Skip to main content
SpringerLink
Log in

The Banzhaf value in the presence of externalities

  • Published:
Social Choice and Welfare Aims and scope Submit manuscript

Abstract

We propose two generalizations of the Banzhaf value for partition function form games. In both cases our approach is based on probability distributions over the set of coalition structures that may arise for any given set of players. First, we introduce a family of values, one for each collection of these latter probability distributions, defined as the Banzhaf value of a coalitional game obtained as the expectation taken according to the given probability distributions of the original partition function form game. For each value of the family we provide two characterization results within the set of all partition function form games. Both results rely on a property of neutrality with respect to the amalgamation of players. Second, we propose another family of values that differ from the previous ones in that the latter values take into account only the information about the most likely coalition structure that may arise according to the given probability distributions. Each value of the second family is also characterized in two results by means of a collusion neutrality property. Unlike the characterizations of the first approach, these characterizations can be restricted to the set of simple games in partition function form.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

Notes

  1. For instance, this has recently been the case in the formation process of the regional governments in the Basque Country and Catalonia.

  2. The difference between these two properties has been studied in Alonso-Meijide et al. (2012).

  3. The asymmetry in the delegation agreement is discussed in more detail in Sect. 3.

  4. We restrict our attention to non-singleton universes of players.

  5. We point out that there are some authors (e.g. Casajus 2012) that use “simple games” to refer to a class of coalitional games where no monotonicity condition is imposed.

  6. We note that \(\mathbb {R}_{+}=[0,\infty )\).

  7. In Sect. 4 we do not impose the consistency requirement given by condition (b).

  8. We opt for consistency, as the property considered relates the probability that players in \(N_{-j} \setminus S\) are organized according to \(P_{-S}\) within two different nested scenarios: one where \(j\) is present, one where \(j\) is absent.

  9. Note that we actually associate a coalitional game with every game in partition function form.

  10. See Example 2.2.

  11. Recall that \(\emptyset \) is a member of any partition.

  12. Note that given \(S\subseteq N\) and \((S,P)\in EC^{N}\), the reference to \(S\) as “active” coalition when we look at \(v(S,P)\) actually originates from an interpretation of the coalition structure \(P\) per se, with no reference to the game whatsoever. Accordingly, it can be thought that \((S,P)\) represents a situation in which players in \(S\) agree to actively participate in \(S\), while players in \(N\setminus S\) simply organize themselves into \(P_{-S}\).

  13. The same is trivially true for one-player games.

  14. We denote by \(\mathsf {Ba}\) the Banzhaf value of coalitional games. We note that it can be defined for every \((N,v)\in \mathcal {CG}\) by \(\mathsf {Ba}(N,v)=\mathsf {Ba}^\Lambda (N,v)\), where \(\Lambda \in \mathcal {L}\) is arbitrary. The \(\{ij\}\)-reduced game for coalitional games is then independent of \(\Lambda \) and coincides with the one used in Casajus (2012).

  15. By definition the summation over the empty set is zero.

  16. For the sake of the notation any reference to \(\Lambda \) has been omitted.

  17. This part is based on the proof of Theorem 7 in Casajus (2012).

  18. This part is based on Lehrer (1988).

  19. The independence of the axioms can be proved analogously as in Casajus (2012).

  20. The independence of the axioms is trivial.

  21. All the results contained in this section remain valid if we dispense with condition (iii) and hence do not impose any monotonicity condition in the definition of \(\mathcal {SG}\).

  22. For instance, consider the following example. There are three parties in a parliament, \(N=\{1,2,3\}\), and the seats are distributed as follows: Parties 1 and 2 have both 20 seats and Party 3 has 15 seats. Obviously, no party has an absolute majority. If absolute majority is required, then the three parties are in fairly symmetric position. This is indeed captured by the simple coalitional game \((N,v)\) where \(v(\{k\})=0\) and \(v(N_{-k})=v(N)=1\) for all \(k\in N\). However, if only a relative majority is needed and we consider \(\Lambda ^{**}\)—i.e. no parties make any agreement when they are in the opposition—then clearly Parties 1 and 2 are in a better position than Party 3. The only way to capture this by means of a game where the values of embedded coalitions are either \(0\) or \(1\), is by letting \(v(\{1\},**)=v(\{2\},**)=1\) and \(v(\{3\},**)=0\). That is, \(\mathcal {SG}\) must necessarily contain games where there are two or more winning coalitions in the same partition.

  23. Without conditions (c) and (d) the interpretation is similar but uniqueness of the most likely configurations is not guaranteed.

  24. Taking the minimum is relevant only when the set \(\left\{ \mathop {\hbox {arg max}}\limits \limits _{T'\in P_{-S}} \lambda ^{N}(S,P_{-T'} \cup \{T'_{+j}\})\right\} \) is not a singleton. This occurs, for instance, in the \(\Lambda \) considered in Examples 2.1 and 2.2.

  25. See Theorem 1 in Casajus (2012). A value on \(\mathcal {SG}, \mathsf {f}\), is symmetric if for every \((N,v)\in \mathcal {SG}, v(S_{+i})=v(S_{+j})\) for all \(S\subseteq N \setminus \{i,j\}\) implies \(\mathsf {f}_i (N,v)=\mathsf {f}_j(N,v)\).

  26. The independence of the axioms is trivial.

  27. We stress that \(\mathcal {L}\cap \tilde{\mathcal {L}} \ne \emptyset \) as both sets contain the \(\Lambda \) defined in Examples 2.1—except if \(p=\frac{1}{2}\)—and 2.2, so a proper comparison between values belonging to the two families can be established. Note that while \(\mathsf {Ba}^\Lambda \) and \(\widetilde{\mathsf {Ba}}^\Lambda \) coincide for \(\Lambda \in \{\Lambda ^{*},\Lambda ^{**}\}\), both values differ in general for \(\Lambda ^{p}\).

  28. The independence of the axioms is trivial.

References

  • Albizuri M, Arin J, Rubio J (2005) An axiom system for games in partition function form. Int Game Theory Rev 7:63–72

    Article  Google Scholar 

  • Alonso-Meijide JM, Álvarez-Mozos M, Fiestras-Janeiro MG (2012) Notes on a comment on 2-efficiency and the Banzhaf value. Appl Math Lett 25(7):1098–1100

    Article  Google Scholar 

  • Banzhaf JF (1964) Weighted voting doesn’t work: a mathematical analysis. Rutgers Law Rev 19:317–343

  • Bolger E (1983) The Banzhaf index for multicandidate presidential elections. SIAM J Algebraic Discrete Methods 4:422–458

    Article  Google Scholar 

  • Bolger E (1986) Power indices for multicandidate voting games. Int J Game Theory 15:175–186

    Article  Google Scholar 

  • Bolger E (1990) A characterization of an extension of the Banzhaf value for multicandidate voting games. SIAM J Discrete Math 3:466–477

    Article  Google Scholar 

  • Bolger E (2002) Characterizations of two power indices for voting games with r alternatives. Soc Choice Welf 19(4):709–721

    Article  Google Scholar 

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

    Article  Google Scholar 

  • de Clippel G, Serrano R (2008) Marginal contributions and externalities in the value. Econometrica 76(6):1413–1436

    Article  Google Scholar 

  • Dutta B, Ehlers L, Kar A (2010) Externalities, potential, value and consistency. J Econ Theory 145:2380–2411

  • Freixas J, Zwicker W (2003) Weighted voting, abstention, and multiple levels of approval. Soc Choice Welf 21:399–431

    Article  Google Scholar 

  • Hafalir I (2007) Efficiency in coalition games with externalities. Games Econ Behav 61:242–258

    Article  Google Scholar 

  • Haller H (1994) Collusion properties of values. Int J Game Theory 23:261–281

    Article  Google Scholar 

  • Hart S, Mas-Colell A (1989) Potential, value, and consistency. Econometrica 57:589–614

    Article  Google Scholar 

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

    Article  Google Scholar 

  • Macho-Stadler I, Pérez-Castrillo D, Wettstein D (2007) Sharing the surplus: an extension of the Shapley value for environments with externalities. J Econ Theory 135:339–356

    Article  Google Scholar 

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

    Article  Google Scholar 

  • May KO (1952) A set of independent necessary and sufficient conditions for simple majority decision. Econometrica 20:680–684

    Article  Google Scholar 

  • Myerson R (1977) Values of games in partition function form. Int J Game Theory 6:23–31

    Article  Google Scholar 

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

    Article  Google Scholar 

  • Owen G (1975) Multilinear extensions and the Banzhaf value. Nav Res Log Q 22:741–750

    Article  Google Scholar 

  • Penrose L (1946) The elementray statistics of majority voting. J R Stat Soc 109:53–57

    Article  Google Scholar 

  • Pham Do K, Norde H (2007) The Shapley value for partition function form games. Int Game Theory Rev 9:353–360

    Article  Google Scholar 

  • Shapley LS, Shubik M (1954) A method for evaluating the distribution of power in a committee system. Am Polit Sci Rev 48(03):787–792

    Article  Google Scholar 

  • Thrall R, Lucas W (1963) n-Person games in partition function form. Nav Res Log Q 10:281–298

    Article  Google Scholar 

Download references

Acknowledgments

We would like to thank two referees for their comments and suggestions which helped to substantially improve a previous version of the document. All remaining errors are our own responsibility. This research received financial support from the Ministerio de Economía y Competitividad through Projects ECO2011-22765 and MTM2011-27731-02, and from Generalitat de Catalunya through project 2014-SGR-40.

Author information

Authors and Affiliations

  1. Departament de Matemàtica Econòmica, Financera i Actuarial, University of Barcelona, Barcelona, Spain

    M. Álvarez-Mozos

  2. CER-ETH Center of Economic Research, ETH Zurich, Zurich, Switzerland

    O. Tejada

Authors
  1. M. Álvarez-Mozos
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. O. Tejada
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to O. Tejada.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Álvarez-Mozos, M., Tejada, O. The Banzhaf value in the presence of externalities. Soc Choice Welf 44, 781–805 (2015). https://doi.org/10.1007/s00355-014-0861-4

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00355-014-0861-4

Search

Navigation