Abstract
This article proposes a value which can be considered an extension of the Banzhaf value for cooperative games. The proposed value is defined on the class of j-cooperative games, i.e., games in which players choose among a finite set of ordered actions and the result depends only on these elections. If the output is binary, only two options are available, then j-cooperative games become j-simple games. The restriction of the value to j-simple games leads to a power index that can be considered an extension of the Banzhaf power index for simple games. The paper provides an axiomatic characterization for the value and the index which is closely related to the first axiomatization of the Banzhaf value and Banzhaf power index in the respective contexts of cooperative and simple games.
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Notes
S precedes T in the lexicographic order if for the smallest \(p \in N\) such that \(p \in S_i\) and \(p \in T_h\) with \(i \ne h\), it holds \(i<h\).
By solving the linear system of equations it can be seen that these coefficients are \(c_S = \sum \nolimits _{T {\subseteq }^j S} (-1)^{\sum \nolimits _{i=1}^j(t_i-s_i)} v(T)\).
References
Banzhaf JF (1965) Weighted voting doesn’t work: a mathematical analysis. Rutgers Law Rev 19:317–343
Bernardi G, Freixas J (2019) An axiomatization for two power indices for (3,2)-simple games. Int Game Theory Rev 21(2):1940001 (24 pages)
Carreras F (2004) \(\alpha \)-Decisiveness in simple games. Theory Decis 56(1–2):77–91 2004
Carreras F (2005) A decisiveness index for simple games. Eur J Oper Res 163(2):370–387
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
Dreher A, Gould M, Rablen MD, Vreeland JR (2014) The determinants of election to the United Nations Security Council. Public Choice 158:51–83
Dubey P, Shapley LS (1979) Mathematical properties of the Banzhaf power index. Math Op Res 4(2):99–131
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, Zwicker WS (2003) Weighted voting, abstention, and multiple levels of approval. Soc Choice Welf 21(3):399–431
Gould M, Rablen MD (2017) Reform of the United Nations Security Council: equity and efficiency. Public Choice 173(1–2):145–168
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
Owen G (1975) Multilinear extensions and the Banzhaf value. Naval Res Logist Q 22(4):741–750
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, Lambo LD (2011) Political influence in multichoice institutions: cyclicity, anonymity and transitivity. Theory Decis 70(2):157–178
Tchantcho B, Lambo LD, Pongou R, Engoulou BM (2008) Voters’ power in voting games with abstention: influence relation and ordinal equivalence of power theories. Games Econ Behav 64(1):335–350
Uleri PV (2002) On referendum voting in Italy: YES, NO, or non-vote? How Italian parties learned to control referendums. Eur J Polit Res 41(6):863–883
Acknowledgements
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).
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Freixas, J. The Banzhaf Value for Cooperative and Simple Multichoice Games. Group Decis Negot 29, 61–74 (2020). https://doi.org/10.1007/s10726-019-09651-4
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DOI: https://doi.org/10.1007/s10726-019-09651-4