A note on associated consistency and linear, symmetric values

Abstract

In the context of cooperative games with transferable utility Hamiache (Int J Game Theory 30:279–289, 2001) utilized continuity, the inessential game property and associated consistency to axiomatize the well-known Shapley value (Ann Math Stud 28:307–317, 1953). The question then arises: “Do there exist linear, symmetric values other than the Shapley value that satisfy associated consistency?”. In this Note we give an affirmative answer to this question by showing that a linear, symmetric value satisfies associated consistency if and only if it is a linear combination of the Shapley value and the equal-division solution. In addition, we offer an explicit formula for generating all such solutions and show how the structure of the null space of the Shapley value contributes to its unique position in Hamiache’s result.

Proof of Propositions 6–8

Note: Below

$$\begin{aligned} {\mathscr {Z}}^0 := {\mathscr {G}} \bigcap \left\{ v: \sum _{\#S = k} v(S) = 0, \ 1 \le k \le n \right\} . \end{aligned}$$

Proposition 6

Proof

Sufficiency follows from the fact that, for each $k, 1 \le k \le n-1$ and every $v \in {\mathscr {Y}}^k$, Equation (5) reduces to

$$\begin{aligned} \phi _i(v) = c^k_1 \gamma (k)^{-1} \sum _{\begin{array}{c} \#S = k \\ i \in S \end{array}} [v(S)-A(v,k)], \quad 1 \le i \le n. \end{aligned}$$

But $v(S) = A(v,k), \forall S \subset N, \#S = k$, so that the right-hand side above is equal to zero. Thus ${\mathscr {Y}}^k \subset {\mathscr {N}}_{\phi }$; since k was arbitrary, we conclude ${\mathscr {Y}} \subset {\mathscr {N}}_{\phi }$.

As for necessity, let k be such that $1 \le k \le n-1$ and $c^{k}_0 \ne 0$. Then from Eq. (5), and for any $v \in {\mathscr {Y}}^k, \ v \ne 0$, we have $A(v,k) \ne 0$ so that $\phi _i(v) = c^{k}_0 k^{-1}A(v,k) \ne 0, 1 \le i \le n$. Thus, ${\mathscr {Y}}^k \not \subset {\mathscr {N}}_{\phi }$, hence ${\mathscr {Y}} \not \subset {\mathscr {N}}_{\phi }$. $\square$

Proposition 7

Proof

Sufficiency: Given any $k, \ 2 \le k \le n-1$, let $v \in {\mathscr {D}}^k$; then, recalling that each ${\mathscr {D}}^k \subset {\mathscr {Z}}^0$, we have

$$\begin{aligned} \phi _i(v)= & {} c^1_1v(i) - \gamma (k)^{-1} c^k_1 \sum _{\begin{array}{c} S \ni i \\ \#S = k \end{array}} \sum _{j \in S}v(j) \nonumber \\= & {} c \left( v(i) - \gamma (k)^{-1} \sum _{\begin{array}{c} S \ni i \\ \#S = k \end{array}} \sum _{j \in S}v(j) \right) , \quad 1 \le i \le n, \end{aligned}$$

(12)

where we have set $c = c^1_1 = c^k_1$. But we observe, in the double sum in Eq. (12), that the term v(i) occurs precisely $\left( \begin{array}{c} {n-1} \\ {k-1} \end{array}\right)$ times, once in each coalition S of size k containing i. On the other hand $v(j), j \ne i$, occurs precisely $\left( \begin{array}{c}{n-2}\\ {k-2} \end{array} \right)$ times, once in each coalition S of size k containing both i and j. Thus, the term in parentheses on the right-hand side of Eq. (12) becomes

$$\begin{aligned}&v(i) - \gamma (k)^{-1} \left( \left[ \left( \begin{array}{c}{n-1}\\ {k-1} \end{array} \right) - \left( \begin{array}{c} {n-2}\\ {k-2} \end{array} \right) \right] v(i) + \left( \begin{array}{c} {n-2} \\ {k-2} \end{array}\right) \sum _{j = 1}^n v(j) \right) \\&\quad = v(i) - \gamma (k)^{-1} \left( \left[ \left( \begin{array}{c}{n-1} \\ {k-1}\end{array} \right) - \left( \begin{array}{c}{n-2} \\ {k-2} \end{array}\right) \right] v(i) \right) \\&\quad = v(i) - v(i) = 0 \end{aligned}$$

where the first equality follows from the definition of ${\mathscr {D}}^k$, the second from our definition of the constant $\gamma (k)$. Thus ${\mathscr {D}}^k \in {\mathscr {N}}_{\phi }$; since k was arbitrary, we conclude ${\mathscr {D}} \subset {\mathscr {N}}_{\phi }$.

Necessity The failure of our hypothesis implies the existence of some $k,\ 2 \le k \le n-2$ such that $c^1_1 \ne c^k_1$. Pick then any non-zero $v \in {\mathscr {D}}^k$; then, Eq. (12) becomes

$$\begin{aligned} \phi _i(v) = c^1_1v(i) - c^k_1 \left( \gamma (k)^{-1} \sum _{\begin{array}{c} S \ni i \\ \#S = k \end{array}} \sum _{j \in S}v(j) \right) , \quad 1 \le i \le n; \end{aligned}$$

(13)

as we observed in the comments immediately following Eq. (12), the term v(i) in the double sum above occurs precisely $\left( \begin{array}{c}{n-1} \\ {k-1} \end{array}\right)$ times, once in each coalition S of size k containing i, while $v(j), j \ne i$, occurs precisely $\left( \begin{array}{c}{n-2} \\ {k-2} \end{array} \right)$ times, once in each coalition S of size k containing both i and j. Thus, as it did in the sufficiency part of the proof, the term in parentheses reduces to v(i) and we conclude $\phi _i(v) = (c^1_1 - c^k_1)v(i)$. Since $v \ne 0$ and $c^1_1 \ne c^k_1$ we infer the existence of an index $i^*$ such that $\phi _{i^*}(v) \ne 0$, hence that ${\mathscr {D}}^k \not \subset {\mathscr {N}}_{\phi }$, thus that ${\mathscr {D}} \not \subset {\mathscr {N}}_{\phi }$. $\square$

Proposition 8

Proof

For each $k,\ 2 \le k \le n-2$ and any $v \in \mathscr {B}^k$, we obtain from Eq. (5)

$$\begin{aligned} \phi _i(v) = c^{k}_1 \gamma (k)^{-1} \sum _{\begin{array}{c} \#S = k \\ i \in S \end{array}} [v(S)-A(v,k)] + c^{k}_0 k^{-1}A(v,k), \quad 1 \le i \le n. \end{aligned}$$

(14)

But, as we observed above, $v \in {\mathscr {Z}}^0$, thus $A(v,k) = 0$, so that Eq. (14) reduces to $\phi _i(v) = c^{k}_1 \gamma (k)^{-1} \sum _{\#S = k,i \in S} v(S) = 0$, the last equality following from the definition of ${\mathscr {B}}^k$. Thus, ${\mathscr {B}}^k \subset {\mathscr {N}}_{\phi }, \ 2 \le k \le n-2$, from which we conclude ${\mathscr {B}} \subset {\mathscr {N}}_{\phi }$. $\square$