Value dividends, the Harsanyi set and extensions, and the proportional Harsanyi solution

Abstract

A new concept for TU-values, called value dividends, is introduced. Similar to Harsanyi dividends, value dividends are defined recursively and provide new characterizations of values from the Harsanyi set. In addition, we generalize the Harsanyi set where each of the TU-values from this set is defined by the distribution of the Harsanyi dividends via sharing function systems and give an axiomatic characterization. As a TU value from the generalized Harsanyi set, we present the proportional Harsanyi solution, a new proportional solution concept. A new characterization of the Shapley value is proposed as a side effect. None of our characterizations uses additivity.

1 Introduction

The concept of Harsanyi dividends was introduced by Harsanyi (1959). They can be defined inductively: the dividend of the empty set is zero and the dividend of any other possible coalition of a player set equals the worth of the coalition minus the sum of all dividends of proper subsets of that coalition. Hence, Harsanyi dividends can be interpreted as “the pure contribution of cooperation in a TU-game” (Billot and Thisse 2005). Harsanyi could show that if the Harsanyi dividends of all possible coalitions are spaced evenly among its members, each player’s payoff equals the Shapley value (Shapley 1953b). The weighted Shapley values (Shapley 1953a) distribute the Harsanyi dividends proportionally to players’ personal given weights. The ratio of the weights of two players is equal for all coalitions containing them. However, sometimes this seems unrealistic. For example, in some coalitions, the influence of one of two players on the other players may be higher than in other coalitions with other players. Hammer et al. (1977) and Vasil’ev (1978) proposed the Harsanyi set, a class of TU-values called Harsanyi solutions, also known as sharing values (Derks et al. 2000) which take this into account. There the players’ weights, assigned to the Harsanyi dividends via a sharing system, can differ for all coalitions.

Myerson (1980) introduced the balanced contributions axiom which allows, along with efficiency, an elegant axiomatization of the Shapley value. It states for two players i and j that j’s presence contributes as much to i’s payoff as i’s presence contributes to j’s payoff. The w-balanced contributions properties, the ratio of the winnings or losses of two players when the other player leaves the game is proportional to their weights, joint with efficiency, characterize the TU-values of an interesting class (Myerson 1980). These values coincide with the weighted Shapley values (Hart and Mas-Colell 1989).

So far, no analogous characterization to the above-mentioned characterization of the weighted Shapley values is known for the Harsanyi solutions. To enable corresponding axiomatizations for the Harsanyi solutions, we present a new concept, called “value dividends.” These are defined inductively: the value dividend of a singleton is the player’s payoff in a single-player game and the value dividend of any (non-empty) other coalition to a player represents that player’s payoff in the game on the player set of this coalition minus all value dividends to that player in all subgames. We can therefore regard a value dividend as the “pure” payoff to a player that has not yet been realized in a subgame.

Similar to the w-balanced contributions properties, we introduce an axiom called $\lambda$-balanced value dividends where $\lambda$ is a sharing system. If $\lambda$ has the characteristic that the ratio of two players sharing weights is equal in all coalitions containing them, the $\lambda$-balanced value dividends property is equivalent to the w-balanced contributions property.

The value dividends allow further axiomatizations of the Harsanyi solutions and lead to an extension of the Harsanyi set: we provide an axiomatization of the class of generalized Harsanyi solutions, called generalized Harsanyi set. These values are generally no longer additive and use sharing function systems to distribute the Harsanyi dividends. A central property of all TU-values, discussed in this paper, is the inessential grand coalition property. This property states that in games where the Harsanyi dividend of the grand coalition is zero a players’ payoff in the subgames determines the player’s payoff.

The generalized Harsanyi set allows proportionality principles in allocation. A common consensus of most proportional sharing rules is the proportional standardness property for two-player games (Ortmann 2000), which means that in two-player games the whole must be divided proportionally to the singleton worths of both players. For more than two players, however, there is absolutely no agreement on how proportionality should be applied. Many possibilities are suggested such as the set-valued proper Shapley value (Vorob’ev and Liapunov 1998; van den Brink et al. 2015) where proportionality is stated by a fixed point argument.

Whereas, as single-valued solutions, the proportional rule (Moriarity 1975) and, as a value from the generalized Harsanyi set, the proportional Shapley value (Gangolly 1981; Besner 2016; Béal et al. 2018) consider only the worths of the singletons as weights, the proportional value (Feldman 1999; Ortmann 2000) uses the worths of all coalitions in a recursive formula. As a new representative of the generalized Harsanyi set, we introduce the proportional Harsanyi solution that also involves for calculating the whole coalition function for a sharing function system in a recursive formula.

The article is organized as follows. Section 2 contains some preliminaries. In Sect.  3, we define the concept of value dividends, present a new characterization of the Harsanyi set via efficiency and $\lambda$-balanced value dividends and contrast w-balanced contributions with $\lambda$-balanced value dividends. In Sect. 4, we introduce the inessential grand coalition property that is crucial for the remainder of the article and propose an axiomatization of the Harsanyi solutions and of the Harsanyi set as a whole. In Sect. 5, we generalize the Harsanyi set and give a class characterization. In Sect. 6, the proportional Harsanyi solution is defined and axiomatized, and the domain, the value, and reasonable axioms are illustrated using an example. Additionally, we give a quick comparison of the proportional Harsanyi solution with the Shapley value by a new axiomatization of the Shapley value. Section 7 concludes and discusses the results. Some extensions of the Harsanyi and the generalized Harsanyi set are briefly presented. Finally, the Appendix (Sect. 8) shows the logical independence of the axioms in our characterizations.

2 Preliminaries

We denote by ${\mathbb {N}}$ the set of natural numbers, by ${\mathbb {R}}$ the set of real numbers, by ${\mathbb {R}}_{+}$ the set of all non-negative real numbers, and by ${\mathbb {R}}_{++}$ the set of all positive real numbers. Let ${\mathfrak {U}}$ be a countably infinite set, the universe of all players. We define by ${\mathcal {N}}$ the set of all non-empty and finite subsets of ${\mathfrak {U}}$. A cooperative game with transferable utility (TU-game) is a pair (N, v) with player set $N\in {\mathcal {N}}$ and a coalition function $v\!: 2^N\!\rightarrow {\mathbb {R}},\,v(\emptyset ) = 0$. We call each subset $S\subseteq N$ a coalition, v(S) represents the worth of coalition S and we denote by $\text{\O}mega ^S$ the set of all non-empty subsets of S. The set of all TU-games with player set N is denoted by ${\mathbb {V}}(N)$, if $v(\{i\})>0$ for all $i \in N$ or if $v(\{i\})<0$ for all $i \in N$, by ${\mathbb {V}}_0(N)$, and if all $v(S)>0$ for all $S\in \text{\O}mega ^N$, by ${\mathbb {V}}_{0+}(N)$. The restriction of (N, v) to a player set $S \in \text{\O}mega ^N$ is the game $(S,v|_S)$, defined by $v|_S(R):=v(R)$ for all $R\subseteq S$ and it is shortly noted by (S, v). An unanimity game $(N,u_S),\, S\in \text{\O}mega ^N\!,$ is defined for all $T\subseteq N$ by $u_S(T)= 1$, if $S\subseteq T$, and $u_S(T)= 0,$ otherwise; the null game $(N,{\mathbf{0 }}^N)$ is given by ${\mathbf{0 }}^N\!(S)=0$ for all $S\subseteq N$.

Let $N\in {\mathcal {N}}$ and $(N,v)\in {\mathbb {V}}(N)$. For all $S\subseteq N$ the Harsanyi dividends $\Delta _v(S)$ (Harsanyi 1959) are defined inductively by

$$\begin{aligned} \Delta _v(S):={\left\{ \begin{array}{ll}0, \text { if }S = \emptyset , \text { and}\\ v(S)- \sum _{R\subsetneq S}\Delta _v(R), \text{ otherwise }. \end{array}\right. } \end{aligned}$$

(1)

A TU-game (N, v) is called almost positive if $\Delta _v(S)\ge 0$ for all $S\subseteq N,\,|S|\ge 2$; it is called totally positive (Vasil’ev 1975) if $\Delta _v(S)\ge 0$ for all $S\subseteq N$. We call a totally positive TU-game (N, v) strongly positive if $v(\{i\})>0$ for all $i \in N$. The set of all totally positive TU-games is denoted by ${\mathbb {V}}_+(N)$, and the set of all strongly positive TU-games by ${\mathbb {V}}_{++}(N)$. A player $i \in N$ is called a dummy player in (N, v) if $v(S\cup \{i\})=v(S)+v(\{i\})$ for all $S\subseteq N\backslash \{i\}$. Two players $i,j\in N,\, i\ne j$, are symmetric in (N, v) if $v(S\cup \{i\})=v(S\cup \{j\})$ for all $S\subseteq N\backslash \{i,j\}$.

We define by $W:=\{f\!:{\mathfrak {U}}\rightarrow {\mathbb {R}}_{++}\}$, $\mathrm {w}_i\!:=\mathrm {w}(i)$ for all $\mathrm {w} \in W,\,i \in {\mathfrak {U}}$, the collection of all positive weight systems on ${\mathfrak {U}}$. The collection $\Lambda$ of all sharing systems $\lambda \in \Lambda$ on ${\mathcal {N}}$ is defined¹ by

$$\begin{aligned} \Lambda :=\Bigg \{\lambda {=}(\lambda _{N,i})_{N \in {\mathcal {N}},\, i \in N}\Bigg |\, \sum _{i \in N}\lambda _{N,i}=1 \text { and } \lambda _{N,i}\ge 0\text { for each }N \in {\mathcal {N}}\text { and all } i {\in } N \Bigg \}. \end{aligned}$$

For all $N\in {\mathcal {N}}$, a TU-value (also called solution) $\varphi$ is an operator that assigns to any $(N,v) \in {\mathbb {V}}(N)$ a payoff vector $\varphi (N,v) \in {\mathbb {R}}^{N}$.

For all $N\in {\mathcal {N}},\,(N,v) \in {\mathbb {V}}(N),$ and each $\mathrm {w} \in W$, the (positively) weighted Shapley Value $Sh^\mathrm {w}$ (Shapley 1953a) is defined by

$$\begin{aligned} Sh_i^\mathrm {w}(N,v) := \sum _{S \subseteq N,\, S \ni i}\frac{\mathrm {w}_i}{\sum _{j \in S}\mathrm {w}_j}\Delta _v(S) \quad \;\text { for all } \,i \in N. \end{aligned}$$

The set of all weighted Shapley values is also known as Shapley set. A special case of a weighted Shapley value, all weights are equal, is the Shapley value Sh (Shapley 1953b), given by

$$\begin{aligned} Sh_i(N,v) := \sum _{S\subseteq N,\, {S\ni i}} \frac{\Delta _v(S)}{|S|} \quad \;\text { for all }N\in {\mathcal {N}},\,(N,v)\in {\mathbb {V}}(N), \text { and } \,i \in N. \end{aligned}$$

Hammer et al. (1977) and Vasil’ev (1978) introduced independently a set of TU-values, called Harsanyi set, also known as selectope (Derks et al. 2000). The payoffs are made by distributing the Harsanyi dividends with the help of a sharing system. Each TU-value $H^\lambda \!,\,\lambda \in \Lambda ,$ in this set, titled Harsanyi solution (or sharing value in Derks et al. (2000)), is defined by

$$\begin{aligned} H^\lambda _i(N,v):=\sum _{S \subseteq N,\, S \ni i}\lambda _{S,i}\Delta _v(S),\quad \text { for all }N \in {\mathcal {N}},\,(N,v)\in {\mathbb {V}}(N), \text { and } i \in N.\nonumber \\ \end{aligned}$$

(2)

Obviously, the Shapley set is a proper subset of the Harsanyi set. The following TU-values are not linear and are defined on subsets of ${\mathbb {V}}(N)$.

For all $N\in {\mathcal {N}},\,(N,v) \in {\mathbb {V}}_0(N)$, the proportional rule $\pi$ (Moriarity 1975) is given by

$$\begin{aligned} \pi _i(N,v): = \frac{ v(\{i\})}{\sum _{j \in N}v(\{j\})}v(N)\;\quad \text { for all } \,i \in N \end{aligned}$$

(3)

and the proportional Shapley value $Sh^{p}$ (Gangolly 1981; Besner 2016; Béal et al. 2018) is defined by

$$\begin{aligned} Sh^{p}_i(N,v) := \sum _{S\subseteq N,\,S\ni i} \frac{ v(\{i\})}{\sum _{j \in S}v(\{j\})}\Delta _v(S)\quad \;\text { for all } \,i \in N. \end{aligned}$$

(4)

For all $N\in {\mathcal {N}},\,(N,v)\in {\mathbb {V}}_{0+}(N)$, the proportional value P (Feldman 1999; Ortmann 2000) is defined inductively for all $i \in S,\,S \subseteq N$, by

$$\begin{aligned} P_i(S,v):={\left\{ \begin{array}{ll} v(\{i\}), \text { if }\;S= \{i\},\\ \dfrac{v(S)}{1+ \sum _{j \in S\backslash \{i\}} \dfrac{P_j(S\backslash \{i\},v)}{P_i(S\backslash \{j\},v)}}, \text { otherwise.} \end{array}\right. } \end{aligned}$$

(5)

We make use of the following axioms for TU-values $\varphi$:

Efficiency, E. For all $N\in {\mathcal {N}},\,(N,v) \in {\mathbb {V}}(N)$, we have $\sum _{i \in N} \varphi _i(N,v) = v(N).$

Dummy, D. For all $N\in {\mathcal {N}},\,(N,v) \in {\mathbb {V}}(N),$ and $i \in N$ such that i is a dummy player in (N, v), we have $\varphi _i(N,v) = v(\{i\})$.

Homogeneity, H. For all $N\in {\mathcal {N}},\,(N,v) \in {\mathbb {V}}(N),\,i \in N,$ and $\alpha \in {\mathbb {R}}$, we have $\varphi _i(N,\alpha v) = \alpha \varphi _i(N,v)$.

Monotonicity², M (Megiddo 1974). For all $N\in {\mathcal {N}},\,(N,v)\in {\mathbb {V}}(N),$ and $\alpha \in {\mathbb {R}}_{++}$, we have $\varphi _i(N,v+\alpha \cdot u_N)\ge \varphi _i(N,v)$ for all $i \in N$.

Positivity, P (Vasil’ev 1975). For all $N\in {\mathcal {N}},\,(N,v) \in {\mathbb {V}}(N)$ such that (N, v) is totally positive, and $i \in N$, we have $\varphi _i(v)\ge 0$.

Symmetry, S. For all $N\in {\mathcal {N}},\,(N,v) \in {\mathbb {V}}(N),$ and $i,j \in N$ such that i and j are symmetric in (N, v), we have $\varphi _i(N,v) = \varphi _j(N,v)$.

Balanced contributions, BC (Myerson 1980). For all $N\in {\mathcal {N}},\,(N,v) \in {\mathbb {V}}(N),$ and $i,j\in N,\, i\ne j$, we have $\varphi _i(N,v)-\varphi _i(N\backslash \{j\},v)=\varphi _j(N,v)-\varphi _j(N\backslash \{i\},v)$.

w-balanced contributions, BC$^\mathrm {w}$ (Myerson 1980). For all $N\in {\mathcal {N}},\,(N,v) \in {\mathbb {V}}(N),\,i,j\in N,\, i\ne j,$ and $\mathrm {w} \in W$, we have

$$\begin{aligned} \frac{\varphi _i(N,v)-\varphi _i(N\backslash \{j\},v)}{\mathrm {w}_i}=\frac{\varphi _j(N,v)-\varphi _j(N\backslash \{i\},v)}{\mathrm {w}_j}. \end{aligned}$$

Proportional standardness, PS (Ortmann 2000). For all $N\in {\mathcal {N}},\,\{i,j\}\subseteq N,\,i\ne j,\,(\{i,j\},v)\in {\mathbb {V}}_0(N),$ we have

$$\begin{aligned} \varphi _i(\{i,j\},v)=\frac{v(\{i\})}{v(\{i\})+v(\{j\})}v(\{i,j\}). \end{aligned}$$

3 Value dividends

Perhaps the most eminent characterization of the TU-values from the Harsanyi set was suggested by Derks et al. (2000, Theorem 4a). They characterized the Harsanyi solutions for the class of TU-games with a fixed player set by efficiency, the null player property, additivity and positivity. Another characterization was proposed by Vasil’ev (1981) [see Vasil’ev and van der Laan (2002, Theorem 5.1)]. This axiomatization additionally requires a convexity and a sign preserving property and, instead of additivity, an additivity property that is restricted to disjoint games. Thus, these characterizations require Harsanyi dividends (within positivity) and at least a specific form of additivity. We can dispense with both in the next axiomatization and with additivity in all subsequent axiomatizations. Therefore, we introduce a new concept that uses variable player sets.

The Harsanyi dividend of a coalition $S\subseteq N$ can be interpreted as the surplus of the worth of the coalition S versus the sum of all the surpluses of the worths of all proper subsets from S. Similarly, we define for a TU-value $\varphi$ the value dividend $\Theta _{\varphi _i(S,v)}$ of a coalition $S\subseteq N$ to a player $i \in S$ as the additional payoff to player i in the subgame (S, v) versus the sum of all additional payoffs to player i in all subgames (R, v), $R \subsetneq S,\; R \ni i$. In detail, we have:

Definition 3.1

For all $N\in {\mathcal {N}}$ and each $(N,v)\in {\mathbb {V}}(N),S \subseteq N,\,i\in S$, and TU-Value $\varphi$, the value dividends $\Theta _{\varphi _i(S,v)}$ are defined inductively by

$$\begin{aligned} \Theta _{\varphi _i(S,v)}:= {\left\{ \begin{array}{ll}\varphi _i({\{i\}},v), \text { if }S = \{i\},\\ \varphi _i(S,v)- \sum _{\begin{array}{c} R\subsetneq S, \, R \ni i \end{array}}\Theta _{\varphi _i(R,v)}, \text{ otherwise. } \end{array}\right. } \end{aligned}$$

(6)

For efficient TU-values, value dividends have a connection to Harsanyi dividends.

Remark 3.2

Let $N\in {\mathcal {N}}$ and $(N,v)\in {\mathbb {V}}(N)$. By (1), (6), and induction on the size $|S|,\,S\subseteq N$, it is easy to show that we have for an efficient TU-value $\varphi$

$$\begin{aligned} \sum _{i\in S}\Theta _{\varphi _i(S,v)}= \Delta _v(S) \text { for all }S \in \text{\O}mega ^N\!. \end{aligned}$$

3.1 $\lambda$-balanced value dividends

We formulate a new axiom for TU-values $\varphi$ which is related to w-balanced contributions. w-balanced contributions means that the ratio of the winnings or losses of two players when the other player leaves the game is proportional to their personal weights. The proportion of these weights is the same for two players for all coalitions containing them. If this ratio of weights can vary from coalition to coalition and we consider a value dividend as the pure payoff to a player that has not yet been realized in a subgame, it makes sense that the value dividends of two players for the same coalition are in the same ratio as the sharing weights for that coalition.

$\lambda$ -balanced value dividends, BVD $^\lambda$. For all $N\in {\mathcal {N}},\,(N,v)\in {\mathbb {V}}(N),\,i,j\in N,$ and $\lambda \in \Lambda ,$ we have $\lambda _{N,\,j}\Theta _{\varphi _i(N,v)}= \lambda _{N,i}\Theta _{\varphi _j(N,v)}.$

Since zero weights are also possible, we do not write the statement with fractions and, by definition of sharing weights, a player with a weight of zero for a coalition receives a zero dividend for that coalition. It turns out that a Harsanyi solution $H^\lambda$ is characterized by E and BVD$^\lambda$.

Theorem 3.3

Let $\lambda \in \Lambda$. $H^\lambda$ is the unique TU-value that satisfies E and BVD$^\lambda$.

Proof

I.:

Let $\lambda \in \Lambda$. It is well-known that $H^\lambda$ satisfies E. Thus, we have only to show that $H^\lambda$ meets BVD$^\lambda$. By (2) and (6), we have $\Theta _{H^\lambda _i(N,v)}=\lambda _{N,i}\Delta _{v}(N)$ for all $i \in N$ and all $N\in {\mathcal {N}}$. Therefore, it is obvious that BVD$^\lambda$ is satisfied too.

II.:

Let $N\in {\mathcal {N}},\,(N,v)\in {\mathbb {V}}(N),\,S\subseteq N,$ and let $\varphi$ and $\phi$ be two arbitrary TU-values which satisfy E and BVD$^\lambda$. We show uniqueness by induction on the size |S|.

Initialization: If $|S|=1$, uniqueness is satisfied by E.

Induction step: Let $|S| \ge 2$. Assume that equality of the two values holds for all $S'\subsetneq S,\,|S'|\ge 1,$ and let $|S|=|S'|+1$ (IH). Then equality of the value dividends of the two values for all $S'\subsetneq S$ holds too. Let $j \in S$ such that $\lambda _{S,\,j}\ne 0$. Such a j always exists. By BVD$^\lambda$, we have for all $i \in S$

$$\begin{aligned} \Theta _{\varphi _i(S,v)}&=\frac{\lambda _{S,i}}{\lambda _{S,\,j}}\Theta _{\varphi _j(S,v)} \nonumber \\&\quad \underset{(6)}{\Leftrightarrow }\;\varphi _i(S,v)- \sum _{\begin{array}{c} R\subsetneq S, \\ R \ni i \end{array}}\Theta _{\varphi _i(R,v)}= \frac{\lambda _{S,i}}{\lambda _{S,\,j}}\bigg [\varphi _j(S,v)- \sum _{\begin{array}{c} R\subsetneq S, \\ R \ni j \end{array}}\Theta _{\varphi _j(R,v)}\bigg ] \end{aligned}$$

(7)

and analogue

$$\begin{aligned} \phi _i(S,v)- \sum _{\begin{array}{c} R\subsetneq S, \\ R \ni i \end{array}}\Theta _{\phi _i(R,v)}&= \frac{\lambda _{S,i}}{\lambda _{S,\,j}}\bigg [\phi _j(S,v)- \sum _{\begin{array}{c} R\subsetneq S, \\ R \ni j \end{array}}\Theta _{\phi _j(R,v)}\bigg ]. \end{aligned}$$

(8)

We subtract (8) from (7)

$$\begin{aligned}&\varphi _i(S,v)- \phi _i(S,v)- \sum _{\begin{array}{c} R\subsetneq S, \\ R \ni i \end{array}}\Theta _{\varphi _i(R,v)}+ \sum _{\begin{array}{c} R\subsetneq S, \\ R \ni i \end{array}}\Theta _{\phi _i(R,v)} \nonumber \\&\quad = \frac{\lambda _{S,i}}{\lambda _{S,\,j}}\bigg [\varphi _j(S,v)- \phi _j(S,v)- \sum _{\begin{array}{c} R\subsetneq S, \\ R \ni j \end{array}}\Theta _{\varphi _j(R,v)}+ \sum _{\begin{array}{c} R\subsetneq S, \\ R \ni j \end{array}}\Theta _{\phi _j(R,v)}\bigg ]\nonumber \\&\quad \underset{(IH)}{\Leftrightarrow }\qquad \varphi _i(S,v)- \phi _i(S,v) = \frac{\lambda _{S,i}}{\lambda _{S,\,j}}\big [\varphi _j(S,v)- \phi _j(S,v)\big ]. \end{aligned}$$

(9)

(9) holds for all $i \in S$. We obtain

$$\begin{aligned} \sum _{i \in S}\big [\varphi _i (S,v) - \phi _i (S,v)\big ]=\sum _{i \in S}\frac{\lambda _{S,i}}{\lambda _{S,\,j}}\big [\varphi _j (S,v) - \phi _j (S,v)\big ] . \end{aligned}$$

(10)

By E, the left side of (10) equals zero. By induction, it follows $\varphi _j(S,v)=\phi _j(S,v)$ for all $j\in S$ and all $S\subseteq N$ with $\lambda _{S,\,j}\ne 0$. By (9), we have $\varphi _i(S,v)=\phi _i(S,v)$ also for all $i\in S$ with $\lambda _{S,i}= 0$ and uniqueness is shown. $\square$

3.2 w-balanced contributions and $\lambda$-balanced value dividends

If players’ sharing weights in all coalitions are in the same ratio, a Harsanyi solution coincides with a weighted Shapley value. For such weights, the w-balanced contributions axiom can be considered as a special case of the $\lambda$-balanced value dividends axiom.

Theorem 3.4

Let $\mathrm {w}\in W$ and $\lambda \in \Lambda$ such that

$$\begin{aligned} \lambda _{N,i}:=\frac{\mathrm {w}_i}{\sum _{j \in N}\mathrm {w}_j},\quad \text { for all }\, N\in {\mathcal {N}},\,|N|\ge 2, \text { and } i \in N. \end{aligned}$$

(11)

Then BVD$^\lambda$ is equivalent to BC$^{\,\mathrm {w}}$.

Proof

Let $N\in {\mathcal {N}},\,(N,v)\in {\mathbb {V}}(N),\,i,j\in N,\,i\ne j,$ and let w and $\lambda$ be defined as in Theorem 3.4.

BVD$^\lambda$ $\Rightarrow$ BC$^{\mathrm {w}}$: By BVD$^\lambda$ and (6), we have

$$\begin{aligned}&\frac{\varphi _i(N,v)-\sum _{\begin{array}{c} S\subsetneq N, \\ S \ni i \end{array}}\Theta _{\phi _i(S,v)}}{\mathrm {w}_i}= \frac{\varphi _j(N,v)-\sum _{\begin{array}{c} S\subsetneq N, \\ S \ni j \end{array}}\Theta _{\phi _j(S,v)}}{\mathrm {w}_j}\\&\qquad \Leftrightarrow \quad \frac{\varphi _i(N,v)-\sum _{\begin{array}{c} S\subseteq N\backslash \{j\}, \\ S \ni i \end{array}}\Theta _{\phi _i(S,v)}-\sum _{\begin{array}{c} S\subsetneq N,\\ \{i,j\}\subseteq S \end{array}}\Theta _{\phi _i(S,v)}}{\mathrm {w}_i}\\&\quad =\frac{\varphi _j(N,v)-\sum _{\begin{array}{c} S\subseteq N\backslash \{i\}, \\ S \ni j \end{array}}\Theta _{\phi _j(S,v)}-\sum _{\begin{array}{c} S\subsetneq N, \\ \{i,j\}\subseteq S \end{array}}\Theta _{\phi _j(S,v)}}{\mathrm {w}_j}\\&\qquad \underset{({{BVD}}{^\lambda })}{\Leftrightarrow }\frac{\varphi _i(N,v)-\sum _{\begin{array}{c} S\subseteq N\backslash \{j\}, \\ S \ni i \end{array}}\Theta _{\phi _i(S,v)}}{\mathrm {w}_i} =\frac{\varphi _j(N,v)-\sum _{\begin{array}{c} S\subseteq N\backslash \{i\}, \\ S \ni j \end{array}}\Theta _{\phi _j(S,v)}}{\mathrm {w}_j} \\&\qquad \;\;\;\underset{(6)}{\Leftrightarrow }\frac{\varphi _i(N,v)-\varphi _i(N\backslash \{j\},v)}{\mathrm {w}_i} =\frac{\varphi _j(N,v)-\varphi _j(N\backslash \{i\},v)}{\mathrm {w}_j}. \end{aligned}$$

BC$^{\mathrm {w}}$ $\Rightarrow$ BVD$^\lambda$: We use induction on the size |N|.

Initialization: Let $N=\{i,j\}$. By BC$^{\mathrm {w}}$, we have

$$\begin{aligned}&\frac{\varphi _i(N,v)-\varphi _i(N\backslash \{j\},v)}{\mathrm {w}_i} =\frac{\varphi _j(N,v)-\varphi _j(N\backslash \{i\},v)}{\mathrm {w}_j}\\&\underset{(6)}{\Leftrightarrow }\,\quad \frac{\Theta _{\varphi _i(N,v)}}{\mathrm {w}_i} =\frac{\Theta _{\varphi _j(N,v)}}{\mathrm {w}_j}\\&\underset{(11)}{\Leftrightarrow }\quad \frac{\Theta _{\varphi _i(N,v)}}{\lambda _{N,i}} =\frac{\Theta _{\varphi _j(N,v)}}{\lambda _{N,\,j}}. \end{aligned}$$

Induction step: Assume that the claim holds true for all player sets $N'\!,\,N'\subsetneq N,$ with $\underset{N'\subsetneq N}{max}|N'|\ge 2$ (IH). By BC$^{\mathrm {w}}$, we get

$$\begin{aligned}&\frac{\varphi _i(N,v)-\varphi _i(N\backslash \{j\},v)}{\mathrm {w}_i} =\frac{\varphi _j(N,v)-\varphi _j(N\backslash \{i\},v)}{\mathrm {w}_j}\\&\qquad \underset{(6)}{\Leftrightarrow }\quad \frac{\Theta _{\varphi _i(N,v)}+\sum _{\begin{array}{c} S\subsetneq N,\\ S\ni i \end{array}}\Theta _{\phi _i(S,v)}-\sum _{\begin{array}{c} S\subseteq N\backslash \{j\}, \\ S \ni i \end{array}}\Theta _{\phi _i(S,v)}}{\mathrm {w}_i}\\&\quad =\frac{\Theta _{\varphi _j(N,v)}+\sum _{\begin{array}{c} S\subsetneq N,\\ S\ni j \end{array}}\Theta _{\phi _j(S,v)}-\sum _{\begin{array}{c} S\subseteq N\backslash \{i\}, \\ S \ni j \end{array}}\Theta _{\phi _j(S,v)}}{\mathrm {w}_j}\\&\quad \underset{(11)}{\Leftrightarrow }\quad \frac{\Theta _{\varphi _i(N,v)}+\sum _{\begin{array}{c} S\subsetneq N,\\ \{i,j\}\subseteq S \end{array}}\Theta _{\phi _i(S,v)}}{\lambda _{N,i}} =\frac{\Theta _{\varphi _j(N,v)}+\sum _{\begin{array}{c} S\subsetneq N,\\ \{i,j\}\subseteq S \end{array}}\Theta _{\phi _j(S,v)}}{\lambda _{N,\,j}}\\&\qquad \underset{(IH)}{\Leftrightarrow }\;\;\;\,\qquad \qquad \qquad \qquad \frac{\Theta _{\varphi _i(N,v)}}{\lambda _{N,i}} =\frac{\Theta _{\varphi _j(N,v)}}{\lambda _{N,\,j}}. \end{aligned}$$

Since the claim holds for all $N\in {\mathcal {N}},\,|N|\ge 2$, equivalence is shown. $\square$

By Theorem 3.4, the following axiom is equivalent to the w-weighted balanced contributions property.

w-weighted balanced value dividends, BVD$^\mathrm {w}$. For all $N\in {\mathcal {N}},\,(N,v)\in {\mathbb {V}}(N),$ $i,j \in N,$ and $\mathrm {w}\in W$, we have

$$\begin{aligned} \frac{\Theta _{\varphi _i(N,v)}}{\mathrm {w}_i}= \frac{\Theta _{\varphi _j(N,v)}}{\mathrm {w}_j}. \end{aligned}$$

Especially, the next property is equivalent to the balanced contributions property.

Balanced value dividends, BVD. For all $N\in {\mathcal {N}},\,(N,v)\in {\mathbb {V}}(N),$ and $i, j \in N,$ we have

$$\begin{aligned} \Theta _{\varphi _i(N,v)}= \Theta _{\varphi _j(N,v)}. \end{aligned}$$

To introduce and justify the balanced contributions property, Myerson (1980) invoked the equal-gains principle: “[...] any two players should enjoy the same gains from their cooperation together, relative to what they would get without cooperation.” We can interpret this as meaning that, in the case of cooperation, all players should have the same share of the cooperation benefit. However, the value dividends are the pure benefit of cooperation to each player and, by the equal-gains principle, the value dividends of all players of the same coalition must be equal. Thus, we can also understand this widely accepted rule as an interpretation of the axiom of balanced value dividends and the balanced value dividends property is nothing more than a “fair” allocation rule. Hart and Mas-Colell (1989) note “[...] weights can be interpreted as ’a-priori measures of importance;’ they are taken to reflect considerations not captured by the characteristic function.” In this sense, we can interpret the w-weighted balanced value dividends property as a w-proportional-gains principle and therefore understand also as a “w-fair” allocation rule. The same shall apply mutatis mutandis to the $\lambda$-balanced value dividends axiom.

By Theorems 3.4 and 3.3, we get a corollary that is equivalent to the well-known axiomatization of the weighted Shapley values by efficiency and w-balanced contributions in Myerson (1980) and Hart and Mas-Colell (1989) and, as a special case, of the Shapley value by efficiency and balanced contributions in Myerson (1980).

Corollary 3.5

Let $\mathrm {w}\in W$. $Sh^\mathrm {w}$ is the unique TU-value that satisfies E and BVD$^\mathrm {w}$. In particular, Sh is the unique TU-value that satisfies E and BVD.

4 Inessential grand coalition

A TU-game (N, v) is called inessential if $v(S)=\sum _{i \in S}v(\{i\})$ for all $S\in \text{\O}mega ^N$. Note that (N, v) is inessential if and only if $v(S)=\sum _{R\subsetneq S}\Delta _v(R)$ for all $S \subseteq N,\,|S|\ge 2$. We weaken this characteristic for games with at least two players so that the last condition must hold only for the grand coalition: a TU-game $(N,v),\,|N|\ge 2,$ is called an inessential grand coalition game if $v(N)=\sum _{S\subsetneq N}\Delta _v(S)$.

The grand coalition is inessential in the sense that v(N) is completely determined by the worths of the proper subsets of N. The following new property for TU-values states that in inessential grand coalition games a player’s payoff is completely determined by the player’s payoff in all proper subgames.

Inessential grand coalition, IGC. For all $N\in {\mathcal {N}}$ and all inessential grand coalition games $(N,v)\in {\mathbb {V}}(N)$, we have $\varphi _i(N,v)=\sum _{S \subsetneq N,\,S\ni i}\Theta _{\varphi _i(S,v)}$ for all $i \in N.$

To axiomatize the proportional Shapley value, Béal et al. (2018) introduced an axiom for two games which only differ in the worth of the grand coalition. Two players’ payoff differentials must be proportional to their singleton worths.

Proportional (aggregate) monotonicity, PM (Béal et al. 2018). For all $N\in {\mathcal {N}}$, $|N|\ge 2$, $(N,v)\in {\mathbb {V}}_0(N)$, $\alpha \in {\mathbb {R}}$, and all $i,j \in N,$ we have

$$\begin{aligned} \frac{\varphi _i(N,v)-\varphi _i(N,v+\alpha \cdot u_N)}{v(\{i\})}=\frac{\varphi _j(N,v)-\varphi _j(N,v+\alpha \cdot u_N)}{v(\{j\})}. \end{aligned}$$

Similarly, the following property requires that two players’ payoff differentials must be proportional to their weights of a sharing system.

$\lambda$ -balanced monotonicity, M $^\lambda$.³ For all $N\in {\mathcal {N}},\,(N,v)\in {\mathbb {V}}(N),\,i,j \in N,\,\lambda \in \Lambda ,$ and $\alpha \in {\mathbb {R}}$, we have

$$\begin{aligned} \lambda _{N,\,j}\big [\varphi _i(N,v)-\varphi _i(N,v+\alpha \cdot u_N)\big ]=\lambda _{N,i}\big [\varphi _j(N,v)-\varphi _j(N,v+\alpha \cdot u_N)\big ]. \end{aligned}$$

$\lambda$-balanced monotonicity is not in itself related to monotonicity, but, along with efficiency, it implies monotonicity, analogous to proportional monotonicity in Béal et al. (2018).

Theorem 4.1

Let $\lambda \in \Lambda$. $H^\lambda$ is the unique TU-value that satisfies E, IGC, and M$^\lambda$.

Proof

I.:

It is well-known that $H^\lambda$ satisfies E and, by (2) and (6), it is clear that $H^\lambda$ meets IGC and M$^\lambda$ and existence is shown.

II.:

For all $N\in {\mathcal {N}},$ let $(N,v)\in {\mathbb {V}}(N),\,\lambda \in \Lambda$, and let $\varphi$ be a TU-value that satisfies E, IGC, and M$^\lambda$. We show uniqueness by induction on the size |N|.

Initialization: If $|N|=1$, uniqueness is satisfied by E.

Inductionstep: Let $|N| \ge 2$. Assume that uniqueness holds for all $N'\subsetneq N,\,|N'|\ge 1,$ (IH). Let $j \in N$ such that $\lambda _{N,\,j}\ne 0$. Note that such a j always exists and that $(N,v-\Delta _v(N)\cdot u_N)$ is an inessential grand coalition game. By M$^\lambda$, we have for all $i \in N$,

$$\begin{aligned} \varphi _i(N,v)-\varphi _i(N,v-\Delta _v(N)\cdot u_N)&=\frac{\lambda _{N,i}}{\lambda _{N,\,j}}\big [\varphi _j(N,v)-\varphi _j(N,v-\Delta _v(N)\cdot u_N)\big ]\\ \Rightarrow \sum _{i \in N}\!\big [\varphi _i (N,v) - \varphi _i (N,v-\Delta _v(N)\cdot u_N)\big ]&=\sum _{i \in N}\!\frac{\lambda _{N,i}}{\lambda _{N,\,j}}\big [\varphi _j (N,v) - \varphi _j (N,v-\Delta _v(N)\cdot u_N)\big ]. \end{aligned}$$

By IGC and (IH), $\varphi _i(N,v-\Delta _v(N)\cdot u_N)$ is unique for all $i\in N$. Therefore, by E, it follows that $\varphi _j (N,v)$ is unique for all $j\in N$ with $\lambda _{N,\,j}\ne 0$. Thus, by M$^\lambda$, $\varphi _i (N,v)$ is unique for all $i\in N$ with $\lambda _{N,\,i}= 0$ too and uniqueness is shown. $\square$

We would like to offer an axiomatic characterization of the Harsanyi set which does not explicitly use the sharing function systems. Nowak and Radzik (1995) used an axiom, called mutual dependence, to characterize the weighted Shapley values. Mutual dependence says that the payoffs to two players who are only jointly productive in two different games are in the same ratio in both games. The next property is implied by mutual dependence and requires that the ratios of two players’ payoff differentials in two different games on the same player set are equal, especially if the payoff differentials are not zero.

Dependent value monotonicity, DVM. For all $N\in {\mathcal {N}},\,(N,v),(N,w)\in {\mathbb {V}}(N),\,i,j \in N,$ and $\alpha ,\beta \in {\mathbb {R}}$, we have

$$\begin{aligned}&\big [\varphi _i(N,v)-\varphi _i(N,v + \alpha \cdot u_N)\big ]\big [\varphi _j(N,w)-\varphi _j(N,w + \beta \cdot u_N)\big ] \\&\quad = \big [\varphi _j(N,v)-\varphi _j(N,v + \alpha \cdot u_N)\big ]\big [\varphi _i(N,w)-\varphi _i(N,w + \beta \cdot u_N)\big ] \end{aligned}$$

We get an axiomatic characterization of the Harsanyi set.

Theorem 4.2

A TU-value $\varphi$ satisfies E, M, DVM, and IGC iff there exists a $\lambda \in \Lambda$, such that $\varphi =H^\lambda \!$.

Proof

I.:

Let $\lambda \in \Lambda$. It is well-known that $H^\lambda$ satisfies E and M. By Theorem 4.1, $H^\lambda$ satisfies IGC and M$^\lambda$. It is obvious that M$^\lambda$ implies DVM and thus existence is shown.

II.:

For all $N\in {\mathcal {N}}$, let $(N,v)\in {\mathbb {V}}(N)$ and let $\varphi$ be a TU-value that satisfies E, M, DVM, and IGC. We show that $\varphi =H^\lambda$ for some $\lambda \in \Lambda$. If $|N|=1$, we have $\varphi =H^\lambda$ for all $\lambda \in \Lambda$ by E. Let now $|N| \ge 2$. By E and M, we have for all such $N\in {\mathcal {N}}$ and all $i \in N$, $\varphi _i(N,u_N)-\varphi _i(N,{\mathbf{0 }}^N)= c_{N,i}\in {\mathbb {R}}_{+}$ and $\sum _{i \in N}c_{N,i}=1$. Thus exists a $\lambda \in \Lambda$ with $\lambda _{N,i}:=c_{N,i}$ for all $N\in {\mathcal {N}},\,|N|\ge 2,$ and all $i \in N$. By DVM, we get for all $N\in {\mathcal {N}},\,(N,v)\in {\mathbb {V}}(N),\,i,j \in N,$ all $\alpha \in {\mathbb {R}},$ and a $\lambda \in \Lambda$ just defined,

$$\begin{aligned} \big [\varphi _i(N,v)-\varphi _i(N,v + \alpha \cdot u_N)\big ]c_{N,j}&=\big [\varphi _j(N,v)-\varphi _j(N,v + \alpha \cdot u_N)\big ]c_{N,i} \\ \Leftrightarrow \; \lambda _{N,\,j}\big [\varphi _i(N,v)-\varphi _i(N,v+\alpha \cdot u_N)\big ]&=\lambda _{N,i}\big [\varphi _j(N,v)-\varphi _j(N,v+\alpha \cdot u_N)\big ]. \end{aligned}$$

Therefore, $\varphi$ satisfies M$^\lambda$ and, due to E, IGC, and Theorem 4.1, $\varphi$ equals a Harsanyi solution $H^\lambda$ with $\lambda \in \Lambda$. $\square$

5 The generalized Harsanyi set

Casajus (2017) introduced a class of TU-values $\varphi ^\omega \!,\,\omega \in \overline{\text{\O}mega }$,   $\overline{\text{\O}mega }:=\{f:{\mathbb {R}}\times {\mathfrak {U}}\rightarrow {\mathbb {R}}_{++}\}$, defined by

$$\begin{aligned} \varphi ^{\omega }_i(N,v) {:=} \sum _{S\subseteq N,\,S\ni i} \frac{\omega ( v(\{i\}),i)}{\sum _{j {\in } S}\omega (v(\{j\}),j)}\Delta _v(S)\;\text { for all }N\in {\mathcal {N}},\,(N,v) \in {\mathbb {V}}(N) ,\text { and }i \in N. \end{aligned}$$

If $\omega$ does not depend on v, $\varphi ^{\omega }$ equals a weighted Shapley value. Therefore, we will call the class of all TU-values $\varphi ^{\omega }$ the generalized Shapley set and we will denote each value in that class by $Sh^\omega$ for each $\omega \in \overline{\text{\O}mega }$. This class obviously contains the weighted Shapley values but also non-linear TU-values like the TU-values $\varphi ^c\!$ which are defined for all $N\in {\mathcal {N}},\,(N,v) \in {\mathbb {V}}(N),$ and all $c>0,$ by

$$\begin{aligned} \varphi ^c_i(N,v) := \sum _{S\subseteq N,\,S\ni i} \frac{ |v(\{i\})|+c}{\sum _{j \in S}(|v(\{j\})|+c)}\Delta _v(S)\;\text { for all } \,i \in N. \end{aligned}$$

(12)

Here the weight function depends on the worth of the singletons. The class of the following TU-values extends the Harsanyi set. The weights for the values of this class may depend on the entire coalition function.

Let $\widetilde{{\mathcal {V}}}:=\{{\widetilde{v}}:{\mathcal {N}}\cup \{\emptyset \}\rightarrow {\mathbb {R}},\,{\widetilde{v}}(\emptyset )=0\}$ and let for all ${\widetilde{v}}\in \widetilde{{\mathcal {V}}}$, ${\mathbb {V}}({\mathcal {N}},{\widetilde{v}}) :=\{(N,v^N)\in {\mathbb {V}}(N)|\,N\in {\mathcal {N}} \text { and } v^N(S)={\widetilde{v}}(S) \text { for all } S\subseteq N\}$ be the set of all TU-games $(N,v^N)\in {\mathbb {V}}(N)$ on all player sets $N\in {\mathcal {N}}$ such that $v^N(S)={\widetilde{v}}(S)$ for all $S\subseteq N$ and all $N\in {\mathcal {N}}$.

Related to a TU-value, we define: for all $N\in {\mathcal {N}}$, a sharing function $\psi ^N$ on N is an operator that assigns to any $(N,v) \in {\mathbb {V}}(N)$ a sharing vector $\psi ^N(v) \in {\mathbb {R}}_+^{N}$ such that $\sum _{i \in N}\psi _i^N(v)=1$. For all ${\widetilde{v}}\in \widetilde{{\mathcal {V}}}$, the collection $\varPsi ({\widetilde{v}})$ of all sharing function systems $\psi \in \varPsi ({\widetilde{v}})$ is defined by

$$\begin{aligned} \varPsi ({\widetilde{v}}):=\Big \{\psi =\big (\psi _i^N(v^N)\big )_{N \in {\mathcal {N}},\, i \in N}\big |\,v^N(S)={\widetilde{v}}(S) \text { for all }S\subseteq N \text { and }N\in {\mathcal {N}}\Big \}. \end{aligned}$$

For each fixed ${\widetilde{v}}$ or if the sharing functions are constants, a sharing function system coincides with a sharing system. This leads to the naming of the following TU-values. For all ${\widetilde{v}}\in \widetilde{{\mathcal {V}}},$ all $(N,v)\in {\mathbb {V}}({\mathcal {N}},{\widetilde{v}})$, and $\psi \in \varPsi ({\widetilde{v}}),$ the generalized Harsanyi solution $H^\psi$ is defined by

$$\begin{aligned} H^\psi _i(N,v) := \sum _{S\subseteq N,\,S\ni i} \psi ^S_i(v)\Delta _v(S) \quad \;\text { for all } \,i \in N. \end{aligned}$$

(13)

We call the class of all generalized Harsanyi solutions generalized Harsanyi set. In the next property the $\lambda \in \Lambda$ in BVD$^\lambda$ will be replaced by a $\psi \in \varPsi ({\widetilde{v}})$.

$\psi$ -balanced value dividends, BVD $^\psi$. For all ${\widetilde{v}}\in \widetilde{{\mathcal {V}}},$ all $(N,v)\in {\mathbb {V}}({\mathcal {N}},{\widetilde{v}})$, and $\psi \in \varPsi ({\widetilde{v}}),$ we have $\psi _{N,\,j}\Theta _{\varphi _i(N,v)}= \psi _{N,i}\Theta _{\varphi _j(N,v)}.$

The following theorem is completely analogous to Theorem 3.3.

Theorem 5.1

Let $\psi \in \varPsi ({\widetilde{v}}),\,{\widetilde{v}}\in \widetilde{{\mathcal {V}}}$. $H^\psi$ is the unique TU-value that satisfies E and BVD$^\psi$.

The proof is omitted since it is straightforward to transfer the proof from Theorem 3.3. We provide a characterization of the generalized Harsanyi set which does not use sharing function systems explicitly. Here, the dependent value monotonicity property in Theorem 4.2 is dropped.

Theorem 5.2

A TU-value $\varphi$ satisfies E, IGC, and M iff there exists for all ${\widetilde{v}}\in \widetilde{{\mathcal {V}}}$ a $\psi \in \varPsi ({\widetilde{v}})$ such that $\varphi =H^\psi$.

Proof

I.:

Let ${\widetilde{v}}\in \widetilde{{\mathcal {V}}},(N,v)\in {\mathbb {V}}({\mathcal {N}},{\widetilde{v}}),$ and $\psi \in \varPsi ({\widetilde{v}})$. By (13) and (6), it is clear that $H^\psi$ satisfies E, IGC, and M.

II.:

Let $(N,v)\in {\mathbb {V}}(N)$ and $\varphi$ a TU-value that satisfies E, IGC, and M. We show uniqueness by induction on the size |N|.

Initialization: If $|N|=1$, uniqueness is satisfied by E.

Induction step: Let $|N| \ge 2$. Assume that uniqueness holds for all $N'\subsetneq N,\,|N'|\ge 1,$ (IH). Let $(N,w)\in {\mathbb {V}}(N)$ such that $w(S):=v(S)$ for all $S\subsetneq N$ and $\Delta _w(N):=0$. Then, by I., (IH), and IGC, we get $\varphi _i(N,w)=H^{\psi '}_i(N,w)$ for all $i \in N$ and some $\psi ' \in \varPsi ({\widetilde{v}})$ and some ${\widetilde{v}}\in \widetilde{{\mathcal {V}}}$ such that $(N,w)\in {\mathbb {V}}({\mathcal {N}},{\widetilde{v}})$. It follows, as a first case, $\varphi (N,v)=H^{\psi '}(N,v)$ if $v(N)=w(N)$.

If, as a second case, $\Delta _v(N)>0$, we have, by E,

$$\begin{aligned} \sum _{i \in N}\varphi _i(N,v) = \sum _{i \in N} H^{\psi '}(N,w)+\Delta _v(N). \end{aligned}$$

By M, we get for all $i \in N$,

$$\begin{aligned} \varphi _i(N,v)&\ge H_i^{\psi '}(N,w) \\ \Rightarrow \quad \varphi _i(N,v)&= H_i^{\psi '}(N,w) {+} \chi _i(N,v),\,\chi _i(N,v) {\ge } 0, \text { and }\sum _{i \in N}\chi _i(N,v)=\Delta _v(N) \\ \Rightarrow \quad \varphi _i(N,v)&= H_i^{\psi '}(N,w) {+} \frac{\chi _i(N,v)}{\sum _{j \in N}\chi _j(N,v)}\Delta _v(N){=} H_i^{\psi '}(N,w) {+} \psi ^N_i(v) \Delta _v(N). \end{aligned}$$

$\psi ^N(v)$ is a sharing function and is part of a sharing function system $\psi \in \varPsi ({\widetilde{v}})$ with $\psi ^S(v)={\psi '}^{S}(v)$ for all $S\subsetneq N$ and sharing functions ${\psi '}^S(v)$ from a sharing function system $\psi ' \in \varPsi ({\widetilde{v}})$. By (13), it follows

$$\begin{aligned} \varphi _i(N,v)= H_i^{\psi }(N,v)\quad \text { for all }i \in N. \end{aligned}$$

If $\Delta _v(N)<0$, equality follows analogously. $\square$

6 The proportional Harsanyi solution

It is clear that the TU-values $\varphi ^c$ from Sect. 5 and the proportional Shapley value $Sh^p$ (on a subset of all TU-games) are also part of the generalized Harsanyi set. One may ask, whether the proportional value P is also a member of the generalized Harsanyi set (on the relevant subset of TU-games). However, it is easy to show that this is not the case.⁴

Many scientific studies about the Harsanyi set deal with totally positive games (see, e. g., Vasil’ev and van der Laan 2002; van den Brink et al. 2014). Also the set-valued proper Shapley value is defined on totally positive games⁵ by Vorob’ev and Liapunov (1998). We introduce a new proportional TU-value from the generalized Harsanyi set, defined on the subset of strictly positive games. It satisfies proportional standardness, in harmony with other proportional TU-values as the proportional rule, the proportional value, or the proportional Shapley value.

Definition 6.1

For all $N\in {\mathcal {N}},\,(N,v)\in {\mathbb {V}}_{++}(N)$, the proportional Harsanyi solution $H^p$ is defined inductively for all $i \in S,\, S\subseteq N,$ by

$$\begin{aligned} H_i^p(S,v):={\left\{ \begin{array}{ll} v(\{i\}), \text { if }\;S= \{i\},\\ \dfrac{\sum _{R \subsetneq S,\,R\ni i}\Theta _{H_i^p(R,v)}}{\sum _{j\in S}\sum _{R \subsetneq S,\,R\ni j}\Theta _{H_j^p(R,v)}}v(S), \text { otherwise.} \end{array}\right. } \end{aligned}$$

(14)

Remark 6.2

One easily shows, by Remark 3.2 and induction on the size |S|, that $H^p$ is well-defined. Moreover, we have $H^p_i(N,v)>0$ for all $i \in N,\,(N,v)\in {\mathbb {V}}_{++}(N)$, and P is satisfied in a strict notion. By (14), it is obvious that $H^p$ satisfies E and PS.

We present a formula for the proportional Harsanyi solution that distributes the Harsanyi dividends proportional to players’ value dividends and confirms the membership to the generalized Harsanyi set.

Proposition 6.3

For all $N\in {\mathcal {N}},\,(N,v)\in {\mathbb {V}}_{++}(N)$, we have

$$\begin{aligned} H_i^p(N,v)= \Delta _v(\{i\})+\sum _{\begin{array}{c} S \subseteq N\\ S\ni i,\,S\ne \{i\} \end{array}}\!\!\!\!\dfrac{\sum _{R \subsetneq S,\,R\ni i}\Theta _{H_i^p(R,v)}}{\sum _{j\in S}\sum _{R \subsetneq S,\,R\ni j}\Theta _{H_j^p(R,v)}}\Delta _v(S)\quad \text { for all }i \in N. \end{aligned}$$

(15)

Proof

If $|N|=1$, the claim is obvious. Let now $|N| \ge 2$. We have for all $i \in N$

$$\begin{aligned}{}\begin{array}{rcl} \Delta _v(\{i\})&{}+&{}\displaystyle {\sum _{\begin{array}{c} S \subseteq N,\\ S\ni i,\,S\ne \{i\} \end{array}}\!\!\!\!\dfrac{\sum _{R \subsetneq S,\,R\ni i}\Theta _{H_i^p(R,v)}}{\sum _{j\in S}\sum _{R \subsetneq S,\,R\ni j}\Theta _{H_j^p(R,v)}}\Delta _v(S)} \\ &{}\underset{\begin{array}{c} (1),\, {{{E}}}\\ {Rem.}\,3.2 \end{array}}{=}&{}\Delta _v(\{i\})+\displaystyle {\sum _{\begin{array}{c} S \subseteq N,\\ S\ni i,\,S\ne \{i\} \end{array}}\!\!\!\!\dfrac{\sum _{R \subsetneq S,\,R\ni i}\Theta _{H_i^p(R,v)}}{\sum _{j\in S}\sum _{R \subsetneq S,\,R\ni j}\Theta _{H_j^p(R,v)}}\Big [v(S)- \sum _{R\subsetneq S}\sum _{j \in R}\Theta _{H_j^p(R,v)}\Big ] } \\ &{}\underset{(14)}{=}&{}\Delta _v(\{i\})+\displaystyle {\!\!\sum _{\begin{array}{c} S\subseteq N,\\ S\ni i,\,S\ne \{i\} \end{array}}\!\!\!\!H_i^p(S,v)-\!\!\sum _{\begin{array}{c} S \subseteq N,\\ S\ni i,\,S\ne \{i\} \end{array}}\sum _{\begin{array}{c} R \subsetneq S,\\ R\ni i \end{array}}\Theta _{H_i^p(R,v)} } \\ &{}\underset{(6)}{=}&{} \Delta _v(\{i\})+\displaystyle {\!\!\sum _{\begin{array}{c} S\subseteq N,\\ S\ni i,\,S\ne \{i\} \end{array}}\sum _{\begin{array}{c} R \subsetneq S,\\ R\ni i \end{array}}\Theta _{H_i^p(R,v)}-\!\!\sum _{\begin{array}{c} S \subseteq N,\\ S\ni i,\,S\ne \{i\} \end{array}}\sum _{\begin{array}{c} R \subsetneq S,\\ R\ni i \end{array}}\Theta _{H_i^p(R,v)}} \\ &{} \underset{(6)}{=}&{} H_i^p(N,v). \end{array} \end{aligned}$$

$\square$

For games which differ only in the grand coalition, the following axiom requires that players’ payoffs remain in the same ratio.

Proportion preservation, PP. For all $N\in {\mathcal {N}},\,(N,v)\in {\mathbb {V}}(N),$ and $\alpha \in {\mathbb {R}}$, we have

$$\begin{aligned} \varphi _i(N,v)\varphi _j(N,v+\alpha \cdot u_N)= \varphi _j(N,v)\varphi _i(N,v+\alpha \cdot u_N)\quad \text { for all }i,j \in N. \end{aligned}$$

As a consequence of the following axiom, each player’s share is independent of the worth of the grand coalition.

Independent share, IS. For all $N\in {\mathcal {N}},\,(N,v)\in {\mathbb {V}}(N),$ and $\alpha \in {\mathbb {R}}$, we have

$$\begin{aligned} \varphi _i(N,v)[v(N)+\alpha ]= \varphi _i(N,v +\alpha \cdot u_N)v(N) \text { for all }i \in N. \end{aligned}$$

This axiom implies, e.g., that if a positive worth of the grand coalition increases, while the worths of all other coalitions remain fixed, then a players’ positive payoff increases proportionally to the increase of the worth of the grand coalition.

Remark 6.4

It is clear that IS implies PP. One also easily checks that if a TU-value satisfies E and PP, then it also satisfies IS.

Remark 6.5

Examination of (3), (5), and (14) shows that the proportional rule $\pi$, the proportional value P, and the proportional Harsanyi solution $H^p$ satisfy PP and IS, but, by (4), this is not the case for the proportional Shapley value $Sh^p$.

The next axiom relates to the $\lambda$-balanced value dividends property: two players’ value dividends of the grand coalition are in the same proportion as the payoffs if the value dividends and the payoffs are not zero.

Value balanced value dividends, VBVD. For all $N\in {\mathcal {N}},\,(N,v)\in {\mathbb {V}}(N),$ and $i, j \in N$, we have

$$\begin{aligned} \varphi _i(N,v)\Theta _{H_j^p(N,v)}= \varphi _j(N,v)\Theta _{H_i^p(N,v)}. \end{aligned}$$

(16)

Remark 6.6

By (6), (16) is equivalent to

$$\begin{aligned} \varphi _i(N,v)\sum _{S \subsetneq N,\,S\ni j}\!\!\!\Theta _{H_j^p(S,v)}&= \varphi _j(N,v)\sum _{S \subsetneq N,\,S\ni i}\!\!\!\Theta _{H_i^p(S,v)}. \end{aligned}$$

Thus, VBVD also states that two players’ payoffs are in the same proportion as the sums of players’ value dividends of all proper subsets of the grand coalition if the value dividends and the payoffs are not zero. Note that VBVD holds for all games whereas PP and IS need games which differ only in the grand coalition. The proportional Harsanyi solution matches VBVD and a lot of other properties.

Proposition 6.7

Let $N\in {\mathcal {N}},\,(N,v)\in {\mathbb {V}}_{++}(N)$. $H^p$ satisfies D, M, H, S, VBVD, and IGC.

Proof

Let $(N,v)\in {\mathbb {V}}_{++}(N)$. It is well-known that we have for a dummy player $i\in N$ in (N, v), $\Delta _v(S)=0$ for all $S\subseteq N,\,|S|\ge 2,\,S\ni i$. Thus, D follows immediately by (15) and, also by (15), M is obviously satisfied. By induction on the size |N| and formula (14) follows H. In addition, also as a well-known fact, we have for two symmetric players $i,j\in N$ in (N, v), $\Delta _v(S\cup \{i\})=\Delta _v(S\cup \{j\})$ for all $S\subseteq N$. Then, it is obvious, by (15), that $H^p$ satisfies S. By (14), we can see that $H^p$ matches VBVD and, by Definition 3.1, Remark 3.2, and E, obviously IGC is satisfied. $\square$

Derks et al. (2000) could show that the Harsanyi set coincides for almost positive games with the core. The core of a TU-game $(N,v)\in {\mathbb {V}}(N)$ is the set $C(N,v):=\{x\in {\mathbb {R}}^N:x(N)=v(N) \text { and } x(S) \ge v(S) \text { for all }S\in \text{\O}mega ^N\}$. It is obvious, by E and (15), that it is a necessary characteristic of the proportional Harsanyi solution to be a member of the core. This can be interpreted in such a way that no coalition of players can improve upon or block the payoff.

Remark 6.8

For all $N\in {\mathcal {N}},\,(N,v)\in {\mathbb {V}}_{++}(N)$, we have $H^p(N,v)\in C(N,v)$.

Our first main result in this section is related to Theorem 3.3 and shows that the value balanced value dividends axiom is a very strong property.

Theorem 6.9

Let $N\in {\mathcal {N}}$ and $(N,v)\in {\mathbb {V}}_{++}(N)$. $H^p$ is the unique TU-value that satisfies E and VBVD.

Proof

By Proposition 6.7 and Remark 6.2, we have only to show uniqueness. Let $(N,v)\in {\mathbb {V}}_{++}(N)$ and let $\varphi$ be a TU-value that satisfies E and VBVD. We use induction on the size $|S|,\,S\subseteq N$.

Initialization: If $|S|=1$, uniqueness of $\varphi (S,v)$ is satisfied by E. Moreover, we have for all $T\subseteq N,\,|T|=|S|+1,$

$$\begin{aligned} \sum _{S\subsetneq T,\,S\ni i,}\Theta _{\varphi _i(S,v)} >0. \end{aligned}$$

(17)

Induction step: Assume uniqueness and (17) hold for $|S|-1, |S|\ge 2$ (IH). Then, by (IH), we have $\sum _{R\subsetneq S,\,R\ni i,}\Theta _{\varphi _i(R,v)} >0$. By Remark 6.6, it follows for all $i \in S$ and a fixed $j \in S$,

$$\begin{aligned} \varphi _i(S,v)&= \frac{\sum _{R\subsetneq S,\,R\ni i,}\Theta _{\varphi _i(R,v)}}{\sum _{R\subsetneq S,\,R\ni j,}\Theta _{\varphi _j(R,v)}}\varphi _j(S,v) \\ \Leftrightarrow \quad \sum _{i \in S}\varphi _i(S,v)&=\sum _{i \in S} \frac{\sum _{R\subsetneq S,\,R\ni i,}\Theta _{\varphi _i(R,v)}}{\sum _{R\subsetneq S,\,R\ni j,}\Theta _{\varphi _j(R,v)}}\varphi _j(S,v). \end{aligned}$$

Thus, by E and induction, $\varphi$ is unique for all $S\subseteq N$ and Theorem  6.9 is shown. $\square$

The next theorem does not need efficiency.

Theorem 6.10

Let $N\in {\mathcal {N}}$ and $(N,v)\in {\mathbb {V}}_{++}(N)$. $H^p$ is the unique TU-value that satisfies D, IGC, and IS.

Proof

By Proposition 6.7 and Remarks 6.2 and 6.5, we have only to show uniqueness. Let $(N,v)\in {\mathbb {V}}_{++}(N)$ and let $\varphi$ be a TU-value that satisfies D, IGC, and IS. We use induction on the size $|S|,\,S\subseteq N$.

Initialization: If $|S|=1$, uniqueness of $\varphi (S,v)$ is satisfied by D.

Induction step: Assume uniqueness holds for $|S|-1, |S|\ge 2$ (IH). Then, by (IH), we have $\sum _{R\subsetneq S,\,R\ni i,}\Theta _{\varphi _i(R,v)}$ is unique for all $i \in S$. Let $(S,w)\in {\mathbb {V}}_{++}(S)$ such that $\Delta _w(S)=0$ and $v(R)=w(R)$ for all $R \subsetneq S$. By IGC, it follows $\varphi _i(S,w)$ is unique for all $i \in S$. By IS, we have for all $i \in S$

$$\begin{aligned} \varphi _i(S,v)= \frac{v(N)}{w(N)}\varphi _i(S,w). \end{aligned}$$

and $\varphi$ is unique for all $S\subseteq N$ and Theorem 6.10 is shown. $\square$

Remark 6.11

The proof shows that in Theorem 6.10D can be replaced by any axiom that guarantees that a players’ payoff in a singleton game is her worth and that is satisfied by $H^P$. Therefore, e. g., E or the inessential game property that states that a player in an inessential game receives her singleton worth can be used instead of D.

We have a last characterization of the proportional Harsanyi solution. It follows immediately from Theorem 6.10 and Remarks 6.4 and 6.11.

Corollary 6.12

Let $N\in {\mathcal {N}}$ and $(N,v)\in {\mathbb {V}}_{++}(N)$. $H^p$ is the unique TU-value that satisfies E, IGC, and PP.

Remark 6.13

By (13), all values from the generalized Harsanyi set satisfy for all $N\in {\mathcal {N}}$ on ${\mathbb {V}}_{++}(N)$, E, D, and IGC. Thus, $H^p$ is the unique value from the Harsanyi set that satisfies PP and IS on ${\mathbb {V}}_{++}(N)$.

Introducing the proportional value, Ortmann (2000) used similar characterization approaches to that of the Shapley value. In particular, he contrasted efficiency and his preservation of ratios axiom with efficiency and the balanced contributions property and standardness in two player games and consistency with proportional standardness and consistency. An analogous proceeding we can see in Béal et al. (2018) with the proportional Shapley value and the Shapley value. For one such comparison, they introduced the proportional monotonicity property and the following axiom, which states that two players’ payoff differentials should be the same for varying worth of the grand coalition.

Equal (aggregate) monotonicity, EM (Béal et al. 2018).⁶ For all $N\in {\mathcal {N}},\,(N,v)\in {\mathbb {V}}(N),$ and $\alpha \in {\mathbb {R}}$, we have

$$\begin{aligned} \varphi _i(N,v)-\varphi _i(N,v+\alpha \cdot u_N)=\varphi _j(N,v)-\varphi _j(N,v+\alpha \cdot u_N) \text { for all }i,j\in N, \end{aligned}$$

Béal et al. (2018) axiomatized the proportional Shapley value by efficiency, dummy player out, weak linearity and proportional monotonicity and contrasted it with an axiomatization of the Shapley value by replacing proportional monotonicity with equal monotonicity.

The question arises whether it is also possible to contrast axiomatizations of the proportional Harsanyi solution with those of the Shapley value which differ only in one axiom. We can show that the answer is yes. The proportional Harsanyi solution is uniquely determined by E, IGC and PP. PP preserves ratios for games which differ only in the grand coalition. For such games EM preserves differences. It follows a “contrasted” axiomatization of the Shapley value.

Theorem 6.14

Sh is the unique TU-value that satisfies E, IGC, and EM.

Proof

Since it is easy to check that all axioms in Theorem 6.14 are satisfied we have only to show uniqueness. Let $(N,v)\in {\mathbb {V}}(N)$ and let $\varphi$ be a TU-value that satisfies E, IGC, and EM. We use induction on the size $|S|,\,S\subseteq N$.

Initialization: If $|S|=1$, uniqueness of $\varphi (S,v)$ is satisfied by E.

Induction step: Assume uniqueness holds for $|S|-1, |S|\ge 2$ (IH). Then, by (IH), we have $\sum _{R\subsetneq S,\,R\ni i,}\Theta _{\varphi _i(R,v)}$ is unique for all $i \in S$. By IGC, it follows $\varphi _k(S,v-\Delta _v(S)\cdot u_S)$ is unique for all $k \in S$. By EM, we have for all $i,j \in S$

$$\begin{aligned} \varphi _i(S,v)&= \varphi _i(S,v-\Delta _v(S)\cdot u_S)+\varphi _j(S,v)-\varphi _j(S,v-\Delta _v(S)\cdot u_S) \\ \Leftrightarrow \;\; \sum _{k \in S}\varphi _k(S,v)&= \sum _{k \in S}\varphi _k(S,v{-}\Delta _v(S)\cdot u_S) {+} |S|\cdot \big [\varphi _j(S,v){-}\varphi _j(S,v{-}\Delta _v(S)\cdot u_S)\big ] \end{aligned}$$

and $\varphi$ is unique for all $S\subseteq N$ by E and Theorem 6.14 is shown. $\square$

The following example justifies the value and illustrates the axioms, used in Theorem 6.10 and Corollary 6.12.

Example

A group N of independent carpenters who are not too busy as a one-man business join forces and work together in different groups $S\subseteq N$. The total quantity and size of the orders depends on a number of external factors, such as the macroeconomic cycle, the general interest rate level, the general state of the buildings and so on.

We assume that the share of each carpenter as a one-man company in the total net profit of all one-man companies depends only on her performance. We also assume that orders for larger coalitions depend only on the efficiency of the subgroups and whether the customers receive good value for money from the subgroups. The order volume is not too large, so that all orders, offered for groups of different sizes, can be executed.

We model the situation as a TU-game. Nobody cooperates with a carpenter who has no net profit and therefore, does not work efficiently. Thus, we only regard the player sets N with $v(\{i\}) > 0$ for all players $i\in N$. The worth v(S) of a coalition $S\subseteq N$ equals the sum of all net profits of all subunits of S. The Harsanyi dividend $\Delta _v(S)$ of coalition S is equal to the net profit of the unit S, but only in cases where the players of S work as a unit. Besides, the carpenters won’t pay on top. Therefore, we have $\Delta _v(S)\ge 0$ for all $S\subseteq N,\,|S|\ge 2,$ and $(N,v)\in {\mathbb {V}}_{++}(N)$.

The carpenters must agree on how to share the worth of the grand coalition v(N). They want to share the whole worth in such a way that E must be satisfied. The carpenters have no way of deciding how to divide the net profit of a unit solely by the performance of the unit itself. They can only take into account the profits of all subunits. Hence, the players’ shares on the grand coalition net profits do not depend on the grand coalition’s profit as a single unit. The carpenters conclude that each player’s share should be independent of the worth of the grand coalition and therefore IS must be satisfied. If a grand coalition does not work as a unit, that unit’s net profit is zero. Thus, the carpenters agree that in this case the payoff should be equal to the sum of players’ payoffs in all proper subunits. Therefore, IGC should also be satisfied. Hence, by Theorem 6.10 and Remark 6.11, the proportional Harsanyi value $H^p$ is the method of choice for distributing the profits. Certainly, the carpenters accept D, so that a carpenter who does not cooperate with partners in other units should only receive her one-man company result. The carpenters can also conclude, that the payoff in the grand coalition unit should be proportional to the sum of their payoffs in all other subunits and therefore agree, by Remark 6.6, that VBVD should be satisfied. After some reflection, all the properties presented in this section seem reasonable to the carpenters.

7 Conclusion and discussion

Most studied TU-values are efficient and a lot of them satisfy monotonicity. Thus, the inessential grand coalition property can be used as a criterion whether such a value has to be in the (generalized) Harsanyi set (see footnote 4).

The article shows a strong connection between values from the (generalized) Harsanyi set and value dividends. Also the non-efficient Banzhaf value has a representation with dividends. For all $N \in {\mathcal {N}}$ and $(N,v) \in {\mathbb {V}}(N)$ the Banzhaf value $\beta$ (Banzhaf 1965) is given by

$$\begin{aligned} \beta _i(N,v) := \sum _{S \subseteq N,\, S \ni i}\frac{\Delta _v(S)}{2^{|S|-1}} \quad \;\text { for all } \,i \in N. \end{aligned}$$

Obviously, the Banzhaf value satisfies the inessential grand coalition property and monotonicity. Thus, the superset of the generalized Harsanyi set that does not require efficiency for its TU-values contains the Banzhaf value. This means that in the definition of a sharing function within a sharing function system the condition $\sum _{i \in N}\psi _i^N(v)=1$ for a sharing vector $\psi ^N(v) \in {\mathbb {R}}_+^{N}$ must be dropped. In Fig. 1/Table 1, the class of these values is marked by $\varphi ^\chi$. However, possible applications of value dividends for axiomatizations of the Banzhaf value are left for further research.

Fig. 1

The subset relationships between some classes of TU-values

Table 1 Properties of some classes of TU-values

If we drop in Theorem 4.2 the monotonicity property, we get an axiomatic characterization of the class of the multiweighted Shapley values⁷MWSV$^{\lambda ^\pm }$ (Dragan 1992). A further removal of the dependent value monotonicity or a removal of the monotonicity property in Theorem 5.1 respectively leads to the characterization of a new class of TU-values (in Fig. 1/Table 1 denoted by $H^{\psi {^\pm }}$). That means that we allow sharing vectors $\psi ^{^\pm N}\!(v) \in {\mathbb {R}}^{N}$ such that $\sum _{i \in N}\psi ^{^\pm N}_i\!(v)=1$. Our last extension, represented in Fig. 1/Table 1 by $\varphi ^{\chi {^\pm }}$, concludes all mentioned classes of TU-values but not the proportional rule and not the proportional Value. This is the class of all TU-values which satisfy the inessential grand coalition property.

Also the TU-values from the Shapley mapping and thus the set-valued proper Shapley value in Vorob’ev and Liapunov (1998) are part (on subsets) of the generalized Harsanyi set [compare Equation (2) in Vorob’ev and Liapunov 1998 with (13)]. Here, too, further research must be shifted into the future.

The usage of the proportional Harsanyi solution is restricted to strongly positive games. This efficient TU-value combines the inessential grand coalition property of the proportional Shapley value with the proportion preservation and independent share property of the proportional rule and the proportional value. The sharing weights are given endogenously and depend on the whole coalition function, not only on the worths of the singletons as by the proportional rule or the proportional Shapley value. Thus, supported by many convincing properties, this value is recommended if the worth of a coalition is dependent from the worths of all subsets of the coalition.

Appendix

We show the logical independence of the axioms used in the characterizations with at least three axioms.

Remark 8.1

For all $N \in {\mathcal {N}}$, let $(N,v)\in {\mathbb {V}}(N)$ and $\lambda \in \Lambda$. The axioms in Theorems 4.1 and 5.2 are logically independent:

• E: The TU-value $\varphi := 2H^\lambda$ satisfies IGC and M$^\lambda$/M but not E in general.

• IGC: The TU-value $\varphi ^\lambda$, defined by $\varphi _i^\lambda (N,v):= \lambda _{N,i}\cdot v(N)$ for all $i \in N$ and all $N\in {\mathcal {N}}$ satisfies E and M$^\lambda$/M but not IGC.

• M$^\lambda$/M: The multiweighted Shapley values MWSV satisfy E and IGC but not M$^\lambda$/M in general.

Remark 8.2

For all $N \in {\mathcal {N}}$, let $(N,v)\in {\mathbb {V}}(N)$. The axioms in Theorem 4.2 are logically independent:

• E: The TU-value $\varphi := 2Sh$ satisfy M, DVM, and IGC but not E.

• M: The multiweighted Shapley values satisfy E, DVM, and IGC but not M in general.

• DVM: The TU-values $\varphi ^c$, defined by (12), satisfy E, M, and IGC but not DVM in general.

• IGC: The equal division value ED, defined by $ED_i(N,v):= \frac{v(N)}{|N|}$ for all $i \in N$, satisfies E, M, and DVM but not IGC.

Remark 8.3

For all $N \in {\mathcal {N}}$, let $(N,v)\in {\mathbb {V}}_{++}(N)$. The axioms in Theorem 6.10 and Corollary 6.12 are logically independent:

• D/E: The TU-value $\varphi := 2H^p$ satisfies IGC and IS/PP but not D/E.

• IGC: The proportional value P satisfies D/E and IS/PP but not IGC.

• PP: The Shapley value Sh satisfies E/D and IGC but not IS/PP.

Remark 8.4

The axioms in Theorem 6.14 are logically independent:

• E: The TU-value $\varphi := 2Sh$ satisfies IGC and EM but not E.

• IGC: The equal division value ED satisfies E and EM but not IGC.

• EM: Each weighted Shapley value $Sh^\mathrm {w}\ne Sh$ satisfies E and IGC but not EM.