Generalized Coleman-Shapley indices and total-power monotonicity

Abstract

We introduce a new axiom for power indices, which requires the total (additively aggregated) power of the voters to be nondecreasing in response to an expansion of the set of winning coalitions; the total power is thereby reflecting an increase in the collective power that such an expansion creates. It is shown that total-power monotonic indices that satisfy the standard semivalue axioms are probabilistic mixtures of generalized Coleman-Shapley indices, where the latter concept extends, and is inspired by, the notion introduced in Casajus and Huettner (Public choice, forthcoming, 2019). Generalized Coleman-Shapley indices are based on a version of the random-order pivotality that is behind the Shapley-Shubik index, combined with an assumption of random participation by players.

Appendix

1.1 Proof of the “if” direction of Theorem 1

Proof

Assume that the c.d.f. $F_{\xi }$ of the distribution corresponding to $\xi$ is concave on $\left[ 0,1\right] .$ The proof of the “only if” part of Theorem 2 shows that, in such a case, the semivalue $\varphi =\phi _{\xi }$ has the representation (26). But then, by the “if” part of Theorem 2, $\varphi$ satisfies TP-Mon. $\square$

1.2 Proof of the uniqueness of a representing measure $\mu$ in Theorem 2

Proof

Assume that a semivalue $\varphi$ possesses a representation (26) for some $\mu \in M\left( \left[ 0,1 \right] \right) .$ Then, as shown in Remark 2, $\varphi$ decomposes $\phi _{\mu }.$ But then $\phi _{\mu }$ is uniquely determined by (30), and $\mu$ is in turn uniquely determined by $\phi _{\mu }$ (due to Proposition 1). $\square$

1.3 Proposition 3

Proposition 3

Consider $\mu \in M\left( [0,1]\right)$ and $v\in \mathcal {SG}$. For any finite carrier N of v and any $i\in N,$

$$\begin{aligned} \phi _{\mu }(v)(i)=v_{\mu }(N)-v_{\mu }(N\backslash i), \end{aligned}$$

(34)

where $v_{\mu }\in \mathcal {G}$ is the game given by (31).

Proof

It is clear from (1), (2) that, for any $q\in (0,1],$ the q-value $\phi _{q}$ is given by

$$\begin{aligned} \phi _{q}\left( v\right) (i)=E\left[ v(\overline{S}_{N}^{q})-v(\overline{S} _{N}^{q}\backslash i) \mid i\in \overline{S}_{N}^{q}\right] , \end{aligned}$$

where $\overline{S}_{N}^{q}$ is the random coalition of q-active players that satisfies (7). Notice that

$$\begin{aligned} E\left[ v(\overline{S}_{N}^{q})-v(\overline{S}_{N}^{q}\backslash i) \mid i\in \overline{S}_{N}^{q}\right]= & {} \frac{1}{q}E\left[ v( \overline{S}_{N}^{q})-v(\overline{S}_{N}^{q}\backslash i) \right] \\= & {} \frac{1}{q}E[v(N\cap \overline{S}_{N}^{q})]-\frac{1}{q}E[v(\left( N\backslash i\right) \cap \overline{S}_{N}^{q})]\\= & {} v_{q}(N)-v_{q}(N\backslash i), \end{aligned}$$

and hence

$$\begin{aligned} \phi _{q}\left( v\right) (i)=v_{q}(N)-v_{q}(N\backslash i). \end{aligned}$$

(35)

When $q=0,$ (35) still holds because $\phi _{0}\left( v\right) (i)=v(i)$ and $v_{0}(N)-v_{0}(N\backslash i)=v(i)$ by (11). By integrating both sides of (35) over q w.r.t. $\mu$ and using (3), the desired equality (34) is obtained. $\square$