## Abstract

We introduce the axiom of composition independence for power indices and value maps. In the context of compound (two-tier) voting, the axiom requires the power attributed to a voter to be independent of the second-tier voting games played in all constituencies other than that of the voter. We show that the Banzhaf power index is uniquely characterized by the combination of composition independence, four semivalue axioms (transfer, positivity, symmetry, and dummy), and a mild efficiency-related requirement. A similar characterization is obtained as a corollary for the Banzhaf value on the space of all finite games (with transfer replaced by additivity).

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

## Notes

- 1.
The name of John F. Banzhaf III has been the one most associated with that power index, due to the number of works he authored on the subject and the legal repercussions of his findings and recommendations. Thus, siding with most of the literature, we will use the term “Banzhaf power index” for brevity, although, and perhaps more appropriately, the index is sometimes referred to as the Penrose-Banzhaf-Coleman power index.

- 2.
Felsenthal and Machover (1998) call this version “the Banzhaf measure.”

- 3.
Our notion of swinger is a slight adaptation of the term used in Dubey and Shapley (1979, p. 103), who defined it in relation to a random coalition that may include the swinger, in which case the effect of his departure from that coalition on its winning status is also considered. The probability of being a swinger is the same under both definitions, and hence both swinger notions may be used in defining the Banzhaf index.

- 4.
The weights given to different representatives may be (roughly) proportional to the population sizes of the counties they represent; but that is often not the case, either by necessity or by design. [See Chapter 4 of Felsenthal and Machover (1998) for many examples of weighted voting in the US.]

- 5.
To be precise, in order for the this property to hold all second-tier games \( w_{j}\) need to be decisive, i.e., constant-sum (as in the scenario where all \(w_{j}\) are simple majority games with an odd number of players).

- 6.
For instance, when the first-tier game is a weighted majority one, the generating functions method suggested in Owen (1982, Chapter X, pp. 226–227) provides an effective way to compute the Banzhaf index of each representative if the number of representatives is not too large (which is typically the case). See also Matsui and Matsui (2000) for specific algorithms for calculating the Banzhaf index in weighted majority games.

- 7.
The underlying assumption behind this principle is that all second-tier voting games are simple majority games, which is the case in most real-life instances of compound voting.

- 8.
The claim is also true for the space of all constant-sum games.

- 9.
- 10.
As in the premise for the composition property, it will be assumed that the second-tier games \(w_{1},\ldots ,w_{k}\) are constant-sum.

- 11.
We will establish this claim in Remark 2. Any efficient power index would satisfy such a claim, but it is not entirely obvious in the case of the non-efficient Banzhaf index.

- 12.
We shall henceforth omit braces when indicating one-element sets.

- 13.
We do not impose the usual efficiency requirement on a value map \(\varphi \) (whereby the equality \(\varphi \left( v\right) (U)=v(U)\) should hold for any \(v\in {\mathcal {G}}\)), and, indeed, the objects of our investigation (namely, the Banzhaf power index and the Banzhaf value, defined next) do not satisfy efficiency. The interpretation of \(\varphi \left( v\right) \left( i\right) \) as

*i*’s “utility of playing the game” is still valid, however, as the framework of Roth allows inefficient subjective valuations when players are averse to strategic risk [see Roth (1988, p. 61)]. - 14.
- 15.
- 16.
That is, player \(\pi ^{-1}\left( i\right) \) has the same role in \(\pi v\) as player

*i*in*v*. - 17.
When

**Pos**is also assumed to hold,**VanPow**implies that \( \lim \inf _{k\rightarrow \infty }\min _{i\in N_{k}}\varphi \left( v_{k}\right) \left( i\right) =0\), as power cannot be negative. - 18.
Notice that \(v_{k}^{q}\) may be the zero game, which is, in principle, excluded from the domain \({\mathcal {SG}}.\) For technical reasons, we will admit the game \(v=0\) as part of \({\mathcal {SG}}\) in our forthcoming considerations, keeping in mind that \(\beta \left( 0\right) =0.\)

- 19.
A reader interested in full details is referred to Haimanko (2017), the original (online) version of the present paper.

## References

Albizuri JM, Ruiz L (2001) A new axiomatization of the Banzhaf semivalue. Spanish Econ Rev 3:97–109

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

Banzhaf JF (1966) Multi-member electoral districts—do they violate the “one man, one vote” principle. Yale Law J 75:1309–1338

Banzhaf JF (1968) One man, 3.312 votes: a mathematical analysis of the electoral college. Vilanova Law Rev 13:304–332

Casajus A (2012) Amalgamating players, symmetry, and the Banzhaf value. Int J Game Theory 41:497–515

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

Dubey P (1975) On the Uniqueness of the Shapley Value. Int J Game Theory 4:131–139

Dubey P, Neyman A, Weber RJ (1981) Value Theory Without Efficiency. Math Oper Res 6:122–128

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

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

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

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

Haimanko O (2017) Composition independence in compound games: a characterization of the Banzhaf power index and the Banzhaf value. DP # 1713 of the Monaster Center for Economic Research. http://in.bgu.ac.il/en/humsos/Econ/WorkingPapers/1713.pdf

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

Laruelle A, Valenciano F (2001) Shapley–Shubik and Banzhaf indices revisited. Math Oper Res 26:89–104

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

Malawski M (2002) Equal treatment, symmetry and Banzhaf value axiomatizations. Int J Game Theory 31:47–67

Matsui T, Matsui Y (2000) A survey of algorithms for calculating power indices of weighted majority games. J Oper Res Soc Jpn 43:71–86

Owen G (1964) The tensor composition of nonnegative games. In: Tucker AW (ed) Annals of Mathematics 52. Princeton University Press, Princeton, pp 307–326

Owen G (1975) Multilinear extensions and the Banzhaf value. Naval Res Log Quart 1975:741–750

Owen G (1978) Characterization of the Banzhaf–Coleman index. SIAM J Appl Math 35:315–327

Owen G (1982) Game theory, 2nd edn. Academic Press, New York

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

Roth AE (1988) The shapley value: essays in honor of Lloyd S. Shapley. In: Roth Alvin E (ed) The expected utility of playing a game. Chapter 4. Cambridge University Press, Cambridge, pp 51–70

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, pp 307–317

Shapley LS (1964) Solutions of compound simple games. In: Tucker AW (ed) Annals of Mathematics 52. Princeton University Press, Princeton, pp 267–306

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

## Author information

### Affiliations

### Corresponding author

## Additional information

### Publisher's Note

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

This work was motivated by a conversation the author had with Sergiu Hart on the composition property of the Banzhaf index in compound games, in which Sergiu observed that each voter’s power is independent of the games played in other constituencies. This fact is the basis for the new axiom of composition independence. The author also acknowledges helpful comments of two anonymous reviewers.

## Rights and permissions

## About this article

### Cite this article

Haimanko, O. Composition independence in compound games: a characterization of the Banzhaf power index and the Banzhaf value.
*Int J Game Theory* **48, **755–768 (2019). https://doi.org/10.1007/s00182-019-00660-w

Accepted:

Published:

Issue Date:

### Keywords

- Simple games
- Compound games
- Banzhaf power index
- Banzhaf value
- Composition property
- Semivalues
- Transfer
- Symmetry
- Positivity
- Dummy

### JEL Classification

- C71
- D72