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.

This is a preview of subscription content, access via your institution.

Notes

  1. 1.

    Defined in Shapley and Shubik (1954).

  2. 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. 3.

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

  4. 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. 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. 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. 7.

    See, e.g., Lehrer (1988), Nowak (1997), Casajus (2012), Haimanko (2018).

  8. 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. 9.

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

  10. 10.

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

  11. 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. 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. 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. 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. 15.

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

  16. 16.

    The term “semivalue” was originally coined in Dubey et al. (1981) in the context of value maps on \(\mathcal {G}\) (see Remark 1).

  17. 17.

    Variants of semivalue axioms have been present in the original axiomatizations of the SSPI and BPI (see Dubey (1975) and Dubey and Shapley (1979)).

  18. 18.

    The term Transfer is due to Weber (1988).

  19. 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. 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. 21.

    Dubey et al. (1981) considered semivalues on \(\mathcal {G}\) and not on \( \mathcal {SG}\) (for further discussion, see Remark 1). The family of power indices with the forthcoming description is obtained by restricting those semivalues to games in \(\mathcal {SG}.\)

  22. 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. 23.

    It is easy to see that the definition of \(v_{q}\) is independent of the choice of a carrier N.

  24. 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. 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. 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. 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. 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. 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. 30.

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

  31. 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. 32.

    Equalities (32) and (33) extend, respectively, Proposition 1 and Theorem 2 of Casajus and Huettner (2019). Our proof of ( 30) follows the argument used by these authors in establishing their Corollary 1.

References

  1. Banzhaf JF (1965) Weighted voting doesn’t work: a mathematical analysis. Rutgers Law Rev 19:317–343

    Google Scholar 

  2. Banzhaf JF (1966) Multi-member electoral districts—Do they violate the “One Man, One Vote” principle. Yale Law J 75:1309–1338

    Article  Google Scholar 

  3. Banzhaf JF (1968) One Man, 3.312 Votes: A Mathematical Analysis of the Electoral College. Vilanova Law Review 13:304–332

    Google Scholar 

  4. Brams SJ (2013) Game Theory and Politics, Dover Books on Mathematics. Dover Publications, Mineola

    Google Scholar 

  5. Casajus A (2012) Amalgamating Players, Symmetry, and the Banzhaf Value. Int J Game Theory 41:497–515

    Article  Google Scholar 

  6. Casajus A, Huettner F (2018) Decomposition of solutions and the shapley value. Games Econ Behav 108:37–48

    Article  Google Scholar 

  7. Casajus A, Huettner F (2019) The Coleman-Shapley-index: being decisive within the coalition of the interested. Public Choice, forthcoming

  8. Coleman JS (1971) Control of collectives and the power of a collectivity to act. In: Lieberman Bernhardt (ed) Social choice. Gordon and Breach, New York, pp 192–225

    Google Scholar 

  9. Dubey P (1975) On the uniqueness of the shapley value. Int J Game Theory 4:131–139

    Article  Google Scholar 

  10. Dubey P, Neyman A, Weber RJ (1981) Value theory without efficiency. Math Oper Res 6:122–128

    Article  Google Scholar 

  11. Dubey P, Einy E, Haimanko O (2005) Compound voting and the Banzhaf index. Games Econo Behav 51:20–30

    Article  Google Scholar 

  12. Dubey P, Shapley LS (1979) Mathematical properties of the Banzhaf power index. Math Oper Res 4:99–131

    Article  Google Scholar 

  13. Einy E (1987) Semivalues of simple games. Math Oper Res 12:185–192

    Article  Google Scholar 

  14. Felsenthal DS, Machover M (1998) The measurement of voting power: theory and practice, problems and paradoxes. Edward Elgar Publishers, London

    Book  Google Scholar 

  15. Haimanko O (2018) The axiom of equivalence to individual power and the Banzhaf index. Games Econ Behav 108:391–400

    Article  Google Scholar 

  16. Hart S, Mas-Colell (1996) Bargaining and Value. Econometrica 64:357–380

    Article  Google Scholar 

  17. Lehrer E (1988) Axiomatization of the Banzhaf value. Int J Game Theory 17:89–99

    Article  Google Scholar 

  18. Nowak AS (1997) On an axiomatization of the Banzhaf value without the additivity axiom. Int J Game Theory 26:137–141

    Article  Google Scholar 

  19. Owen G (1968) A note on the shapley value. Manag Sci 14:731–732

    Article  Google Scholar 

  20. Penrose LS (1946) The elementary statistics of majority voting. J R Stat Soc 109:53–57

    Article  Google Scholar 

  21. Shapley LS (1953) A value for \(n\)-person games. In: Kuhn HW, Tucker AW (eds) Contributions to the theory of Games II (annals of mathematical studies 28). Princeton University Press, Princeton

    Google Scholar 

  22. Shapley LS, Shubik M (1954) A method for evaluating the distribution of power in a committee system. Am Polit Sci Rev 48:787–792

    Article  Google Scholar 

  23. Weber RJ (1988) Probabilistic values for games. In: Roth AE (ed) The Shapley value: essays in honor of Lloyd S. Shapley. Cambridge University Press, Cambridge, pp 101–121

    Google Scholar 

Download references

Author information

Affiliations

Authors

Corresponding author

Correspondence to Ori Haimanko.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

It is a pleasure to thank two anonymous reviewers for their comments, which helped to improve the paper.

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 \(i\in N,\)

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

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

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

Download citation

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