Cross invariance, the Shapley value, and the Shapley–Shubik power index

Abstract

In this paper we propose a simple axiom which, along with the axioms of additivity (transfer) and dummy player, characterizes the Shapley value (the Shapley–Shubik power index) on the domain of TU (simple) games. The new axiom, cross invariance, demands payoff invariance on symmetric players across “quasi-symmetric games,” that is, games where excluding null players, all players are symmetric. Additionally, we demonstrate that the axiom of additivity can be replaced by a new axiom called strong monotonicity, or it can be completely dropped if a stronger version of cross invariance is employed. We also show that the weighted Shapley values can be characterized using a weighted variant of cross invariance. Efficiency is derived rather than assumed in our characterizations. This fresh perspective contributes to a deeper understanding of the Shapley value and its applicability.

1 Introduction

This paper contributes to advancing our understanding of the Shapley value (Shapley 1951, 1953a), one of the most prominent solution concepts in cooperative game theory. A cooperative game specifies for each coalition of players the highest surplus they can obtain through cooperation. A value in a cooperative game makes a recommendation on allocating the gains from cooperation to players in a fair way. The axiomatic approach appeals to normative rationality postulates (axioms) to determine the value of a game. In the classical characterization of the Shapley value, for example, the Shapley value is shown to be the unique value that satisfies the following four axioms: dummy player (or null player), additivity, symmetry, and efficiency. Various characterizations of the Shapley value based on different fairness desiderata have been proposed since Shapley’s seminal work. Our paper contributes to the literature in that vein.

The rational foundation of a value is manifested by the axioms used to characterize it. Consider the four axioms used in the classical characterization of the Shapley value. Dummy player requires that if a player does not contribute to the creation of surplus in any coalition, then he should not benefit from participating in the game. Additivity requires that the value of the sum of two games is the sum of the values of the two games. Symmetry requires that players who contribute to the game in the same manner should receive equal payoffs. It also implies that no information beyond the game’s description is relevant in determining the distribution of the surplus. Lastly, efficiency mandates that the sum of the payoffs of all the players equals the worth of the grand coalition.

Using efficiency to characterize a value raises several issues. First, efficiency is an assumption on collective rationality. Imposing a collective rationality property on a value directly, to some extent, beats the purpose of the exercise, as the axiomatic approach aims to derive collective behavior based on fundamental rationalities. Shapley himself criticizes the use of efficiency in his characterization (Shapley 1951, p. 5):

“In the general-sum case, this axiom involves an assumption about the “group rationality” of the players—namely that they will behave so as to maximize their common gain. In general it is better to steer clear of such assumptions in the axiom schema since our goal is, after all, to derive (or delimit) an as yet unknown principle of group rationality. Thus, the generality of our present work is again weakened for the non-constant-sum case.”

The second issue is that for cooperative games outside the realm of surplus division, values can have diverse and context-dependent meanings. In such environments, the implication of efficiency becomes nebulous. For example, on the domain of simple (voting) games, a value (called a power index) measures players’ a priori voting powers in a game, and efficiency normalizes the total voting power to 1 in all voting games. As pointed out by Einy and Haimanko (2011), such a normalization is innocuous if it is to assess the relative power of each player in a game, but imposing it uniformly across all games would imply rescaling individual power differently in different games, making comparisons of individual voting power across games unclear.

The third issue is that the Shapley value is frequently used as a fair allocation rule in strategic games. For instance, the Myerson value, defined as the Shapley value of the graph-restricted game, is commonly employed to determine how surplus should be divided in strategic network formation games. It is the unique value that satisfies two axioms: efficiency and balanced contributions (Myerson 1980). However, using an efficiency-based rational foundation to justify the Shapley value as a fair division rule in strategic games is unconvincing, as many strategic games can result in inefficient equilibrium outcomes, such as the prisoner’s dilemma.

Nevertheless, dropping efficiency from Shapley’s system of axioms significantly expands the set of feasible values. Dubey et al. (1981) introduce the concept of semivalues, a class of symmetric positive linear operators on a space of games. Each semivalue satisfies dummy player, additivity, and symmetry, and can be supported (i.e., uniquely characterized) by a probability measure on coalition formation. There exist continuously many semivalues, and the Shapley value, being the only efficient one, corresponds to just one particular probability measure on coalition formation. In other words, without the efficiency axiom, the Shapley value no longer remains a unique recommendation.

One may ask, without relying on efficiency, whether the uniqueness of the Shapley value can be restored by making minimal changes to Shapley’s system of axioms. Answering this question would provide a deeper understanding of the rational foundation of the Shapley value and its distinguishing features compared to other solution concepts. It would also help evaluate the robustness of the Shapley value. Clearly this question is not trivial in light of Dubey et al. (1981)’s characterization of semivalues. One has to find a compelling new axiom to distinguish the Shapley value from the continuously many feasible semivalues.

We show that the answer to the above question is positive. In particular, we demonstrate that to establish the uniqueness of the Shapley value, it is not so much an issue of whether efficiency is assumed or not, but more about how symmetric players should be treated across games. The axiom of symmetry, which is a property of equal-treatment-of-equals, is probably one of the least controversial axioms ever proposed in cooperative game theory. But it is hardly the only feasible property of equal-treatment-of-equals. We propose a new axiom to replace the axiom of symmetry to characterize the Shapley value.

Define a quasi-symmetric game as a game with a binary coalition partition structure where one coalition consists of symmetric players and the other consists of null players. The new axiom, termed cross invariance, demands that the payoffs of symmetric players remain invariant across quasi-symmetric games with the same coalition partition structure and grand-coalition worth. While both symmetry and cross invariance are based on the principle of equal treatment of equals, they are two distinctly different properties: symmetry focuses on within-game invariance, while cross invariance addresses between-game invariance.

We show that on the domain of transferable utility (TU) games, the Shapley value is the unique value that satisfies dummy player, additivity, and cross invariance (Proposition 1). The axiom of additivity can be dropped when a stronger version of cross invariance is employed (Proposition 2), or it can be replaced by a marginality axiom (Propositions 3 and 4). Additionally, we establish that the weighted Shapley values can be characterized using a weight-adjusted version of cross invariance (Proposition 5). When the focus is on the domain of simple games, we show that the Shapley–Shubik power index, i.e., the Shapley value restricted to the domain of simple games, is the unique power index that satisfies dummy player, transfer, and cross invariance (Proposition 6).

It is noteworthy that the efficiency axiom is not used in all of our characterizations. As a matter of fact, dummy player is the only axiom in our characterizations that encompasses a mild aspect of efficiency. Thus, replacing symmetry by cross invariance in Shapley’s original characterization, we are able to derive rather than assume the efficiency property in the Shapley value. Or, putting it differently, while symmetry fails to differentiate between efficient and inefficient semivalues, cross invariance successfully accomplishes this task.

There are a few papers that characterize the Shapley value or the Shapley–Shubik power index without using the axiom of efficiency. Einy and Haimanko (2011) introduce the gain–loss axiom as a replacement for efficiency in characterizing the Shapley–Shubik power index. This axiom stipulates that any gain in power for a player in a voting game must come at the expense of another player, making it a weaker form of efficiency. Casajus (2014) shows that the gain–loss axiom can be used to characterize the Shapley value along with dummy player and either the fairness axiom (van den Brink 2001) or differential marginality (Casajus 2011). Blair and McLean (1990) propose an axiom that requires players to be indifferent between all symmetric simple games where the minimal winning coalitions have the same size. The axiom is weaker than cross invariance. It is also weaker than the combination of symmetry and the gain–loss axiom. However, besides dummy player and transfer, the axiom has to be combined with symmetry to characterize the Shapley–Shubik power index. In contrast, cross invariance is the only equal-treatment-of-equals property utilized in our characterizations. Other characterizations of the Shapley value without efficiency include Roth (1977), Hamiache (2001), Laruelle and Valenciano (2001), and Béal et al. (2015).

2 Basic definitions

Let $N\equiv \{1,\ldots ,n\}$ be the set of players. A coalition is a subset $S\subset N.$ $\emptyset$ is the empty coalition. The cardinality of S is denoted by |S|. Denote the collection of all coalitions by $2^{N}$. Then a game on N is given by a (characteristic) function $v:2^{N}\rightarrow {\mathbb {R}}$ with $v(\emptyset )=0$. v(S) is called the worth of coalition S. The space of all games on N is denoted by ${\mathbb {V}}\equiv \{ v:2^{N}\rightarrow {\mathbb {R}} |v(\emptyset )=0\}$. Given v,  $w\in {\mathbb {V}}$ and $\alpha \in {\mathbb {R}},$ the games $v+w$ and $\alpha v$ are given by $(v+w)(S)=v(S)+w(S)$ and $(\alpha v)(S)=\alpha \cdot v(S)$ for all $S\in 2^{N}.$ The marginal contribution of $i\in N$ to $S\subset N\backslash \{i\}$ in game $v\in {\mathbb {V}}$ is $v(S\cup \{i\})-v(S).$ Player $i\in N$ is called a dummy player in $v\in {\mathbb {V}}$ if $v(S\cup \{i\})-v(S)=v(\{i\})$ for all $S\subset N\backslash \{i\}.$ Player $i\in N$ is called a null player in $v\in {\mathbb {V}}$ if $v(S\cup \{i\})=v(S)$ for all $S\subset N\backslash \{i\},$ i.e., i is a dummy player in v and $v(\{i\})=0.$ Players i,  $j\in N$ are called symmetric in v if $v(S\cup \{i\})=v(S\cup \{j\})$ for all $S\subset N\backslash \{i,j\}.$ $S\in 2^{N}$ is called a symmetric coalition in v if all players in S are symmetric in v. $S\in 2^{N}$ is called a null coalition in v if all players in S are null players in v. A game $v\in {\mathbb {V}}$ is called quasi-symmetric if there exists a coalition $S\in 2^{N}$ such that S is symmetric and $N\backslash S$ is null in v. A coalition $S\in 2^{N}$ is called a carrier in v if $v(T)=v(S\cap T)$ for any coalition $T\in 2^{N}.$

Given $T\in 2^{N}\backslash \{\emptyset \},$ the unanimity game $u_{T}$ is such that

$$\begin{aligned} u_{T}(S)=\left\{ \begin{array}{cc} 1 &{} \text {if }T\subset S \\ 0 &{} \text {otherwise} \end{array} \right. . \end{aligned}$$

(1)

A value is a function $\Phi :{\mathbb {V}}\rightarrow {\mathbb {R}} ^{n},$ which associates a vector in ${\mathbb {R}} ^{n}$ to every game $v\in {\mathbb {V}}$. $\Phi _{i}(v)$ is player i’s value or payoff in the game v.

In the following we list some standard axioms in the cooperative game literature.

Efficiency (EFF): For all $v\in {\mathbb {V}},$ $\sum _{i\in N}\Phi _{i}(v)=v(N).$

Dummy Player (DP): For all $v\in {\mathbb {V}}$ and all $i\in N$ such that i is a dummy player in v,  $\Phi _{i}(v)=v(\{i\}).$

Null Player (NP): For all $v\in {\mathbb {V}}$ and all $i\in N$ such that i is a null player in v,  $\Phi _{i}(v)=0.$

Symmetry (SYM): For all $v\in {\mathbb {V}}$ and all symmetric players $i,j\in N$ in v,  $\Phi _{i}(v)=\Phi _{j}(v).$

Additivity (ADD): For all $v,w\in {\mathbb {V}}$, $\Phi (v+w)=\Phi (v)+\Phi (w).$

Carrier (CA): For all $v\in {\mathbb {V}}$ and all carrier $S\in 2^{N}$ in v,  $\sum _{i\in S}\Phi _{i}(v)=v(S).$

Note that NP is weaker than DP, and CA is equivalent to the combination of EFF and NP.

3 The Shapley value

The Shapley value, $Sh:{\mathbb {V}}\rightarrow {\mathbb {R}} ^{n},$ is a weighted sum of the players’ marginal contributions:

$$\begin{aligned} Sh_{i}(v)=\sum _{S\subset N\backslash \{i\}}\frac{| S| !(n-|S|-1)!}{n!}(v(S\cup \{i\})-v(S)) \end{aligned}$$

(2)

for each game $v\in {\mathbb {V}}$ and for each player $i\in N.$

As alluded to earlier, the classical characterization of the Shapley value involves four axioms: DP or NP, ADD, SYM, and EFF. Let us take a look of the normative interpretation of SYM. Recall that the Aristotelian principle of equality, one of the fundamental principles on distributive justice, states that like cases should be treated alike. SYM requires that symmetric players in a game should be treated equally, and therefore it can be viewed as an application of the Aristotelian principle of equality to players in a given game-symmetric players are like cases here. We would like to introduce an axiom that applies the Aristotelian principle of equality from a different perspective. Specifically, we take a bird’s eye view on the space of all games and look for like games to apply the principle. Fix the set of players and consider two games with the same grand-coalition worth. Suppose in each of the two games, all players are symmetric (their marginal contributions may differ in these two games). Given that these two games are symmetric in the sense that from each player’s perspective, his contributions relative to others remain unchanged in these two games, he should be treated equally in these two games. Stretch the logic a little further and consider again two games with the same grand-coalition worth. Suppose the two games can be commonly partitioned into two coalitions where in one coalition all players are symmetric while in the other coalition all players are null players; i.e., the two games are quasi-symmetric with the same binary coalition partition structure. Given that null players contribute nothing to any coalition and symmetric players play the same relative role in these two games, each symmetric player again should receive the same value in these two games. This leads to our main axiom:

Cross Invariance (CI): For all $v,w\in {\mathbb {V}}$ and $S\in 2^{N}$ such that $v(N)=w(N)$ and S is symmetric and $N\backslash S$ is null in both v and w, $\Phi _{i}(v)=\Phi _{i}(w)$ for all $i\in S$.¹

CI demands payoff invariance on symmetric players across quasi-symmetric games with identical grand-coalition worth and coalition structure. On the other hand, SYM demands payoff equivalence on symmetric players within a game. This distinction highlights that CI and SYM capture different aspects of fairness and do not necessarily align. Nevertheless, it can be observed that SYM, EFF and NP collectively imply CI. To see this, suppose the premise of CI holds. EFF and NP imply that v(N) is fully divided among the players in S in game v,  and SYM then implies that players in S share the grand-coalition worth equally, resulting in each player in S receiving v(N)/|S| . Similarly, each player in S receives w(N)/|S| in game w. As $v(N)=w(N),$ the payoffs for players in S remain invariant across games v and w. On the other hand, it can be readily checked that by CI and DP, the payoffs of symmetric players are equivalent in unanimity games $u_{T}.$ However, to extend this result to all games, ADD is needed (see the proof of Proposition 1 below). Thus, to imply SYM, CI needs to be combined with both DP and ADD.

It is also worth noting that unless all players are null, the partition $(S,N\backslash S)$ is unique whenever it exists. No such partition exists if there is more than one type of non-null players. This further emphasizes that CI alone is not a strong requirement. CI is satisfied by several prominent solution concepts, including the Shapley value, the equal division solution, and the dictatorial solutions.²

Example 1

Let $N=\{1,2,3\},$ $v=u_{\{1,2\}}$ and $w=\frac{1}{2}u_{\{1\}}+\frac{1}{2} u_{\{2\}}.$ Then we have

$$\begin{aligned} \begin{array}{cccc} S &{} v(S) &{} &{} w(S) \\ \{1\} &{} 0 &{} &{} \frac{1}{2} \\ \{2\} &{} 0 &{} &{} \frac{1}{2} \\ \{3\} &{} 0 &{} &{} 0 \\ \{1,2\} &{} 1 &{} &{} 1 \\ \{1,3\} &{} 0 &{} &{} \frac{1}{2} \\ \{2,3\} &{} 0 &{} &{} \frac{1}{2} \\ \{1,2,3\} &{} 1 &{} &{} 1 \end{array} \end{aligned}$$

We observe that $v(N)=w(N)=1,$ and $\{1,2\}$ is symmetric and $\{3\}$ is null in both v and w. CI requires that $\Phi _{1}(v)=\Phi _{1}(w)$ and $\Phi _{2}(v)=\Phi _{2}(w)$.

Proposition 1

The Shapley value is the unique value that satisfies DP, ADD, and CI.

Proof

It can be readily seen that the Shapley value satisfies DP, ADD, and CI. We show that if a value satisfies DP, ADD, and CI, then it is the Shapley value. We observe that ${\mathbb {V}}$ is a linear space and the class of unanimity games $U\equiv \{u_{T} \vert T\in 2^{N}\backslash \{\emptyset \}\}$ is a basis of ${\mathbb {V}}$. Suppose a value function $\Phi$ satisfies DP, ADD, and CI. Given $v\in {\mathbb {V}}$, as U is a basis of ${\mathbb {V}}$, there are unique numbers $\alpha _{T}$ such that $v=\sum _{T\in 2^{N}\backslash \{\emptyset \}}\alpha _{T}u_{T}.$ By ADD,

$$\begin{aligned} \Phi (v)=\Phi \left( \sum _{T\in 2^{N}\backslash \{\emptyset \}}\alpha _{T}u_{T}\right) =\sum _{T\in 2^{N}\backslash \{\emptyset \}}\Phi (\alpha _{T}u_{T}). \end{aligned}$$

(3)

Let us compare two games, $\alpha _{T}u_{T}$ and $\sum _{j\in T}\frac{\alpha _{T}}{| T| }u_{\{j\}}.$ Note first that T is symmetric and $N\backslash T$ is null in both $\alpha _{T}u_{T}$ and $\sum _{j\in T} \frac{\alpha _{T}}{| T| }u_{\{j\}}.$ Moreover, we have

$$\begin{aligned} \alpha _{T}u_{T}(N)=\left( \sum _{j\in T}\frac{\alpha _{T}}{| T| }u_{\{j\}}\right) (N)=\alpha _{T}. \end{aligned}$$

(4)

By CI and DP,

$$\begin{aligned} \Phi _{i}(\alpha _{T}u_{T})=\Phi _{i}\left( \sum _{j\in T}\frac{\alpha _{T}}{ | T| }u_{\{j\}}\right) =\frac{\alpha _{T}}{| T| }\text { for all }i\in T,\text { } \end{aligned}$$

(5)

and by DP,

$$\begin{aligned} \Phi _{i}(\alpha _{T}u_{T})=\Phi _{i}\left( \sum _{j\in T}\frac{\alpha _{T}}{ | T| }u_{\{j\}}\right) =0\text { for all }i\in N\backslash T. \end{aligned}$$

(6)

Therefore, the value of $\Phi (\alpha _{T}u_{T})$ is uniquely determined, and so does the value of $\Phi (v).$ As the Shapley value satisfies DP, ADD, and CI, $\Phi$ is the Shapley value. $\square$

Compared to the classical characterization of the Shapley value, we replace SYM by CI as a fundamental property that a value should possess. This change renders EFF unnecessary. Our alternative characterization not only highlights the important role DP plays as a foundation in constructing the Shapley value, but it also shows that the Shapley value is robust as a fair allocation rule in the following sense.³ Given a game v,  a set of players N,  and a probability measure $\xi$ on [0, 1],  the function $\Psi ^{\xi }$ is defined as

$$\begin{aligned} \Psi _{i}^{\xi }(v)=\sum _{S\subset N\backslash \{i\}}p_{S}^{N}(v(S\cup \{i\})-v(S)),\text { where }p_{S}^{N}=\mathop {\int }\limits _{0}^{1}t^{| S| }(1-t)^{| N| -| S| -1}d\xi (t). \end{aligned}$$

(7)

Dubey et al. (1981) axiomatically define semivalues and show that the class of semivalues can be characterized as $SV=\{\Psi ^{\xi }|$ $\xi$ is a probability measure on $[0,1]\}.$ All semivalues satisfy DP, SYM, and ADD. Two well-known semivalues are the Shapley value and the Banzhaf value (Banzhaf 1965). The Shapley value corresponds to the case when $\xi$ is the uniform distribution on [0, 1], and the Banzhaf value corresponds to the case when $\xi$ assigns probability one to $t=\frac{1}{2}.$

The characterization of semivalues seems to suggest that in the absence of efficiency, we may not be able to distinguish the Shapley value from other semivalues, thus rendering all semivalues seemingly equally reasonable. Consider the following example:

Example 2

Let $N=\{1,2,3\}$ and consider two games v and w such that

$$\begin{aligned} \begin{array}{cccc} S &{} v(S) &{} &{} w(S) \\ \{1\} &{} 0 &{} &{} 0 \\ \{2\} &{} 0 &{} &{} 0 \\ \{3\} &{} 0 &{} &{} 0 \\ \{1,2\} &{} 0 &{} &{} 1 \\ \{1,3\} &{} 0 &{} &{} 1 \\ \{2,3\} &{} 0 &{} &{} 1 \\ \{1,2,3\} &{} 1 &{} &{} 1 \end{array} \end{aligned}$$

We observe that all players are symmetric in both v and w,  and $v(N)=w(N)=1.$ The Shapley value assigns $\frac{1}{3}$ to each player in both games. On the other hand, the Banzhaf value assigns $\frac{1}{4}$ to each player in v,  and $\frac{1}{2}$ to each player in w.

Example 2 serves as an illustration of the distinction between the Shapley value and other semivalues, such as the Banzhaf value. It highlights a crucial characteristic of the Shapley value that sets it apart from alternative solutions. In two symmetric games with the same grand-coalition worth, the Shapley value ensures that the assigned value to each player remains unaltered. Conversely, other semivalues may yield different values for the players in these games. This discrepancy underscores the limitations of SYM as a within-game invariance property, as it does not possess sufficient discriminatory power to differentiate between semivalues. Proposition 1 provides further insight by demonstrating that by embracing CI as a substitute for SYM, the Shapley value reestablishes its distinctive position even in the absence of efficiency. Thus, we may say that the Shapley value demonstrates robustness in terms of providing an unparalleled solution for fair allocation.

Remark 1 Independence of the axioms. The three axioms in Proposition 1 are independent. The equal division solution ED, which assigns $\frac{v(N)}{n}$ to each player for each $v\in {\mathbb {V}}$, satisfies ADD and CI but not DP. The Banzhaf value satisfies DP and ADD but not CI. Given $v\in {\mathbb {V}}$, denote by d(v) the number of dummy players in v. The dummy-adjusted equal division solution is defined as

$$\begin{aligned} \Phi _{i}^{DAED}(v)=\left\{ \begin{array}{cc} v(\{i\})+\frac{v(N)-\sum _{j\in N}v(\{j\})}{N-d(v)} &{} \text {if }i\text { is not a dummy player,} \\ v(\{i\}) &{} \text {if }i\text { is a dummy player.} \end{array} \right. \end{aligned}$$

It can be readily seen that $\Phi ^{DAED}$ satisfies DP and CI. However, $\Phi ^{DAED}$ violates ADD (see, for example, Maschler et al. 2013, p. 752).

Remark 1 shows that the Shapley value cannot be uniquely determined by CI and DP alone without the inclusion of ADD. However, ADD imposes an additive structure on a value, which is a strong restriction. In the following we present two alternative characterizations of the Shapley value without using ADD.

First, consider the following strengthening of CI:

Strong Cross Invariance (SCI): For all $w,w^{\prime }\in {\mathbb {V}}$ and $S\in 2^{N}$ such that $w(N)=w^{\prime }(N)$ and S is symmetric and $N\backslash S$ is null in both w and $w^{\prime }$, $\Phi _{i}(v+w)=\Phi _{i}(v+w^{\prime })$ for all $i\in S$ and for all $v\in {\mathbb {V}}.$

SCI extends CI by requiring that the payoff invariance property continues to hold if a game is added to quasi-symmetric games that share the same grand-coalition worth and coalition structure. It is evident that ADD and CI imply SCI, and hence the Shapley value satisfies SCI. Additionally, akin to CI, both the equal division and the dictatorial solutions satisfy SCI. $Sh^{2},$ the squared Shapley value, is a solution that satisfies SCI but not ADD, demonstrating that SCI does not imply ADD.

Proposition 2

The Shapley value is the unique value that satisfies DP and SCI.

Proof

Suppose a value function $\Phi$ satisfies DP and SCI. Given $v\in {\mathbb {V}}$, as U is a basis of ${\mathbb {V}}$, there are unique numbers $\alpha _{T}$ such that $v=\sum _{T\in 2^{N}\backslash \{\emptyset \}}\alpha _{T}u_{T}.$ Fix $i\in N.$ We can write

$$\begin{aligned} v=\sum _{T\in 2^{N}\backslash \{\emptyset \}}\alpha _{T}u_{T}=\sum _{T\in 2^{N}\backslash \{\emptyset \},\text { }i\notin T}\alpha _{T}u_{T}+\sum _{T\in 2^{N}\backslash \{\emptyset \},\text { }i\in T}\alpha _{T}u_{T}. \end{aligned}$$

(8)

Let $w_{T}\equiv \alpha _{T}u_{T}$ and $w_{T}^{\prime }\equiv \sum _{j\in T} \frac{\alpha _{T}}{| T| }u_{\{j\}}.$ We observe that $w_{T}(N)=w_{T}^{\prime }(N),$ T is symmetric in both $w_{T}$ and $w_{T}^{\prime },$ and $N\backslash T$ is null in both $w_{T}$ and $w_{T}^{\prime }.$ By repeatedly applying SCI, we get

$$\begin{aligned} \Phi _{i}(v)= & {} \Phi _{i}\left( \sum _{T\in 2^{N}\backslash \{\emptyset \},\text { } i\notin T}\alpha _{T}u_{T}+\sum _{T\in 2^{N}\backslash \{\emptyset \},\text { } i\in T}\alpha _{T}u_{T}\right) \nonumber \\= & {} \Phi _{i}\left( \sum _{T\in 2^{N}\backslash \{\emptyset \},\text { }i\notin T}\alpha _{T}u_{T}+\sum _{T\in 2^{N}\backslash \{\emptyset \},\text { }i\in T}w_{T}\right) \nonumber \\= & {} \Phi _{i}\left( \sum _{T\in 2^{N}\backslash \{\emptyset \},\text { }i\notin T}\alpha _{T}u_{T}+\sum _{T\in 2^{N}\backslash \{\emptyset \},\text { }i\in T}w_{T}^{\prime }\right) \nonumber \\= & {} \Phi _{i}\left( \sum _{T\in 2^{N}\backslash \{\emptyset \},\text { }i\notin T}\alpha _{T}u_{T}+\sum _{T\in 2^{N}\backslash \{\emptyset \},\text { }i\in T}\sum _{j\in T}\frac{\alpha _{T}}{| T| }u_{\{j\}}\right) \end{aligned}$$

(9)

$$\begin{aligned}= & {} \Phi _{i}({\widehat{\omega }}),\text { where} \nonumber \\ {\widehat{\omega }}\equiv & {} \sum _{T\in 2^{N}\backslash \{\emptyset \},\text { } i\notin T}\alpha _{T}u_{T}+\sum _{T\in 2^{N}\backslash \{\emptyset \},\text { } i\in T}\sum _{j\in T}\frac{\alpha _{T}}{| T| }u_{\{j\}}. \end{aligned}$$

(10)

As i is a dummy player in ${\widehat{\omega }},$ by DP,

$$\begin{aligned} \Phi _{i}(v)=\Phi _{i}({\widehat{\omega }})={\widehat{\omega }}(i)=\sum _{T\in 2^{N}\backslash \{\emptyset \},\text { }i\in T}\frac{\alpha _{T}}{| T| }, \end{aligned}$$

(11)

which is exactly $Sh_{i}(v).$ $\square$

Proposition 2 provides a two-axiom characterization of the Shapley value. Based on the criterion of equal relative contributions, SCI establishes that a player’s value in a given game remains unchanged when considering a corresponding game where the player serves as a dummy. Subsequently, DP is applied to the corresponding game to precisely determine the value for the player. It is worth noting that neither EFF nor ADD is used in this characterization. In other words, among all the linear and non-linear values, regardless of their efficiency properties, the Shapley value stands out as the sole solution that satisfies both DP and SCI.

In the literature, two prominent axioms have been proposed as substitutes for ADD in characterizing the Shapley value. Young (1985) characterizes the Shapley value using EFF, SYM and the marginality axiom:

Marginality (M): For all $v,w\in {\mathbb {V}}$ and $i\in N$ such that $v(S\cup \{i\})-v(S)=w(S\cup \{i\})-w(S)$ for all $S\subset N\backslash \{i\},$ $\Phi _{i}(v)=\Phi _{i}(w).$

M requires that if a player has the same marginal contributions in two games, then he should receive the same payoff in these two games. Similar to CI, M is a payoff invariance property. However, the distinction lies in the application of the payoff invariance requirement. CI focuses on games where players make the same relative contributions, while M pertains to games where an individual player makes the same absolute contributions.

Chun (1989) characterizes the Shapley value with the following axiom:

Coalitional Strategic Equivalence (CSE): For all $v\in {\mathbb {V}}$, $T\in 2^{N}\backslash \{\emptyset \}$ and $\alpha \in {\mathbb {R}},$ $\Phi _{i}(v)=\Phi _{i}(v+\alpha u_{T})$ for all $i\in N\backslash T.$

CSE dictates that the addition of a constant to the worths of coalitions containing a specific coalition T should not impact the payoffs of players outside of T. It is evident that M implies CSE. Interestingly, it turns out that CSE also implies M, and hence the two axioms are equivalent (Casajus 2014).

The next proposition shows that M can replace ADD in Proposition 1 to characterize the Shapley value in two-person games.

Proposition 3

Let $N=\{1,2\}.$ The Shapley value is the unique value that satisfies DP, M, and CI.

Proof

Suppose a value function $\Phi$ satisfies DP, M, and CI. Given $v\in {\mathbb {V}}$, as U is a basis of ${\mathbb {V}}$, there are unique numbers $\alpha _{T}$ such that $v=\sum _{T\in 2^{N}\backslash \{\emptyset \}}\alpha _{T}u_{T}=\alpha _{\{1\}}u_{\{1\}}+\alpha _{\{2\}}u_{\{2\}}+\alpha _{\{1,2\}}u_{\{1,2\}}.$ We show that $\Phi _{i}(v)=Sh_{i}(v)$ $\forall i\in N.$ Without loss of generality, let $i=1.$ Consider the games w and $w^{\prime }$ such that

$$\begin{aligned} w= & {} \alpha _{\{1\}}u_{\{1\}}+\alpha _{\{1\}}u_{\{2\}}+\alpha _{\{1,2\}}u_{\{1,2\}} \\ w^{\prime }= & {} \frac{2\alpha _{\{1\}}+\alpha _{\{1,2\}}}{2}u_{\{1\}}+\frac{ 2\alpha _{\{1\}}+\alpha _{\{1,2\}}}{2}u_{\{2\}} \end{aligned}$$

Note that player 1 has identical marginal contributions in v and w. By M, $\Phi _{1}(v)=\Phi _{1}(w).$ Moreover, we observe that player 1 and player 2 are symmetric in both w and $w^{\prime },$ and $w(N)=w^{\prime }(N)=2\alpha _{\{1\}}+\alpha _{\{1,2\}}.$ By CI, $\Phi _{1}(w)=\Phi _{1}(w^{\prime }).$ By DP, $\Phi _{1}(w^{\prime })=\frac{2\alpha _{\{1\}}+\alpha _{\{1,2\}}}{2}.$ Thus, we have $\Phi _{1}(v)=\Phi _{1}(w)=\Phi _{1}(w^{\prime })=\frac{2\alpha _{\{1\}}+\alpha _{\{1,2\}}}{2} =Sh_{1}(v).$ $\square$

In cases where there are more than two players, it is important to note that the combination of DP, M, and CI do not uniquely identify the Shapley value, as demonstrated by the subsequent example.

Example 3

Suppose $N=\{1,2,3\}.$ Let

$$ w= u_{\{1\}}+2u_{\{1,2\}}+3u_{\{1,3\}}$$

$$\begin{aligned} {\mathbb {W}}= & {} \{v\in {\mathbb {V}}|\text {Player 1's marginal contributions in }v \text { are identical to that in }w\} \\= & {} \left\{ \sum _{T\in 2^{N}\backslash \{\emptyset \}}\alpha _{T}u_{T}|\alpha _{\{1\}}=1,\alpha _{\{1,2\}}=2,\alpha _{\{1,3\}}=3,\alpha _{N}=0,\alpha _{\{2\}}\in {\mathbb {R}},\alpha _{\{3\}}\in {\mathbb {R}},\alpha _{\{2,3\}}\in {\mathbb {R}} \right\} \\ \overline{{\mathbb {W}}}= & {} {\mathbb {V}}\backslash {\mathbb {W}}. \end{aligned}$$

Note that in the games in ${\mathbb {W}},$ there are no dummy players, and not all players are symmetric. Thus, DP and CI play no role in ${\mathbb {W}}.$ Consider the value $\Phi ^{{\mathbb {W}}}$ such that

$$\begin{aligned} \Phi ^{{\mathbb {W}}}(v)=\left\{ \begin{array}{cc} (1,Sh_{2}(v),Sh_{3}(v))\text { } &{} \text {if }v\in {\mathbb {W}} \\ Sh(v) &{} \text {if }v\in \overline{{\mathbb {W}}} \end{array} \right. . \end{aligned}$$

It can be readily checked that $\Phi ^{{\mathbb {W}}}$ satisfies DP, CI, and M, but $\Phi ^{{\mathbb {W}}}\ne Sh$ as $\Phi _{1}^{{\mathbb {W}}}(v)=1\ne Sh_{1}(v)=3.5$ $\forall v\in {\mathbb {W}}$.

In order to characterize the Shapley value in n-person games with $n\ge 3,$ we introduce the following strengthened version of M. Let us denote $\pi :N\rightarrow N$ as a permutation of N,  and $\pi _{i}:N\rightarrow N$ as a permutation of N such that $\pi _{i}(i)=i.$ For every $v\in {\mathbb {V}},$ the permutation of v by $\pi$ is denoted by $\pi v,$ where $\pi v(S)=v(\pi (S))$ for all $S\in 2^{N}.$ Let $\Delta _{i}v$ denote i’s marginal contribution vector in v. Given $v,w\in {\mathbb {V}},$ define the equivalence relation $=_{M}$ to be such that $\Delta _{i}v=_{M}\Delta _{i}w$ if and only if $v(S\cup \{i\})-v(S)=w(S\cup \{i\})-w(S)$ for all $S\subset N\backslash \{i\}.$

Strong Marginality (SM): For all $i\in N$ and $v,w,w^{\prime }\in {\mathbb {V}},$ if $\Delta _{i}(v+w)=_{M}\Delta _{i}(v+\pi _{i}w^{\prime })$ for some $\pi _{i},$ then $\Phi _{i}(v+w)=\Phi _{i}(v+w^{\prime }).$⁴

SM enhances the concept of M by incorporating an impartiality principle, which eliminates the influence of personal identity when assessing a player’s marginal contributions. It is noteworthy that all semivalues satisfy SM.

Proposition 4

The Shapley value is the unique value that satisfies DP, SM, and CI.

Proof

Clearly the Shapley value satisfies DP, SM, and CI. Suppose a value function $\Phi$ satisfies DP, SM, and CI. We show that $\Phi =Sh.$ Given $v\in {\mathbb {V}}$, as U is a basis of ${\mathbb {V}}$, there are unique numbers $\alpha _{T}$ such that $v=\sum _{T\in 2^{N}\backslash \{\emptyset \}}\alpha _{T}u_{T}.$ Fix $i\in N.$ Let $z\in {\mathbb {V}}$ be the zero game, i.e., $z(S)=0$ for all $S\in 2^{N}.$ By SM, we have

$$\begin{aligned} \Phi _{i}(v)= & {} \Phi _{i}\left( \sum _{T\in 2^{N}\backslash \{\emptyset \}}\alpha _{T}u_{T}\right) \nonumber \\= & {} \Phi _{i}\left( \sum _{T\in 2^{N}\backslash \{\emptyset \},\text { }i\in T}\alpha _{T}u_{T}+\sum _{T\in 2^{N}\backslash \{\emptyset \},\text { }i\notin T}\alpha _{T}u_{T}\right) \nonumber \\= & {} \Phi _{i}\left( \sum _{T\in 2^{N}\backslash \{\emptyset \},\text { }i\in T}\alpha _{T}u_{T}+z\right) \nonumber \\= & {} \Phi _{i}\left( \sum _{T\in 2^{N}\backslash \{\emptyset \},\text { }i\in T}\alpha _{T}u_{T}\right) \nonumber \\= & {} \Phi _{i}\left( \sum _{k=1}^{n}\sum _{T\in 2^{N}\backslash \{\emptyset \},\text { } i\in T,\text { }| T| =k}\alpha _{T}u_{T}\right) . \end{aligned}$$

(12)

Denote by ${\mathbb {T}}_{k}=\{T\in 2^{N}\backslash \{\emptyset \}|$ $i\in T$ and $| T| =k\}.$ Pick an arbitrary element in ${\mathbb {T}} _{k}$ and denote it by $T_{k}^{*}.$ We observe that for any T $\in {\mathbb {T}}_{k},$ there exists a permutation $\pi _{i}$ such that $\Delta _{i}u_{T_{k}^{*}}=_{M}\Delta _{i}\pi _{i}u_{T}.$ Applying SM repeatedly, we get

$$\begin{aligned} \Phi _{i}(v)= & {} \Phi _{i}\left( \sum _{k=1}^{n}\sum _{T\in {\mathbb {T}}_{k}}\alpha _{T}u_{T}\right) \nonumber \\= & {} \Phi _{i}\left( \sum _{k=1}^{n}\left( \sum _{T\in {\mathbb {T}}_{k}}\alpha _{T}\right) u_{T_{k}^{*}}\right) \nonumber \\= & {} \Phi _{i}\left( \sum _{k=1}^{n}\frac{\sum _{T\in {\mathbb {T}}_{k}}\alpha _{T}}{ \left( {\begin{array}{c}n-1\\ k-1\end{array}}\right) }\sum _{T\in {\mathbb {T}}_{k}}u_{T}\right) \nonumber \\= & {} \Phi _{i}\left( \sum _{k=1}^{n}\frac{\sum _{T\in {\mathbb {T}}_{k}}\alpha _{T}}{ \left( {\begin{array}{c}n-1\\ k-1\end{array}}\right) }\sum _{T\in 2^{N}\backslash \{\emptyset \},\text { } | T| =k}u_{T}\right) . \end{aligned}$$

(13)

Let

$$\begin{aligned} v^{\prime }\equiv \sum _{k=1}^{n}\frac{\sum _{T\in {\mathbb {T}}_{k}}\alpha _{T}}{ \left( {\begin{array}{c}n-1\\ k-1\end{array}}\right) }\sum _{T\in 2^{N}\backslash \{\emptyset \},\text { } | T| =k}u_{T}\text { and }v^{\prime \prime }\equiv \sum _{j=1}^{n}\frac{v^{\prime }(N)}{n}u_{\{j\}}. \end{aligned}$$

(14)

We observe that all players are symmetric in both $v^{\prime }$ and $v^{\prime \prime },$ and

$$\begin{aligned} v^{\prime \prime }(N)= & {} v^{\prime }(N)=\sum _{k=1}^{n}\frac{\sum _{T\in {\mathbb {T}}_{k}}\alpha _{T}}{\left( {\begin{array}{c}n-1\\ k-1\end{array}}\right) }\sum _{T\in 2^{N}\backslash \{\emptyset \},\text { }| T| =k}u_{T}(N) \nonumber \\= & {} \sum _{k=1}^{n}\frac{\sum _{T\in {\mathbb {T}}_{k}}\alpha _{T}}{\left( {\begin{array}{c}n-1\\ k-1 \end{array}}\right) }\left( {\begin{array}{c}n\\ k\end{array}}\right) \nonumber \\= & {} \sum _{k=1}^{n}\left( \frac{n}{k}\sum _{T\in {\mathbb {T}}_{k}}\alpha _{T}\right) . \end{aligned}$$

(15)

By CI and DP,

$$\begin{aligned} \Phi _{i}(v)=\Phi _{i}(v^{\prime })=\Phi _{i}(v^{\prime \prime })=\frac{ v^{\prime }(N)}{n}=\sum _{k=1}^{n}\frac{\sum _{T\in {\mathbb {T}}_{k}}\alpha _{T} }{k}=Sh_{i}(v). \end{aligned}$$

(16)

$\square$

The characterization of the Shapley value in Proposition 4 neither assumes collective rationality (EFF) nor imposes linear restrictions on the functional form of a value (ADD). Instead, this characterization hinges on two payoff invariance properties: SM and CI. Based on the criterion of absolute contributions, SM ensures that each player’s payoff remains unchanged when games where they make no contributions are filtered out. On the other hand, CI establishes payoff invariance for each player based on the relative contributions of all players involved. By combining these two axioms, we can establish payoff invariance for each player between any given game and a game where they assume the role of a dummy player. The determination of the value is then based on DP.

To conclude the section, we present Table 1 above summarizing the key properties satisfied by the aforermentioned solution concepts. The table also serves to demonstrate the independence of the axioms in Propositions 1–4.

Table 1 Solutions and their properties

4 The weighted Shapley values

The Shapley value is intended to guarantee that players who make the same marginal contributions are rewarded equally in terms of payoffs. However, it is essential to acknowledge that there can be inherent asymmetries among players that are not reflected in the characteristic function alone. These differences may arise from variations in effort levels, contributions in distinct forms (such as labor versus capital), or divergent opportunity costs associated with cooperation.

In scenarios where such variations exist, the weighted Shapley values (Shapley 1953b) offer a more suitable approach for fair allocation. By incorporating weights that account for the aforementioned asymmetries, the weighted Shapley values provide a more nuanced and balanced distribution of payoffs among the players. In the following we propose a variation of CI to characterize the weighted Shapley values.

Given $\lambda \in {\mathbb {R}}_{++}^{n}$ such that $\sum _{i\in N}\lambda _{i}=1$, players $i,j\in N$ are called $\lambda$-weighted symmetric in v if there exists a constant $c_{v}\in {\mathbb {R}}$ such that

$$\begin{aligned} \frac{v(S\cup \{i\})-v(S)}{\lambda _{i}}=\frac{v(S\cup \{j\})-v(S)}{\lambda _{j}}=c_{v} \end{aligned}$$

for all $S\subset N\setminus \{i,j\}$. Note that if players i and j are $\lambda$-weighted symmetric in v,  then $v(\{i\})/\lambda _{i}=v(\{j\})/\lambda _{j}=c_{v}$. $S\in 2^{N}$ is called a $\lambda$-weighted symmetric coalition if all players in S are $\lambda$-weighted symmetric. The following axiom adjusts CI by taking players’ weights $\lambda$ into consideration.

$\lambda$-Weighted Cross Invariance ($\lambda$-CI): For all $v,w\in {\mathbb {V}}$ and $S\in 2^{N}$ such that $v(N)=w(N)$ and S is $\lambda$-weighted symmetric and $N\backslash S$ is null in both v and w, $\Phi _{i}(v)=\Phi _{i}(w)$ for all $i\in S.$

Given $\lambda \in {\mathbb {R}}_{++}^{n}$ such that $\sum _{i\in N}\lambda _{i}=1,$ the $\lambda$-weighted Shapley value, $Sh^{\lambda },$ is defined as the unique linear value which assigns to each unanimity game $u_{T}$:

$$\begin{aligned} Sh_{i}^{\lambda }(u_{T})=\left\{ \begin{array}{cc} \frac{\lambda _{i}}{\sum _{j\in T}\lambda _{j}} &{} \text {if }i\in T \\ 0 &{} \text {if }i\notin T \end{array} \right. . \end{aligned}$$

Alternatively, $Sh^{\lambda }$ can be defined based on probabilistic random arrivals. Given a permutation $\pi =(i_{1},\ldots ,i_{n}),$ let $B(\pi ,i)$ represent the set of agents preceding i in $\pi$ and $P_{\lambda }(\pi )=\Pi _{k=1}^{n}\left( \lambda _{i_{k}}/\Sigma _{t=1}^{k}\lambda _{i_{t}}\right) .$ Let $\Omega$ be the set of permutations $\pi$ on N. Then

$$\begin{aligned} Sh_{i}^{\lambda }(v)=\mathop {\sum }\limits _{\pi \in \Omega }P_{\lambda }(\pi )[v(B(\pi ,i)\cup \{i\})-v(B(\pi ,i))]. \end{aligned}$$

Proposition 5

The weighted Shapley value with weights $\lambda ,$ $Sh^{\lambda },$ is the unique value that satisfies DP, ADD, and $\lambda$-CI.

Proof

Clearly $Sh^{\lambda }$ satisfies DP and ADD. We show that $Sh^{\lambda }$ satisfies $\lambda$-CI. Let $v,w\in {\mathbb {V}}$ and $S\in 2^{N}$ be such that $v(N)=w(N)$ and S is $\lambda$-weighted symmetric and $N\backslash S$ is null in both v and w. Let $c_{v}\in {\mathbb {R}}$ and $c_{w}\in \mathbb { R}$ be the constants defined in $\lambda$-weighted symmetry. We first establish that

$$\begin{aligned} {{\textbf {Claim 1.}} }\quad v(R)=\sum _{i\in R}v(\{i\})\text { for all strict subsets }R\subsetneq S\text {.} \end{aligned}$$

(17)

The statement holds trivially for all $R\subsetneq S$ with $| R| =1,$ as $R=\{i\}$ for some $i\in S$ and $v(R)=v(\{i\}).$ We also know that $v(\{i\})=\lambda _{i}c_{v}$ by $\lambda$-weighted symmetry. The proof proceeds for the cases $1<| R| <| S|$ by induction. Suppose $v(R)=\sum _{i\in R}v(\{i\})$ for all $R\subsetneq S$ with $1\le | R| =r<$ $| S| -1.$ We show that $v(R)=\sum _{i\in R}v(\{i\})$ for all $R\subsetneq S$ with $| R| =r+1.$ Let $R\subsetneq S$ with $| R| =r+1.$ R can be decomposed as $R=R^{\prime }\cup \{i\},$ where $i\in R$ and $R^{\prime }=R\backslash \{i\}\subsetneq S.$ As $| R^{\prime }| =r,$ $v(R^{\prime })=\sum _{k\in R^{\prime }}v(\{k\})$. Pick $j\in S\backslash R$. As S is $\lambda$-weighted symmetric in v, 

$$\begin{aligned} \frac{v(R^{\prime }\cup \{i\})-v(R^{\prime })}{\lambda _{i}}=\frac{ v(R^{\prime }\cup \{j\})-v(R^{\prime })}{\lambda _{j}}=c_{v}. \end{aligned}$$

(18)

Then we have

$$\begin{aligned} v(R)=v(R^{\prime }\cup \{i\})=\lambda _{i}c_{v}+v(R^{\prime })=v(\{i\})+\sum _{k\in R^{\prime }}v(\{k\})=\sum _{k\in R}v(\{k\}), \end{aligned}$$

(19)

and Claim 1 is established.

Fix $v\in {\mathbb {V}}.$ There are unique numbers $\alpha _{T}$ such that $v=\sum _{T\in 2^{N}\backslash \{\emptyset \}}\alpha _{T}u_{T}.$ $\alpha _{T}$ is the Harsanyi dividend of T in v and can be calculated as $\alpha _{T}=\sum _{R\subset T}(-1)^{| T\backslash R| }v(R).$ It is evident that $\alpha _{T}=0$ if T has a null player. Since $N\backslash S$ is null, the nonzero terms of $\alpha _{T}$ occur only when $T\subset S.$ We may express $\alpha _{T}$ as

$$\begin{aligned} \alpha _{T}=(-1)^{0}\sum _{R\subset T,\text { }| T\backslash R| =0}v(R)+\ldots +(-1)^{| T| -1}\sum _{R\subset T, \text { }| T\backslash R| =| T| -1}v(R). \end{aligned}$$

(20)

By Claim 1, we have that for all $T\subsetneq S,$

$$\begin{aligned} \sum _{R\subset T,\text { }| T\backslash R| =a}v(R)= & {} \sum _{R\subset T,\text { }| T\backslash R| =a}\sum _{i\in R}v(\{i\})\text { } \nonumber \\= & {} \left( {\begin{array}{c}| T| -1\\ | R| -1\end{array}}\right) \sum _{i\in T}v(\{i\}) \nonumber \\= & {} \left( {\begin{array}{c}| T| -1\\ | T| -a-1\end{array}}\right) \sum _{i\in T}v(\{i\}).\text { } \end{aligned}$$

(21)

Therefore, for all $T\subsetneq S$ such that $| T| \ne 1,$ we have

$$\begin{aligned} \alpha _{T}= & {} (-1)^{0}\left( {\begin{array}{c}| T| -1\\ | T| -1\end{array}}\right) \sum _{i\in T}v(\{i\})+\ldots +(-1)^{| T| -1} \left( {\begin{array}{c}| T| -1\\ 0\end{array}}\right) \sum _{i\in T}v(\{i\}) \\= & {} (1-1)^{| T| -1}\sum _{i\in T}v(\{i\})=0. \nonumber \end{aligned}$$

(22)

For all $T\subsetneq S$ with $| T| =1,$ $\alpha _{T}=v(\{i\})$ for some $i\in S.$ When $T=S,$ we have

$$\begin{aligned} \alpha _{S}= & {} \sum _{i\in S}v(\{i\})-\sum _{i\in S}v(\{i\})+(-1)^{0}\sum _{R\subset S,\text { }| S\backslash R| =0}v(R)+\ldots +(-1)^{| S| -1}\sum _{R\subset S, \text { }| S\backslash R| =| S| -1}v(R)\\= & {} v(S)-\sum _{i\in S}v(\{i\})+\left( \sum _{i\in S}v(\{i\})+\ldots +(-1)^{| S| -1}\left( {\begin{array}{c}| S| -1\\ 0\end{array}}\right) \sum _{i\in S}v(\{i\})\right) \\= & {} v(S)-\sum _{i\in S}v(\{i\}). \end{aligned}$$

To sum up, v can be expressed as

$$\begin{aligned} v=\sum _{j\in S}v(\{j\})u_{\{j\}}+(v(S)-\sum _{j\in S}v(\{j\}))u_{S}. \end{aligned}$$

(23)

We then have for all $i\in S,$

$$\begin{aligned} Sh_{i}^{\lambda }(v)= & {} Sh_{i}^{\lambda }\left( \sum _{j\in S}v(\{j\})u_{\{j\}}+(v(S)-\sum _{j\in S}v(\{j\}))u_{S}\right) \nonumber \\= & {} v(\{i\})+(v(S)-\sum _{j\in S}v(\{j\}))\frac{\lambda _{i}}{\sum _{j\in S}\lambda _{j}} \nonumber \\= & {} v(S)\frac{\lambda _{i}}{\sum _{j\in S}\lambda _{j}}+v(\{i\})-\sum _{j\in S}\lambda _{j}c_{v}\frac{\lambda _{i}}{\sum _{j\in S}\lambda _{j}} \nonumber \\= & {} v(S)\frac{\lambda _{i}}{\sum _{j\in S}\lambda _{j}}. \end{aligned}$$

(24)

Similarly, w can be expressed as

$$\begin{aligned} w=\sum _{j\in S}w(\{j\})u_{\{j\}}+(w(S)-\sum _{j\in S}w(\{j\}))u_{S}, \end{aligned}$$

(25)

and for all $i\in S,$

$$\begin{aligned} Sh_{i}^{\lambda }(w)=w(S)\frac{\lambda _{i}}{\sum _{j\in S}\lambda _{j}}. \end{aligned}$$

(26)

As $v(N)=w(N)$ and $N\backslash S$ is null in both v and w,  $v(S)=v(N)=w(N)=w(S)$ and hence $Sh_{i}^{\lambda }(v)=Sh_{i}^{\lambda }(w)$ for all $i\in S$.

Next we show that if a value function $\Phi$ satisfies DP, ADD, and $\lambda$-CI, then $\Phi =Sh^{\lambda }.$ By ADD, it is sufficient for us to show that $\Phi (\alpha _{T}u_{T})=Sh^{\lambda }(\alpha _{T}u_{T})$ for any $\alpha _{T}\in {\mathbb {R}}$ and $T\in 2^{N}\backslash \{\emptyset \}.$ Let

$$\begin{aligned} v= & {} \alpha _{T}u_{T}\text { } \\ w= & {} \sum _{j\in T}\alpha _{T}\frac{\lambda _{j}}{\sum _{k\in T}\lambda _{k}} u_{\{j\}}. \end{aligned}$$

It can be readily seen that T is $\lambda$-weighted symmetric and $N\backslash T$ is null in both v and w. Moreover, we have $v(N)=w(N)=\alpha _{T}.$ By $\lambda$-CI and DP,

$$\begin{aligned} \Phi _{i}(v)=\Phi _{i}(w)=\Phi _{i}\left( \sum _{j\in T}\alpha _{T}\frac{\lambda _{j}}{\sum _{k\in T}\lambda _{k}}u_{\{j\}}\right) =\alpha _{T}\frac{\lambda _{i}}{ \sum _{k\in T}\lambda _{k}}=Sh_{i}^{\lambda }(v)\text { for all }i\in T,\text { } \end{aligned}$$

(27)

and by DP,

$$\begin{aligned} \Phi _{i}(v)=0=Sh_{i}^{\lambda }(v)\text { for all }i\in N\backslash T. \end{aligned}$$

(28)

Combining Eqs. (27) and (28), we conclude that $\Phi =Sh^{\lambda }.$ $\square$

In contrast to the classical characterizations of the weighted Shapley values where EFF is indispensable (Chun 1991; Kalai and Samet 1987; Nowak and Radzik 1995), Proposition 5 offers a unique perspective by providing a characterization of the weighted Shapley values that doesn’t require invoking EFF. The rational foundation of the weighted Shapley values provided here avoids the need for assuming collective rationality. Instead, the key axiom in this characterization, $\lambda$-CI, focuses on the fair allocation of gains from cooperation, taking into account the relative weights of players across various circumstances where players assume the same role.

Remark 2. Independence of the axioms.The three axioms in Proposition 5 are independent. The dictatorial solutions satisfy ADD and $\lambda$-CI but not DP. The Banzhaf value satisfies DP and ADD but not $\lambda$-CI. Given $v\in {\mathbb {V}}$, denote by D(v) the set of dummy players in v. The $\lambda$-dummy-adjusted equal division solution is defined as

$$\begin{aligned} \Phi _{i}^{\lambda -DAED}(v)=\left\{ \begin{array}{cc} v(\{i\})+\frac{\lambda _{i}}{\mathop {\sum }\nolimits _{j\in N\backslash D(v)}\lambda _{j}}\left( v(N)-\sum _{j\in N}v(\{j\})\right) &{} \text {if }i\text { is not a dummy player, } \\ v(\{i\}) &{} \text {if }i\text { is a dummy player.} \end{array} \right. \end{aligned}$$

It can be readily seen that $\Phi ^{\lambda -DAED}$ satisfies DP and $\lambda$-CI. However, $\Phi ^{\lambda -DAED}$ violates ADD.

5 The Shapley–Shubik power index

Next we consider an interesting subclass of games in ${\mathbb {V}}$. The domain ${\mathbb {S}}\subset {\mathbb {V}}$ of (monotonic) simple games on N consists of all $v\in {\mathbb {V}}$ such that

• (i) $v(S)\in \{0,1\}$ for all $S\in 2^{N},$

• (ii) $v(N)=1,$ and

• (iii) $v(S)\le v(T)$ whenever $S\subset T,$ S,  $T\in 2^{N}$.

A game $v\in {\mathbb {S}}$ is a mathematical representation of a voting system, and players in N are constituents (legislative bodies, states, senators, etc.) of the voting system. There are only two possible values for each coalition. The value of 1 represents winning or passing a bill, and the value of 0 represents losing or failing to pass a bill. $S\in 2^{N}$ is called a winning coalition in v if $v(S)=1.$ S is called a minimal winning coalition in v if $v(S)=1$ and $v(T)=0$ for each proper subset $T\subset S.$

To determine the distribution of power among constituents in a voting system, define a power index as a function $\Phi :{\mathbb {S}} \rightarrow {\mathbb {R}} ^{n}.$ $\Phi _{i}(v)$ measures player i’s a priori voting power in the game $v\in {\mathbb {S}}$. The Shapley–Shubik power index (SSPI) is simply the Shapley value restricted to the domain of simple games, i.e., $SSPI=Sh|_{{\mathbb {S}}}$. According to the SSPI, a player’s voting power is his chance of being critical to the success of a winning coalition, assuming players join in a random order.

Given v,  $w\in {\mathbb {S}}$, define join and meet operations, $v\vee w\in {\mathbb {S}}$ and $v\wedge w\in {\mathbb {S}},$ as follows:

$(v\vee w)(S)=\max \{v(S),w(S)\}$ and $(v\wedge w)(S)=\min \{v(S),w(S)\}$ for all $S\in 2^{N}.$

Thus S is a winning coalition in $v\vee w$ if and only if it is winning in either v or w,  and it is a winning coalition in $v\wedge w$ if and only if it is winning in both v and w. As the domain of simple games $\mathbb { S}$ is not closed under addition due to its nonlinear (lattice) structure, ADD is no longer suitable to be used to characterize a power index on ${\mathbb {S}}$. The following axiom suggested by Dubey (1975) is a replacement of ADD to characterize a power index on the domain of simple games:

Transfer (TR): For all $v,w\in {\mathbb {S}}$, $\Phi (v\vee w)+\Phi (v\wedge w)=\Phi (v)+\Phi (w).$

TR requires that the change in voting power depends only on the change in the voting game. The following proposition characterizes the SSPI with our CI axiom.⁵

Proposition 6

The SSPI is the unique power index on ${\mathbb {S}}$ that satisfies DP, TR, and CI.

Proof

Clearly the SSPI satisfies DP, TR, and CI. We show that if a power index $\Phi$ satisfies DP, TR, and CI on ${\mathbb {S}}$, then $\Phi$ is the SSPI. We first note that the class of unanimity games $U\equiv \{u_{T} \vert T\in 2^{N}\backslash \{\emptyset \}\}\subset {\mathbb {S}}$. For each $v\in {\mathbb {S}}$, denote by $\left\{ T_{j}\right\} _{j=1}^{k}$ the minimal winning coalitions in v. Then we have:

$$\begin{aligned} v=u_{T_{1}}\vee u_{T_{2}}\vee \ldots \vee u_{T_{k}}. \end{aligned}$$

(29)

By TR,  $\Phi (v)$ can be obtained as (see Lemma 2.3 in Einy (1987)):

$$\begin{aligned} \Phi (v)=\sum _{I\subset \{1,\ldots ,k\},I\ne \emptyset }(-1)^{| I| +1}\Phi (u_{\cup _{j\in I}T_{j}}). \end{aligned}$$

(30)

Thus the power index $\Phi$ on ${\mathbb {S}}$ is uniquely determined by the values of $\Phi$ on unanimity games. In the following, we show by induction that DP, TR and CI uniquely pin down the values of $\Phi$ on unanimity games. As the SSPI satisfies DP, TR and CI, $\Phi$ is the SSPI.

Step 1. Let $T=\{i\},$ $i\in N.$ We observe that by DP,

$$\begin{aligned} \Phi _{j}(u_{T})=\left\{ \begin{array}{cc} 1 &{} \text {if }j=i \\ 0 &{} \text {otherwise} \end{array} \right. . \end{aligned}$$

(31)

Step 2. Let $T\subset N$ with $| T| =m,$ $m\in \{2,\ldots ,| N| \}.$ Without loss of generality, let $T=\{1,\ldots ,m\}.$ Denote by $T_{-i}=T\backslash \{i\},$ $i=1,\ldots ,m.$ We observe that T is symmetric and $N\backslash T$ is null in both $u_{T}$ and $u_{T_{-1}}\vee \ldots \vee u_{T_{-m}}.$ By CI,

$$\begin{aligned} \Phi _{i}(u_{T})=\Phi _{i}(u_{T_{-1}}\vee \ldots \vee u_{T_{-m}})\text { for all } i\in T, \end{aligned}$$

(32)

and by DP,

$$\begin{aligned} \Phi _{i}(u_{T})=\Phi _{i}(u_{T_{-1}}\vee \ldots \vee u_{T_{-m}})=0\text { for all }i\in N\backslash T. \end{aligned}$$

(33)

Thus we have

$$\begin{aligned} \Phi (u_{T})=\Phi (u_{T_{-1}}\vee \ldots \vee u_{T_{-m}}). \end{aligned}$$

(34)

By TR,

$$\begin{aligned}{} & {} \Phi (u_{T_{-1}}\vee \ldots \vee u_{T_{-m}})+\Phi ((u_{T_{-1}}\vee \ldots \vee u_{T_{-(m-1)}})\wedge u_{T_{-m}})\\ \nonumber{} & {} \qquad =\Phi (u_{T_{-1}}\vee \ldots \vee u_{T_{-(m-1)}})+\Phi (u_{T_{-m}}). \end{aligned}$$

(35)

It can be readily checked that $u_{T_{-i}}\wedge u_{T_{-m}}=u_{T},$ $i=1,\ldots ,m-1.$ Accordingly, we have

$$\begin{aligned} (u_{T_{-1}}\vee \ldots \vee u_{T_{-(m-1)}})\wedge u_{T_{-m}}=u_{T}. \end{aligned}$$

(36)

Combining Eqs. (34)–(36), we get

$$\begin{aligned} 2\Phi (u_{T})=\Phi (u_{T_{-1}}\vee \ldots \vee u_{T_{-(m-1)}})+\Phi (u_{T_{-m}}). \end{aligned}$$

(37)

Continuing in this fashion and utilizing the fact that $(u_{T_{-1}}\vee \ldots \vee u_{T_{-(k-1)}})\wedge u_{T_{-k}}=u_{T}$ for $k=2,\ldots ,m-1$, we obtain

$$\begin{aligned} \Phi (u_{T})=\frac{1}{m}\sum _{i=1}^{m}\Phi (u_{T_{-i}}). \end{aligned}$$

(38)

Note that for every $i=1,\ldots ,m,$ $T_{-i}$ is a coalition with size $m-1.$ Thus, we have established that for any coalition T with $| T| =m\in \{2,\ldots ,| N| \}$, $\Phi (u_{T})$ is a summation of values of $\Phi$ on unanimity games with coalition size $m-1.$

Step 3. Step 1 gives us the "initial condition" on the values of $\Phi$ on unanimity games. By inductively applying the summation formula in Step 2, $\Phi (u_{T})$ can be uniquely determined for every $u_{T}\in U.$ $\square$

In our characterization, as demonstrated in the proof, DP gives the distribution of power in games where only one player is pivotal. Build upon this basis, TR and CI are employed to extend the analysis and recursively determine the power distribution for every simple game uniquely. The examples provided in Remark 1 demonstrate the independence of the axioms in Proposition 6.

Dubey (1975) shows that the SSPI is the unique power index that satisfies CA (NP + EFF), TR, and SYM. Einy and Haimanko (2011) suggest to use the following gain–loss axiom as a replacement for EFF:

Gain–loss (GL): For all $v,w\in {\mathbb {S}}$, $i\in N$ such that $\Phi _{i}(v)>\Phi _{i}(w),$ there is some $j\in N$ such that $\Phi _{j}(v)<\Phi _{j}(w).$

GL is an appealing property which states that if the power of some player increases as a result of changes in the game, the power cannot concomitantly increase for all players. In other words, it is not possible that all constituents gain more power when switcing from one voting system to another. Clearly GL is much weaker than EFF as quantitatively it does not require the gain and loss to be the same.

Einy and Haimanko (2011) demonstrate that the SSPI is the unique power index that satisfies DP, TR, SYM and GL.⁶ We observe that their characterization involves two efficiency-type axioms: DP and GL. In contrast, DP is the only axiom used in our characterization that has some indirect efficiency implications on a subset of games. CI, which replaces SYM and GL in their characterization, is a pure symmetry property that focuses solely on equal treatment of players across different games, without any explicit efficiency implications. In this regard, our characterization highlights the distinctive rational foundation of the SSPI, showcasing its unique position among power indices and reinforcing its suitability as an equitable solution for power allocation in voting systems.

6 Concluding remarks

We have proposed a simple invariance property called CI, which serves to characterize the Shapley value in TU games and the Shapley–Shubik power index in simple games. CI is motivated by the application of the Aristotelian principle of equality to games where players make the same relative contributions. Furthermore, we have introduced a weighted version of CI to characterize the weighted Shapley values.

By considering a strengthened version of CI, we can eliminate the need for the axiom of additivity in our characterization of the Shapley value. Notably, all of our characterizations do not rely on the axiom of efficiency, making them well-suited to establish a rational foundation for the utilization of the Shapley value in strategic games.

An intriguing extension of our work involves relaxing the requirements of CI and employing it to characterize a new family of semivalues. This extension could provide valuable insights into the distinctions among various solution concepts.