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Composition independence in compound games: a characterization of the Banzhaf power index and the Banzhaf value

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).

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Notes

  1. 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. 2.

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

  3. 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. 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. 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. 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. 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. 8.

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

  9. 9.

    See Theorems 7 and 8 in Owen (1978). Although some other indices, such as the useless null index and the simplistic “dictatorial” index, also emerge from his axiomatization, they are easily removed by adding the dummy and strict positivity axioms; see Section 5 in Owen (1978).

  10. 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. 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. 12.

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

  13. 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. 14.

    Variants of these axioms have been present in the original axiomatizations of the Shapley–Shubik and the Banzhaf power indices [see Dubey (1975) and Dubey and Shapley (1979)]. The term “semivalue” comes from Einy (1987); see Remark 1 below.

  15. 15.

    The possibility of such a restatement has been mentioned in Dubey and Shapley (1979, p. 106). A special version of the restatement also appeared in Laruelle and Valenciano (2001).

  16. 16.

    That is, player \(\pi ^{-1}\left( i\right) \) has the same role in \(\pi v\) as player i in v.

  17. 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. 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. 19.

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

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Correspondence to Ori Haimanko.

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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.

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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

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Keywords

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

JEL Classification

  • C71
  • D72