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.

CONSISTENCY FOR PROPORTIONAL SOLUTIONS

2002 ◽  
Vol 04 (03) ◽  
pp. 343-356 ◽  
Author(s):  
ELENA YANOVSKAYA
Keyword(s):  
Cooperative Game ◽  
Solution Set ◽  
Original Game ◽  
Payoff Vector ◽  
Tu Games ◽  
Reduced Game ◽  
Reduced Games ◽  
Consistency Property ◽  
Finite Set

One of the properties characterizing cooperative game solutions is consistency connecting solution vectors of a cooperative game with finite set of players and its reduced game defined by removing one or more players and by assigning them the payoffs according to some specific principle (e.g., a proposed payoff vector). Consistency of a solution means that any part (defined by a coalition of the original game) of a solution payoff vector belongs to the solution set of the corresponding reduced game. In the paper the proportional solutions for TU-games are defined as those depending only on the proportional excess vectors in the same manner as translation covariant solutions depend on the usual Davis–Maschler excess vectors. The general form of the reduced games defining consistent proportional solutions is given. The efficient, anonymous, proportional TU cooperative game solutions meeting the consistency property with respect to any reduced game are described.

THE EXTENDED CORE OF A COOPERATIVE NTU GAME

2010 ◽  
Vol 12 (03) ◽  
pp. 263-274 ◽  
Author(s):  
HANS KEIDING ◽  
YAROSLAVNA PANKRATOVA
Keyword(s):  
Solution Concept ◽  
Tu Games ◽  
Ntu Games ◽  
The Core ◽  
Ntu Game ◽  
Straightforward Generalization

In this paper we propose an extension of the core of NTU games from its domain to a larger set of games satisfying a few conditions of well-behavedness. The solution concept is a rather straightforward generalization of the extended core of TU games introduced by Gomez [2003] and is shown to have similar properties. Also, a set of axioms for solutions of NTU games is presented which characterizes the extended core.

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.

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.

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.

scholarly journals The Fairness of Solidarity Bills under the Solidarity Value of Nowak and Radzik

Game Theory ◽  
2015 ◽  
Vol 2015 ◽  
pp. 1-12
Author(s):  
Lawrence Diffo Lambo ◽  
Pierre Wambo
Keyword(s):  
Shapley Value ◽  
Real World ◽  
Side Effect ◽  
Tu Games ◽  
Unwanted Side Effect ◽  
The Disabled ◽  
Threshold Position ◽  
To Receive

The solidarity value is a variant of the well-known Shapley value in which some sense of solidarity between the players is implemented allowing the disabled to receive help from the fortunate ones. We investigate on how fairly solidarity expenses are shared. We discuss the unwanted side effect of someone paying undue solidarity contributions as far as reversing his condition from a privileged to a needy person. A deeper case study is conducted for two classes of TU games that we obtain by modeling two real world business contexts. Here, we trace all player to player transfers of funds that arise when solidarity actions are processed, and we answer the question of who settles the solidarity bills. Also, we obtain the threshold position of a player below which he gets solidarity help, but above which he instead pays out donation.

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)

On a family of values for TU-games generalizing the Shapley value

2013 ◽  
Vol 65 (2) ◽  
pp. 105-111 ◽  
Author(s):  
Tadeusz Radzik ◽  
Theo Driessen
Keyword(s):  
Shapley Value ◽  
Tu Games ◽  
The Shapley Value

Peer Group Situations and Games With Grey Uncertainty

2018 ◽  
pp. 265-278
Author(s):  
Sırma Zeynep Alparslan-Gök ◽  
Osman Palancı ◽  
Züleyha Yücesan
Keyword(s):  
Shapley Value ◽  
Operations Research ◽  
Peer Group ◽  
Solution Concept ◽  
Important Group ◽  
The Social ◽  
The Given

In many economic and Operations Research (OR) situations the social configuration of the organization, influences the potential possibilities of all the groups of agents. The important group for an agent consists of the leader, the agent himself and all the intermediate agents that exist in the given hierarchy (peer group). In this study, peer group situations where grey uncertainty is included. Peer group grey situations are introduced and related peer group grey games are constructed. The grey Shapley value is specified as a solution concept with a proposition. As an application auction situations are considered. Finally, a conclusion is given.

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