The Core, Shapley Value and Weber Set

2018 ◽  
pp. 92-105
Keyword(s):  
Shapley Value ◽  
Weber Set ◽  
The Core

scholarly journals Cooperative optimality principals in differential games on networks

2020 ◽  
Vol 12 (4) ◽  
pp. 93-111
Author(s):  
Анна Тур ◽  
Anna Tur ◽  
Леон Аганесович Петросян ◽  
Leon Petrosyan
Keyword(s):  
Characteristic Function ◽  
Differential Games ◽  
Network Structure ◽  
Shapley Value ◽  
The Core ◽  
Investment Game ◽  
The Shapley Value ◽  
Research Investment ◽  
Optimality Principles

The paper describes a class of differential games on networks. The construction of cooperative optimality principles using a special type of characteristic function that takes into account the network structure of the game is investigated. The core, the Shapley value and the tau-value are used as cooperative optimality principles. The results are demonstrated on a model of a differential research investment game, where the Shapley value and the tau-value are explicitly constructed.

scholarly journals Synergy within the West African Triple Helix innovation systems as measured with game theory

2019 ◽  
Vol 1 (2) ◽  
pp. 96-114
Author(s):  
Eustache Mêgnigbêto
Keyword(s):  
Game Theory ◽  
Information Theory ◽  
Shapley Value ◽  
Triple Helix ◽  
Innovation Systems ◽  
West African ◽  
The West ◽  
Content Type ◽  
The Core ◽  
University Industry

Purpose University, industry and government relationships, known under the Triple Helix, have been studied under various aspects. The West African region and countries have been analysed with mutual information and transmission power, two information theory-based indicators. The purpose of this paper is to portray the landscape of West African Triple Helix innovation systems using three main game theory indicators (core, Shapley value and nucleolus) with the objective to measure the synergy within the selected innovation systems. Design/methodology/approach The collaboration between university, industry and government is modelled as a three-person coalitional game. Bibliographical data of selected countries were collected from Web of Science and organised according to collaboration patterns between the three actors. The characteristic functions of the games were computed, the cores plotted, the Shapley values and the nucleoli computed. Findings Either university or government has more power to create and lead to synergy; government shows solidarity towards university and industry in most of countries; and they are joined in their efforts by industry in two countries. The core exists in all the countries meaning that all the selected innovation systems present synergy; however, the extent is limited and varies over countries. Research limitations/implications Innovation includes all research products; however, this study focuses on publications only. Originality/value Synergy within a Triple Helix innovation system is studied up to now with information theory indicators. The paper portrays the landscape of West African Triple Helix innovation systems using three main game theory indicators: the core, the Shapley value and the nucleolus and gives a new way to study university, industry and government relationships.

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 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 A reply to Ponte et al (2016) Supply chain collaboration: Some comments on the nucleolus of the beer game

2018 ◽  
Vol 11 (3) ◽  
pp. 528 ◽  
Author(s):  
David Mueller
Keyword(s):  
Supply Chain ◽  
Shapley Value ◽  
Theoretical Knowledge ◽  
Supply Chain Collaboration ◽  
Lorenz Dominance ◽  
The Core ◽  
Core Solution ◽  
Aggregate Monotonicity ◽  
Beer Game ◽  
Published Paper

Purpose: The aim of the paper is to pick up the result of a previously published paper in order to deepen the discussion. We analyze the solution against the background of some well-known concepts and we introduce a newer one. In doing so we would like to inspire the further discussion of supply chain collaborationDesign/methodology/approach: Based on game theoretical knowledge we present and compare seven properties of fair profit sharing.Findings: We show that the nucleolus is a core-solution, which does not fulfil aggregate monotonicity. In contrast the Shapley value is an aggregate monotonic solution but does not belong to the core of every cooperative game. Moreover, we present the Lorenz dominance as an additional fairness criteria.Originality/value: We discuss the very involved procedure of establishing lexicographic orders of excess vectors for games with many players.

Multiplayer Allocations in the Presence of Diminishing Marginal Contributions: Cooperative Game Analysis and Applications in Management Science

2020 ◽  
Author(s):  
Mingming Leng ◽  
Chunlin Luo ◽  
Liping Liang
Keyword(s):  
Cooperative Game ◽  
Shapley Value ◽  
Extreme Points ◽  
Game Analysis ◽  
The Core ◽  
The Shapley Value ◽  
Code Sharing ◽  
Marginal Contributions ◽  
Least Core ◽  
Necessary And Sufficient

We use cooperative game theory to investigate multiplayer allocation problems under the almost diminishing marginal contributions (ADMC) property. This property indicates that a player’s marginal contribution to a non-empty coalition decreases as the size of the coalition increases. We develop ADMC games for such problems and derive a necessary and sufficient condition for the non-emptiness of the core. When the core is non-empty, at least one extreme point exists, and the maximum number of extreme points is the total number of players. The Shapley value may not be in the core, which depends on the gap of each coalition. A player can receive a higher allocation based on the Shapley value in the core than based on the nucleolus, if the gap of the player is no greater than the gap of the complementary coalition. We also investigate the least core value for ADMC games with an empty core. To illustrate the applications of our results, we analyze a code-sharing game, a group buying game, and a scheduling profit game. This paper was accepted by Chung Piaw Teo, optimization.

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