scholarly journals CONVEXITY IN STOCHASTIC COOPERATIVE SITUATIONS

2005 ◽  
Vol 07 (01) ◽  
pp. 25-42 ◽  
Author(s):  
JUDITH TIMMER ◽  
PETER BORM ◽  
STEF TIJS
Keyword(s):  
Shapley Value ◽  
Cooperative Games ◽  
Sufficient Conditions ◽  
Random Variables ◽  
Convex Game ◽  
Tu Games ◽  
New Model ◽  
The Core ◽  
Random Payoffs ◽  
Marginal Vectors

This paper introduces a new model concerning cooperative situations in which the payoffs are modeled by random variables. We analyze these situations by means of cooperative games with random payoffs. Special attention is paid to three types of convexity, namely coalitional-merge, individual-merge and marginal convexity. The relations between these types are studied and in particular, as opposed to their deterministic counterparts for TU games, we show that these three types of convexity are not equivalent. However, all types imply that the core of the game is nonempty. Sufficient conditions on the preferences are derived such that the Shapley value, defined as the average of the marginal vectors, is an element of the core of a convex game.

scholarly journals CORE CONCEPTS FOR DYNAMIC TU GAMES

2005 ◽  
Vol 07 (01) ◽  
pp. 43-61 ◽  
Author(s):  
LAURENCE KRANICH ◽  
ANDRÉS PEREA ◽  
HANS PETERS
Keyword(s):  
Cooperative Games ◽  
Sufficient Conditions ◽  
Finite Sequence ◽  
Essential Difference ◽  
Tu Games ◽  
The Core ◽  
Dynamic Setting ◽  
Core Concepts ◽  
Provided Examples

This paper is concerned with the question of how to define the core when cooperation takes place in a dynamic setting. The focus is on dynamic cooperative games in which the players face a finite sequence of exogenously specified TU-games. Three different core concepts are presented: the classical core, the strong sequential core and the weak sequential core. The differences between the concepts arise from different interpretations of profitable deviations by coalitions. Sufficient conditions are given for nonemptiness of the classical core in general and of the weak sequential core for the case of two players. Simplifying characterizations of the weak and strong sequential core are provided. Examples highlight the essential difference between these core concepts.

scholarly journals The core and superdifferential of a fuzzy TU-cooperative game

2020 ◽  
Vol 12 (2) ◽  
pp. 20-35
Author(s):  
Валерий Александрович Васильев ◽  
Valery Vasil'ev
Keyword(s):  
Cooperative Game ◽  
Cooperative Games ◽  
Sufficient Conditions ◽  
Subdifferential Calculus ◽  
The Core ◽  
Side Payments ◽  
Weak Homogeneity ◽  
Fuzzy Game

In the paper, we consider conditions providing coincidence of the cores and superdifferentials of fuzzy cooperative games with side payments. It turned out that one of the most simple sufficient conditions consists of weak homogeneity. Moreover, by applying so-called S*-representation of a fuzzy game introduced by the author, we show that for any vwith nonempty core C(v) there exists some game u such that C(v) coincides with the superdifferential of u. By applying subdifferential calculus we describe a structure of the core forboth classic fuzzy extensions of the ordinary cooperative game (e.g., Aubin and Owen extensions) and for some new continuations, like Harsanyi extensions and generalized Airport game.

scholarly journals Hart–Mas-Colell consistency and the core in convex games

2021 ◽  
Author(s):  
Bas Dietzenbacher ◽  
Peter Sudhölter
Keyword(s):  
Shapley Value ◽  
Cooperative Games ◽  
Individual Rationality ◽  
Transferable Utility ◽  
Convex Games ◽  
The Core ◽  
The Shapley Value ◽  
Scale Covariance

AbstractThis paper formally introduces Hart–Mas-Colell consistency for general (possibly multi-valued) solutions for cooperative games with transferable utility. This notion is used to axiomatically characterize the core on the domain of convex games. Moreover, we characterize all nonempty solutions satisfying individual rationality, anonymity, scale covariance, superadditivity, weak Hart–Mas-Colell consistency, and converse Hart–Mas-Colell consistency. This family consists of (a) the Shapley value, (b) all homothetic images of the core with the Shapley value as center of homothety and with positive ratios of homothety not larger than one, and (c) their relative interiors.

Irrational-Behavior-Proof Conditions Based on Limit Characteristic Functions

2019 ◽  
Vol 7 (1) ◽  
pp. 1-16
Author(s):  
Cui Liu ◽  
Hongwei Gao ◽  
Ovanes Petrosian ◽  
Juan Xue ◽  
Lei Wang
Keyword(s):  
Characteristic Function ◽  
Cooperative Game ◽  
Shapley Value ◽  
Cooperative Games ◽  
Characteristic Functions ◽  
The Core ◽  
The Shapley Value ◽  
Irrational Behavior ◽  
The Individual

Abstract Irrational-behavior-proof (IBP) conditions are important aspects to keep stable cooperation in dynamic cooperative games. In this paper, we focus on the establishment of IBP conditions. Firstly, the relations of three kinds of IBP conditions are described. An example is given to show that they may not hold, which could lead to the fail of cooperation. Then, based on a kind of limit characteristic function, all these conditions are proved to be true along the cooperative trajectory in a transformed cooperative game. It is surprising that these facts depend only upon the individual rationalities of players for the Shapley value and the group rationalities of players for the core. Finally, an illustrative example is given.

scholarly journals Allocation Problems, by the Method of Alternative Representation of the Inverse Set, for Values of Cooperative TU Games

2021 ◽  
Vol 6 (3) ◽  
pp. 173-177
Author(s):  
Irinel Dragan
Keyword(s):  
Shapley Value ◽  
Rational Solution ◽  
Tu Games ◽  
Alternative Representation ◽  
The Core ◽  
The Shapley Value ◽  
Cooperative Tu Games ◽  
The Family ◽  
Unique Parameter ◽  
Definition Of

In earlier works, we introduced the Inverse Problem, relative to the Shapley Value, as follows: for a given n-dimensional vector L, find out the transferable utilities’ games , such that  The same problem has been discussed further for Semivalues. A connected problem has been considered more recently: find out TU-games for which the Shapley Value equals L, and this value is coalitional rational, that is belongs to the Core of the game . Then, the same problem was discussed for other two linear values: the Egalitarian Allocation and the Egalitarian Nonseparable Contribution, even though these are not Semivalues. To solve such problems, we tried to find a solution in the family of so called Almost Null Games of the Inverse Set, relative to the Shapley Value, by imposing to games in the family, the coalitional rationality conditions. In the present paper, we use the same idea, but a new tool, an Alternative Representation of Semivalues. To get such a representation, the definition of the Binomial Semivalues due to A. Puente was extended to all Semivalues. Then, we looked for a coalitional rational solution in the Family of Almost Null games of the Inverse Set, relative to the Shapley Value. In each case, such games depend on a unique parameter, so that the coalitional rationality will be expressed by a simple inequality, determined by a number, the coalitional rationality threshold. The relationships between the three numbers corresponding to the above three efficient values have been found. Some numerical examples of the method are given.

RANKING AUCTIONS: A COOPERATIVE APPROACH

2012 ◽  
Vol 14 (01) ◽  
pp. 1250003 ◽  
Author(s):  
JUAN APARICIO ◽  
NATIVIDAD LLORCA ◽  
JOAQUIN SANCHEZ-SORIANO ◽  
MANUEL A. PULIDO ◽  
JULIA SANCHO
Keyword(s):  
Shapley Value ◽  
Service Provider ◽  
Internet Search ◽  
Assignment Game ◽  
Tu Games ◽  
Cooperative Approach ◽  
The Core ◽  
The Shapley Value ◽  
Search Service

In this paper, we deal with situations arising from markets where an Internet search service provider offers a service of listing firms in decreasing order according to what they have bid. We call these ranking auction situations and introduce the corresponding TU-games. The core, as well as the two friendly solutions for the corners of the market, in this class of games can be easily described using a related assignment game. We study the Alexia value and the Shapley value of this type of games. Using these solutions, we show which circumstances in the game are in favor of the provider and which are beneficial to the bidders.

APPLICATIONS OF GAME THEORY TO ECONOMIC EQUILIBRIUM

1999 ◽  
Vol 01 (01) ◽  
pp. 1-8 ◽  
Author(s):  
GUILLERMO OWEN
Keyword(s):  
Game Theory ◽  
Shapley Value ◽  
Cooperative Games ◽  
Competitive Equilibrium ◽  
Random Variable ◽  
Market Games ◽  
The Core ◽  
Thin Markets ◽  
Shapley Values ◽  
Game Theoretic

One of the original expectations for the theory of cooperative games was that it would give us results valid for thin markets (where the number of traders is too small for an equilibrium to be reached). Over a period of years, however, it has been shown that, for market games, both the core and the Shapley values converge, in some sense, to the competitive equilibrium. Thus, the feeling arises that for large market games, the game-theoretic concepts yield nothing other than the equilibrium. In this article, we study the question of convergence of the Shapley value to the equilibrium and show that in some cases the convergence can be extremely slow. A very simple example (the "shoe" game) suggests that replacing the value by the equilibrium is in some sense akin to replacing a random variable by its mean.

scholarly journals Cooperative and axiomatic approaches to the knapsack allocation problem

2021 ◽  
Author(s):  
R. Pablo Arribillaga ◽  
G. Bergantiños
Keyword(s):  
Knapsack Problem ◽  
Shapley Value ◽  
Operations Research ◽  
Cooperative Games ◽  
Optimal Solution ◽  
Axiomatic Approach ◽  
The Other ◽  
Total Revenue ◽  
The Core ◽  
Axiomatic Approaches

AbstractIn the knapsack problem a group of agents want to fill a knapsack with several goods. Two issues must be considered. The first is to decide optimally what goods to select for the knapsack. This issue has been studied in many papers in the literature on Operations Research and Management Science. The second issue is to divide the total revenue among the agents. This issue has been studied in only a few papers, and this is one of them. For each knapsack problem we consider three associated cooperative games. One of them (the pessimistic game) has already been considered in the literature. The other two (realistic and optimistic games) are defined in this paper. The pessimistic and realistic games have non-empty cores but the core of the optimistic game could be empty. We then follow the axiomatic approach. We propose two rules: The first is based on the optimal solution of the knapsack problem. The second is the Shapley value of the so called optimistic game. We offer axiomatic characterizations of both rules.

Cooperative games with additive multiple attributes

2021 ◽  
Vol 41 (1) ◽  
pp. 1135-1150
Author(s):  
Haitao Liu ◽  
Qiang Zhang
Keyword(s):  
Finite Number ◽  
Cooperative Game ◽  
Shapley Value ◽  
Cooperative Games ◽  
Polynomial Form ◽  
The Core ◽  
The Shapley Value ◽  
Efficient Resource ◽  
Multiple Attributes ◽  
Special Case

This paper studies cooperative games in which players have multiple attributes. Such games are applicable to situations in which each player has a finite number of independent additive attributes in cooperative games and the payoffs of coalitions are endogenous functions of these attributes. The additive attributes cooperative game, which is a special case of the multiattribute cooperative game, is studied with respect to the core, the conditions for existence and boundedness and methods of transformation regarding a general cooperative game. A coalitional polynomial form is also proposed to discuss the structure of coalition. Moreover, a Shapley-like solution called the efficient resource (ER) solution for additive attributes cooperative games is studied via the axiomatical method, and the ER solution of two additive attribute games with equivalent total resources coincides with the Shapley value. Finally, some examples of additive attribute games are given.

Normal approximations to discrete unimodal distributions

1986 ◽  
Vol 23 (04) ◽  
pp. 1013-1018
Author(s):  
B. G. Quinn ◽  
H. L. MacGillivray
Keyword(s):  
Stationary Distribution ◽  
Sufficient Conditions ◽  
Random Variables ◽  
Hypergeometric Distribution ◽  
Death Process ◽  
Normal Approximations ◽  
Discrete Random Variables ◽  
Birth Death

Sufficient conditions are presented for the limiting normality of sequences of discrete random variables possessing unimodal distributions. The conditions are applied to obtain normal approximations directly for the hypergeometric distribution and the stationary distribution of a special birth-death process.

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