Dynamic Approaches for Multichoice Solutions

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 𝑈 be the universe of players and 𝑁⊆𝑈 be a set of players. Suppose each player 𝑖 has 𝑚_𝑖∈ℕ levels at which he can actively participate. Let 𝑚=(𝑚_𝑖)_𝑖∈𝑁 be the vector that describes the number of activity levels for each player, at which he can actively participate. For 𝑖∈𝑈, we set 𝑀^𝑖={0,1,…,𝑚_𝑖} as the action space of player 𝑖, where the action 0 means not participating, and 𝑀^𝑖+=𝑀^𝑖⧵{0}. For 𝑁⊆U, 𝑁≠∅, let 𝑀^𝑁=∏_𝑖∈𝑁𝑀^𝑖 be the product set of the action spaces for players 𝑁 and 𝑀^𝑁+=∏_𝑖∈𝑁𝑀^𝑖+. Denote the zero vector in ℝ^𝑁 by 0_𝑁.

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 𝑣∶𝑀^𝑁→ℝ is a characteristic function which assigns to each action vector 𝛼=(𝛼_𝑖)_𝑖∈𝑁∈𝑀^𝑁 the worth that the players can jointly obtain when each player 𝑖 plays at activity level 𝛼_𝑖∈𝑀_𝑖 with 𝑣(0_𝑁)=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 𝛼∈𝑀^𝑁, we write (𝑁,𝛼,𝑣) for the multichoice TU subgame obtained by restricting 𝑣 to {𝛽∈𝑀^𝑁∣𝛽_𝑖≤𝛼_𝑖∀𝑖∈𝑁} only. Denote the class of all multichoice TU games by 𝑀𝐶.

Given (𝑁,𝑚,𝑣)∈𝑀𝐶, let 𝐿^𝑁,𝑚={(𝑖,𝑘_𝑖)∣𝑖∈𝑁,𝑘_𝑖∈𝑀^𝑖+}. A solution on 𝑀𝐶 is a map 𝜓 assigning to each (𝑁,𝑚,𝑣)∈𝑀𝐶 an element𝜓(𝑁,𝑚,𝑣)=𝜓_{𝑖,𝑘𝑖}(𝑁,𝑚,𝑣)_{(𝑖,𝑘𝑖)∈𝐿𝑁,𝑚}∈ℝ^{𝐿𝑁,𝑚}.(2.1) Here 𝜓_{𝑖,𝑘𝑖}(𝑁,𝑚,𝑣) is the power index or the value of the player 𝑖 when he takes action 𝑘_𝑖 to play game 𝑣. For convenience, given (𝑁,𝑚,𝑣)∈𝑀𝐶 and a solution 𝜓 on 𝑀𝐶, we define 𝜓_𝑖,0(𝑁,𝑚,𝑣)=0 for all 𝑖∈𝑁.

To state the three extended Shapley values, some more notations will be needed. Given 𝑆⊆𝑁, let |S| be the number of elements in 𝑆 and let 𝑒^𝑆(𝑁) be the binary vector in ℝ^𝑁 whose component 𝑒^𝑆𝑖(𝑁) satisfies𝑒^𝑆𝑖(𝑁)=10iofth𝑖e∈rwSi,se.(2.2) Note that if no confusion can arise 𝑒^𝑆𝑖(𝑁) will be denoted by 𝑒^𝑆𝑖.

Given (𝑁,𝑚,𝑣)∈𝑀𝐶 and 𝛼∈𝑀^𝑁, we define 𝑆(𝛼)={𝑘∈𝑁∣𝛼_𝑘≠0} and ‖𝛼‖=∑_𝑖∈𝑁𝛼_𝑖. Let 𝛼,𝛽∈ℝ^𝑁, we say 𝛽≤𝛼 if 𝛽_𝑖≤𝛼_𝑖 for all 𝑖∈𝑁.

The analogue of unanimity games for multichoice games are minimal effort games (𝑁,𝑚,𝑢^𝛼𝑁), where 𝛼∈𝑀^𝑁, 𝛼≠0_𝑁, defined by for all 𝛽∈𝑀^𝑁,𝑢^𝛼𝑁(𝛽)=10iofth𝛽er≥w𝛼is;e.(2.3) It is known that for (𝑁,𝑚,𝑣)∈𝑀𝐶 it holds that 𝑣=∑_{𝛼∈𝑀𝑁⧵{0𝑁}}𝑎^𝛼(𝑣)𝑢^𝛼𝑁, where 𝑎^𝛼(𝑣)=∑_{𝑆⊆𝑆(𝛼)}(−1)^|𝑆|𝑣(𝛼−𝑒^𝑆).

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 𝑖∈𝑁 and each 𝑘_𝑖∈𝑀^+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) Λ_{𝑖,𝑘𝑖}(𝑁,𝑚,𝑣)=_{𝛼∈𝑀𝑁𝛼𝑖≤𝑘𝑖}𝑎^𝛼(𝑣)||𝑆(𝛼)||.(2.4) Note that the so-called dividend 𝑎^𝛼(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 𝑖∈𝑁 and each 𝑘_𝑖∈𝑀^+𝑖 the value Θ_{𝑖,𝑘𝑖}(𝑁,𝑚,𝑣)=_{𝛼∈𝑀𝑁𝛼𝑖≥𝑘𝑖}𝑎^𝛼(𝑣)‖𝛼‖.(2.5) Note that the so-called dividend 𝑎^𝛼(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 𝑖∈𝑁 and each 𝑘_𝑖∈𝑀^+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) Γ_{𝑖,𝑘𝑖}(𝑁,𝑚,𝑣)=_{𝛼∈𝑀𝑁𝛼𝑖=𝑘𝑖}𝑎^𝛼(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,k_𝑖)∈𝐿^𝑁,𝑚, Λ_{𝑖,𝑘𝑖}(𝑁,𝑚,𝑣)=_{𝛼∈𝑀𝑁𝛼𝑖≤𝑘𝑖}𝑎^𝛼(𝑣)||𝑆(𝛼)||=^𝑘𝑖_𝑡𝑖=1_{𝛼∈𝑀𝑁𝛼𝑖=𝑡𝑖}𝑎^𝛼(𝑣)||𝑆(𝛼)||=^𝑘𝑖_𝑡𝑖=1Γ_{𝑖,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, ∑_{𝑖∈𝑆(𝑚)}𝜓_{𝑖,𝑚𝑖}(𝑁,𝑚,𝑣)=𝑣(𝑚).(ii)2-efficiency (2EFF) if for each (𝑁,𝑚,𝑣)∈𝑀𝐶, ∑_{𝑖∈𝑆(𝑚)}∑^𝑚𝑖_𝑘𝑖=1𝜓_{𝑖,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,𝑘_𝑖),(𝑗,𝑘_𝑗)∈𝐿^𝑁,𝑚,𝑖≠𝑗, 𝜓_{𝑖,𝑘𝑖}=𝑁𝜓,𝑚_{𝑁⧵{𝑗}𝑗,𝑘𝑗},𝑘_𝑗,𝑣−𝜓_{𝑖,𝑘𝑖}𝑁,𝑚_{𝑁⧵{𝑗}},0,𝑣𝑁,𝑚_{𝑁⧵{𝑖}},𝑘_𝑖,𝑣−𝜓_{𝑗,𝑘𝑗}𝑁,𝑚_{𝑁⧵{𝑖}},0,𝑣.(3.1)(ii)2-strong balanced contributions (2SBC) if for each (𝑁,𝑚,𝑣)∈𝑀𝐶 and (𝑖,𝑘_𝑖),(𝑗,𝑘_𝑗)∈𝐿^𝑁,𝑚,𝑖≠𝑗, 𝜓_{𝑖,𝑘𝑖}=(𝑁𝜓,𝑚,_{𝑗,𝑘𝑗}𝑣)−𝜓_{𝑖,𝑘𝑖}𝑁,𝑚_{𝑁⧵{𝑗}},𝑘_𝑗−1,𝑣(𝑁,𝑚,𝑣)−𝜓_{𝑗,𝑘𝑗}𝑁,𝑚_{𝑁⧵{𝑖}},𝑘_𝑖−1,𝑣.(3.2)(iii)3-strong balanced contributions (3SBC) if for each (𝑁,𝑚,𝑣)∈𝑀𝐶 and (𝑖,𝑘_𝑖),(𝑗,𝑘_𝑗)∈𝐿^𝑁,𝑚,𝑖≠𝑗, 𝜓_{𝑖,𝑘𝑖}=𝑁𝜓,𝑚_{𝑁⧵{𝑗}𝑗,𝑘𝑗},𝑘_𝑗,𝑣−𝜓_{𝑖,𝑘𝑖}𝑁,𝑚_{𝑁⧵{𝑗}},𝑘_𝑗−1,𝑣𝑁,𝑚_{𝑁⧵{𝑖}},𝑘_𝑖,𝑣−𝜓_{𝑗,𝑘𝑗}𝑁,𝑚_{𝑁⧵{𝑖}},𝑘_𝑖−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,𝑘_𝑖)∈𝐿^𝑁,𝑚, 𝑗≠𝑚_𝑖, 𝜓_{𝑖,𝑘𝑖}𝑁,𝑚_{𝑁⧵{𝑖}},𝑘_𝑖,𝑣=𝜓_{𝑖,𝑘𝑖}𝑁,𝑚_{𝑁⧵{𝑖}},𝑘_𝑖+1,𝑣=⋯=𝜓_{𝑖,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,𝑘_𝑖)∈𝐿^𝑁,𝑚, 𝑘_𝑖≠𝑚_𝑖, 𝜓_{𝑖,𝑘𝑖}(𝑁,𝑚,𝑣)−𝜓_{𝑖,𝑘𝑖}𝑁,𝑚−𝑒^{𝑖},𝑣=𝜓_{𝑖,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 𝑘_𝑖=𝑚_𝑖 and 𝑘_𝑗=𝑚_𝑗. 2-upper balanced contributions (2UBC) only requires that 2SBC or 3SBC holds if 𝑘_𝑖=𝑚_𝑖 and 𝑘_𝑗=𝑚_𝑗. 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 [8–10]. Let (𝑁,𝑚,𝑣)∈𝑀𝐶, 𝑆⊆𝑁⧵{∅} and 𝜓 be a solution. (i)The 1-reduced game (𝑆,𝑚_𝑆,𝑣^𝜓1,𝑆) with respect to 𝜓 and 𝑆 is defined as, for all 𝛼∈𝑀^𝑆, 𝑣^𝜓1,𝑆(𝛼)=𝑣𝛼,𝑚_𝑁⧵𝑆−_{𝑖∈𝑁⧵𝑆}𝜓_{𝑖,𝑚𝑖}𝑁,𝛼,𝑚_𝑁⧵𝑆,𝑣.(3.6)(ii)The 2-reduced game (𝑆,𝑚_𝑆,𝑣^𝜓2,𝑆) with respect to 𝜓 and 𝑆 is defined as, for all 𝛼∈𝑀^𝑆, 𝑣^𝜓2,𝑆(𝛼)=𝑣𝛼,𝑚_𝑁⧵𝑆−_{𝑚𝑖∈𝑁⧵𝑆}^𝑖_𝑘_𝑖=1𝜓_{𝑖,𝑘𝑖}𝑁,𝛼,𝑚_𝑁⧵𝑆,𝑣.(3.7)(iii)𝜓 on 𝑀𝐶 satisfies 1-consistency (1CON) if for all (𝑁,𝑚,𝑣)∈𝑀𝐶, for all 𝑆⊆𝑁 and for all (𝑖,𝑘_𝑖)∈𝐿^{𝑆,𝑚𝑆}, 𝜓_{𝑖,𝑘𝑖}(𝑁,𝑚,𝑣)=𝜓_{𝑖,𝑘𝑖}(𝑆,𝑚_𝑆,𝑣^𝜓1,𝑆). (iv)𝜓 on 𝑀𝐶 satisfies 2-consistency (2CON) if for all (𝑁,𝑚,𝑣)∈𝑀𝐶, for all 𝑆⊆𝑁 and for all (𝑖,𝑘_𝑖)∈𝐿^{𝑆,𝑚𝑆}, 𝜓_{𝑖,𝑘𝑖}(𝑁,𝑚,𝑣)=𝜓_{𝑖,𝑘𝑖}(𝑆,𝑚_𝑆,𝑣^𝜓2,𝑆).

Remark 3.1. Hwang and Liao [8–10] 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 (𝑥_{𝑖,𝑘𝑖})_{(𝑖,𝑘𝑖)∈𝐿}^𝑁,𝑚∈ℝ^{𝐿𝑁,𝑚} where 𝑥_{𝑖,𝑘𝑖} denotes the payoff to player 𝑖 corresponding to his activity level 𝑘_𝑖 for all (𝑖,𝑘_𝑖)∈𝐿^𝑁,𝑚. A payoff vector 𝑥 of (𝑁,𝑚,𝑣) is 1-efficient (1EFF) if ∑_𝑖∈𝑁𝑥_{𝑖,𝑚𝑖}=𝑣(𝑚). 𝑥 is 2-efficient (2EFF) if ∑_𝑖∈𝑁∑_{𝑘𝑖∈𝑀+𝑖}𝑥_{𝑖,𝑘𝑖}=𝑣(𝑚). Moreover, the sets of 1-preimputations and 2-preimputations of (𝑁,𝑚,𝑣) are denoted by 𝑋𝑋¹(𝑁,𝑚,𝑣)=𝑥∈ℝ^{𝐿𝑁,𝑚}∣𝑥is1EFFin(𝑁,𝑚,𝑣)₂,(𝑁,𝑚,𝑣)=𝑥∈ℝ^{𝐿𝑁,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, 𝑆⊆𝑁, and let 𝜓 be a solution and 𝑥 a payoff vector. (i)The (1,𝜓)-reduced game (𝑆,𝑚_𝑆,𝑣^{𝑥,𝜓1,𝑆}) with respect to 𝑆, 𝑥, and 𝜓 is defined as, for all 𝛼∈𝑀^𝑆, 𝑣^{𝑥,𝜓1,𝑆}(𝛼)=⎧⎪⎨⎪⎩𝑣(𝑚)−^{𝑖∈𝑁⧵𝑆}𝑣_𝜓1,𝑆𝑥_{𝑖,𝑚𝑖},𝛼=𝑚_𝑆,(𝛼),otherwise.(3.9)(ii)The (2,𝜓)-reduced game (𝑆,𝑚_𝑆,𝑣^{𝑥,𝜓2,𝑆}) with respect to 𝑆, 𝑥, and 𝜓 is defined as, for all 𝛼∈𝑀^𝑆, 𝑣^{𝑥,𝜓2,𝑆}(𝛼)=⎧⎪⎨⎪⎩𝑣(𝑚)−^{𝑖∈𝑁⧵𝑆}_𝑘𝑖_∈𝑀+𝑣_𝜓2,𝑆𝑥_{𝑖,𝑘𝑖},𝛼=𝑚_𝑆,(𝛼),otherwise.(3.10)

Let (𝑁,𝑚,𝑣)∈𝑀𝐶, 𝑁≥3 and (𝑖,𝑘_𝑖)∈𝐿^𝑁,𝑚. Inspired by Maschler and Owen [7], we define 𝑓_{𝑖,𝑘𝑖}∶𝑋¹(𝑁,𝑚,𝑣)→ℝ, 𝑔_{𝑖,𝑘𝑖}∶𝑋²(𝑁,𝑚,𝑣)→ℝ, ℎ_{𝑖,𝑘𝑖}∶𝑋²(𝑁,𝑚,𝑣)→ℝ to be as follows:(i)𝑓_{𝑖,𝑘𝑖}(𝑥)=𝑥_{𝑖,𝑘𝑖}+𝑡⋅∑_{𝑗∈𝑁⧵{𝑖}}(Λ_{𝑖,𝑘𝑖}({𝑖,𝑗},𝑚_{𝑖,𝑗},𝑣^{𝑥,Λ1,{𝑖,𝑗}})−𝑥_{𝑖,𝑘𝑖}), (ii)𝑔_{𝑖,𝑘𝑖}(𝑥)=𝑥_{𝑖,𝑘𝑖}+𝑡⋅∑_{𝑗∈𝑁⧵{𝑖}}(Θ_{𝑖,𝑘𝑖}({𝑖,𝑗},𝑚_{𝑖,𝑗},𝑣^{𝑥,Θ2,{𝑖,𝑗}})−𝑥_{𝑖,𝑘𝑖}), (iii)ℎ_{𝑖,𝑘𝑖}(𝑥)=𝑥_{𝑖,𝑘𝑖}+𝑡⋅∑_{𝑗∈𝑁⧵{𝑖}}(Γ_{𝑖,𝑘𝑖}({𝑖,𝑗},𝑚_{𝑖,𝑗},𝑣^{𝑥,Γ2,{𝑖,𝑗}})−𝑥_{𝑖,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 (𝑓_{𝑖,𝑘𝑖}(𝑥))_{(𝑖,𝑘𝑖)∈𝐿}^𝑁,𝑚∈𝑋¹(𝑁,𝑚,𝑣) if 𝑥∈𝑋¹(𝑁,𝑚,𝑣), (𝑔_{𝑖,𝑘𝑖}(𝑥))_{(𝑖,𝑘𝑖)∈𝐿}^𝑁,𝑚∈𝑋²(𝑁,𝑚,𝑣), and (ℎ_{𝑖,𝑘𝑖}(𝑥))_{(𝑖,𝑘𝑖)∈𝐿}^𝑁,𝑚∈𝑋²(𝑁,𝑚,𝑣) if 𝑥∈𝑋²(𝑁,𝑚,𝑣).

Inspired by Maschler and Owen [7], we define correction functions 𝑓_{𝑖,𝑘𝑖},𝑔_{𝑖,𝑘𝑖},ℎ_{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 satisfyΨ𝑃,𝑚^,𝑢=Λ𝑃,𝑚^𝑥,,𝑢,(𝑁𝑃,,𝑚𝑚^,,𝑣𝑢)=∈𝑀𝑃,𝐶𝑚,_||𝑃||<||𝑁||,,𝑢.(3.11) On the basis of this Ψ, the members of a coalition 𝑆={𝑖,𝑗} will examine 𝑣^{𝑥,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 Λ_{𝑖,𝑘𝑖}(𝑆,𝑚_𝑆,𝑣^{𝑥,Λ1,𝑆}) in a manner which will be explained subsequently (see the definition of 𝑓_{𝑖,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 𝑔_{𝑖,𝑘𝑖} and ℎ_{𝑖,𝑘𝑖}.

Theorem 3.2. Let (𝑁,𝑚,𝑣)∈𝑀𝐶 and 𝑥∈𝑋¹(𝑁,𝑚,𝑣). Define 𝑥⁰=𝑥,𝑥¹=(𝑓_{𝑖,𝑘𝑖}(𝑥⁰))_{(𝑖,𝑘𝑖)∈𝐿}^𝑁,𝑚,…,𝑥^𝑞=(𝑓_{𝑖,𝑘𝑖}(𝑥^𝑞−1))_{(𝑖,𝑘𝑖)∈𝐿}^𝑁,𝑚 for all 𝑞∈ℕ. (1)If 0<𝑡<4/|𝑁|, then for all 𝑖∈𝑁 and for all 𝑥∈𝑋¹(𝑁,𝑚,𝑣), {𝑥^{𝑞𝑖,𝑚}_𝑖}^∞𝑞=1 converges to Λ_{𝑖,𝑚𝑖}(𝑁,𝑚,𝑣).(2)If 0<𝑡<4/|𝑁|, then for all (𝑖,𝑘_𝑖)∈𝐿^𝑁,𝑚 and for all 𝑥∈𝑋¹(𝑁,(𝑚_{𝑁⧵{𝑖}},𝑘_𝑖),𝑣), {𝑥^{𝑞𝑖,𝑘}_𝑖}^∞𝑞=1 converges to Λ_{𝑖,𝑘𝑖}(𝑁,𝑚,𝑣).

Proof. Fix (𝑁,𝑚,𝑣)∈𝑀𝐶 and 𝑥∈𝑋¹(𝑁,𝑚,𝑣). To prove (1), let 𝑖,𝑗∈𝑆(𝑚) and 𝑆={𝑖,𝑗}. By 1EFF and 1UBC of Λ, and definitions of 𝑣^Λ1,𝑆 and 𝑣^{𝑥,Λ1,𝑆}, ΛΛ_{𝑖,𝑚𝑖}𝑆,𝑚^𝑆,𝑣^{𝑥,Λ1,𝑆}+Λ^𝑗,𝑚_𝑗𝑆,𝑚^𝑆,𝑣^{𝑥,Λ1,𝑆}=𝑥^𝑖,𝑚_𝑖+𝑥_{𝑗,𝑚𝑗},(by1EFFofΛ),(3.12)_{𝑖,𝑚𝑖}=Λ𝑆,𝑚^𝑆_{𝑖,𝑚𝑖},𝑣^{𝑥,Λ1,𝑆}−Λ^𝑗,𝑚_𝑗𝑆,𝑚^𝑆,𝑣^{𝑥,Λ1,𝑆}=Λ_{𝑖,𝑚𝑖}𝑆,𝑚^{𝑆⧵{𝑗}},0,𝑣^{𝑥,Λ1,𝑆}−Λ^𝑗,𝑚_𝑗𝑆,𝑚^{𝑆⧵{𝑖}},0,𝑣^{𝑥,Λ1,𝑆}(by1UBCofΛ)=Λ_{𝑖,𝑚𝑖}𝑆,𝑚^{𝑆⧵{𝑗}},0,𝑣^Λ1,𝑆−Λ^𝑗,𝑚_𝑗𝑆,𝑚^{𝑆⧵{𝑖}},0,𝑣^Λ1,𝑆bydefinitionof𝑣^Λ1,𝑆𝑆,𝑚^𝑆,𝑣^Λ1,𝑆−Λ^𝑗,𝑚_𝑗𝑆,𝑚^𝑆,𝑣^Λ1,𝑆.(by1UBCofΛ).(3.13) Therefore, 2⋅Λ_{𝑖,𝑚𝑖}𝑆,𝑚^𝑆,𝑣^{𝑥,Λ1,𝑆}−𝑥^𝑖,𝑚_𝑖=Λ^𝑖,𝑚_𝑖𝑆,𝑚^𝑆,𝑣^Λ1,𝑆−Λ^𝑗,𝑚_𝑗𝑆,𝑚^𝑆,𝑣^Λ1,𝑆−𝑥^𝑖,𝑚_𝑖+𝑥_{𝑗,𝑚𝑗}.(3.14)
By definition of 𝑓, 1CON and 1EFF of Λ and (3.14), 𝑓_{𝑖,𝑚𝑖}(𝑥)=𝑥_{𝑖,𝑚𝑖}+𝑡2⋅⎡⎢⎢⎣_{𝑗∈𝑁⧵{𝑖}}Λ_{𝑖,𝑚𝑖}{𝑖,𝑗},𝑚^{𝑖,𝑗},𝑣^{Λ1,{𝑖,𝑗}}−_{𝑗∈𝑁⧵{𝑖}}𝑥_{𝑖,𝑚𝑖}−_{𝑗∈𝑁⧵{𝑖}}Λ_{𝑗,𝑚𝑗}{𝑖,𝑗},𝑚^{𝑖,𝑗},𝑣^{Λ1,{𝑖,𝑗}}+_{𝑗∈𝑁⧵{𝑖}}𝑥_{𝑗,𝑚𝑗}bydefinitionof𝑓_{𝑖,𝑚𝑖}⎤⎥⎥⎦=𝑥_{𝑖,𝑚𝑖}andequation(3.14)+𝑡2⋅⎡⎢⎢⎣_{𝑘∈𝑁⧵{𝑖}}Λ_{𝑖,𝑚𝑖}(𝑁,𝑚,𝑣)−||𝑁||−1𝑥_{𝑖,𝑚𝑖}=𝑥_{𝑖,𝑚𝑖}−_{𝑗∈𝑁⧵{𝑖}}Λ_{𝑗,𝑚𝑗}(𝑁,𝑚,(𝑣b)y+1C𝑣O(N𝑚)o−fΛ𝑥an_{𝑖,𝑚𝑖}d⎤⎥⎥⎦1EFFof𝑥)+𝑡2⋅||𝑁||−1Λ^{𝑖,𝑚𝑖}(𝑁,𝑚,𝑣)−||𝑁||−1𝑥_{𝑖,𝑚𝑖}=𝑥_{𝑖,𝑚𝑖}−𝑣(𝑚)−Λ_{𝑖,𝑚𝑖}(𝑁,𝑚,𝑣)+(𝑣by(𝑚1)E−FF𝑥of_{𝑖,𝑚𝑖}Λ)+||𝑁||⋅𝑡2⋅Λ^{𝑖,𝑚𝑖}(𝑁,𝑚,𝑣)−𝑥_{𝑖,𝑚𝑖}.(3.15) Hence, for all 𝑞∈ℕ, 1−||𝑁2||⋅𝑡^𝑞+1Λ^𝑖,𝑚_𝑖(𝑁,𝑚,𝑣)−𝑥^{𝑞𝑖,𝑚}_𝑖==ΛΛ_{𝑖,𝑚𝑖}(_{𝑖,𝑚𝑖}𝑁,𝑚,𝑣)−𝑓_{𝑖,𝑚𝑖}(𝑥^𝑞)(𝑁,𝑚,𝑣)−𝑥^{𝑞+1𝑖,𝑚}_𝑖.(3.16) If 0<𝑡<4/|𝑁|, then −1<(1−(|𝑁|⋅𝑡)/2)<1 and {𝑥^{𝑞𝑖,𝑚}_𝑖}^∞𝑞=1 converges to Λ_{𝑖,𝑚𝑖}(𝑁,𝑚,𝑣).
To prove (2), by 1 of this theorem, if 0<𝑡<4/|𝑁|, then for all (𝑖,𝑘_𝑖)∈𝐿^𝑁,𝑚 and for all 𝑥∈𝑋¹(𝑁,(𝑚_{𝑁⧵{𝑖}},𝑘_𝑖),𝑣), {𝑥^{𝑞𝑖,𝑘}_𝑖}^∞𝑞=1 converges to Λ_{𝑖,𝑘𝑖}(𝑁,(𝑚_{𝑁⧵{𝑖}},𝑘_𝑖),𝑣). By IIE of Λ, {𝑥^{𝑞𝑖,𝑘}_𝑖}^∞𝑞=1 converges to Λ_{𝑖,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 𝑦∈𝑋²(𝑁,𝑚,v). Define 𝑦⁰=𝑦,𝑦¹=(ℎ_{𝑖,𝑘𝑖}(𝑦⁰))_{(𝑖,𝑘𝑖)∈𝐿}^𝑁,𝑚,…,𝑦^𝑞=(ℎ_{𝑖,𝑘𝑖}(𝑦^𝑞−1))_{(𝑖,𝑘𝑖)∈𝐿}^𝑁,𝑚 for all 𝑞∈ℕ.
(1)If 0<𝛼<4/|𝑁|, then for all 𝑖∈𝑁 and for all 𝑦∈𝑋²(𝑁,𝑚,𝑣), {∑^𝑚𝑖_𝑘𝑖=1𝑦^{𝑞𝑖,𝑘}_𝑖}^∞𝑞=1 converges to ∑^𝑚𝑖_𝑘𝑖=1Γ_{𝑖,𝑘𝑖}(𝑁,𝑚,𝑣).(2)If 0<𝑡<4/|𝑁|, then for all (𝑖,𝑘_𝑖)∈𝐿^𝑁,𝑚 and for all 𝑦∈𝑋²(𝑁,(𝑚_{𝑁⧵{𝑖}},𝑘_𝑖),𝑣), {∑^𝑘𝑖_𝑡𝑖=1𝑦^{𝑞𝑖,𝑡}_𝑖}^∞𝑞=1 converges to ∑^𝑘𝑖_𝑡𝑖=1Γ_{𝑖,𝑡𝑖}(𝑁,𝑚,𝑣).(3)If 0<𝛼<4/|𝑁|, then for all (𝑖,𝑘_𝑖)∈𝐿^𝑁,𝑚 and for all payoff vectors 𝑦 with 𝑦_{𝑖,𝑘𝑖}=𝑣(𝑚_{𝑁⧵{𝑖}},𝑘_𝑖)−𝑣(𝑚_{𝑁⧵{𝑖}},𝑘_𝑖−1), {𝑦^{𝑞𝑖,𝑘}_𝑖}^∞𝑞=1 converges to Γ_{𝑖,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 𝑧∈𝑋²(𝑁,𝑚,𝑣). Define 𝑧⁰=𝑧,𝑧¹=(𝑔_{𝑖,𝑘𝑖}(𝑧⁰))_{(𝑖,𝑘𝑖)∈𝐿}^𝑁,𝑚,…,𝑧^𝑞=(𝑔_{𝑖,𝑘𝑖}(𝑧^𝑞−1))_{(𝑖,𝑘𝑖)∈𝐿}^𝑁,𝑚 for all 𝑞∈ℕ.
(1)If 0<𝛼<4/|𝑁|, then for all 𝑖∈𝑁 and for all 𝑧∈𝑋²(𝑁,𝑚,𝑣), {∑^𝑚𝑖_𝑘𝑖=1𝑧^{𝑞𝑖,𝑘}_𝑖}^∞𝑞=1 converges to ∑^𝑚𝑖_𝑘𝑖=1Θ_{𝑖,𝑘𝑖}(𝑁,𝑚,𝑣).(2)If 0<𝑡<4/|𝑁|, then for all (𝑖,𝑘_𝑖)∈𝐿^𝑁,𝑚 and for all 𝑧∈𝑋²(𝑁,(𝑚_{𝑁⧵{𝑖}},𝑘_𝑖),𝑣), {∑^𝑘𝑖_𝑡𝑖=1𝑧^{𝑞𝑖,𝑡}_𝑖}^∞𝑞=1 converges to ∑^𝑘𝑖_𝑡𝑖=1Θ_{𝑖,𝑡𝑖}(𝑁,𝑚,𝑣).(3)If 0<𝛼<4/|𝑁|, then for all (𝑖,𝑘_𝑖)∈𝐿^𝑁,𝑚 and for all payoff vectors 𝑧 with 𝑧_{𝑖,𝑘𝑖}=𝑣(𝑚_{𝑁⧵{𝑖}},𝑘_𝑖)−𝑣(𝑚_{𝑁⧵{𝑖}},𝑘_𝑖−1), {𝑧^{𝑞𝑖,𝑘}_𝑖}^∞𝑞=1 converges to Θ_{𝑖,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 [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,𝑚,𝑣)∈𝑀𝐶, 𝑆⊆𝑁⧵{∅}, 𝜓 be a solution and 𝛾∈𝑀^+𝑁⧵𝑆. (i)The 1-player-action reduced game (𝑆,𝑚_𝑆,𝑣^{1,𝜓𝑆,𝛾}) with respect to 𝑆, 𝛾 and 𝜓 is defined as for all 𝛼∈𝑀^𝑆, 𝑣^{1,𝜓𝑆,𝛾}(𝛼)=𝑣(𝛼,𝛾)−_{𝑗∈𝑁⧵𝑆}𝜓_{𝑗,𝛾𝑗}(𝑁,(𝛼,𝛾),𝑣).(4.1)(ii)The 2-player-action reduced game (𝑆,𝑚_𝑆,𝑣^{2,𝜓𝑆,𝛾}) with respect to 𝑆, 𝛾 and 𝜓 is defined as for all 𝛼∈𝑀^𝑆, 𝑣^{2,𝜓𝑆,𝛾}(𝛼)=𝑣(𝛼,𝛾)−_{𝛾𝑗∈𝑁⧵𝑆}^𝑗_𝑘_𝑗=1𝜓_{𝑗,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 𝛾∈𝑀^+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 𝜓 be a solution on 𝑀𝐶. (i)𝜓 satisfies 1-player-action consistency (1PACON) if for all (𝑁,𝑚,𝑣)∈𝑀𝐶, for all 𝑆⊆𝑁⧵{∅}, for all (𝑖,𝑘_𝑖)∈𝐿^{𝑆,𝑚𝑆} and for all 𝛾∈𝑀^+𝑁⧵𝑆, 𝜓_{𝑖,𝑘𝑖}(𝑁,(𝑚_𝑆,𝛾),𝑣)=𝜓_{𝑖,𝑘𝑖}(𝑆,𝑚_𝑆,𝑣^{1,𝜓𝑆,𝛾}). (ii)𝜓 satisfies 2-player-action consistency (2PACON) if for all (𝑁,𝑚,𝑣)∈𝑀𝐶, for all 𝑆⊆𝑁⧵{∅}, for all (𝑖,𝑘_𝑖)∈𝐿^{𝑆,𝑚𝑆} and for all 𝛾∈𝑀^+𝑁⧵𝑆, 𝜓_{𝑖,𝑘𝑖}(𝑁,(𝑚_𝑆,𝛾),𝑣)=𝜓_{𝑖,𝑘𝑖}(𝑆,𝑚_𝑆,𝑣^{2,𝜓𝑆,𝛾}).

Remark 4.2. Let (𝑁,𝑚,𝑣)∈𝑀𝐶, 𝑆⊆𝑁⧵{∅} and 𝜓 be a solution. Let 𝛾=𝑚_𝑁⧵𝑆, by definitions of reduced games and player-action reduced games, 𝑣^{1,𝜓𝑆,𝑚}_𝑁⧵𝑆=𝑣^𝜓1,𝑆 and 𝑣^{2,𝜓𝑆,𝑚}_𝑁⧵𝑆=𝑣^𝜓2,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,𝑚_𝑆,𝑣^{1,Λ𝑆,𝛾}), (𝑆,𝑚_𝑆,𝑣^{2,Γ𝑆,𝛾}), and (𝑆,𝑚_𝑆,𝑣^{2,Θ𝑆,𝛾}).

Lemma 4.3. Let (𝑁,𝑚,𝑣)∈𝑀𝐶, (𝑆,𝑚_𝑆,𝑣^{1,Λ𝑆,𝛾}) be a 1-player-action reduced game of 𝑣 with respect to 𝑆, 𝛾, and the solution Λ and let (𝑆,𝑚_𝑆,𝑣^{2,Γ𝑆,𝛾}) be a 2-player-action reduced game of 𝑣 with respect to 𝑆, 𝛾 and the solution Γ. If 𝑣=∑_{𝛼∈𝑀𝑁⧵{0𝑁}}𝑎^𝛼(𝑣)⋅𝑢^𝛼𝑁, then 𝑣^{1,Λ𝑆,𝛾} can be expressed as 𝑣^{1,Λ𝑆,𝛾}=𝑣^{2,Γ𝑆,𝛾}=∑_{𝛼∈𝑀𝑆⧵{0𝑆}}𝑎^𝛼(𝑣^{2,Γ𝑆,𝛾})⋅𝑢^𝛼𝑆, where for all 𝛼∈𝑀^𝑆, 𝑎^𝛼𝑣^{1,Λ𝑆,𝛾}=𝑎^𝛼𝑣^{2,Γ𝑆,𝛾}=_𝜆≤𝛾||𝑆(𝛼)||||𝑆(𝛼)||+||𝑆(𝜆)||⋅𝑎^(𝛼,𝜆)(𝑣).(4.3)

Proof. Let (𝑁,𝑚,𝑣)∈𝑀𝐶, 𝑆⊆𝑁 with 𝑆≠∅ and 𝛾∈𝑀^+𝑁⧵𝑆. For all 𝛼∈𝑀^𝑆, 𝑣^{2,Γ𝑆,𝛾}(𝛼)=𝑣(𝛼,𝛾)−_{𝛾𝑗∈𝑁⧵𝑆}^𝑗_𝑘_𝑗=1Γ_{𝑗,𝑘𝑗}(𝑁,(𝛼,𝛾),𝑣).(4.4) By 2EFF of Γ, 𝑣^{2,Γ𝑆,𝛾}(0_𝑆)=0. For all 𝛼∈𝑀^𝑆 with 𝛼≠0_𝑆, (4.2)=_{𝛼𝑗∈𝑆(𝛼)}^𝑗_𝑘_𝑗=1=Γ_{𝑗,𝑘𝑗}(𝑁,(𝛼,𝛾),𝑣)_{𝛼𝑗∈𝑆(𝛼)𝑗𝑘}_𝑗=1_{𝛽𝛽≤(𝛼,𝛾)𝑗=𝑘𝑗}𝑎^𝛽(𝑣)=||𝑆(𝛽)_{𝑗∈𝑆(𝛼)}||⎡⎢⎢⎢⎢⎣_{𝛽𝛽≤(𝛼,𝛾)𝑗=1}𝑎^𝛽(𝑣)||𝑆(𝛽)||+⋯+_{𝛽𝛽≤(𝛼,𝛾)𝑗=𝛼𝑗}𝑎^𝛽(𝑣)=||𝑆(𝛽)_{𝑗∈𝑆(𝛼)}||⎤⎥⎥⎥⎥⎦⎡⎢⎢⎢⎢⎣_{𝜂𝜂≤𝛼𝑗=1}_𝜆≤𝛾𝑎^(𝜂,𝜆)(𝑣)||𝑆(𝜂)||+||𝑆(𝜆)||+⋯+_{𝜂𝜂≤𝛼𝑗=𝛼𝑗}_𝜆≤𝛾𝑎^(𝜂,𝜆)(𝑣)=||𝑆(𝜂)||+||𝑆(𝜆)_𝜂≤𝛼||⎤⎥⎥⎥⎥⎦_𝜆≤𝛾||𝑆(𝜂)||||𝑆(𝜂)||+||𝑆(𝜆)||⋅𝑎^(𝜂,𝜆)(𝑣).(4.5) Set 𝑎^𝜂𝑣^{2,Γ𝑆,𝛾}=_𝜆≤𝛾||𝑆(𝜂)||||𝑆(𝜂)||+||𝑆(𝜆)||⋅𝑎^(𝜂,𝜆)(𝑣).(4.6) By (4.5), for all 𝛼∈𝑀^𝑆, 𝑣^{2,Γ𝑆,𝛾}(𝛼)=_𝜂≤𝛼_𝜆≤𝛾||𝑆(𝜂)||||𝑆(𝜂)||+||𝑆(𝜆)||⋅𝑎^(𝜂,𝜆)(𝑣)=_𝜂≤𝛼𝑎^𝜂𝑣^{2,Γ𝑆,𝛾}.(4.7) Hence 𝑣^{2,Γ𝑆,𝛾} can be expressed to be 𝑣^{2,Γ𝑆,𝛾}=∑_{𝛼∈𝑀𝑆⧵{0𝑆}}𝑎^𝛼(𝑣^{1,Γ𝑆,𝛾})⋅𝑢^𝛼𝑆. By Definition 2.1 and the definitions of 𝑣^{1,Λ𝑆,𝛾} and 𝑣^{2,Γ𝑆,𝛾}, for all 𝛼∈𝑀^𝑁, 𝑣^{2,Γ𝑆,𝛾}(𝛼)=𝑣(𝛼,𝛾)−_{𝛾𝑗∈𝑁⧵𝑆}^𝑗_𝑘_𝑗=1=𝑣(𝛼,𝛾)−_{𝑗∈𝑁⧵𝑆}Γ_{𝑗,𝑘𝑗}(𝑁,(𝛼,𝛾),𝑣)=𝑣_{1,Λ𝑆,𝛾}Λ_{𝑗,𝛾𝑗}(𝑁,(𝛼,𝛾),𝑣)(𝛼).(4.8) Hence, for each 𝛼∈𝑀^𝑆, 𝑣^{1,Λ𝑆,𝛾}(𝛼)=𝑣^{2,Γ𝑆,𝛾}(𝛼) and 𝑎^𝛼(𝑣^{1,Λ𝑆,𝛾})=𝑎^𝛼(𝑣^{2,Γ𝑆,𝛾}).

Lemma 4.4. Let (𝑁,𝑚,𝑣)∈𝑀𝐶 and (𝑆,𝑚_𝑆,𝑣^{2,Θ𝑆,𝛾}) be a 2-player-action reduced game of 𝑣 with respect to 𝑆, 𝛾, and the solution Θ. If 𝑣=∑_{𝛼∈𝑀𝑁⧵{0𝑁}}𝑎^𝛼(𝑣)⋅𝑢^𝛼𝑁, then 𝑣^{2,Θ𝑆,𝛾} can be expressed to be 𝑣^{2,Θ𝑆,𝛾}=∑_{𝛼∈𝑀𝑆⧵{0𝑆}}𝑎^𝛼(𝑣^{2,Θ𝑆,𝛾})⋅𝑢^𝛼𝑆, where for all 𝛼∈𝑀^𝑆, 𝑎^𝛼𝑣^{2,Θ𝑆,𝛾}=_𝜆≤𝛾‖𝛼‖‖𝛼‖+‖𝜆‖⋅𝑎^(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, 𝑆⊆𝑁 with 𝑆≠∅ and 𝛾∈𝑀^+𝑁⧵𝑆. First, we show that the solution Λ satisfies 1PACON. By Definition 2.1 and Lemma 4.3, for all (𝑖,𝑘_𝑖)∈𝐿^{𝑆,𝑚𝑆}, Λ_{𝑖,𝑘𝑖}𝑆,𝑚^𝑆,𝑣^{1,Λ𝑆,𝛾}=_{𝛼∈𝑀𝑆𝛼𝑖≤𝑘𝑖}𝑎^𝛼𝑣^{1,Λ𝑆,𝛾}=||𝑆(𝛼)_{𝛼∈𝑀𝑆𝛼𝑖≤𝑘𝑖}||1||𝑆(𝛼)||⋅_𝜆≤𝛾||𝑆(𝛼)||||𝑆(𝛼)||+||𝑆(𝜆)||⋅𝑎^(𝛼,𝜆)=(𝑣)_{𝛽≤(𝑚𝑆𝛽,𝛾)𝑖≤𝑘𝑖}𝑎^𝛽(𝑣)=Λ_{𝑖,𝑘𝑖}||𝑆(𝛽)||𝑁,𝑚_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 [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.