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Advances in Operations Research
Volume 2011 (2011), Article ID 283978, 13 pages
http://dx.doi.org/10.1155/2011/283978
Research Article

Dynamic Approaches for Multichoice Solutions

Department of Applied Mathematics, National Pingtung University of Education, Pingtung 900, Taiwan

Received 3 September 2010; Revised 3 April 2011; Accepted 23 May 2011

Academic Editor: H. A. Eiselt

Copyright © 2011 Yu-Hsien Liao. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

Based on alternative reduced games, several dynamic approaches are proposed to show how the three extended Shapley values can be reached dynamically from arbitrary efficient payoff vectors on multichoice games.

1. Introduction

A multichoice transferable-utility (TU) game, introduced by Hsiao and Raghavan [1], is a generalization of a standard coalition TU game. In a standard coalition TU game, each player is either fully involved or not involved at all in participation with some other agents, while in a multichoice TU game, each player is allowed to participate with many finite different activity levels. Solutions on multichoice TU games could be applied in many fields such as economics, political sciences, management, and so forth. Van den Nouweland et al. [2] referred to several applications of multichoice TU games, such as a large building project with a deadline and a penalty for every day if this deadline is overtime. The date of completion depends on the effort of how all of the people focused on the project: the harder they exert themselves, the sooner the project will be completed. This situation gives rise to a multichoice TU game. The worth of a coalition resulted from the players working in certain levels to a project is defined as the penalty for their delay of the project completion with the same efforts. Another application appears in a large company with many divisions, where the profit-making depends on their performance. This situation also gives rise to a multichoice TU game. The players are the divisions, and the worth of a coalition resulted from the divisions functioning in certain levels is the corresponding profit produced by the company.

Here we apply three solutions for multichoice TU games due to Hsiao and Raghavan [1], Derks and Peters [3], and Peters and Zank [4], respectively. Two main results are as follows. (1)A solution concept can be given axiomatic justification. Oppositely, dynamic processes can be described that lead the players to that solution, starting from an arbitrary efficient payoff vector (the foundation of a dynamic theory was laid by Stearns [5]. Related dynamic results may be found in, for example, Billera [6], Maschler and Owen [7], etc.). In Section 3, we firstly define several alternative reductions on multichoice TU games. Further, we adopt these reductions and some axioms introduced by Hsiao and Raghavan [1], Hwang and Liao [810], and Klijn et al. [11] to show how the three extended Shapley values can be reached dynamically from arbitrary efficient payoff vectors. In the proofs of Theorems 3.2 and 3.4, we will point out how these axioms would be used in the dynamic approaches. (2)There are two important factors, the players and their activity levels, for multichoice games. Inspired by Hart and Mas-Colell [12], Hwang and Liao [810] proposed two types of reductions by only reducing the number of the players. In Section 4, we propose two types of player-action reduced games by reducing both the number of the players and the activity levels. Based on the potential, Hart and Mas-Colell [12] showed that the Shapley value [13] satisfies consistency. Different from Hart and Mas-Colell [12], we show that the three extended Shapley values satisfy related properties of player-action consistency by applying alternative method.

2. Preliminaries

Let 𝑈 be the universe of players and 𝑁𝑈 be a set of players. Suppose each player 𝑖 has 𝑚𝑖 levels at which he can actively participate. Let 𝑚=(𝑚𝑖)𝑖𝑁 be the vector that describes the number of activity levels for each player, at which he can actively participate. For 𝑖𝑈, we set 𝑀𝑖={0,1,,𝑚𝑖} as the action space of player 𝑖, where the action 0 means not participating, and 𝑀𝑖+=𝑀𝑖{0}. For 𝑁U, 𝑁, let 𝑀𝑁=𝑖𝑁𝑀𝑖 be the product set of the action spaces for players 𝑁 and 𝑀𝑁+=𝑖𝑁𝑀𝑖+. Denote the zero vector in 𝑁 by 0𝑁.

A multichoice TU game is a triple (𝑁,𝑚,𝑣), where 𝑁 is a nonempty and finite set of players, 𝑚 is the vector that describes the number of activity levels for each player, and 𝑣𝑀𝑁 is a characteristic function which assigns to each action vector 𝛼=(𝛼𝑖)𝑖𝑁𝑀𝑁 the worth that the players can jointly obtain when each player 𝑖 plays at activity level 𝛼𝑖𝑀𝑖 with 𝑣(0𝑁)=0. If no confusion can arise, a game (𝑁,𝑚,𝑣) will sometimes be denoted by its characteristic function 𝑣. Given a multichoice game (𝑁,𝑚,𝑣) and 𝛼𝑀𝑁, we write (𝑁,𝛼,𝑣) for the multichoice TU subgame obtained by restricting 𝑣 to {𝛽𝑀𝑁𝛽𝑖𝛼𝑖𝑖𝑁} only. Denote the class of all multichoice TU games by 𝑀𝐶.

Given (𝑁,𝑚,𝑣)𝑀𝐶, let 𝐿𝑁,𝑚={(𝑖,𝑘𝑖)𝑖𝑁,𝑘𝑖𝑀𝑖+}. A solution on 𝑀𝐶 is a map 𝜓 assigning to each (𝑁,𝑚,𝑣)𝑀𝐶 an element𝜓𝜓(𝑁,𝑚,𝑣)=𝑖,𝑘𝑖(𝑁,𝑚,𝑣)(𝑖,𝑘𝑖)𝐿𝑁,𝑚𝐿𝑁,𝑚.(2.1) Here 𝜓𝑖,𝑘𝑖(𝑁,𝑚,𝑣) is the power index or the value of the player 𝑖 when he takes action 𝑘𝑖 to play game 𝑣. For convenience, given (𝑁,𝑚,𝑣)𝑀𝐶 and a solution 𝜓 on 𝑀𝐶, we define 𝜓𝑖,0(𝑁,𝑚,𝑣)=0 for all 𝑖𝑁.

To state the three extended Shapley values, some more notations will be needed. Given 𝑆𝑁, let |S| be the number of elements in 𝑆 and let 𝑒𝑆(𝑁) be the binary vector in 𝑁 whose component 𝑒𝑆𝑖(𝑁) satisfies𝑒𝑆𝑖(𝑁)=1if𝑖S,0otherwise.(2.2) Note that if no confusion can arise 𝑒𝑆𝑖(𝑁) will be denoted by 𝑒𝑆𝑖.

Given (𝑁,𝑚,𝑣)𝑀𝐶 and 𝛼𝑀𝑁, we define 𝑆(𝛼)={𝑘𝑁𝛼𝑘0} and 𝛼=𝑖𝑁𝛼𝑖. Let 𝛼,𝛽𝑁, we say 𝛽𝛼 if 𝛽𝑖𝛼𝑖 for all 𝑖𝑁.

The analogue of unanimity games for multichoice games are minimal effort games (𝑁,𝑚,𝑢𝛼𝑁), where 𝛼𝑀𝑁, 𝛼0𝑁, defined by for all 𝛽𝑀𝑁,𝑢𝛼𝑁(𝛽)=1if𝛽𝛼;0otherwise.(2.3) It is known that for (𝑁,𝑚,𝑣)𝑀𝐶 it holds that 𝑣=𝛼𝑀𝑁{0𝑁}𝑎𝛼(𝑣)𝑢𝛼𝑁, where 𝑎𝛼(𝑣)=𝑆𝑆(𝛼)(1)|𝑆|𝑣(𝛼𝑒𝑆).

Here we apply three extensions of the Shapley value for multichoice games due to Hsiao and Raghavan [1], Derks and Peters [3], and Peters and Zank [4].

Definition 2.1. (i) (Hsiao and Raghavan, [1]).
The H&R Shapley value Λ is the solution on 𝑀𝐶 which associates with each (𝑁,𝑚,𝑣)𝑀𝐶 and each player 𝑖𝑁 and each 𝑘𝑖𝑀+𝑖 the value (Hsiao and Raghavan [1] provided an alternative formula of the H&R Shapley value. Hwang and Liao [9] defined the H&R Shapley value in terms of the dividends) Λ𝑖,𝑘𝑖(𝑁,𝑚,𝑣)=𝛼𝑀𝑁𝛼𝑖𝑘𝑖𝑎𝛼(𝑣)||||𝑆(𝛼).(2.4) Note that the so-called dividend 𝑎𝛼(𝑣) is divided equally among the necessary players.
(ii) (Derks and Peters, [3]).
The D&P Shapley value Θ is the solution on 𝑀𝐶 which associates with each (𝑁,𝑚,𝑣)𝑀𝐶 and each player 𝑖𝑁 and each 𝑘𝑖𝑀+𝑖 the value Θ𝑖,𝑘𝑖(𝑁,𝑚,𝑣)=𝛼𝑀𝑁𝛼𝑖𝑘𝑖𝑎𝛼(𝑣)𝛼.(2.5) Note that the so-called dividend 𝑎𝛼(𝑣) is divided equally among the necessary levels.
(iii) (Peters and Zank, [4]).
The P&Z Shapley value Γ is the solution on 𝑀𝐶 which associates with each (𝑁,𝑚,𝑣)𝑀𝐶 and each player 𝑖𝑁 and each 𝑘𝑖𝑀+𝑖 the value (Peters and Zank [4] defined the P&Z Shapley value by fixing its values on minimal effort games and imposing linearity. Hwang and Liao [8] defined the P&Z Shapley value based on the dividends) Γ𝑖,𝑘𝑖(𝑁,𝑚,𝑣)=𝛼𝑀𝑁𝛼𝑖=𝑘𝑖𝑎𝛼(𝑣)||||𝑆(𝛼).(2.6) Clearly, the P&Z Shapley value is a subdivision of the H&R Shapley value. For all (𝑁,𝑚,𝑣)𝑀𝐶 and for all (𝑖,k𝑖)𝐿𝑁,𝑚, Λ𝑖,𝑘𝑖(𝑁,𝑚,𝑣)=𝛼𝑀𝑁𝛼𝑖𝑘𝑖𝑎𝛼(𝑣)||||=𝑆(𝛼)𝑘𝑖𝑡𝑖=1𝛼𝑀𝑁𝛼𝑖=𝑡𝑖𝑎𝛼(𝑣)||||=𝑆(𝛼)𝑘𝑖𝑡𝑖=1Γ𝑖,𝑡𝑖(𝑁,𝑚,𝑣).(2.7)

3. Axioms and Dynamic Approaches

In this section, we propose dynamic processes to illustrate that the three extended Shapley values can be reached by players who start from an arbitrary efficient solution.

In order to provide several dynamic approaches, some more definitions will be needed. Let 𝜓 be a solution on 𝑀𝐶. 𝜓 satisfies the following. (i)1-efficiency (1EFF) if for each (𝑁,𝑚,𝑣)𝑀𝐶, 𝑖𝑆(𝑚)𝜓𝑖,𝑚𝑖(𝑁,𝑚,𝑣)=𝑣(𝑚).(ii)2-efficiency (2EFF) if for each (𝑁,𝑚,𝑣)𝑀𝐶, 𝑖𝑆(𝑚)𝑚𝑖𝑘𝑖=1𝜓𝑖,𝑘𝑖(𝑁,𝑚,𝑣)=𝑣(𝑚).

The following axioms are analogues of the balanced contributions property due to Myerson [14]. The solution 𝜓 satisfies the following. (i)1-strong balanced contributions (1SBC) if for each (𝑁,𝑚,𝑣)𝑀𝐶 and (𝑖,𝑘𝑖),(𝑗,𝑘𝑗)𝐿𝑁,𝑚,𝑖𝑗, 𝜓𝑖,𝑘𝑖𝑚𝑁,𝑁{𝑗},𝑘𝑗,𝑣𝜓𝑖,𝑘𝑖𝑚𝑁,𝑁{𝑗},0,𝑣=𝜓𝑗,𝑘𝑗𝑚𝑁,𝑁{𝑖},𝑘𝑖,𝑣𝜓𝑗,𝑘𝑗𝑚𝑁,𝑁{𝑖}.,0,𝑣(3.1)(ii)2-strong balanced contributions (2SBC) if for each (𝑁,𝑚,𝑣)𝑀𝐶 and (𝑖,𝑘𝑖),(𝑗,𝑘𝑗)𝐿𝑁,𝑚,𝑖𝑗, 𝜓𝑖,𝑘𝑖(𝑁,𝑚,𝑣)𝜓𝑖,𝑘𝑖𝑚𝑁,𝑁{𝑗},𝑘𝑗1,𝑣=𝜓𝑗,𝑘𝑗(𝑁,𝑚,𝑣)𝜓𝑗,𝑘𝑗𝑚𝑁,𝑁{𝑖},𝑘𝑖.1,𝑣(3.2)(iii)3-strong balanced contributions (3SBC) if for each (𝑁,𝑚,𝑣)𝑀𝐶 and (𝑖,𝑘𝑖),(𝑗,𝑘𝑗)𝐿𝑁,𝑚,𝑖𝑗, 𝜓𝑖,𝑘𝑖𝑚𝑁,𝑁{𝑗},𝑘𝑗,𝑣𝜓𝑖,𝑘𝑖𝑚𝑁,𝑁{𝑗},𝑘𝑗1,𝑣=𝜓𝑗,𝑘𝑗𝑚𝑁,𝑁{𝑖},𝑘𝑖,𝑣𝜓𝑗,𝑘𝑗𝑚𝑁,𝑁{𝑖},𝑘𝑖.1,𝑣(3.3)

The following axiom was introduced by Hwang and Liao ([9]). The solution 𝜓 satisfies the following. (i)Independence of individual expansions (IIE) if for each (𝑁,𝑚,𝑣)𝑀𝐶 and each (𝑖,𝑘𝑖)𝐿𝑁,𝑚, 𝑗𝑚𝑖, 𝜓𝑖,𝑘𝑖𝑚𝑁,𝑁{𝑖},𝑘𝑖,𝑣=𝜓𝑖,𝑘𝑖𝑚𝑁,𝑁{𝑖},𝑘𝑖+1,𝑣==𝜓𝑖,𝑘𝑖(𝑁,𝑚,𝑣).(3.4)

In the framework of multichoice games, IIE asserts that whenever a player gets available higher activity level, the payoff for all original levels should not be changed under condition that other players are fixed.

The following axiom was introduced by Klijn et al. [11]. The solution 𝜓 satisfies the following. (i)Equal loss (EL) if for each (𝑁,𝑚,𝑣)𝑀𝐶 and each (𝑖,𝑘𝑖)𝐿𝑁,𝑚, 𝑘𝑖𝑚𝑖, 𝜓𝑖,𝑘𝑖(𝑁,𝑚,𝑣)𝜓𝑖,𝑘𝑖𝑁,𝑚𝑒{𝑖},𝑣=𝜓𝑖,𝑚𝑖(𝑁,𝑚,𝑣).(3.5)

Klijn et al. [11] provided an interpretation of the equal loss property as follows. EL is also inspired by the balanced contributions property of Myerson [14]. In the framework of multichoice games, EL says that whenever a player gets available higher activity level the payoff for all original levels changes with an amount equal to the payoff for the highest level in the new situation. Note that EL is a vacuous property for standard coalition TU games.

Some considerable weakenings of the previous axioms are as follows. Weak 1-efficiency (1WEFF) simply says that for all (𝑁,𝑚,𝑣)𝑀𝐶 with |𝑆(𝑚)|=1, 𝜓 satisfies 1EFF. Weak 2-efficiency (2WEFF) simply says that for all (𝑁,𝑚,𝑣)𝑀𝐶 with |𝑆(𝑚)|=1, 𝜓 satisfies 2EFF. 1-upper balanced contributions (1UBC) only requires that 1SBC holds if 𝑘𝑖=𝑚𝑖 and 𝑘𝑗=𝑚𝑗. 2-upper balanced contributions (2UBC) only requires that 2SBC or 3SBC holds if 𝑘𝑖=𝑚𝑖 and 𝑘𝑗=𝑚𝑗. Weak independence of individual expansions (WIIE) simply says that for all (𝑁,𝑚,𝑣)𝑀𝐶 with |𝑆(𝑚)|=1, 𝜓 satisfies IIE. Weak equal loss (WEL) simply says that for all (𝑁,𝑚,𝑣)𝑀𝐶 with |𝑆(𝑚)|=1, 𝜓 satisfies EL.

Subsequently, we recall the reduced games and related consistency properties introduced by Hwang and Liao [810]. Let (𝑁,𝑚,𝑣)𝑀𝐶, 𝑆𝑁{} and 𝜓 be a solution. (i)The 1-reduced game (𝑆,𝑚𝑆,𝑣𝜓1,𝑆) with respect to 𝜓 and 𝑆 is defined as, for all 𝛼𝑀𝑆, 𝑣𝜓1,𝑆(𝛼)=𝑣𝛼,𝑚𝑁𝑆𝑖𝑁𝑆𝜓𝑖,𝑚𝑖𝑁,𝛼,𝑚𝑁𝑆.,𝑣(3.6)(ii)The 2-reduced game (𝑆,𝑚𝑆,𝑣𝜓2,𝑆) with respect to 𝜓 and 𝑆 is defined as, for all 𝛼𝑀𝑆, 𝑣𝜓2,𝑆(𝛼)=𝑣𝛼,𝑚𝑁𝑆𝑚𝑖𝑁𝑆𝑖𝑘𝑖=1𝜓𝑖,𝑘𝑖𝑁,𝛼,𝑚𝑁𝑆,𝑣.(3.7)(iii)𝜓 on 𝑀𝐶 satisfies 1-consistency (1CON) if for all (𝑁,𝑚,𝑣)𝑀𝐶, for all 𝑆𝑁 and for all (𝑖,𝑘𝑖)𝐿𝑆,𝑚𝑆, 𝜓𝑖,𝑘𝑖(𝑁,𝑚,𝑣)=𝜓𝑖,𝑘𝑖(𝑆,𝑚𝑆,𝑣𝜓1,𝑆). (iv)𝜓 on 𝑀𝐶 satisfies 2-consistency (2CON) if for all (𝑁,𝑚,𝑣)𝑀𝐶, for all 𝑆𝑁 and for all (𝑖,𝑘𝑖)𝐿𝑆,𝑚𝑆, 𝜓𝑖,𝑘𝑖(𝑁,𝑚,𝑣)=𝜓𝑖,𝑘𝑖(𝑆,𝑚𝑆,𝑣𝜓2,𝑆).

Remark 3.1. Hwang and Liao [810] characterized the solutions Λ, Γ, and Θ by means of 1CON and 2CON as follows.(i)The solution Λ is the only solution satisfying 1WEFF (1EFF), WIIE (IIE), 1UBC (1SBC), and 1CON.(ii)The solution Θ is the only solution satisfying 2WEFF (2EFF), WEL (EL), 2UBC (2SBC), and 2CON.(iii)The solution Γ is the only solution satisfying 2WEFF (2EFF), WIIE (IIE), 2UBC (3SBC), and 2CON.

Next, we will find dynamic processes that lead the players to solutions, starting from arbitrary efficient payoff vectors.

Let (𝑁,𝑚,𝑣)𝑀𝐶. A payoff vector of (𝑁,𝑚,𝑣) is a vector (𝑥𝑖,𝑘𝑖)(𝑖,𝑘𝑖)𝐿𝑁,𝑚𝐿𝑁,𝑚 where 𝑥𝑖,𝑘𝑖 denotes the payoff to player 𝑖 corresponding to his activity level 𝑘𝑖 for all (𝑖,𝑘𝑖)𝐿𝑁,𝑚. A payoff vector 𝑥 of (𝑁,𝑚,𝑣) is 1-efficient (1EFF) if 𝑖𝑁𝑥𝑖,𝑚𝑖=𝑣(𝑚). 𝑥 is 2-efficient (2EFF) if 𝑖𝑁𝑘𝑖𝑀+𝑖𝑥𝑖,𝑘𝑖=𝑣(𝑚). Moreover, the sets of 1-preimputations and 2-preimputations of (𝑁,𝑚,𝑣) are denoted by 𝑋1(𝑁,𝑚,𝑣)=𝑥𝐿𝑁,𝑚,𝑋𝑥is1EFFin(𝑁,𝑚,𝑣)2(𝑁,𝑚,𝑣)=𝑥𝐿𝑁,𝑚.𝑥is2EFFin(𝑁,𝑚,𝑣)(3.8)

In order to exhibit such processes, let us define two alternative reduced games as follows. Let (𝑁,𝑚,𝑣)𝑀𝐶, 𝑆𝑁, and let 𝜓 be a solution and 𝑥 a payoff vector. (i)The (1,𝜓)-reduced game (𝑆,𝑚𝑆,𝑣𝑥,𝜓1,𝑆) with respect to 𝑆, 𝑥, and 𝜓 is defined as, for all 𝛼𝑀𝑆, 𝑣𝑥,𝜓1,𝑆(𝛼)=𝑣(𝑚)𝑖𝑁𝑆𝑥𝑖,𝑚𝑖,𝛼=𝑚𝑆,𝑣𝜓1,𝑆(𝛼),otherwise.(3.9)(ii)The (2,𝜓)-reduced game (𝑆,𝑚𝑆,𝑣𝑥,𝜓2,𝑆) with respect to 𝑆, 𝑥, and 𝜓 is defined as, for all 𝛼𝑀𝑆, 𝑣𝑥,𝜓2,𝑆(𝛼)=𝑣(𝑚)𝑖𝑁𝑆𝑘𝑖𝑀+𝑥𝑖,𝑘𝑖,𝛼=𝑚𝑆,𝑣𝜓2,𝑆(𝛼),otherwise.(3.10)

Let (𝑁,𝑚,𝑣)𝑀𝐶, 𝑁3 and (𝑖,𝑘𝑖)𝐿𝑁,𝑚. Inspired by Maschler and Owen [7], we define 𝑓𝑖,𝑘𝑖𝑋1(𝑁,𝑚,𝑣), 𝑔𝑖,𝑘𝑖𝑋2(𝑁,𝑚,𝑣), 𝑖,𝑘𝑖𝑋2(𝑁,𝑚,𝑣) to be as follows:(i)𝑓𝑖,𝑘𝑖(𝑥)=𝑥𝑖,𝑘𝑖+𝑡𝑗𝑁{𝑖}(Λ𝑖,𝑘𝑖({𝑖,𝑗},𝑚{𝑖,𝑗},𝑣𝑥,Λ1,{𝑖,𝑗})𝑥𝑖,𝑘𝑖), (ii)𝑔𝑖,𝑘𝑖(𝑥)=𝑥𝑖,𝑘𝑖+𝑡𝑗𝑁{𝑖}(Θ𝑖,𝑘𝑖({𝑖,𝑗},𝑚{𝑖,𝑗},𝑣𝑥,Θ2,{𝑖,𝑗})𝑥𝑖,𝑘𝑖), (iii)𝑖,𝑘𝑖(𝑥)=𝑥𝑖,𝑘𝑖+𝑡𝑗𝑁{𝑖}(Γ𝑖,𝑘𝑖({𝑖,𝑗},𝑚{𝑖,𝑗},𝑣𝑥,Γ2,{𝑖,𝑗})𝑥𝑖,𝑘𝑖),

where 𝑡 is a fixed positive number, which reflects the assumption that player 𝑖 does not ask for adequate correction (when 𝑡=1) but only (usually) a fraction of it. It is easy to check that (𝑓𝑖,𝑘𝑖(𝑥))(𝑖,𝑘𝑖)𝐿𝑁,𝑚𝑋1(𝑁,𝑚,𝑣) if 𝑥𝑋1(𝑁,𝑚,𝑣), (𝑔𝑖,𝑘𝑖(𝑥))(𝑖,𝑘𝑖)𝐿𝑁,𝑚𝑋2(𝑁,𝑚,𝑣), and (𝑖,𝑘𝑖(𝑥))(𝑖,𝑘𝑖)𝐿𝑁,𝑚𝑋2(𝑁,𝑚,𝑣) if 𝑥𝑋2(𝑁,𝑚,𝑣).

Inspired by Maschler and Owen [7], we define correction functions 𝑓𝑖,𝑘𝑖,𝑔𝑖,𝑘𝑖,𝑖,𝑘𝑖 on multichoice games. In the following, we provided some discussions which are analogues to the discussion of Maschler and Owen [7]. Let (𝑁,𝑚,𝑣)𝑀𝐶 and 𝑥 be a 1-efficient payoff vector. By a process of induction we assume that the players have already agreed on the solution Λ for all 𝑝-person games, 1<𝑝<|𝑁|. In particular, we assume that they agreed on Λ for 1-person games (involving only Pareto optimality) and for 2-person games (which are side-payment games after an appropriate change in the utility scale of one player). Now somebody suggests that 𝑥 should be the solution for an 𝑛-person game (N,𝑚,𝑣), thus suggesting a solution concept Ψ, which should satisfyΨ𝑃,𝑚=Λ,𝑢𝑃,𝑚,,𝑢𝑃,𝑚||𝑃||<||𝑁||,,𝑢𝑀𝐶,𝑥,(𝑁,𝑚,𝑣)=𝑃,𝑚.,𝑢(3.11) On the basis of this Ψ, the members of a coalition 𝑆={𝑖,𝑗} will examine 𝑣𝑥,Λ1,𝑆 for related 1-consistency. If the solution turns out to be inconsistent, they will modify 𝑥 “in the direction” which is dictated by Λ𝑖,𝑘𝑖(𝑆,𝑚𝑆,𝑣𝑥,Λ1,𝑆) in a manner which will be explained subsequently (see the definition of 𝑓𝑖,𝑘𝑖). These modifications, done simultaneously by all 2-person coalitions, will lead to a new payoff vector 𝑥 and the process will repeat. The hope is that it will converge and, moreover, converge to Λ(𝑁,𝑚,𝑣). Similar discussions could be used to 𝑔𝑖,𝑘𝑖 and 𝑖,𝑘𝑖.

Theorem 3.2. Let (𝑁,𝑚,𝑣)𝑀𝐶 and 𝑥𝑋1(𝑁,𝑚,𝑣). Define 𝑥0=𝑥,𝑥1=(𝑓𝑖,𝑘𝑖(𝑥0))(𝑖,𝑘𝑖)𝐿𝑁,𝑚,,𝑥𝑞=(𝑓𝑖,𝑘𝑖(𝑥𝑞1))(𝑖,𝑘𝑖)𝐿𝑁,𝑚 for all 𝑞. (1)If 0<𝑡<4/|𝑁|, then for all 𝑖𝑁 and for all 𝑥𝑋1(𝑁,𝑚,𝑣), {𝑥𝑞𝑖,𝑚𝑖}𝑞=1 converges to Λ𝑖,𝑚𝑖(𝑁,𝑚,𝑣).(2)If 0<𝑡<4/|𝑁|, then for all (𝑖,𝑘𝑖)𝐿𝑁,𝑚 and for all 𝑥𝑋1(𝑁,(𝑚𝑁{𝑖},𝑘𝑖),𝑣), {𝑥𝑞𝑖,𝑘𝑖}𝑞=1 converges to Λ𝑖,𝑘𝑖(𝑁,𝑚,𝑣).

Proof. Fix (𝑁,𝑚,𝑣)𝑀𝐶 and 𝑥𝑋1(𝑁,𝑚,𝑣). To prove (1), let 𝑖,𝑗𝑆(𝑚) and 𝑆={𝑖,𝑗}. By 1EFF and 1UBC of Λ, and definitions of 𝑣Λ1,𝑆 and 𝑣𝑥,Λ1,𝑆, Λ𝑖,𝑚𝑖𝑆,𝑚𝑆,𝑣𝑥,Λ1,𝑆+Λ𝑗,𝑚𝑗𝑆,𝑚𝑆,𝑣𝑥,Λ1,𝑆=𝑥𝑖,𝑚𝑖+𝑥𝑗,𝑚𝑗Λ,(by1EFFofΛ),(3.12)𝑖,𝑚𝑖𝑆,𝑚𝑆,𝑣𝑥,Λ1,𝑆Λ𝑗,𝑚𝑗𝑆,𝑚𝑆,𝑣𝑥,Λ1,𝑆=Λ𝑖,𝑚𝑖𝑚𝑆,𝑆{𝑗},0,𝑣𝑥,Λ1,𝑆Λ𝑗,𝑚𝑗𝑚𝑆,𝑆{𝑖},0,𝑣𝑥,Λ1,𝑆(by1UBCofΛ)=Λ𝑖,𝑚𝑖𝑚𝑆,𝑆{𝑗},0,𝑣Λ1,𝑆Λ𝑗,𝑚𝑗𝑚𝑆,𝑆{𝑖},0,𝑣Λ1,𝑆bydenitionof𝑣Λ1,𝑆=Λ𝑖,𝑚𝑖𝑆,𝑚𝑆,𝑣Λ1,𝑆Λ𝑗,𝑚𝑗𝑆,𝑚𝑆,𝑣Λ1,𝑆.(by1UBCofΛ).(3.13) Therefore, Λ2𝑖,𝑚𝑖𝑆,𝑚𝑆,𝑣𝑥,Λ1,𝑆𝑥𝑖,𝑚𝑖=Λ𝑖,𝑚𝑖𝑆,𝑚𝑆,𝑣Λ1,𝑆Λ𝑗,𝑚𝑗𝑆,𝑚𝑆,𝑣Λ1,𝑆𝑥𝑖,𝑚𝑖+𝑥𝑗,𝑚𝑗.(3.14)
By definition of 𝑓, 1CON and 1EFF of Λ and (3.14), 𝑓𝑖,𝑚𝑖(𝑥)=𝑥𝑖,𝑚𝑖+𝑡2𝑗𝑁{𝑖}Λ𝑖,𝑚𝑖{𝑖,𝑗},𝑚{𝑖,𝑗},𝑣Λ1,{𝑖,𝑗}𝑗𝑁{𝑖}𝑥𝑖,𝑚𝑖𝑗𝑁{𝑖}Λ𝑗,𝑚𝑗{𝑖,𝑗},𝑚{𝑖,𝑗},𝑣Λ1,{𝑖,𝑗}+𝑗𝑁{𝑖}𝑥𝑗,𝑚𝑗bydenitionof𝑓𝑖,𝑚𝑖andequation(3.14)=𝑥𝑖,𝑚𝑖+𝑡2𝑘𝑁{𝑖}Λ𝑖,𝑚𝑖||𝑁||𝑥(𝑁,𝑚,𝑣)1𝑖,𝑚𝑖𝑗𝑁{𝑖}Λ𝑗,𝑚𝑗(𝑁,𝑚,𝑣)+𝑣(𝑚)𝑥𝑖,𝑚𝑖(by1CONofΛand1EFFof𝑥)=𝑥𝑖,𝑚𝑖+𝑡2||𝑁||Λ1𝑖,𝑚𝑖||𝑁||𝑥(𝑁,𝑚,𝑣)1𝑖,𝑚𝑖𝑣(𝑚)Λ𝑖,𝑚𝑖+(𝑁,𝑚,𝑣)𝑣(𝑚)𝑥𝑖,𝑚𝑖(by1EFFofΛ)=𝑥𝑖,𝑚𝑖+||𝑁||𝑡2Λ𝑖,𝑚𝑖(𝑁,𝑚,𝑣)𝑥𝑖,𝑚𝑖.(3.15) Hence, for all 𝑞, ||𝑁||1𝑡2𝑞+1Λ𝑖,𝑚𝑖(𝑁,𝑚,𝑣)𝑥𝑞𝑖,𝑚𝑖=Λ𝑖,𝑚𝑖(𝑁,𝑚,𝑣)𝑓𝑖,𝑚𝑖(𝑥𝑞)=Λ𝑖,𝑚𝑖(𝑁,𝑚,𝑣)𝑥𝑞+1𝑖,𝑚𝑖.(3.16) If 0<𝑡<4/|𝑁|, then 1<(1(|𝑁|𝑡)/2)<1 and {𝑥𝑞𝑖,𝑚𝑖}𝑞=1 converges to Λ𝑖,𝑚𝑖(𝑁,𝑚,𝑣).
To prove (2), by 1 of this theorem, if 0<𝑡<4/|𝑁|, then for all (𝑖,𝑘𝑖)𝐿𝑁,𝑚 and for all 𝑥𝑋1(𝑁,(𝑚𝑁{𝑖},𝑘𝑖),𝑣), {𝑥𝑞𝑖,𝑘𝑖}𝑞=1 converges to Λ𝑖,𝑘𝑖(𝑁,(𝑚𝑁{𝑖},𝑘𝑖),𝑣). By IIE of Λ, {𝑥𝑞𝑖,𝑘𝑖}𝑞=1 converges to Λ𝑖,𝑘𝑖(𝑁,𝑚,𝑣).

Remark 3.3. Huang et al. [15] provided dynamic processes for the P&Z Shapley value as follows. Let (𝑁,𝑚,𝑣)𝑀𝐶 and 𝑦𝑋2(𝑁,𝑚,v). Define 𝑦0=𝑦,𝑦1=(𝑖,𝑘𝑖(𝑦0))(𝑖,𝑘𝑖)𝐿𝑁,𝑚,,𝑦𝑞=(𝑖,𝑘𝑖(𝑦𝑞1))(𝑖,𝑘𝑖)𝐿𝑁,𝑚 for all 𝑞.
(1)If 0<𝛼<4/|𝑁|, then for all 𝑖𝑁 and for all 𝑦𝑋2(𝑁,𝑚,𝑣), {𝑚𝑖𝑘𝑖=1𝑦𝑞𝑖,𝑘𝑖}𝑞=1 converges to 𝑚𝑖𝑘𝑖=1Γ𝑖,𝑘𝑖(𝑁,𝑚,𝑣).(2)If 0<𝑡<4/|𝑁|, then for all (𝑖,𝑘𝑖)𝐿𝑁,𝑚 and for all 𝑦𝑋2(𝑁,(𝑚𝑁{𝑖},𝑘𝑖),𝑣), {𝑘𝑖𝑡𝑖=1𝑦𝑞𝑖,𝑡𝑖}𝑞=1 converges to 𝑘𝑖𝑡𝑖=1Γ𝑖,𝑡𝑖(𝑁,𝑚,𝑣).(3)If 0<𝛼<4/|𝑁|, then for all (𝑖,𝑘𝑖)𝐿𝑁,𝑚 and for all payoff vectors 𝑦 with 𝑦𝑖,𝑘𝑖=𝑣(𝑚𝑁{𝑖},𝑘𝑖)𝑣(𝑚𝑁{𝑖},𝑘𝑖1), {𝑦𝑞𝑖,𝑘𝑖}𝑞=1 converges to Γ𝑖,𝑘𝑖(𝑁,𝑚,𝑣).
In fact, the proofs of (1), (2), and (3) are similar to Theorem 3.2.

Theorem 3.4. Let (𝑁,𝑚,𝑣)𝑀𝐶 and 𝑧𝑋2(𝑁,𝑚,𝑣). Define 𝑧0=𝑧,𝑧1=(𝑔𝑖,𝑘𝑖(𝑧0))(𝑖,𝑘𝑖)𝐿𝑁,𝑚,,𝑧𝑞=(𝑔𝑖,𝑘𝑖(𝑧𝑞1))(𝑖,𝑘𝑖)𝐿𝑁,𝑚 for all 𝑞.
(1)If 0<𝛼<4/|𝑁|, then for all 𝑖𝑁 and for all 𝑧𝑋2(𝑁,𝑚,𝑣), {𝑚𝑖𝑘𝑖=1𝑧𝑞𝑖,𝑘𝑖}𝑞=1 converges to 𝑚𝑖𝑘𝑖=1Θ𝑖,𝑘𝑖(𝑁,𝑚,𝑣).(2)If 0<𝑡<4/|𝑁|, then for all (𝑖,𝑘𝑖)𝐿𝑁,𝑚 and for all 𝑧𝑋2(𝑁,(𝑚𝑁{𝑖},𝑘𝑖),𝑣), {𝑘𝑖𝑡𝑖=1𝑧𝑞𝑖,𝑡𝑖}𝑞=1 converges to 𝑘𝑖𝑡𝑖=1Θ𝑖,𝑡𝑖(𝑁,𝑚,𝑣).(3)If 0<𝛼<4/|𝑁|, then for all (𝑖,𝑘𝑖)𝐿𝑁,𝑚 and for all payoff vectors 𝑧 with 𝑧𝑖,𝑘𝑖=𝑣(𝑚𝑁{𝑖},𝑘𝑖)𝑣(𝑚𝑁{𝑖},𝑘𝑖1), {𝑧𝑞𝑖,𝑘𝑖}𝑞=1 converges to Θ𝑖,𝑘𝑖(𝑁,𝑚,𝑣).

Proof. “EL” instead of “IIE”, the proofs of this theorem are immediate analogues Theorem 3.2 and Remark 3.3, hence we omit them.

4. Player-Action Reduction and Related Consistency

By reducing the number of the players, Hwang and Liao [810] proposed 1-reduction and 2-reduction on multichoice games. Here we define two types of player-action reduced games by reducing both the number of the players and the activity levels. Let (𝑁,𝑚,𝑣)𝑀𝐶, 𝑆𝑁{}, 𝜓 be a solution and 𝛾𝑀+𝑁𝑆. (i)The 1-player-action reduced game (𝑆,𝑚𝑆,𝑣1,𝜓𝑆,𝛾) with respect to 𝑆, 𝛾 and 𝜓 is defined as for all 𝛼𝑀𝑆, 𝑣1,𝜓𝑆,𝛾(𝛼)=𝑣(𝛼,𝛾)𝑗𝑁𝑆𝜓𝑗,𝛾𝑗(𝑁,(𝛼,𝛾),𝑣).(4.1)(ii)The 2-player-action reduced game (𝑆,𝑚𝑆,𝑣2,𝜓𝑆,𝛾) with respect to 𝑆, 𝛾 and 𝜓 is defined as for all 𝛼𝑀𝑆, 𝑣2,𝜓𝑆,𝛾(𝛼)=𝑣(𝛼,𝛾)𝛾𝑗𝑁𝑆𝑗𝑘𝑗=1𝜓𝑗,𝑘𝑗(𝑁,(𝛼,𝛾),𝑣).(4.2)

The player-action reduced games are based on the idea that, when renegotiating the payoff distribution within 𝑆, the condition 𝛾𝑀+𝑁𝑆 means that the members of 𝑁𝑆 continue to cooperate with the members of 𝑆. All members in 𝑁𝑆 take nonzero levels based on the participation vector 𝛾 to cooperate. Then in the player-action reduced games, the coalition 𝑆 with activity level 𝛼 cooperates with all the members of 𝑁𝑆 with activity level 𝛾.

Definition 4.1. Let 𝜓 be a solution on 𝑀𝐶. (i)𝜓 satisfies 1-player-action consistency (1PACON) if for all (𝑁,𝑚,𝑣)𝑀𝐶, for all 𝑆𝑁{}, for all (𝑖,𝑘𝑖)𝐿𝑆,𝑚𝑆 and for all 𝛾𝑀+𝑁𝑆, 𝜓𝑖,𝑘𝑖(𝑁,(𝑚𝑆,𝛾),𝑣)=𝜓𝑖,𝑘𝑖(𝑆,𝑚𝑆,𝑣1,𝜓𝑆,𝛾). (ii)𝜓 satisfies 2-player-action consistency (2PACON) if for all (𝑁,𝑚,𝑣)𝑀𝐶, for all 𝑆𝑁{}, for all (𝑖,𝑘𝑖)𝐿𝑆,𝑚𝑆 and for all 𝛾𝑀+𝑁𝑆, 𝜓𝑖,𝑘𝑖(𝑁,(𝑚𝑆,𝛾),𝑣)=𝜓𝑖,𝑘𝑖(𝑆,𝑚𝑆,𝑣2,𝜓𝑆,𝛾).

Remark 4.2. Let (𝑁,𝑚,𝑣)𝑀𝐶, 𝑆𝑁{} and 𝜓 be a solution. Let 𝛾=𝑚𝑁𝑆, by definitions of reduced games and player-action reduced games, 𝑣1,𝜓𝑆,𝑚𝑁𝑆=𝑣𝜓1,𝑆 and 𝑣2,𝜓𝑆,𝑚𝑁𝑆=𝑣𝜓2,𝑆. Clearly, if a solution satisfies 1PACON, then it also satisfies 1CON. Similarly, if a solution satisfies 2PACON, then it also satisfies 2CON.

As we knew, each (𝑁,𝑚,𝑣)𝑀𝐶 can be expressed as a linear combination of minimal effort games and this decomposition exists uniquely. The following lemmas point out the relations of coefficients of expressions among (𝑁,𝑚,𝑣), (𝑆,𝑚𝑆,𝑣1,Λ𝑆,𝛾), (𝑆,𝑚𝑆,𝑣2,Γ𝑆,𝛾), and (𝑆,𝑚𝑆,𝑣2,Θ𝑆,𝛾).

Lemma 4.3. Let (𝑁,𝑚,𝑣)𝑀𝐶, (𝑆,𝑚𝑆,𝑣1,Λ𝑆,𝛾) be a 1-player-action reduced game of 𝑣 with respect to 𝑆, 𝛾, and the solution Λ and let (𝑆,𝑚𝑆,𝑣2,Γ𝑆,𝛾) be a 2-player-action reduced game of 𝑣 with respect to 𝑆, 𝛾 and the solution Γ. If 𝑣=𝛼𝑀𝑁{0𝑁}𝑎𝛼(𝑣)𝑢𝛼𝑁, then 𝑣1,Λ𝑆,𝛾 can be expressed as 𝑣1,Λ𝑆,𝛾=𝑣2,Γ𝑆,𝛾=𝛼𝑀𝑆{0𝑆}𝑎𝛼(𝑣2,Γ𝑆,𝛾)𝑢𝛼𝑆, where for all 𝛼𝑀𝑆, 𝑎𝛼𝑣1,Λ𝑆,𝛾=𝑎𝛼𝑣2,Γ𝑆,𝛾=𝜆𝛾||||𝑆(𝛼)||||+||||𝑆(𝛼)𝑆(𝜆)𝑎(𝛼,𝜆)(𝑣).(4.3)

Proof. Let (𝑁,𝑚,𝑣)𝑀𝐶, 𝑆𝑁 with 𝑆 and 𝛾𝑀+𝑁𝑆. For all 𝛼𝑀𝑆, 𝑣2,Γ𝑆,𝛾(𝛼)=𝑣(𝛼,𝛾)𝛾𝑗𝑁𝑆𝑗𝑘𝑗=1Γ𝑗,𝑘𝑗(𝑁,(𝛼,𝛾),𝑣).(4.4) By 2EFF of Γ, 𝑣2,Γ𝑆,𝛾(0𝑆)=0. For all 𝛼𝑀𝑆 with 𝛼0𝑆, (4.2)=𝛼𝑗𝑆(𝛼)𝑗𝑘𝑗=1Γ𝑗,𝑘𝑗=(𝑁,(𝛼,𝛾),𝑣)𝛼𝑗𝑆(𝛼)𝑗𝑘𝑗=1𝛽𝛽(𝛼,𝛾)𝑗=𝑘𝑗𝑎𝛽(𝑣)||||=𝑆(𝛽)𝑗𝑆(𝛼)𝛽𝛽(𝛼,𝛾)𝑗=1𝑎𝛽(𝑣)||||𝑆(𝛽)++𝛽𝛽(𝛼,𝛾)𝑗=𝛼𝑗𝑎𝛽(𝑣)||||=𝑆(𝛽)𝑗𝑆(𝛼)𝜂𝜂𝛼𝑗=1𝜆𝛾𝑎(𝜂,𝜆)(𝑣)||𝑆||+||𝑆||(𝜂)(𝜆)++𝜂𝜂𝛼𝑗=𝛼𝑗𝜆𝛾𝑎(𝜂,𝜆)(𝑣)||𝑆||+||𝑆||=(𝜂)(𝜆)𝜂𝛼𝜆𝛾||||𝑆(𝜂)||𝑆||+||𝑆||(𝜂)(𝜆)𝑎(𝜂,𝜆)(𝑣).(4.5) Set 𝑎𝜂𝑣2,Γ𝑆,𝛾=𝜆𝛾||||𝑆(𝜂)||||+||||𝑆(𝜂)𝑆(𝜆)𝑎(𝜂,𝜆)(𝑣).(4.6) By (4.5), for all 𝛼𝑀𝑆, 𝑣2,Γ𝑆,𝛾(𝛼)=𝜂𝛼𝜆𝛾||||𝑆(𝜂)||||+||||𝑆(𝜂)𝑆(𝜆)𝑎(𝜂,𝜆)(𝑣)=𝜂𝛼𝑎𝜂𝑣2,Γ𝑆,𝛾.(4.7) Hence 𝑣2,Γ𝑆,𝛾 can be expressed to be 𝑣2,Γ𝑆,𝛾=𝛼𝑀𝑆{0𝑆}𝑎𝛼(𝑣1,Γ𝑆,𝛾)𝑢𝛼𝑆. By Definition 2.1 and the definitions of 𝑣1,Λ𝑆,𝛾 and 𝑣2,Γ𝑆,𝛾, for all 𝛼𝑀𝑁, 𝑣2,Γ𝑆,𝛾(𝛼)=𝑣(𝛼,𝛾)𝛾𝑗𝑁𝑆𝑗𝑘𝑗=1Γ𝑗,𝑘𝑗(𝑁,(𝛼,𝛾),𝑣)=𝑣(𝛼,𝛾)𝑗𝑁𝑆Λ𝑗,𝛾𝑗(𝑁,(𝛼,𝛾),𝑣)=𝑣1,Λ𝑆,𝛾(𝛼).(4.8) Hence, for each 𝛼𝑀𝑆, 𝑣1,Λ𝑆,𝛾(𝛼)=𝑣2,Γ𝑆,𝛾(𝛼) and 𝑎𝛼(𝑣1,Λ𝑆,𝛾)=𝑎𝛼(𝑣2,Γ𝑆,𝛾).

Lemma 4.4. Let (𝑁,𝑚,𝑣)𝑀𝐶 and (𝑆,𝑚𝑆,𝑣2,Θ𝑆,𝛾) be a 2-player-action reduced game of 𝑣 with respect to 𝑆, 𝛾, and the solution Θ. If 𝑣=𝛼𝑀𝑁{0𝑁}𝑎𝛼(𝑣)𝑢𝛼𝑁, then 𝑣2,Θ𝑆,𝛾 can be expressed to be 𝑣2,Θ𝑆,𝛾=𝛼𝑀𝑆{0𝑆}𝑎𝛼(𝑣2,Θ𝑆,𝛾)𝑢𝛼𝑆, where for all 𝛼𝑀𝑆, 𝑎𝛼𝑣2,Θ𝑆,𝛾=𝜆𝛾𝛼𝛼+𝜆𝑎(𝛼,𝜆)(𝑣).(4.9)

Proof. The proof is similar to Lemma 4.3; hence, we omit it.

By applying Lemmas 4.3 and 4.4, we show that the three extended Shapley values satisfy related properties of player-action consistency.

Proposition 4.5. The solution Λ satisfies 1PACON. The solutions Γ and Θ satisfy 2PACON.

Proof. Let (𝑁,𝑚,𝑣)𝑀𝐶, 𝑆𝑁 with 𝑆 and 𝛾𝑀+𝑁𝑆. First, we show that the solution Λ satisfies 1PACON. By Definition 2.1 and Lemma 4.3, for all (𝑖,𝑘𝑖)𝐿𝑆,𝑚𝑆, Λ𝑖,𝑘𝑖𝑆,𝑚𝑆,𝑣1,Λ𝑆,𝛾=𝛼𝑀𝑆𝛼𝑖𝑘𝑖𝑎𝛼𝑣1,Λ𝑆,𝛾||||=𝑆(𝛼)𝛼𝑀𝑆𝛼𝑖𝑘𝑖1||||𝑆(𝛼)𝜆𝛾||𝑆||(𝛼)||||+||||𝑆(𝛼)𝑆(𝜆)𝑎(𝛼,𝜆)=(𝑣)𝛽(𝑚𝑆𝛽,𝛾)𝑖𝑘𝑖𝑎𝛽(𝑣)||||𝑆(𝛽)=Λ𝑖,𝑘𝑖𝑚𝑁,𝑆.,𝛾,𝑣(4.10) Hence, the solution Λ satisfies 1PACON. Similarly, by Definition 2.1, Lemmas 4.3, 4.4 and previous proof, we can show that the solutions Θ and Γ satisfy 2PACON.

Hwang and Liao [810] characterized the three extended Shapley values by means of 1CON and 2CON. By Proposition 4.5 and Remarks 3.1 and 4.2, it is easy to check that 1CON and 2CON could be replaced by 1PACON and 2PACON in axiomatizations proposed by Hwang and Liao [810].

Acknowledgment

The author is very grateful to the editor, the associate editor, and the referees who proposed very helpful suggestions and comments to improve the paper.

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