Abstract
The Banzhaf power index for games admits several extensions if the players have more than two ordered voting options. In this paper we prove that the most intuitive and recognized extension of the index fails to preserve the desirability relation for games with more than three ordered input levels of approval, a failure that undermines the index to be a good measure of power. This leads us to think of an alternative to the Banzhaf index for several input levels of approval. We propose a candidate for which it is proved that: (1) coincides with the Banzhaf index for simple games, (2) it is proportional to its known extension for three levels of approval, and (3) preserves the desirability relation regardless of the number of input levels of approval. This new index is based on measuring the total capacity the player has to alter the outcome. In addition, it can be expressed through a very appropriate mathematical formulation that greatly facilitates its computation. Defining extensions of well-established notions in a wider context requires a careful analysis. Different extensions can provide complementary nuances and, when this occurs, none of them can be considered to be ‘the’ extension. As shown in this paper, this situation applies when trying to extend the Banzhaf power index from simple games to the broader context of games with several ordered input levels of approval.
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References
Albizuri MJ, Ruiz LM (2001) A new axiomatization of the Banzhaf semivalue. Span Econ Rev 3:97–109
Alonso-Meijide JM, Álvarez-Mozos M, Fiestras-Janeiro MG (2017) Power indices and minimal winning coalitions for simple games in partition function form. Group Decis Negot 26:1231–1245
Amer R, Carreras F, Magaña A (1998) The Banzhaf-Coleman index for games with r alternatives. Optimization 44:175–198
Aubin JP (1974) Coeur et valeur des jeux flous à paiements latéraux. C R Hebdomad D 279–A:891–894
Banzhaf JF (1965) Weighted voting doesn’t work: a mathematical analysis. Rutgers Law Rev 19:317–343
Barua R, Chakravarty SR, Roy S (2005) A new characterization of the Banzhaf index of power. Int Game Theory Rev 7:545–553
Bernardi G (2018) A new axiomatization of the Banzhaf index for games with abstention. Group Decis Negot 27:165–177
Bhaumik A, Roy SK, Weber GW (2020) Hesitant interval-valued intuitionistic fuzzy-liguistic term set approach in Prisoners dilemma game theory using TOPSIS: a case study on Human-trafficking. CEJOR 28:797–816
Bolger E (1986) Power indices for multicandidate voting games. Int J Game Theory 15:175–186
Bolger E (1990) A characterization of an extension of the Banzhaf value for multicandidate voting games. SIAM J Discret Math 3:466–477
Bolger E (2002) Characterizations of two power indices for voting games with r alternatives. Soc Choice Welf 19:709–721
Borkotokey S (2008) Cooperative games with fuzzy coalitions and fuzzy characteristic functions. Fuzzy Sets Syst 159:138–151
Casajus A (2011) Marginality, differential marginality, and the Banzhaf value. Theory Decis 71(3):365–372
Casajus A (2012) Amalgamating players, symmetry, and the Banzhaf value. Int J Game Theory 41(3):497–515
Coleman JS (1971) Control of collectivities and the power of a collectivity to act. In: Lieberman B (ed) Social choice. Gordon and Breach, New York, pp 269–300
Courtin S, Tchantcho B (2020) Public good indices for games with several levels of approval. B.E. J Theor Econ 20(1):1–20
Dubey P, Einy E, Haimanko O (2005) Compound voting and the Banzhaf index. Game Econ Behav 51(1):20–30
Dubey P, Shapley LS (1979) Mathematical properties of the Banzhaf power index. Mathe Oper Res 4(2):99–131
Einy E (1985) The desirability relation of simple games. Math Soc Sci 10(2):155–168
Felsenthal DS, Machover M (1997) Ternary voting games. Int J Game Theory 26(3):335–351
Felsenthal DS, Machover M (1998) The measurament of voting power: theory and practice, problems and paradoxes. Edward Elgar, Cheltenham
Feltkamp V (1995) Alternative axiomatic characterizations of the Shapley and the Banzhaf values. Int J Game Theory 24(2):179–186
Freixas J (2005) Banzhaf measures for games with several levels of approval in the input and output. Ann Oper Res 137(1):45–66
Freixas J (2012) Probabilistic power indices for voting rules with abstention. Math Soc Sci 64(1):89–99
Freixas J (2020) The Banzhaf value for cooperative and simple multichoice games. Group Decis Negot 29:61–74
Freixas J, Lucchetti R (2016) Power in voting rules with abstention: an axiomatization of two components power index. Ann Oper Res 244(2):455–474
Freixas J, Zwicker WS (2003) Weighted voting, abstention, and multiple levels of approval. Soc Choice Welf 21(3):399–431
Freixas J, Tchantcho B, Tedjeugang N (2014) Achievable hierarchies in voting games with abstention. Eur J Oper Res 236(1):254–260
Freixas J, Tchantcho B, Tedjeugang N (2014) Voting games with abstention: Linking completeness and weightedness. Decis Support Syst 57:172–177
Gallego I, Fernández A, Jiménez-Losada A, Ordóñez M (2014) A Banzhaf value for games with fuzzy communication structure: Computing the power of the political groups in the European Parliament. Fuzzy Sets Syst 255:128–145
Haller H (1994) Collusion properties of values. Int J Game Theory 23:261–281
Holler MJ (1982) Forming coalitions and measuring voting power. Political Stud 30(2):262–271
Hsiao CR, Raghavan TES (1992) Monotonicity and dummy free property for multi-choice cooperative games. Int J Game Theory 21(1):301–302
Hsiao CR, Raghavan TES (1993) Shapley value for multichoice cooperative games, I. Games Econ Behav 5(2):240–256
Isbell JR (1958) A class of simple games. Duke Math J 25(3):423–439
Jana J, Roy SK (2019) Dual hesitant fuzzy matrix games: based on new similarity measure. Soft Comput 23:8873–8886
Johnston RJ (1978) On the measurement of power: some reactions to Laver. Environ Plan A 10(8):907–914
Kenfack JAM, Tchantcho B, Tsague BP (2019) On the ordinal equivalence of the Johnston, Banzhaf and Shapley–Shubik power indices for voting games with abstention. Int J Game Theory 48(2):647–671
Kurz S (2020) A note on limit results for the Penrose–Banzhaf index. Theory Decis 88:191–203
Lehrer E (1998) An axiomatization of the Banzhaf value. Int J Game Theory 17:89–99
Lidner I (2008) A special case of Penrose’s limit theorem when abstention is allowed. Theory Decis 64(4):495–518
Lidner I, Machover M (2004) L.S. Penrose’s limit theorem: proof of some special cases. Math Soc Sci 47(1):37–49
Lidner I, Owen G (2007) Cases where the Penrose limit theorem does not hold. Math Soc Sci 53(3):232–238
Meng F, Zhang Q, Chen X (2017) Fuzzy multichoice games with fuzzy characteristic functions. Group Decis Negot 26:565–595
Ono R (2001) Values for multialternative games and multilinear extensions. In: Holler MJ, Owen G (eds) Power indices and coalition formation. Springer, Boston
Owen G (1975) Multilinear extensions and the Banzhaf value. Naval Res Logist Q 22(4):741–750
Owen G (1978) Characterization of the Banzhaf–Coleman index. SIAM J Appl Math 35:315–327
Parker C (2012) The influence relation for ternary voting games. Games Econ Behav 75(2):867–881
Penrose LS (1946) The elementary statistics of majority voting. J R Stat Soc 109(1):53–57
Pongou R, Tchantcho B, Diffo Lambo L (2011) Political influence in multichoice institutions: cyclicity, anonymity and transitivity. Theory Decis 70(2):157–178
Taylor AD, Zwicker WS (1999) Simple games: desirability relations, trading, and pseudoweightings. Princeton University Press, New Jersey
Tchantcho B, Diffo Lambo L, Pongou R, Mbama Engoulou B (2008) Voters’ power in voting games with abstention: influence relation and ordinal equivalence of power theories. Games Econ Behav 64(1):335–350
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This research was partially supported by funds from: the Spanish Ministry of Economy and Competitiveness (MINECO) and from the European Union (FEDER funds) under grant MTM2015-66818-P (MINECO/FEDER), and the Spanish Ministry of Science and Innovation grant PID2019-I04987GB-I00.
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Freixas, J., Pons, M. An Appropriate Way to Extend the Banzhaf Index for Multiple Levels of Approval. Group Decis Negot 30, 447–462 (2021). https://doi.org/10.1007/s10726-020-09718-7
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DOI: https://doi.org/10.1007/s10726-020-09718-7