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.

A NOTE ON CHARACTERIZING CORE STABILITY WITH FUZZY GAMES

2011 ◽  
Vol 13 (01) ◽  
pp. 105-118 ◽  
Author(s):  
EVAN SHELLSHEAR
Keyword(s):  
Cooperative Game ◽  
Core Stability ◽  
Original Game ◽  
Tu Games ◽  
The Core ◽  
Cooperative Tu Games ◽  
Fuzzy Games ◽  
The Stability

This paper investigates core stability of cooperative (TU) games via a fuzzy extension of the totally balanced cover of a cooperative game. The stability of the core of the fuzzy extension of a game, the concave extension, is shown to reflect the core stability of the original game and vice versa. Stability of the core is then shown to be equivalent to the existence of an equilibrium of a certain correspondence.

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 THE POSITIVE PREKERNEL OF A COOPERATIVE GAME

2000 ◽  
Vol 02 (04) ◽  
pp. 287-305 ◽  
Author(s):  
PETER SUDHÖLTER ◽  
BEZALEL PELEG
Keyword(s):  
Cooperative Game ◽  
Weak Version ◽  
Transferable Utility ◽  
Bargaining Set ◽  
The Core ◽  
Reduced Game ◽  
Positive Kernel ◽  
Game Property ◽  
Transferable Utility Games

The positive prekernel, a solution of cooperative transferable utility games, is introduced. We show that this solution inherits many properties of the prekernel and of the core, which are both sub-solutions. It coincides with its individually rational variant, the positive kernel, when applied to any zero-monotonic game. The positive (pre)kernel is a sub-solution of the reactive (pre)bargaining set. We prove that the positive prekernel on the set of games with players belonging to a universe of at least three possible members can be axiomatized by non-emptiness, anonymity, reasonableness, the weak reduced game property, the converse reduced game property, and a weak version of unanimity for two-person games.

THE CORE AND CONSISTENCY PROPERTIES: A GENERAL CHARACTERISATION

2001 ◽  
Vol 03 (02n03) ◽  
pp. 175-187 ◽  
Author(s):  
YUKIHIKO FUNAKI ◽  
TAKEHIKO YAMATO
Keyword(s):  
General Theorem ◽  
General Definition ◽  
Tu Games ◽  
The Core ◽  
Special Cases ◽  
Reduced Games ◽  
Definition Of ◽  
Consistency Properties

In this paper, we unify various axiomatisations of the core of TU games by means of consistency with respect to different definitions of reduced games. First, we introduce a general definition of reduced games including the reduced games due to Davis and Maschler (1965), Moulin (1985), and Funaki (1995) as special cases. Then, we provide a general theorem from which the characterisations due to Peleg (1986), Tadenuma (1992), and Funaki (1995) can be obtained. Our general theorem clarifies how the three characterisations of the core differ and are related.

The Need for Permission, the Power to Enforce, and Duality in Cooperative Games with a Hierarchy

2016 ◽  
Vol 18 (04) ◽  
pp. 1650015 ◽  
Author(s):  
Frank Huettner ◽  
Harald Wiese
Keyword(s):  
Cooperative Game ◽  
Cooperative Games ◽  
Transferable Utility ◽  
Tu Game ◽  
Tu Games ◽  
Permission Value

A cooperative game with transferable utility (TU game) captures a situation in which players can achieve certain payoffs by cooperating. We assume that the players are part of a hierarchy. In the literature, this invokes the assumption that subordinates cannot cooperate without the permission of their superiors. Instead, we assume that superiors can force their subordinates to cooperate. We show how both notions correspond to each other by means of dual TU games. This way, we capture the idea that a superiors’ ability to enforce cooperation can be seen as the ability to neutralize her subordinate’s threat to abstain from cooperation. Moreover, we introduce the coercion value for games with a hierarchy and provide characterizations thereof that reveal the similarity to the permission value.

The core of a strategic game

2018 ◽  
Vol 19 (1) ◽  
Author(s):  
Parkash Chander
Keyword(s):  
Partition Function ◽  
Cooperative Game ◽  
Unique Equilibrium ◽  
Strategic Game ◽  
Function Form ◽  
Equilibrium Outcome ◽  
Cooperative Solution ◽  
Payoff Vector ◽  
The Core ◽  
Partition Function Form

AbstractIn this paper, I introduce and study the $\gamma$-core of a general strategic game. I first show that the $\gamma$-core of an arbitrary strategic game is smaller than the conventional $\alpha$- and $\beta$- cores. I then consider the partition function form of a general strategic game and show that a prominent class of partition function games admit nonempty $\gamma$-cores. Finally, I show that each $\gamma$-core payoff vector (a cooperative solution) can be supported as an equilibrium outcome of an intuitive non-cooperative game and the grand coalition is the unique equilibrium outcome if and only if the $\gamma$-core is non-empty.

UNIONS AND UNEMPLOYMENT BENEFITS: SOME INSIGHTS FROM A SIMPLE THREE-PLAYER EXAMPLE

2011 ◽  
Vol 13 (04) ◽  
pp. 383-402
Author(s):  
HARALD WIESE
Keyword(s):  
Game Theory ◽  
Cooperative Game ◽  
Shapley Value ◽  
Cooperative Game Theory ◽  
Theory Building ◽  
Unemployment Benefits ◽  
Tu Games ◽  
Coalition Structures ◽  
A Value ◽  
Basic Input

The aim of this paper is to analyze the interconnections between employment and unionization. We will also see how unemployment benefits drive the interplay of employment and unionization. The basic input into our model stems from cooperative game theory. Building on the Shapley value, several values for TU games with coalition structures have been presented in the literature, most notably by Aumann and Drèze and Owen. We present a value that is capable of dealing with unemployment and unionization. We show that unemployment benefits increase wages but contribute to unemployment, that unemployment can be voluntary, and that unions tend to be beneficial for employed workers if there is overstaffing.

scholarly journals Reducing two person, zero sum games with underlying symmetry

1982 ◽  
Vol 33 (2) ◽  
pp. 152-161 ◽  
Author(s):  
K. R. Pearson
Keyword(s):  
Optimal Strategies ◽  
Original Game ◽  
Zero Sum Games ◽  
Reduced Game ◽  
Zero Sum

AbstractWe consider two person, zero sum games with several symmetries. Where such symmetries are present there is a group acting on the strategies of the game. We show how to use this action to produce a reduced game with a smaller matrix, but having the same value as the original game, and how to obtain optimal strategies for the original game from optimal strategies of the reduced game. An analysis of a simplified version of the popular game Mastermind is given to illustrate the theory developed.

scholarly journals Two-Level Cooperative Game on Hypergraph

2021 ◽  
Vol 14 ◽  
pp. 227-235
Author(s):  
David A. Kosian ◽  
◽  
Leon A. Petrosyan ◽  
Keyword(s):  
Cooperative Game ◽  
Allocation Rule ◽  
Characteristic Functions ◽  
Convexity Property ◽  
Cooperative Behaviour ◽  
Original Game ◽  
Communication Structure

In the paper, the cooperative game with a hypergraph communication structure is considered. For this class of games, a new allocation rule was proposed by splitting the original game into a game between hyperlinks and games within them. The communication possibilities are described by the hypergraph in which the nodes are players and hyperlinks are the communicating subgroups of players. The game between hyperlinks and between players in each hyperlink is described. The payoff of each player is influenced by the actions of other players dependent on the distance between them on hypergraph. Constructed characteristic functions based on cooperative behaviour satisfy the convexity property. The results are shown by the example.

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