## 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.

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## Notes

- 1.
Defined in Shapley and Shubik (1954).

- 2.
As is often done in the literature, we use the term “Banzhaf power index” for brevity, although the origin of this power index lies in multiple works (Penrose (1946), Banzhaf (1965, 1966, 1968), Coleman (1971)). The specific variant of the BPI used in this work is referred to as the “Banzhaf measure” in Felsenthal and Machover (1998).

- 3.
All quotations in this sentence are taken from Dubey and Shapley (1979, p. 103).

- 4.
This may be weakened by noting that voting “yes” is not expected from players less enthusiastic than the pivot. Indeed, the proposal will be collectively approved (by virtue of the votes of the pivot and his predecessors in the order)

*before*the less enthusiastic players will be asked to join in support. - 5.
This view of the SSPI may also be (more formally) supported by the fact that the underlying Shapley value arises as an equilibrium outcome of certain natural bargaining procedures (see, e.g., Hart and Mas-Colell (1996)).

- 6.
The notion of pivotality here still implies the ability to affect the content of a proposal, but the passage of the proposal is now uncertain because the set of

*q*-active voters may be losing. - 7.
- 8.
See Dubey et al. (1979, p. 103). A swinger is defined w.r.t. a random set of yes-voters (with the uniform distribution over all subsets of the voting body) by the requirement that the change in his vote affects the the voting outcome.

- 9.
See Felsenthal and Machover (1998, p. 52).

- 10.
Under the degenerate 0-CSPI, the total power aslo responds in a (weakly) monotonic fashion to an addition of winning sets.

- 11.
This is because no coalition of yes-voters can turn from winning to losing, and at least one such coalition turns from losing to winning, under such a change in the decision rule.

- 12.
It may be argued that if the winning coalitions become too numerous, then in some contexts (such as under symmetric majority rules with quotas below \( \frac{1}{2}\)) the collective power could suffer because any proposal that may be easy to pass with just a minority approval, can be subsequently overturned by a counter-proposal supported by an opposing minority. However, we take the view that the measurement of power concerns a

*single decision*, namely, passage of a single (anticipated but a priori unknown) proposal. Under a scope restricted to the proposal at hand, it is natural to regard the power of collectivity as commensurate with the ease of passing the proposal. - 13.
In the context of a simple game, a pivot for a coalition is a player whose presence switches that coalition from losing to winning.

- 14.
This definition of simple games follows the convention set forth in Dubey and Shapley (1979), and used in much of the subsequent research.

- 15.
We shall henceforth omit braces when indicating one-player sets.

- 16.
- 17.
- 18.
The term

**Transfer**is due to Weber (1988). - 19.
\(\max \), \(\min \) in the statement of

**Transfer**refer to the maximum/minimum of functions on \(2^{U},\) and hence both \(\max \{v,w\}\) and \( \min \{v,w\}\) are well-defined games in \(\mathcal {SG}.\) - 20.
Specifically, if \(v,w,v^{\prime },w^{\prime }\in \mathcal {SG}\) are such that \(v\ge v^{\prime },\)\(w\ge w^{\prime }\) and \(v-v^{\prime }=w-w^{\prime },\) then \(\varphi (v)-\varphi (v^{\prime })=\varphi (w)-\varphi (w^{\prime }).\)

- 21.
- 22.
In (8), \(\mathcal {R}_{N}\) can be replaced by \(\mathcal {R} _{\overline{S}_{N}^{q}}\) (a random, uniformly distributed order of players in \(\overline{S}_{N}^{q}\)), i.e., it suffices to rank only the active players. Such an equation would have been the reduced form of both (5) and (8), consistent with our description of the

*q*-CSPI in the Introduction. The current (8) is preferable, however, as it is used in the proof of our upcoming Proposition 2. - 23.
It is easy to see that the definition of \(v_{q}\) is independent of the choice of a carrier

*N*. - 24.
A more general (and easily verifiable) version of this property is the following: if \(w\in \mathcal {SG}\) is obtained from \(v\in \mathcal {SG}\) by adding a single minimal winning coalition \(T^{\prime }\) (that is, \(w=\max (v,u_{T^{\prime }})\)), then \(\phi _{\xi }\left( w\right) \left( i\right) \ge \phi _{\xi }\left( v\right) \left( i\right) \) for every \(i\in T^{\prime }.\)

- 25.
Notice that manifestations of

*individual power*monotonicity expressed by the inequality (15) or, more generally, the statement in Footnote 24, are not selective, in contrast to**TP-Mon**: they are satisfied by*all*semvalues. - 26.
Notice that this use of Theorem 1 is legitimate because it relies on the “only if” part of that theorem, which has already been established in the previous section. It is the “if” part that still awaits proof, given in the Appendix.

- 27.
Continuity of \(F_{\xi }\) was established in the proof of Theorem 1, but we did not need to claim both continuity and concavity in the statement of that theorem because concavity of \(F_{\xi }\) on \(\left[ 0,1\right] \) implies its continuity on that interval. Indeed, the only discontinuity of a concave function on \(\left[ 0,1\right] \) might occur at the end-points, but that is impossible because \(F_{\xi }\) is right-continuous and nondecreasing as a c.d.f.

- 28.
One may take \(f_{\xi }\) to be the left-hand derivative of \(F_{\xi }\) on (0, 1]. If \(\lim _{x\rightarrow 0+}f_{\xi }(x)=\infty ,\) then all integrals in the proof that have the form \(\int _{0}^{t}...dx\) (for \(0<t\le 1\)), and in which the integrand involves \(f_{\xi }(x),\) should be regarded as improper integrals.

- 29.
The limit \(a_{\xi }\) exists because \(f_{\xi }\) is nondecreasing, and its positivity follows from the assumption that \(F_{\xi }(0)<1.\) It may, furthermore, be equal to \(\infty .\)

- 30.
I.e., \(\varphi \) acts as the identity map when restricted to \(\mathcal {AG}\).

- 31.
Formally, \(v_{-i}\left( S\right) :=v\left( S\backslash i\right) \) for every \( S\in 2^{U}.\) Notice that \(v_{-i}\) may be the zero game, which is excluded from our definition of simple games. In such a case, \(\varphi \left( v_{-i}\right) \) is also taken to be the zero game.

- 32.

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## Appendix

### Appendix

### 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**. \(\square \)

### 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). \(\square \)

### 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 \(i\in N,\)

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

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

and hence

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. \(\square \)

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Haimanko, O. Generalized Coleman-Shapley indices and total-power monotonicity.
*Int J Game Theory* **49, **299–320 (2020). https://doi.org/10.1007/s00182-019-00692-2

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### Keywords

- Simple games
- Voting power
- Shapley-Shubik index
- Banzhaf index
- Coleman-Shapley index
- Semivalues
- Power of collectivity to act
- Total-power monotonicity axiom
- Probabilistic mixtures

### JEL classification numbers

- C71
- D72