scholarly journals The Shapley value of coalitions to other coalitions

2020 ◽  
Vol 7 (1) ◽  
Author(s):  
Kjell Hausken
Keyword(s):  
Shapley Value ◽  
Symmetric Matrix ◽  
Positive Zero ◽  
Multiple Perspectives ◽  
Left Part ◽  
The Shapley Value ◽  
Matrix Properties ◽  
Shapley Values ◽  
Person Game

Abstract The Shapley value for an n-person game is decomposed into a 2n × 2n value matrix giving the value of every coalition to every other coalition. The cell ϕIJ(v, N) in the symmetric matrix is positive, zero, or negative, dependent on whether row coalition I is beneficial, neutral, or unbeneficial to column coalition J. This enables viewing the values of coalitions from multiple perspectives. The n × 1 Shapley vector, replicated in the bottom row and right column of the 2n × 2n matrix, follows from summing the elements in all columns or all rows in the n × n player value matrix replicated in the upper left part of the 2n × 2n matrix. A proposition is developed, illustrated with an example, revealing desirable matrix properties, and applicable for weighted Shapley values. For example, the Shapley value of a coalition to another coalition equals the sum of the Shapley values of each player in the first coalition to each player in the second coalition.

COPING WITH UNCERTAINTY IN n-PERSON GAMES

2000 ◽  
Vol 08 (05) ◽  
pp. 503-523 ◽  
Author(s):  
SILVIU GUIASU
Keyword(s):  
Characteristic Function ◽  
Security Council ◽  
Shapley Value ◽  
Statistical Equilibrium ◽  
Individual Rationality ◽  
Von Neumann ◽  
Minimum Deviation ◽  
The Shapley Value ◽  
Un Security Council ◽  
Person Game

A solution of n-person games is proposed, based on the minimum deviation from statistical equilibrium subject to the constraints imposed by the group rationality and individual rationality. The new solution is compared with the Shapley value and von Neumann-Morgenstern's core of the game in the context of the 15-person game of passing and defeating resolutions in the UN Security Council involving five permanent members and ten nonpermanent members. A coalition classification, based on the minimum ramification cost induced by the characteristic function of the game, is also presented.

Shapley Ranking of Wines

2012 ◽  
Vol 7 (2) ◽  
pp. 169-180 ◽  
Author(s):  
Victor Ginsburgh ◽  
Israël Zang
Keyword(s):  
Game Theory ◽  
Unique Solution ◽  
Cooperative Game ◽  
Shapley Value ◽  
Rank Order ◽  
Jel Classification ◽  
Ranking Method ◽  
The Shapley Value ◽  
Shapley Values

AbstractWe suggest a new game-theory-based ranking method for wines, in which the Shapley Value of each wine is computed, and wines are ranked according to their Shapley Values. Judges should find it simpler to use, since they are not required to rank order or grade all the wines, but merely to choose the group of those that they find meritorious. Our ranking method is based on the set of reasonable axioms that determine the Shapley Value as the unique solution of an underlying cooperative game. Unlike in the general case, where computing the Shapley Value could be complex, here the Shapley Value and hence the final ranking, are straightforward to compute. (JEL Classification: C71, D71, D78)

scholarly journals Exchange de Risques entre Assureurs et Theorie des Jeux

1977 ◽  
Vol 9 (1-2) ◽  
pp. 155-180 ◽  
Author(s):  
Jean Lemaire
Keyword(s):  
Shapley Value ◽  
Sufficient Conditions ◽  
Pareto Optimal ◽  
Existence Of A Solution ◽  
The Shapley Value ◽  
Théorie Des Jeux ◽  
Person Game ◽  
Reinsurance Market ◽  
Value Concept

SummaryA theorem of Borch characterizing Pareto-optimal treaties in a reinsurance market is extended to the case of non-differentiable utilities. Sufficient conditions for the existence of a solution to the equations are established. The problem is then shown to be identical to the determination of the value of a cooperative non-transferable m-person game. We show how to compute the Shapley value of this game, then we introduce a new value concept. An example illustrates both methods.

CONSISTENCY IN VALUES

2004 ◽  
Vol 06 (04) ◽  
pp. 461-473 ◽  
Author(s):  
GUILLERMO OWEN
Keyword(s):  
Shapley Value ◽  
Solution Concept ◽  
Tu Games ◽  
Ntu Games ◽  
Reduced Game ◽  
Reduced Games ◽  
Shapley Values ◽  
Consistent Solution ◽  
Person Game ◽  
Reasonable Behavior

Given an n-person game (N, v), a reduced game (T, vT) is the game obtained if some subset T of the players assumes reasonable behavior on the part of the remaining players and uses that as a given so as to bargain within T. This "reasonable" behavior on the part of N-T must be defined in terms of some solution concept, ϕ, and so the reduced game depends on ϕ. Then, the solution concept ϕ is said to be consistent if it gives the same result to the reduced games as it does to the original game. It turns out that, given a symmetry condition on two-person games, the Shapley value is the only consistent solution on the space of TU games. Modification of some definitions will instead give the prekernel, the prenucleolus, or the weighted Shapley values. A generalization to NTU games is given. This works well for the class of hyperplane games, but not quite so well for general games.

scholarly journals Meaningful and Meaningless Solutions for Cooperative N-person Games

1996 ◽  
Vol 3 (47) ◽  
Author(s):  
Aleksandar Pekec
Keyword(s):  
Shapley Value ◽  
Game Model ◽  
Solution Concepts ◽  
Measurement Sensitivity ◽  
The Core ◽  
The Shapley Value ◽  
Person Game

<p>Game values often represent data that can be measured in more<br />than one acceptable way (e.g. monetary amounts). We point out that<br />in such a case a statement about cooperative n-person game model<br />might be "meaningless" in the sense that its truth or falsity depends on<br />the choice of an acceptable way to measure game values. In particular<br />we analyze statements about solution concepts such as the core, stable<br />sets, the nucleolus, the Shapley value (and its generalizations).</p><p>Keywords: Cooperative n-person Games, Measurement, Sensitivity<br />Analysis.</p>

scholarly journals Revenue Distribution of Hybrid Channel Supply Chain Based on Modified Shapley Value with Cost

2021 ◽  
Vol 2021 ◽  
pp. 1-8
Author(s):  
Jin Sha ◽  
Sisi Zheng
Keyword(s):  
Supply Chain ◽  
Shapley Value ◽  
Chain Model ◽  
The Shapley Value ◽  
Online Retailers ◽  
Supply Chain Model ◽  
Shapley Values ◽  
Object Of Study ◽  
The Cost ◽  
The Impact

The reasonable distribution of revenue determines not only the coordination but also the development of supply chain. A hybrid channel supply chain model composed of manufacturer and traditional and online retailers was identified as the object of study, decision models under different cooperation modes were built, different revenues of supply chain members and overall were analyzed, satisfaction of characteristic function required by the Shapley value was proved, and the Shapley values of revenue distribution were calculated. In view of the shortcoming of the Shapley value method which takes contribution margin as the only consideration and ignores the impact of other factors such as cost on revenue distribution, the cost correction model was built by using production and sales cost to modify the traditional Shapley value; the reasonable distribution of individual benefits under the premise of maximizing the overall revenue of hybrid channel supply chain was realized.

A novel q-rung orthopair fuzzy TODIM approach for multi-criteria group decision making based on Shapley value and relative entropy

2021 ◽  
Vol 40 (1) ◽  
pp. 235-250
Author(s):  
Liuxin Chen ◽  
Nanfang Luo ◽  
Xiaoling Gou
Keyword(s):  
Decision Making ◽  
Relative Entropy ◽  
Group Decision Making ◽  
Shapley Value ◽  
Group Decision ◽  
Similarity Measures ◽  
Entropy Measure ◽  
Shapley Values ◽  
Interactive Relationship ◽  
The Difference

In the real multi-criteria group decision making (MCGDM) problems, there will be an interactive relationship among different decision makers (DMs). To identify the overall influence, we define the Shapley value as the DM’s weight. Entropy is a measure which makes it better than similarity measures to recognize a group decision making problem. Since we propose a relative entropy to measure the difference between two systems, which improves the accuracy of the distance measure.In this paper, a MCGDM approach named as TODIM is presented under q-rung orthopair fuzzy information.The proposed TODIM approach is developed for correlative MCGDM problems, in which the weights of the DMs are calculated in terms of Shapley values and the dominance matrices are evaluated based on relative entropy measure with q-rung orthopair fuzzy information.Furthermore, the efficacy of the proposed Gq-ROFWA operator and the novel TODIM is demonstrated through a selection problem of modern enterprises risk investment. A comparative analysis with existing methods is presented to validate the efficiency of the approach.

Query Games in Databases

2021 ◽  
Vol 50 (1) ◽  
pp. 78-85
Author(s):  
Ester Livshits ◽  
Leopoldo Bertossi ◽  
Benny Kimelfeld ◽  
Moshe Sebag
Keyword(s):  
Game Theory ◽  
Cooperative Game ◽  
Shapley Value ◽  
Cooperative Game Theory ◽  
The Shapley Value ◽  
Computational Aspects ◽  
Aggregate Query ◽  
Boolean Query

Database tuples can be seen as players in the game of jointly realizing the answer to a query. Some tuples may contribute more than others to the outcome, which can be a binary value in the case of a Boolean query, a number for a numerical aggregate query, and so on. To quantify the contributions of tuples, we use the Shapley value that was introduced in cooperative game theory and has found applications in a plethora of domains. Specifically, the Shapley value of an individual tuple quantifies its contribution to the query. We investigate the applicability of the Shapley value in this setting, as well as the computational aspects of its calculation in terms of complexity, algorithms, and approximation.

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