Advances in Operations Research

Advances in Operations Research / 2011 / Article

Research Article | Open Access

Volume 2011 |Article ID 283978 | https://doi.org/10.1155/2011/283978

Yu-Hsien Liao, "Dynamic Approaches for Multichoice Solutions", Advances in Operations Research, vol. 2011, Article ID 283978, 13 pages, 2011. https://doi.org/10.1155/2011/283978

Dynamic Approaches for Multichoice Solutions

Academic Editor: H. A. Eiselt
Received03 Sep 2010
Revised03 Apr 2011
Accepted23 May 2011
Published21 Jul 2011

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 [8–10], 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 [8–10] 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 π‘ˆ be the universe of players and π‘βŠ†π‘ˆ be a set of players. Suppose each player 𝑖 has π‘šπ‘–βˆˆβ„• levels at which he can actively participate. Let π‘š=(π‘šπ‘–)π‘–βˆˆπ‘ be the vector that describes the number of activity levels for each player, at which he can actively participate. For π‘–βˆˆπ‘ˆ, we set 𝑀𝑖={0,1,…,π‘šπ‘–} as the action space of player 𝑖, where the action 0 means not participating, and 𝑀𝑖+=𝑀𝑖⧡{0}. For π‘βŠ†U, π‘β‰ βˆ…, let 𝑀𝑁=βˆπ‘–βˆˆπ‘π‘€π‘– be the product set of the action spaces for players 𝑁 and 𝑀𝑁+=βˆπ‘–βˆˆπ‘π‘€π‘–+. Denote the zero vector in ℝ𝑁 by 0𝑁.

A multichoice TU game is a triple (𝑁,π‘š,𝑣), where 𝑁 is a nonempty and finite set of players, π‘š is the vector that describes the number of activity levels for each player, and π‘£βˆΆπ‘€π‘β†’β„ is a characteristic function which assigns to each action vector 𝛼=(𝛼𝑖)π‘–βˆˆπ‘βˆˆπ‘€π‘ the worth that the players can jointly obtain when each player 𝑖 plays at activity level π›Όπ‘–βˆˆπ‘€π‘– with 𝑣(0𝑁)=0. If no confusion can arise, a game (𝑁,π‘š,𝑣) will sometimes be denoted by its characteristic function 𝑣. Given a multichoice game (𝑁,π‘š,𝑣) and π›Όβˆˆπ‘€π‘, we write (𝑁,𝛼,𝑣) for the multichoice TU subgame obtained by restricting 𝑣 to {π›½βˆˆπ‘€π‘βˆ£π›½π‘–β‰€π›Όπ‘–βˆ€π‘–βˆˆπ‘} only. Denote the class of all multichoice TU games by 𝑀𝐢.

Given (𝑁,π‘š,𝑣)βˆˆπ‘€πΆ, let 𝐿𝑁,π‘š={(𝑖,π‘˜π‘–)βˆ£π‘–βˆˆπ‘,π‘˜π‘–βˆˆπ‘€π‘–+}. A solution on 𝑀𝐢 is a map πœ“ assigning to each (𝑁,π‘š,𝑣)βˆˆπ‘€πΆ an elementξ€·πœ“πœ“(𝑁,π‘š,𝑣)=𝑖,π‘˜π‘–ξ€Έ(𝑁,π‘š,𝑣)(𝑖,π‘˜π‘–)βˆˆπΏπ‘,π‘šβˆˆβ„πΏπ‘,π‘š.(2.1) Here πœ“π‘–,π‘˜π‘–(𝑁,π‘š,𝑣) is the power index or the value of the player 𝑖 when he takes action π‘˜π‘– to play game 𝑣. For convenience, given (𝑁,π‘š,𝑣)βˆˆπ‘€πΆ and a solution πœ“ on 𝑀𝐢, we define πœ“π‘–,0(𝑁,π‘š,𝑣)=0 for all π‘–βˆˆπ‘.

To state the three extended Shapley values, some more notations will be needed. Given π‘†βŠ†π‘, let |S| be the number of elements in 𝑆 and let 𝑒𝑆(𝑁) be the binary vector in ℝ𝑁 whose component 𝑒𝑆𝑖(𝑁) satisfies𝑒𝑆𝑖(𝑁)=1ifπ‘–βˆˆS,0otherwise.(2.2) Note that if no confusion can arise 𝑒𝑆𝑖(𝑁) will be denoted by 𝑒𝑆𝑖.

Given (𝑁,π‘š,𝑣)βˆˆπ‘€πΆ and π›Όβˆˆπ‘€π‘, we define 𝑆(𝛼)={π‘˜βˆˆπ‘βˆ£π›Όπ‘˜β‰ 0} and βˆ‘β€–π›Όβ€–=π‘–βˆˆπ‘π›Όπ‘–. Let 𝛼,π›½βˆˆβ„π‘, we say 𝛽≀𝛼 if 𝛽𝑖≀𝛼𝑖 for all π‘–βˆˆπ‘.

The analogue of unanimity games for multichoice games are minimal effort games (𝑁,π‘š,𝑒𝛼𝑁), where π›Όβˆˆπ‘€π‘, 𝛼≠0𝑁, defined by for all π›½βˆˆπ‘€π‘,𝑒𝛼𝑁(𝛽)=1if𝛽β‰₯𝛼;0otherwise.(2.3) It is known that for (𝑁,π‘š,𝑣)βˆˆπ‘€πΆ it holds that βˆ‘π‘£=π›Όβˆˆπ‘€π‘β§΅{0𝑁}π‘Žπ›Ό(𝑣)𝑒𝛼𝑁, where π‘Žπ›Όβˆ‘(𝑣)=π‘†βŠ†π‘†(𝛼)(βˆ’1)|𝑆|𝑣(π›Όβˆ’π‘’π‘†).

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 Ξ› is the solution on 𝑀𝐢 which associates with each (𝑁,π‘š,𝑣)βˆˆπ‘€πΆ and each player π‘–βˆˆπ‘ and each π‘˜π‘–βˆˆπ‘€+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,π‘˜π‘–ξ“(𝑁,π‘š,𝑣)=π›Όβˆˆπ‘€π‘π›Όπ‘–β‰€π‘˜π‘–π‘Žπ›Ό(𝑣)||||𝑆(𝛼).(2.4) Note that the so-called dividend π‘Žπ›Ό(𝑣) is divided equally among the necessary players.
(ii) (Derks and Peters, [3]).
The D&P Shapley value Θ is the solution on 𝑀𝐢 which associates with each (𝑁,π‘š,𝑣)βˆˆπ‘€πΆ and each player π‘–βˆˆπ‘ and each π‘˜π‘–βˆˆπ‘€+𝑖 the value Ξ˜π‘–,π‘˜π‘–ξ“(𝑁,π‘š,𝑣)=π›Όβˆˆπ‘€π‘π›Όπ‘–β‰₯π‘˜π‘–π‘Žπ›Ό(𝑣)‖𝛼‖.(2.5) Note that the so-called dividend π‘Žπ›Ό(𝑣) is divided equally among the necessary levels.
(iii) (Peters and Zank, [4]).
The P&Z Shapley value Ξ“ is the solution on 𝑀𝐢 which associates with each (𝑁,π‘š,𝑣)βˆˆπ‘€πΆ and each player π‘–βˆˆπ‘ and each π‘˜π‘–βˆˆπ‘€+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,π‘˜π‘–ξ“(𝑁,π‘š,𝑣)=π›Όβˆˆπ‘€π‘π›Όπ‘–=π‘˜π‘–π‘Žπ›Ό(𝑣)||||𝑆(𝛼).(2.6) Clearly, the P&Z Shapley value is a subdivision of the H&R Shapley value. For all (𝑁,π‘š,𝑣)βˆˆπ‘€πΆ and for all (𝑖,k𝑖)βˆˆπΏπ‘,π‘š, Λ𝑖,π‘˜π‘–ξ“(𝑁,π‘š,𝑣)=π›Όβˆˆπ‘€π‘π›Όπ‘–β‰€π‘˜π‘–π‘Žπ›Ό(𝑣)||||=𝑆(𝛼)π‘˜π‘–ξ“π‘‘π‘–=1ξ“π›Όβˆˆπ‘€π‘π›Όπ‘–=π‘‘π‘–π‘Žπ›Ό(𝑣)||||=𝑆(𝛼)π‘˜π‘–ξ“π‘‘π‘–=1Γ𝑖,𝑑𝑖(𝑁,π‘š,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 πœ“ be a solution on 𝑀𝐢. πœ“ satisfies the following. (i)1-efficiency (1EFF) if for each (𝑁,π‘š,𝑣)βˆˆπ‘€πΆ, βˆ‘π‘–βˆˆπ‘†(π‘š)πœ“π‘–,π‘šπ‘–(𝑁,π‘š,𝑣)=𝑣(π‘š).(ii)2-efficiency (2EFF) if for each (𝑁,π‘š,𝑣)βˆˆπ‘€πΆ, βˆ‘π‘–βˆˆπ‘†(π‘š)βˆ‘π‘šπ‘–π‘˜π‘–=1πœ“π‘–,π‘˜π‘–(𝑁,π‘š,𝑣)=𝑣(π‘š).

The following axioms are analogues of the balanced contributions property due to Myerson [14]. The solution πœ“ satisfies the following. (i)1-strong balanced contributions (1SBC) if for each (𝑁,π‘š,𝑣)βˆˆπ‘€πΆ and (𝑖,π‘˜π‘–),(𝑗,π‘˜π‘—)βˆˆπΏπ‘,π‘š,𝑖≠𝑗, πœ“π‘–,π‘˜π‘–ξ€·ξ€·π‘šπ‘,𝑁⧡{𝑗},π‘˜π‘—ξ€Έξ€Έ,π‘£βˆ’πœ“π‘–,π‘˜π‘–ξ€·ξ€·π‘šπ‘,𝑁⧡{𝑗}ξ€Έξ€Έ,0,𝑣=πœ“π‘—,π‘˜π‘—ξ€·ξ€·π‘šπ‘,𝑁⧡{𝑖},π‘˜π‘–ξ€Έξ€Έ,π‘£βˆ’πœ“π‘—,π‘˜π‘—ξ€·ξ€·π‘šπ‘,𝑁⧡{𝑖}ξ€Έξ€Έ.,0,𝑣(3.1)(ii)2-strong balanced contributions (2SBC) if for each (𝑁,π‘š,𝑣)βˆˆπ‘€πΆ and (𝑖,π‘˜π‘–),(𝑗,π‘˜π‘—)βˆˆπΏπ‘,π‘š,𝑖≠𝑗, πœ“π‘–,π‘˜π‘–(𝑁,π‘š,𝑣)βˆ’πœ“π‘–,π‘˜π‘–ξ€·ξ€·π‘šπ‘,𝑁⧡{𝑗},π‘˜π‘—ξ€Έξ€Έβˆ’1,𝑣=πœ“π‘—,π‘˜π‘—(𝑁,π‘š,𝑣)βˆ’πœ“π‘—,π‘˜π‘—ξ€·ξ€·π‘šπ‘,𝑁⧡{𝑖},π‘˜π‘–ξ€Έξ€Έ.βˆ’1,𝑣(3.2)(iii)3-strong balanced contributions (3SBC) if for each (𝑁,π‘š,𝑣)βˆˆπ‘€πΆ and (𝑖,π‘˜π‘–),(𝑗,π‘˜π‘—)βˆˆπΏπ‘,π‘š,𝑖≠𝑗, πœ“π‘–,π‘˜π‘–ξ€·ξ€·π‘šπ‘,𝑁⧡{𝑗},π‘˜π‘—ξ€Έξ€Έ,π‘£βˆ’πœ“π‘–,π‘˜π‘–ξ€·ξ€·π‘šπ‘,𝑁⧡{𝑗},π‘˜π‘—ξ€Έξ€Έβˆ’1,𝑣=πœ“π‘—,π‘˜π‘—ξ€·ξ€·π‘šπ‘,𝑁⧡{𝑖},π‘˜π‘–ξ€Έξ€Έ,π‘£βˆ’πœ“π‘—,π‘˜π‘—ξ€·ξ€·π‘šπ‘,𝑁⧡{𝑖},π‘˜π‘–ξ€Έξ€Έ.βˆ’1,𝑣(3.3)

The following axiom was introduced by Hwang and Liao ([9]). The solution πœ“ satisfies the following. (i)Independence of individual expansions (IIE) if for each (𝑁,π‘š,𝑣)βˆˆπ‘€πΆ and each (𝑖,π‘˜π‘–)βˆˆπΏπ‘,π‘š, π‘—β‰ π‘šπ‘–, πœ“π‘–,π‘˜π‘–ξ€·ξ€·π‘šπ‘,𝑁⧡{𝑖},π‘˜π‘–ξ€Έξ€Έ,𝑣=πœ“π‘–,π‘˜π‘–ξ€·ξ€·π‘šπ‘,𝑁⧡{𝑖},π‘˜π‘–ξ€Έξ€Έ+1,𝑣=β‹―=πœ“π‘–,π‘˜π‘–(𝑁,π‘š,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 πœ“ satisfies the following. (i)Equal loss (EL) if for each (𝑁,π‘š,𝑣)βˆˆπ‘€πΆ and each (𝑖,π‘˜π‘–)βˆˆπΏπ‘,π‘š, π‘˜π‘–β‰ π‘šπ‘–, πœ“π‘–,π‘˜π‘–(𝑁,π‘š,𝑣)βˆ’πœ“π‘–,π‘˜π‘–ξ€·π‘,π‘šβˆ’π‘’{𝑖}ξ€Έ,𝑣=πœ“π‘–,π‘šπ‘–(𝑁,π‘š,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,π‘š,𝑣)βˆˆπ‘€πΆ with |𝑆(π‘š)|=1, πœ“ satisfies 1EFF. Weak 2-efficiency (2WEFF) simply says that for all (𝑁,π‘š,𝑣)βˆˆπ‘€πΆ with |𝑆(π‘š)|=1, πœ“ satisfies 2EFF. 1-upper balanced contributions (1UBC) only requires that 1SBC holds if π‘˜π‘–=π‘šπ‘– and π‘˜π‘—=π‘šπ‘—. 2-upper balanced contributions (2UBC) only requires that 2SBC or 3SBC holds if π‘˜π‘–=π‘šπ‘– and π‘˜π‘—=π‘šπ‘—. Weak independence of individual expansions (WIIE) simply says that for all (𝑁,π‘š,𝑣)βˆˆπ‘€πΆ with |𝑆(π‘š)|=1, πœ“ satisfies IIE. Weak equal loss (WEL) simply says that for all (𝑁,π‘š,𝑣)βˆˆπ‘€πΆ with |𝑆(π‘š)|=1, πœ“ satisfies EL.

Subsequently, we recall the reduced games and related consistency properties introduced by Hwang and Liao [8–10]. Let (𝑁,π‘š,𝑣)βˆˆπ‘€πΆ, π‘†βŠ†π‘β§΅{βˆ…} and πœ“ be a solution. (i)The 1-reduced game (𝑆,π‘šπ‘†,π‘£πœ“1,𝑆) with respect to πœ“ and 𝑆 is defined as, for all π›Όβˆˆπ‘€π‘†, π‘£πœ“1,𝑆(𝛼)=𝑣𝛼,π‘šπ‘β§΅π‘†ξ€Έβˆ’ξ“π‘–βˆˆπ‘β§΅π‘†πœ“π‘–,π‘šπ‘–ξ€·ξ€·π‘,𝛼,π‘šπ‘β§΅π‘†ξ€Έξ€Έ.,𝑣(3.6)(ii)The 2-reduced game (𝑆,π‘šπ‘†,π‘£πœ“2,𝑆) with respect to πœ“ and 𝑆 is defined as, for all π›Όβˆˆπ‘€π‘†, π‘£πœ“2,𝑆(𝛼)=𝑣𝛼,π‘šπ‘β§΅π‘†ξ€Έβˆ’ξ“π‘šπ‘–βˆˆπ‘β§΅π‘†π‘–ξ“π‘˜π‘–=1πœ“π‘–,π‘˜π‘–ξ€·ξ€·π‘,𝛼,π‘šπ‘β§΅π‘†ξ€Έξ€Έ,𝑣.(3.7)(iii)πœ“ on 𝑀𝐢 satisfies 1-consistency (1CON) if for all (𝑁,π‘š,𝑣)βˆˆπ‘€πΆ, for all π‘†βŠ†π‘ and for all (𝑖,π‘˜π‘–)βˆˆπΏπ‘†,π‘šπ‘†, πœ“π‘–,π‘˜π‘–(𝑁,π‘š,𝑣)=πœ“π‘–,π‘˜π‘–(𝑆,π‘šπ‘†,π‘£πœ“1,𝑆). (iv)πœ“ on 𝑀𝐢 satisfies 2-consistency (2CON) if for all (𝑁,π‘š,𝑣)βˆˆπ‘€πΆ, for all π‘†βŠ†π‘ and for all (𝑖,π‘˜π‘–)βˆˆπΏπ‘†,π‘šπ‘†, πœ“π‘–,π‘˜π‘–(𝑁,π‘š,𝑣)=πœ“π‘–,π‘˜π‘–(𝑆,π‘šπ‘†,π‘£πœ“2,𝑆).

Remark 3.1. Hwang and Liao [8–10] characterized the solutions Ξ›, Ξ“, and Θ by means of 1CON and 2CON as follows.(i)The solution Ξ› 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 Ξ“ 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,π‘š,𝑣)βˆˆπ‘€πΆ. A payoff vector of (𝑁,π‘š,𝑣) is a vector (π‘₯𝑖,π‘˜π‘–)(𝑖,π‘˜π‘–)βˆˆπΏπ‘,π‘šβˆˆβ„πΏπ‘,π‘š where π‘₯𝑖,π‘˜π‘– denotes the payoff to player 𝑖 corresponding to his activity level π‘˜π‘– for all (𝑖,π‘˜π‘–)βˆˆπΏπ‘,π‘š. A payoff vector π‘₯ of (𝑁,π‘š,𝑣) is 1-efficient (1EFF) if βˆ‘π‘–βˆˆπ‘π‘₯𝑖,π‘šπ‘–=𝑣(π‘š). π‘₯ is 2-efficient (2EFF) if βˆ‘π‘–βˆˆπ‘βˆ‘π‘˜π‘–βˆˆπ‘€+𝑖π‘₯𝑖,π‘˜π‘–=𝑣(π‘š). Moreover, the sets of 1-preimputations and 2-preimputations of (𝑁,π‘š,𝑣) are denoted by 𝑋1(𝑁,π‘š,𝑣)=π‘₯βˆˆβ„πΏπ‘,π‘šξ‚‡,π‘‹βˆ£π‘₯is1EFFin(𝑁,π‘š,𝑣)2(𝑁,π‘š,𝑣)=π‘₯βˆˆβ„πΏπ‘,π‘šξ‚‡.∣π‘₯is2EFFin(𝑁,π‘š,𝑣)(3.8)

In order to exhibit such processes, let us define two alternative reduced games as follows. Let (𝑁,π‘š,𝑣)βˆˆπ‘€πΆ, π‘†βŠ†π‘, and let πœ“ be a solution and π‘₯ a payoff vector. (i)The (1,πœ“)-reduced game (𝑆,π‘šπ‘†,𝑣π‘₯,πœ“1,𝑆) with respect to 𝑆, π‘₯, and πœ“ is defined as, for all π›Όβˆˆπ‘€π‘†, 𝑣π‘₯,πœ“1,𝑆(⎧βŽͺ⎨βŽͺβŽ©ξ“π›Ό)=𝑣(π‘š)βˆ’π‘–βˆˆπ‘β§΅π‘†π‘₯𝑖,π‘šπ‘–,𝛼=π‘šπ‘†,π‘£πœ“1,𝑆(𝛼),otherwise.(3.9)(ii)The (2,πœ“)-reduced game (𝑆,π‘šπ‘†,𝑣π‘₯,πœ“2,𝑆) with respect to 𝑆, π‘₯, and πœ“ is defined as, for all π›Όβˆˆπ‘€π‘†, 𝑣π‘₯,πœ“2,π‘†βŽ§βŽͺ⎨βŽͺβŽ©ξ“(𝛼)=𝑣(π‘š)βˆ’π‘–βˆˆπ‘β§΅π‘†ξ“π‘˜π‘–βˆˆπ‘€+π‘₯𝑖,π‘˜π‘–,𝛼=π‘šπ‘†,π‘£πœ“2,𝑆(𝛼),otherwise.(3.10)

Let (𝑁,π‘š,𝑣)βˆˆπ‘€πΆ, 𝑁β‰₯3 and (𝑖,π‘˜π‘–)βˆˆπΏπ‘,π‘š. Inspired by Maschler and Owen [7], we define 𝑓𝑖,π‘˜π‘–βˆΆπ‘‹1(𝑁,π‘š,𝑣)→ℝ, 𝑔𝑖,π‘˜π‘–βˆΆπ‘‹2(𝑁,π‘š,𝑣)→ℝ, β„Žπ‘–,π‘˜π‘–βˆΆπ‘‹2(𝑁,π‘š,𝑣)→ℝ to be as follows:(i)𝑓𝑖,π‘˜π‘–(π‘₯)=π‘₯𝑖,π‘˜π‘–βˆ‘+π‘‘β‹…π‘—βˆˆπ‘β§΅{𝑖}(Λ𝑖,π‘˜π‘–({𝑖,𝑗},π‘š{𝑖,𝑗},𝑣π‘₯,Ξ›1,{𝑖,𝑗})βˆ’π‘₯𝑖,π‘˜π‘–), (ii)𝑔𝑖,π‘˜π‘–(π‘₯)=π‘₯𝑖,π‘˜π‘–βˆ‘+π‘‘β‹…π‘—βˆˆπ‘β§΅{𝑖}(Ξ˜π‘–,π‘˜π‘–({𝑖,𝑗},π‘š{𝑖,𝑗},𝑣π‘₯,Θ2,{𝑖,𝑗})βˆ’π‘₯𝑖,π‘˜π‘–), (iii)β„Žπ‘–,π‘˜π‘–(π‘₯)=π‘₯𝑖,π‘˜π‘–βˆ‘+π‘‘β‹…π‘—βˆˆπ‘β§΅{𝑖}(Γ𝑖,π‘˜π‘–({𝑖,𝑗},π‘š{𝑖,𝑗},𝑣π‘₯,Ξ“2,{𝑖,𝑗})βˆ’π‘₯𝑖,π‘˜π‘–),

where 𝑑 is a fixed positive number, which reflects the assumption that player 𝑖 does not ask for adequate correction (when 𝑑=1) but only (usually) a fraction of it. It is easy to check that (𝑓𝑖,π‘˜π‘–(π‘₯))(𝑖,π‘˜π‘–)βˆˆπΏπ‘,π‘šβˆˆπ‘‹1(𝑁,π‘š,𝑣) if π‘₯βˆˆπ‘‹1(𝑁,π‘š,𝑣), (𝑔𝑖,π‘˜π‘–(π‘₯))(𝑖,π‘˜π‘–)βˆˆπΏπ‘,π‘šβˆˆπ‘‹2(𝑁,π‘š,𝑣), and (β„Žπ‘–,π‘˜π‘–(π‘₯))(𝑖,π‘˜π‘–)βˆˆπΏπ‘,π‘šβˆˆπ‘‹2(𝑁,π‘š,𝑣) if π‘₯βˆˆπ‘‹2(𝑁,π‘š,𝑣).

Inspired by Maschler and Owen [7], we define correction functions 𝑓𝑖,π‘˜π‘–,𝑔𝑖,π‘˜π‘–,β„Žπ‘–,π‘˜π‘– on multichoice games. In the following, we provided some discussions which are analogues to the discussion of Maschler and Owen [7]. Let (𝑁,π‘š,𝑣)βˆˆπ‘€πΆ and π‘₯ be a 1-efficient payoff vector. By a process of induction we assume that the players have already agreed on the solution Ξ› for all 𝑝-person games, 1<𝑝<|𝑁|. In particular, we assume that they agreed on Ξ› 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 π‘₯ should be the solution for an 𝑛-person game (N,π‘š,𝑣), thus suggesting a solution concept Ξ¨, which should satisfyΨ𝑃,π‘šξ…žξ€Έ=ξƒ―Ξ›ξ€·,𝑒𝑃,π‘šξ…žξ€Έ,ξ€·,𝑒𝑃,π‘šξ…žξ€Έ||𝑃||<||𝑁||,ξ€·,π‘’βˆˆπ‘€πΆ,π‘₯,(𝑁,π‘š,𝑣)=𝑃,π‘šξ…žξ€Έ.,𝑒(3.11) On the basis of this Ξ¨, the members of a coalition 𝑆={𝑖,𝑗} will examine 𝑣π‘₯,Ξ›1,𝑆 for related 1-consistency. If the solution turns out to be inconsistent, they will modify π‘₯ β€œin the direction” which is dictated by Λ𝑖,π‘˜π‘–(𝑆,π‘šπ‘†,𝑣π‘₯,Ξ›1,𝑆) in a manner which will be explained subsequently (see the definition of 𝑓𝑖,π‘˜π‘–). These modifications, done simultaneously by all 2-person coalitions, will lead to a new payoff vector π‘₯βˆ— and the process will repeat. The hope is that it will converge and, moreover, converge to Ξ›(𝑁,π‘š,𝑣). Similar discussions could be used to 𝑔𝑖,π‘˜π‘– and β„Žπ‘–,π‘˜π‘–.

Theorem 3.2. Let (𝑁,π‘š,𝑣)βˆˆπ‘€πΆ and π‘₯βˆˆπ‘‹1(𝑁,π‘š,𝑣). Define π‘₯0=π‘₯,π‘₯1=(𝑓𝑖,π‘˜π‘–(π‘₯0))(𝑖,π‘˜π‘–)βˆˆπΏπ‘,π‘š,…,π‘₯π‘ž=(𝑓𝑖,π‘˜π‘–(π‘₯π‘žβˆ’1))(𝑖,π‘˜π‘–)βˆˆπΏπ‘,π‘š for all π‘žβˆˆβ„•. (1)If 0<𝑑<4/|𝑁|, then for all π‘–βˆˆπ‘ and for all π‘₯βˆˆπ‘‹1(𝑁,π‘š,𝑣), {π‘₯π‘žπ‘–,π‘šπ‘–}βˆžπ‘ž=1 converges to Λ𝑖,π‘šπ‘–(𝑁,π‘š,𝑣).(2)If 0<𝑑<4/|𝑁|, then for all (𝑖,π‘˜π‘–)βˆˆπΏπ‘,π‘š and for all π‘₯βˆˆπ‘‹1(𝑁,(π‘šπ‘β§΅{𝑖},π‘˜π‘–),𝑣), {π‘₯π‘žπ‘–,π‘˜π‘–}βˆžπ‘ž=1 converges to Λ𝑖,π‘˜π‘–(𝑁,π‘š,𝑣).

Proof. Fix (𝑁,π‘š,𝑣)βˆˆπ‘€πΆ and π‘₯βˆˆπ‘‹1(𝑁,π‘š,𝑣). To prove (1), let 𝑖,π‘—βˆˆπ‘†(π‘š) and 𝑆={𝑖,𝑗}. By 1EFF and 1UBC of Ξ›, and definitions of 𝑣Λ1,𝑆 and 𝑣π‘₯,Ξ›1,𝑆, Λ𝑖,π‘šπ‘–ξ‚€π‘†,π‘šπ‘†,𝑣π‘₯,Ξ›1,𝑆+Λ𝑗,π‘šπ‘—ξ‚€π‘†,π‘šπ‘†,𝑣π‘₯,Ξ›1,𝑆=π‘₯𝑖,π‘šπ‘–+π‘₯𝑗,π‘šπ‘—Ξ›,(by1EFFofΞ›),(3.12)𝑖,π‘šπ‘–ξ‚€π‘†,π‘šπ‘†,𝑣π‘₯,Ξ›1,π‘†ξ‚βˆ’Ξ›π‘—,π‘šπ‘—ξ‚€π‘†,π‘šπ‘†,𝑣π‘₯,Ξ›1,𝑆=Λ𝑖,π‘šπ‘–ξ‚€ξ€·π‘šπ‘†,𝑆⧡{𝑗}ξ€Έ,0,𝑣π‘₯,Ξ›1,π‘†ξ‚βˆ’Ξ›π‘—,π‘šπ‘—ξ‚€ξ€·π‘šπ‘†,𝑆⧡{𝑖}ξ€Έ,0,𝑣π‘₯,Ξ›1,𝑆(by1UBCofΞ›)=Λ𝑖,π‘šπ‘–ξ‚€ξ€·π‘šπ‘†,𝑆⧡{𝑗}ξ€Έ,0,𝑣Λ1,π‘†ξ‚βˆ’Ξ›π‘—,π‘šπ‘—ξ‚€ξ€·π‘šπ‘†,𝑆⧡{𝑖}ξ€Έ,0,𝑣Λ1,𝑆bydefinitionof𝑣Λ1,𝑆=Λ𝑖,π‘šπ‘–ξ‚€π‘†,π‘šπ‘†,𝑣Λ1,π‘†ξ‚βˆ’Ξ›π‘—,π‘šπ‘—ξ‚€π‘†,π‘šπ‘†,𝑣Λ1,𝑆.(by1UBCofΞ›).(3.13) Therefore, Λ2⋅𝑖,π‘šπ‘–ξ‚€π‘†,π‘šπ‘†,𝑣π‘₯,Ξ›1,π‘†ξ‚βˆ’π‘₯𝑖,π‘šπ‘–ξ‚„=Λ𝑖,π‘šπ‘–ξ‚€π‘†,π‘šπ‘†,𝑣Λ1,π‘†ξ‚βˆ’Ξ›π‘—,π‘šπ‘—ξ‚€π‘†,π‘šπ‘†,𝑣Λ1,π‘†ξ‚βˆ’π‘₯𝑖,π‘šπ‘–+π‘₯𝑗,π‘šπ‘—.(3.14)
By definition of 𝑓, 1CON and 1EFF of Ξ› and (3.14), 𝑓𝑖,π‘šπ‘–(π‘₯)=π‘₯𝑖,π‘šπ‘–+𝑑2β‹…βŽ‘βŽ’βŽ’βŽ£ξ“π‘—βˆˆπ‘β§΅{𝑖}Λ𝑖,π‘šπ‘–ξ‚€{𝑖,𝑗},π‘š{𝑖,𝑗},𝑣Λ1,{𝑖,𝑗}ξ‚βˆ’ξ“π‘—βˆˆπ‘β§΅{𝑖}π‘₯𝑖,π‘šπ‘–βˆ’ξ“π‘—βˆˆπ‘β§΅{𝑖}Λ𝑗,π‘šπ‘—ξ‚€{𝑖,𝑗},π‘š{𝑖,𝑗},𝑣Λ1,{𝑖,𝑗}+ξ“π‘—βˆˆπ‘β§΅{𝑖}π‘₯𝑗,π‘šπ‘—βŽ€βŽ₯βŽ₯βŽ¦ξ€·bydefinitionof𝑓𝑖,π‘šπ‘–ξ€Έandequation(3.14)=π‘₯𝑖,π‘šπ‘–+𝑑2β‹…βŽ‘βŽ’βŽ’βŽ£ξ“π‘˜βˆˆπ‘β§΅{𝑖}Λ𝑖,π‘šπ‘–ξ€·||𝑁||ξ€Έπ‘₯(𝑁,π‘š,𝑣)βˆ’βˆ’1𝑖,π‘šπ‘–βˆ’ξ“π‘—βˆˆπ‘β§΅{𝑖}Λ𝑗,π‘šπ‘—(𝑁,π‘š,𝑣)+𝑣(π‘š)βˆ’π‘₯𝑖,π‘šπ‘–ξ€ΈβŽ€βŽ₯βŽ₯⎦(by1CONofΞ›and1EFFofπ‘₯)=π‘₯𝑖,π‘šπ‘–+𝑑2β‹…||𝑁||ξ€ΈΞ›ξ€Ίξ€·βˆ’1𝑖,π‘šπ‘–ξ€·||𝑁||ξ€Έπ‘₯(𝑁,π‘š,𝑣)βˆ’βˆ’1𝑖,π‘šπ‘–βˆ’ξ€·π‘£(π‘š)βˆ’Ξ›π‘–,π‘šπ‘–ξ€Έ+ξ€·(𝑁,π‘š,𝑣)𝑣(π‘š)βˆ’π‘₯𝑖,π‘šπ‘–ξ€Έξ€»(by1EFFofΞ›)=π‘₯𝑖,π‘šπ‘–+||𝑁||⋅𝑑2⋅Λ𝑖,π‘šπ‘–(𝑁,π‘š,𝑣)βˆ’π‘₯𝑖,π‘šπ‘–ξ€».(3.15) Hence, for all π‘žβˆˆβ„•, ξ‚΅||𝑁||1βˆ’β‹…π‘‘2ξ‚Άπ‘ž+1Λ𝑖,π‘šπ‘–(𝑁,π‘š,𝑣)βˆ’π‘₯π‘žπ‘–,π‘šπ‘–ξ‚„=Λ𝑖,π‘šπ‘–(𝑁,π‘š,𝑣)βˆ’π‘“π‘–,π‘šπ‘–(π‘₯π‘ž)ξ€»=Λ𝑖,π‘šπ‘–(𝑁,π‘š,𝑣)βˆ’π‘₯π‘ž+1𝑖,π‘šπ‘–ξ‚„.(3.16) If 0<𝑑<4/|𝑁|, then βˆ’1<(1βˆ’(|𝑁|⋅𝑑)/2)<1 and {π‘₯π‘žπ‘–,π‘šπ‘–}βˆžπ‘ž=1 converges to Λ𝑖,π‘šπ‘–(𝑁,π‘š,𝑣).
To prove (2), by 1 of this theorem, if 0<𝑑<4/|𝑁|, then for all (𝑖,π‘˜π‘–)βˆˆπΏπ‘,π‘š and for all π‘₯βˆˆπ‘‹1(𝑁,(π‘šπ‘β§΅{𝑖},π‘˜π‘–),𝑣), {π‘₯π‘žπ‘–,π‘˜π‘–}βˆžπ‘ž=1 converges to Λ𝑖,π‘˜π‘–(𝑁,(π‘šπ‘β§΅{𝑖},π‘˜π‘–),𝑣). By IIE of Ξ›, {π‘₯π‘žπ‘–,π‘˜π‘–}βˆžπ‘ž=1 converges to Λ𝑖,π‘˜π‘–(𝑁,π‘š,𝑣).

Remark 3.3. Huang et al. [15] provided dynamic processes for the P&Z Shapley value as follows. Let (𝑁,π‘š,𝑣)βˆˆπ‘€πΆ and π‘¦βˆˆπ‘‹2(𝑁,π‘š,v). Define 𝑦0=𝑦,𝑦1=(β„Žπ‘–,π‘˜π‘–(𝑦0))(𝑖,π‘˜π‘–)βˆˆπΏπ‘,π‘š,…,π‘¦π‘ž=(β„Žπ‘–,π‘˜π‘–(π‘¦π‘žβˆ’1))(𝑖,π‘˜π‘–)βˆˆπΏπ‘,π‘š for all π‘žβˆˆβ„•.
(1)If 0<𝛼<4/|𝑁|, then for all π‘–βˆˆπ‘ and for all π‘¦βˆˆπ‘‹2(𝑁,π‘š,𝑣), {βˆ‘π‘šπ‘–π‘˜π‘–=1π‘¦π‘žπ‘–,π‘˜π‘–}βˆžπ‘ž=1 converges to βˆ‘π‘šπ‘–π‘˜π‘–=1Γ𝑖,π‘˜π‘–(𝑁,π‘š,𝑣).(2)If 0<𝑑<4/|𝑁|, then for all (𝑖,π‘˜π‘–)βˆˆπΏπ‘,π‘š and for all π‘¦βˆˆπ‘‹2(𝑁,(π‘šπ‘β§΅{𝑖},π‘˜π‘–),𝑣), {βˆ‘π‘˜π‘–π‘‘π‘–=1π‘¦π‘žπ‘–,𝑑𝑖}βˆžπ‘ž=1 converges to βˆ‘π‘˜π‘–π‘‘π‘–=1Γ𝑖,𝑑𝑖(𝑁,π‘š,𝑣).(3)If 0<𝛼<4/|𝑁|, then for all (𝑖,π‘˜π‘–)βˆˆπΏπ‘,π‘š and for all payoff vectors 𝑦 with 𝑦𝑖,π‘˜π‘–=𝑣(π‘šπ‘β§΅{𝑖},π‘˜π‘–)βˆ’π‘£(π‘šπ‘β§΅{𝑖},π‘˜π‘–βˆ’1), {π‘¦π‘žπ‘–,π‘˜π‘–}βˆžπ‘ž=1 converges to Γ𝑖,π‘˜π‘–(𝑁,π‘š,𝑣).
In fact, the proofs of (1), (2), and (3) are similar to Theorem 3.2.

Theorem 3.4. Let (𝑁,π‘š,𝑣)βˆˆπ‘€πΆ and π‘§βˆˆπ‘‹2(𝑁,π‘š,𝑣). Define 𝑧0=𝑧,𝑧1=(𝑔𝑖,π‘˜π‘–(𝑧0))(𝑖,π‘˜π‘–)βˆˆπΏπ‘,π‘š,…,π‘§π‘ž=(𝑔𝑖,π‘˜π‘–(π‘§π‘žβˆ’1))(𝑖,π‘˜π‘–)βˆˆπΏπ‘,π‘š for all π‘žβˆˆβ„•.
(1)If 0<𝛼<4/|𝑁|, then for all π‘–βˆˆπ‘ and for all π‘§βˆˆπ‘‹2(𝑁,π‘š,𝑣), {βˆ‘π‘šπ‘–π‘˜π‘–=1π‘§π‘žπ‘–,π‘˜π‘–}βˆžπ‘ž=1 converges to βˆ‘π‘šπ‘–π‘˜π‘–=1Ξ˜π‘–,π‘˜π‘–(𝑁,π‘š,𝑣).(2)If 0<𝑑<4/|𝑁|, then for all (𝑖,π‘˜π‘–)βˆˆπΏπ‘,π‘š and for all π‘§βˆˆπ‘‹2(𝑁,(π‘šπ‘β§΅{𝑖},π‘˜π‘–),𝑣), {βˆ‘π‘˜π‘–π‘‘π‘–=1π‘§π‘žπ‘–,𝑑𝑖}βˆžπ‘ž=1 converges to βˆ‘π‘˜π‘–π‘‘π‘–=1Ξ˜π‘–,𝑑𝑖(𝑁,π‘š,𝑣).(3)If 0<𝛼<4/|𝑁|, then for all (𝑖,π‘˜π‘–)βˆˆπΏπ‘,π‘š and for all payoff vectors 𝑧 with 𝑧𝑖,π‘˜π‘–=𝑣(π‘šπ‘β§΅{𝑖},π‘˜π‘–)βˆ’π‘£(π‘šπ‘β§΅{𝑖},π‘˜π‘–βˆ’1), {π‘§π‘žπ‘–,π‘˜π‘–}βˆžπ‘ž=1 converges to Ξ˜π‘–,π‘˜π‘–(𝑁,π‘š,𝑣).

Proof. β€œEL” instead of β€œIIE”, the proofs of this theorem are immediate analogues Theorem 3.2 and Remark 3.3, hence we omit them.

By reducing the number of the players, Hwang and Liao [8–10] 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,π‘š,𝑣)βˆˆπ‘€πΆ, π‘†βŠ†π‘β§΅{βˆ…}, πœ“ be a solution and π›Ύβˆˆπ‘€+𝑁⧡𝑆. (i)The 1-player-action reduced game (𝑆,π‘šπ‘†,𝑣1,πœ“π‘†,𝛾) with respect to 𝑆, 𝛾 and πœ“ is defined as for all π›Όβˆˆπ‘€π‘†, 𝑣1,πœ“π‘†,𝛾(𝛼)=𝑣(𝛼,𝛾)βˆ’π‘—βˆˆπ‘β§΅π‘†πœ“π‘—,𝛾𝑗(𝑁,(𝛼,𝛾),𝑣).(4.1)(ii)The 2-player-action reduced game (𝑆,π‘šπ‘†,𝑣2,πœ“π‘†,𝛾) with respect to 𝑆, 𝛾 and πœ“ is defined as for all π›Όβˆˆπ‘€π‘†, 𝑣2,πœ“π‘†,𝛾(𝛼)=𝑣(𝛼,𝛾)βˆ’π›Ύπ‘—βˆˆπ‘β§΅π‘†π‘—ξ“π‘˜π‘—=1πœ“π‘—,π‘˜π‘—(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 π›Ύβˆˆπ‘€+𝑁⧡𝑆 means that the members of 𝑁⧡𝑆 continue to cooperate with the members of 𝑆. All members in 𝑁⧡𝑆 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 𝑁⧡𝑆 with activity level 𝛾.

Definition 4.1. Let πœ“ be a solution on 𝑀𝐢. (i)πœ“ satisfies 1-player-action consistency (1PACON) if for all (𝑁,π‘š,𝑣)βˆˆπ‘€πΆ, for all π‘†βŠ†π‘β§΅{βˆ…}, for all (𝑖,π‘˜π‘–)βˆˆπΏπ‘†,π‘šπ‘† and for all π›Ύβˆˆπ‘€+𝑁⧡𝑆, πœ“π‘–,π‘˜π‘–(𝑁,(π‘šπ‘†,𝛾),𝑣)=πœ“π‘–,π‘˜π‘–(𝑆,π‘šπ‘†,𝑣1,πœ“π‘†,𝛾). (ii)πœ“ satisfies 2-player-action consistency (2PACON) if for all (𝑁,π‘š,𝑣)βˆˆπ‘€πΆ, for all π‘†βŠ†π‘β§΅{βˆ…}, for all (𝑖,π‘˜π‘–)βˆˆπΏπ‘†,π‘šπ‘† and for all π›Ύβˆˆπ‘€+𝑁⧡𝑆, πœ“π‘–,π‘˜π‘–(𝑁,(π‘šπ‘†,𝛾),𝑣)=πœ“π‘–,π‘˜π‘–(𝑆,π‘šπ‘†,𝑣2,πœ“π‘†,𝛾).

Remark 4.2. Let (𝑁,π‘š,𝑣)βˆˆπ‘€πΆ, π‘†βŠ†π‘β§΅{βˆ…} and πœ“ be a solution. Let 𝛾=π‘šπ‘β§΅π‘†, by definitions of reduced games and player-action reduced games, 𝑣1,πœ“π‘†,π‘šπ‘β§΅π‘†=π‘£πœ“1,𝑆 and 𝑣2,πœ“π‘†,π‘šπ‘β§΅π‘†=π‘£πœ“2,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,π‘š,𝑣)βˆˆπ‘€πΆ 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,π‘š,𝑣), (𝑆,π‘šπ‘†,𝑣1,Λ𝑆,𝛾), (𝑆,π‘šπ‘†,𝑣2,Γ𝑆,𝛾), and (𝑆,π‘šπ‘†,𝑣2,Ξ˜π‘†,𝛾).

Lemma 4.3. Let (𝑁,π‘š,𝑣)βˆˆπ‘€πΆ, (𝑆,π‘šπ‘†,𝑣1,Λ𝑆,𝛾) be a 1-player-action reduced game of 𝑣 with respect to 𝑆, 𝛾, and the solution Ξ› and let (𝑆,π‘šπ‘†,𝑣2,Γ𝑆,𝛾) be a 2-player-action reduced game of 𝑣 with respect to 𝑆, 𝛾 and the solution Ξ“. If βˆ‘π‘£=π›Όβˆˆπ‘€π‘β§΅{0𝑁}π‘Žπ›Ό(𝑣)⋅𝑒𝛼𝑁, then 𝑣1,Λ𝑆,𝛾 can be expressed as 𝑣1,Λ𝑆,𝛾=𝑣2,Γ𝑆,𝛾=βˆ‘π›Όβˆˆπ‘€π‘†β§΅{0𝑆}π‘Žπ›Ό(𝑣2,Γ𝑆,𝛾)⋅𝑒𝛼𝑆, where for all π›Όβˆˆπ‘€π‘†, π‘Žπ›Όξ‚€π‘£1,Λ𝑆,𝛾=π‘Žπ›Όξ‚€π‘£2,Γ𝑆,𝛾=ξ“πœ†β‰€π›Ύ||||𝑆(𝛼)||||+||||𝑆(𝛼)𝑆(πœ†)β‹…π‘Ž(𝛼,πœ†)(𝑣).(4.3)

Proof. Let (𝑁,π‘š,𝑣)βˆˆπ‘€πΆ, π‘†βŠ†π‘ with π‘†β‰ βˆ… and π›Ύβˆˆπ‘€+𝑁⧡𝑆. For all π›Όβˆˆπ‘€π‘†, 𝑣2,Γ𝑆,𝛾(𝛼)=𝑣(𝛼,𝛾)βˆ’π›Ύπ‘—βˆˆπ‘β§΅π‘†π‘—ξ“π‘˜π‘—=1Γ𝑗,π‘˜π‘—(𝑁,(𝛼,𝛾),𝑣).(4.4) By 2EFF of Ξ“, 𝑣2,Γ𝑆,𝛾(0𝑆)=0. For all π›Όβˆˆπ‘€π‘† with 𝛼≠0𝑆, (4.2)=π›Όπ‘—βˆˆπ‘†(𝛼)π‘—ξ“π‘˜π‘—=1Γ𝑗,π‘˜π‘—=(𝑁,(𝛼,𝛾),𝑣)π›Όπ‘—βˆˆπ‘†(𝛼)π‘—ξ“π‘˜π‘—=1𝛽𝛽≀(𝛼,𝛾)𝑗=π‘˜π‘—π‘Žπ›½(𝑣)||||=𝑆(𝛽)π‘—βˆˆπ‘†(𝛼)βŽ‘βŽ’βŽ’βŽ’βŽ’βŽ£ξ“π›½π›½β‰€(𝛼,𝛾)𝑗=1π‘Žπ›½(𝑣)||||𝑆(𝛽)+β‹―+𝛽𝛽≀(𝛼,𝛾)𝑗=π›Όπ‘—π‘Žπ›½(𝑣)||||⎀βŽ₯βŽ₯βŽ₯βŽ₯⎦=𝑆(𝛽)π‘—βˆˆπ‘†(𝛼)βŽ‘βŽ’βŽ’βŽ’βŽ’βŽ£ξ“πœ‚πœ‚β‰€π›Όπ‘—=1ξ“πœ†β‰€π›Ύπ‘Ž(πœ‚,πœ†)(𝑣)||𝑆||+||𝑆||(πœ‚)(πœ†)+β‹―+πœ‚πœ‚β‰€π›Όπ‘—=π›Όπ‘—ξ“πœ†β‰€π›Ύπ‘Ž(πœ‚,πœ†)(𝑣)||𝑆||+||𝑆||⎀βŽ₯βŽ₯βŽ₯βŽ₯⎦=(πœ‚)(πœ†)πœ‚β‰€π›Όξ“πœ†β‰€π›Ύ||||𝑆(πœ‚)||𝑆||+||𝑆||(πœ‚)(πœ†)β‹…π‘Ž(πœ‚,πœ†)(𝑣).(4.5) Set π‘Žπœ‚ξ‚€π‘£2,Γ𝑆,𝛾=ξ“πœ†β‰€π›Ύ||||𝑆(πœ‚)||||+||||𝑆(πœ‚)𝑆(πœ†)β‹…π‘Ž(πœ‚,πœ†)(𝑣).(4.6) By (4.5), for all π›Όβˆˆπ‘€π‘†, 𝑣2,Γ𝑆,𝛾(𝛼)=πœ‚β‰€π›Όξ“πœ†β‰€π›Ύ||||𝑆(πœ‚)||||+||||𝑆(πœ‚)𝑆(πœ†)β‹…π‘Ž(πœ‚,πœ†)(𝑣)=πœ‚β‰€π›Όπ‘Žπœ‚ξ‚€π‘£2,Γ𝑆,𝛾.(4.7) Hence 𝑣2,Γ𝑆,𝛾 can be expressed to be 𝑣2,Γ𝑆,𝛾=βˆ‘π›Όβˆˆπ‘€π‘†β§΅{0𝑆}π‘Žπ›Ό(𝑣1,Γ𝑆,𝛾)⋅𝑒𝛼𝑆. By Definition 2.1 and the definitions of 𝑣1,Λ𝑆,𝛾 and 𝑣2,Γ𝑆,𝛾, for all π›Όβˆˆπ‘€π‘, 𝑣2,Γ𝑆,𝛾(𝛼)=𝑣(𝛼,𝛾)βˆ’π›Ύπ‘—βˆˆπ‘β§΅π‘†π‘—ξ“π‘˜π‘—=1Γ𝑗,π‘˜π‘—ξ“(𝑁,(𝛼,𝛾),𝑣)=𝑣(𝛼,𝛾)βˆ’π‘—βˆˆπ‘β§΅π‘†Ξ›π‘—,𝛾𝑗(𝑁,(𝛼,𝛾),𝑣)=𝑣1,Λ𝑆,𝛾(𝛼).(4.8) Hence, for each π›Όβˆˆπ‘€π‘†, 𝑣1,Λ𝑆,𝛾(𝛼)=𝑣2,Γ𝑆,𝛾(𝛼) and π‘Žπ›Ό(𝑣1,Λ𝑆,𝛾)=π‘Žπ›Ό(𝑣2,Γ𝑆,𝛾).

Lemma 4.4. Let (𝑁,π‘š,𝑣)βˆˆπ‘€πΆ and (𝑆,π‘šπ‘†,𝑣2,Ξ˜π‘†,𝛾) be a 2-player-action reduced game of 𝑣 with respect to 𝑆, 𝛾, and the solution Θ. If βˆ‘π‘£=π›Όβˆˆπ‘€π‘β§΅{0𝑁}π‘Žπ›Ό(𝑣)⋅𝑒𝛼𝑁, then 𝑣2,Ξ˜π‘†,𝛾 can be expressed to be 𝑣2,Ξ˜π‘†,𝛾=βˆ‘π›Όβˆˆπ‘€π‘†β§΅{0𝑆}π‘Žπ›Ό(𝑣2,Ξ˜π‘†,𝛾)⋅𝑒𝛼𝑆, where for all π›Όβˆˆπ‘€π‘†, π‘Žπ›Όξ‚€π‘£2,Ξ˜π‘†,𝛾=ξ“πœ†β‰€π›Ύβ€–π›Όβ€–β€–π›Όβ€–+β€–πœ†β€–β‹…π‘Ž(𝛼,πœ†)(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 Ξ› satisfies 1PACON. The solutions Ξ“ and Θ satisfy 2PACON.

Proof. Let (𝑁,π‘š,𝑣)βˆˆπ‘€πΆ, π‘†βŠ†π‘ with π‘†β‰ βˆ… and π›Ύβˆˆπ‘€+𝑁⧡𝑆. First, we show that the solution Ξ› satisfies 1PACON. By Definition 2.1 and Lemma 4.3, for all (𝑖,π‘˜π‘–)βˆˆπΏπ‘†,π‘šπ‘†, Λ𝑖,π‘˜π‘–ξ‚€π‘†,π‘šπ‘†,𝑣1,Λ𝑆,𝛾=ξ“π›Όβˆˆπ‘€π‘†π›Όπ‘–β‰€π‘˜π‘–π‘Žπ›Όξ‚€π‘£1,Λ𝑆,𝛾||||=𝑆(𝛼)π›Όβˆˆπ‘€π‘†π›Όπ‘–β‰€π‘˜π‘–1||||⋅𝑆(𝛼)πœ†β‰€π›Ύ||𝑆||(𝛼)||||+||||𝑆(𝛼)𝑆(πœ†)β‹…π‘Ž(𝛼,πœ†)=(𝑣)𝛽≀(π‘šπ‘†π›½,𝛾)π‘–β‰€π‘˜π‘–π‘Žπ›½(𝑣)||||𝑆(𝛽)=Λ𝑖,π‘˜π‘–ξ€·ξ€·π‘šπ‘,𝑆.,𝛾,𝑣(4.10) Hence, the solution Ξ› satisfies 1PACON. Similarly, by Definition 2.1, Lemmas 4.3, 4.4 and previous proof, we can show that the solutions Θ and Ξ“ satisfy 2PACON.

Hwang and Liao [8–10] 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 [8–10].

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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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.


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