On membership and marginal values

Abstract

In Kleinberg and Weiss, Math Soc Sci 12:21–30 (1986b), the authors used the representation theory of the symmetric groups to characterize the space of linear and symmetric values. We call such values “membership” values, as a player’s payoff depends on the worths of the coalitions to which he belongs and not necessarily on his marginal contributions. This could mean that the player would get some share of $v(N)$ regardless of whether or not he makes a marginal contribution to the welfare of society. In this paper it is demonstrated that the set of (non-marginal) membership values include those that embody numerous widely held notions of fairness, such as partial “benefit equalization”, individual rationality and “greater rewards follow from greater contributions”, where one’s contributions are not measured marginally. We also present a very simple and revealing way of interpreting all values, including those having a marginal interpretation. Finally, we obtain a mapping which effectively embeds the space of marginal values in the space of all membership values.

Appendix

As preparation for the results below, we shall require some tools and definitions. Define, for each $k, \; 1 \le k \le n$, the map:

$$\begin{aligned} L^k: \mathfrak G \longrightarrow \mathfrak G := L^kv(S) = \left\{ \begin{array}{cll} v(S)&\,&\quad \text{ if} \; \#S = k \\ 0&\,&\quad \text{ otherwise} \\ \end{array}\right. \end{aligned}$$

and then set $\mathfrak{M }^k = L^k \mathfrak G$. Clearly, each $\mathfrak{M }^k$ is isomorphic to $R^{{\small \left( \begin{array}{l} {n} \\ {k} \end{array}\right) }}$, and may be viewed as a sub-module of $\mathfrak G$ under the action (3); note that then $\mathfrak G = \oplus \sum _{k=1}^n \mathfrak{M }^k$. Next let, for each $k, \; 1 \le k \le n$, and any MV $T, \; {\tilde{T}}^k \equiv T|_{\mathfrak{M }^k}$; observe that, for any $v \in \mathfrak G$,

$$\begin{aligned} Tv = \sum \limits _{k=1}^n {\tilde{T}}^k(L^kv), \end{aligned}$$

(8)

and set $T^k = L^k \circ {\tilde{T}}^k$. Finally, let $\Pi _0^{k}$ denote the orthogonal projection of $\mathfrak{M }^k$ onto $L^k(\mathfrak{I _0}^\perp )$ and $\Pi _1^k$ the orthogonal projection of $\mathfrak{M }^k$ onto $L^k \left( \mathfrak{I }_0 \right)$. The maps

$$\begin{aligned} \phi ^k: R^n \longrightarrow R^{\left( \begin{array}{l} {n} \\ {k} \end{array}\right) } := \phi _S^k(x_1,\ldots ,x_n) = \left\{ \begin{array}{cll} \sum \nolimits _{i\in S}x_i&\,&\quad \text{ if} \; \#S = k \\ 0&\,&\quad \text{ otherwise} \\ \end{array}\right. \end{aligned}$$

are each univalent and, for given $k$, take the diagonal $\Delta \subset R^n$ onto $L^k(\mathfrak{I _0}^\perp )$ and $\Delta ^\perp$ (the vectors whose coordinates sum to zero) onto $L^k(\mathfrak{I _0})$. Since, for each vector $v \in \Delta ^\perp ,\,v(i)$ appears exactly $\left( \begin{array}{l} {n-1} \\ {k-1} \end{array}\right)$ times in $\phi ^k(v)$, the sum of the coordinates of $\phi ^k(v)$ is zero as well, showing that $L^k(\mathfrak{I _0}^\perp ) = L^k(\mathfrak{I _0})^\perp$. Thus, in particular, for each coalition size $k$ and every $w \in L^k(\mathfrak I ),\;w = (\Pi _0^{k} + \Pi _1^k)w$ (orthogonal sum). We then have

Lemma 1

For each $k, \; 1 \le k \le n-1$ and $v \in \mathfrak{M }^k$, there exist unique games $\xi ^0(v,k) \in \mathfrak{I _0}^\perp$ and $\xi ^1(v,k) \in \mathfrak{I }_0$ such that

$$\begin{aligned} \Pi _0^{k} v = L^k\big ( \xi ^0(v,k)\big ), \; \Pi _1^{k}v = L^k\big ( \xi ^1(v,k)\big ). \end{aligned}$$

The maps

$$\begin{aligned} \xi ^0(\cdot ,k): \mathfrak{M }^k \longrightarrow \mathfrak{I _0}^\perp , \xi ^1(\cdot ,k): \mathfrak{M }^k \longrightarrow \mathfrak{I }_0 \end{aligned}$$

are each $R \mathfrak{S }_n$-homomorphisms determined by the formulae:

$$\begin{aligned} \xi ^0(v,k)(i)&= k^{-1}A(v,k) \\ \xi ^1(v,k)(i)&= \gamma (k)^{-1} \mathop {\mathop \sum \limits _{ {\#S=k}}}\limits _{{i \in S}} \Big [ v(S)-A(v,k) \Big ], \quad 1 \le i \le n. \end{aligned}$$

Proof

Define, for each $k, \; 1 \le k \le n-1$,

$$\begin{aligned} \theta ^k : \mathfrak{M }^1 \longrightarrow L^k \left( \mathfrak I \right) := \left( \theta ^k v \right) (S) = \sum \limits _{i \in S} v(i). \end{aligned}$$

We observe that $\theta ^k$ is an $R \mathfrak{S }_n$-isomorphism carrying $L^1 \left( \mathfrak{I }_0 \right)$ onto $L^k \left( \mathfrak{I }_0 \right)$ and $L^1\left( \mathfrak{I _0}^\perp \right)$ onto $L^k\left( \mathfrak{I _0}^\perp \right)$ (Kleinberg and Weiss 1985a, Proposition 4). This shows that, for each $k$, the maps $\xi ^0(\cdot ,k): \mathfrak{M }^k \longrightarrow L^k(\mathfrak{I _0}^\perp )$ and $\xi ^1(\cdot ,k): \mathfrak{M }^k \longrightarrow L^k(\mathfrak{I }_0)$ exist and are the unique $R \mathfrak{S }_n$-homomorphisms determined by:

$$\begin{aligned} \xi ^0(v,k)(i)&= \left[ \left( \theta ^k \right) ^{-1}\Pi _0^{k}v \right] (i) \end{aligned}$$

(9)

$$\begin{aligned} \xi ^1(v,k)(i)&= \left[ \left( \theta ^k \right) ^{-1}\Pi _1^{k}v \right] (i) \end{aligned}$$

(10)

(note that $\Pi _0^{k}$ and $\Pi _1^{k}$ are $R\mathfrak{S }_n$-homomorphisms as orthogonal projections onto the invariant subspaces $\mathfrak{I _0}^\perp$ and $\mathfrak{I }_0$, respectively). Thus it remains only to demonstrate the formulae.

Towards this end we first observe that, if $v \in L^k \left( \mathfrak{I _0}^\perp \right)$, then all coalitions of size $k$ receive the same worth in $v$ so that, clearly,

$$\begin{aligned} \left[ \left( \theta ^k \right) ^{-1}v \right] (i) = k^{-1}A(v,k), \quad 1 \le i \le n. \end{aligned}$$

(11)

Next, suppose that $v \in L^k \left( \mathfrak{I }_0 \right)$. Then, for any $i, \; 1 \le i \le n$,

$$\begin{aligned} \mathop {\mathop \sum \limits _{ {\#S=k}}}\limits _{{i \in S}} v(S)&= \mathop {\mathop \sum \limits _{ {\#S=k}}}\limits _{{i \in S}}\left( \sum \limits _{j \in S} \left[ \left( \theta ^k \right) ^{-1}v \right] (j) \right) \!=\! \left[ \left( \begin{array}{l} {n-1} \\ {k-1} \end{array}\right) - \left( \begin{array}{l} {n-2} \\ {k-2} \end{array}\right) \right] \left[ \left( \theta ^k \right) ^{-1}v \right] (i) \\&\quad +\left( \begin{array}{l} {n-2} \\ {k-2} \end{array}\right) \sum \limits _{j=1}^n \left[ \left( \theta ^k \right) ^{-1}v \right] (j) = \gamma (k)\left[ \left( \theta ^k \right) ^{-1}v \right] (i), \end{aligned}$$

where the last equality follows from the fact observed above that $v \in L^k \left( \mathfrak{I }_0 \right)$ if and only if $(\theta ^k)^{-1}v \in L^1 \left( \mathfrak{I }_0 \right)$. Thus, for $v \in L^k\left( \mathfrak{I }_0 \right)$,

$$\begin{aligned} \left[ \left( \theta ^k\right) ^{-1}v\right] (i) = \gamma (k)^{-1} \mathop {\mathop \sum \limits _{ {\#S=k}}}\limits _{{i \in S}} v(S), \quad 1 \le i \le n. \end{aligned}$$

(12)

Now let $v \in \mathfrak{M }^k$, and define the game $A_k(v) = \Pi _0^{k}v$. For clearly $A_k(v) \in L^k \left( \mathfrak{I _0}^\perp \right)$, and $\sum _{\#S=k} [ v(S) - A_k(v)(S) ] = 0$, so that $(v - A_k(v)) \perp L^k \left( \mathfrak{I _0}^\perp \right)$. Hence, by (10) and (11),

$$\begin{aligned} \xi ^0(v,k)(i) = k^{-1}A \left( \Pi _0^{k}v,k \right) = k^{-1}A \big (A_k(v),k\big ) = k^{-1}A(v,k). \end{aligned}$$

Next, by (10) and (12)

$$\begin{aligned} \xi ^1(v,k)(i) = \left[ \left( \theta ^k \right) ^{-1}\Pi _1^{k}v \right] (i) = {\gamma (k)}^{-1}\mathop {\mathop \sum \limits _{ {\#S=k}}}\limits _{{i \in S}} \Pi _1^{k}v(S). \end{aligned}$$

Since, as was observed above, $w = (\Pi _0^{k} + \Pi _1^{k})w$ for every $w \in L^k(\mathfrak I )$, we can set $v = w + z$, where $w \equiv (\Pi _0^{k} + \Pi _1^{k})v$ and $z \equiv v - w = v - \Pi _0^{k}v - \Pi _1^{k}v = \Pi _{ \mathfrak{B }}^{k}v$ (here $\Pi ^k_{ \mathfrak{B }}$ is the orthogonal projection of $\mathfrak{M }^k$ onto $\mathfrak{M }^k \bigcap L^k\left( \mathfrak{I }\right) ^{\perp } \equiv \mathfrak{B }^k$, see the discussion after the statement of the Theorem). Thus $\Pi _1^{k}v = v - \Pi _0^{k}v - \Pi _{ \mathfrak{B }}^{k} v$.

Now by the definition of the $\mathfrak{B }^k,\,{\sum _{\begin{array}{l}{\#S=k}\\ {i \in S}\end{array}}} \Pi ^k_\mathfrak{B } v(S) = 0$; hence

$$\begin{aligned} \mathop {\mathop \sum \limits _{ {\#S=k}}}\limits _{{i \in S}}\Pi _1^{k}v(S) = \mathop {\mathop \sum \limits _{ {\#S=k}}}\limits _{{i \in S}}\left( v - \Pi _0^{k}v \right) (S) = \mathop {\mathop \sum \limits _{ {\#S=k}}}\limits _{{i \in S}} \big (v(S) - A(v,k)\big ). \end{aligned}$$

Therefore we may conclude

$$\begin{aligned} \xi ^1(v,k)(i) = \gamma (k)^{-1} \mathop {\mathop \sum \limits _{ {\#S=k}}}\limits _{{i \in S}}\big (v(S)-A(v,k)\big ). \end{aligned}$$

$\square$