is located in the common area of operational research and economics, with a close relation and joint future potential with optimization: game theory. We focus on collaborative game theory under uncertainty. This study is on a new class of cooperative games where the set of players is finite and the coalition values are interval grey numbers. An interesting solution concept, the grey Shapley value, is introduced and characterized with the properties of additivity, efficiency, symmetry and dummy player, which are straightforward generalizations of the corresponding properties in the classical cooperative game theory. The paper ends with a conclusion and an outlook to future studies.

Distributed Cooperative Sensing in Cognitive Radio Networks: An Overlapping Coalition Formation Approach

Cooperative spectrum sensing has been shown to yield a significant performance improvement in cognitive radio networks. In this paper, we consider distributed cooperative sensing (DCS) in which secondary users (SUs) exchange data with one another instead of reporting to a common fusion center. In most existing DCS algorithms, the SUs are grouped into disjoint cooperative groups or coalitions, and within each coalition the local sensing data is exchanged. However, these schemes do not account for the possibility that an SU can be involved in multiple cooperative coalitions thus forming overlapping coalitions. Here, we address this problem using novel techniques from a class of cooperative games, known as overlapping coalition formation games, and based on the game model, we propose a distributed DCS algorithm in which the SUs self-organize into a desirable network structure with overlapping coalitions. Simulation results show that the proposed overlapping algorithm yields significant performance improvements, decreasing the total error probability up to 25% in the Q <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">m</sub> + Q <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">f</sub> criterion, the missed detection probability up to 20% in the Q <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">m</sub> /Q <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">f</sub> criterion, the overhead up to 80%, and the total report number up to 10%, compared with the state-of-the-art non-overlapping algorithm.

Cooperative games under bubbly uncertainty

The allocation problem of rewards/costs is a basic question for players, namely, individuals and companies that are planning cooperation under uncertainty. The involvement of uncertainty in cooperative game theory is motivated by the real world in which noise in observation and experimental design, incomplete information and vagueness in preference structures and decision-making play an important role. In this study, a new class of cooperative games, namely, the cooperative bubbly games, where the worth of each coalition is a bubble instead of a real number, is presented. Furthermore, a new solution concept, the bubbly core, is defined. Finally, the properties and the conditions for the non-emptiness of the bubbly core are given. The paper ends with a conclusion and an outlook to related and future studies.