The Banzhaf Value and General Semivalues for Differentiable Mixed Games

References

• [1] Aumann RJ, Shapley LS (1974) Values of Non Atomic Games (Princeton University Press, Princeton, NJ).Google Scholar
• [2] Banzhaf JF (1965) Weighted voting doesn’t work: A mathematical analysis. Rutgers Law Rev. 19(2):317–343.Google Scholar
• [3] Banzhaf JF (1966) Multi-member electoral districts—do they violate the “one man, one vote” principle. Yale Law J. 75(8):1309–1338.Crossref, Google Scholar
• [4] Banzhaf JF (1968) One man, 3.312 votes: A mathematical analysis of the electoral College. Vilanova Law Rev. 13(2):304–332.Google Scholar
• [5] Coleman JS (1971) Control of collectives and the power of a collectivity to act. Lieberman B, ed. Social Choice (Gordon and Breach, New York), 192–225.Google Scholar
• [6] Diaconis P (1988) Recent progress on de Finetti’s notions of exchangeability. Bernardo JM, DeGroot MH, Lindley DV, Smith AFM, eds. Bayesian Statistics, vol. 3 (Oxford University Press, Oxford, UK), 111–125.Google Scholar
• [7] Dubey P (1975) On the uniqueness of the Shapley value. Internat. J. Game Theory 4(3):131–139.Crossref, Google Scholar
• [8] Dubey P, Shapley LS (1979) Mathematical properties of the Banzhaf power index. Math. Oper. Res. 4(2):99–131.Link, Google Scholar
• [9] Dubey P, Einy E, Haimanko O (2005) Compound voting and the Banzhaf index. Games Econom. Behav. 51(1):20–30.Crossref, Google Scholar
• [10] Dubey P, Neyman A, Weber RJ (1981) Value theory without efficiency. Math. Oper. Res. 6(1):122–128.Link, Google Scholar
• [11] Einy E (1987) Semivalues of simple games. Math. Oper. Res. 12(2):185–192.Link, Google Scholar
• [12] Haimanko O (2000) Partially symmetric values. Math. Oper. Res. 25(4):573–590.Link, Google Scholar
• [13] Hart S (1973) Values of mixed games. Internat. J. Game Theory 2(1):69–85.Crossref, Google Scholar
• [14] Hart S, Monderer D (1997) Potential and weighted values of nonatomic games. Math. Oper. Res. 22(3):619–630.Link, Google Scholar
• [15] Judd KL (1985) The law of large numbers with a continuum of IID random variables. J. Econom. Theory 35(1):19–25.Crossref, Google Scholar
• [16] Lehrer E (1988) An axiomatization of the Banzhaf value. Internat. J. Game Theory 17(2):89–99.Crossref, Google Scholar
• [17] Mertens JF (1988) The Shapley value in the non-differentiable case. Internat. J. Game Theory 17(1):1–65.Crossref, Google Scholar
• [18] Monderer D (1990) A Milnor condition for non-atomic Lipschitz games and its applications. Math. Oper. Res. 15(4):714–723.Link, Google Scholar
• [19] Monderer D, Neyman A (1988) Values of smooth nonatomic games: The method of multilinear approximation. Roth AE, ed. The Shapley Value: Essays in Honor of Lloyd S. Shapley (Cambridge University Press, Cambridge, UK), 217–234.Google Scholar
• [20] Owen G (1964) The tensor composition of nonnegative games. Dresher M, Shapley LS, Tucker AW, eds. Annals of Mathematics, vol. 52 (Princeton University Press, Princeton, NJ), 307–326.Crossref, Google Scholar
• [21] Owen G (1975) Multilinear extensions and the Banzhaf value. Naval Res. Logist. Quart. 1975(4):741–750.Crossref, Google Scholar
• [22] Owen G (1978) Characterization of the Banzhaf-Coleman index. SIAM J. Appl. Math. 35(2):315–327.Crossref, Google Scholar
• [23] Penrose LS (1946) The elementary statistics of majority voting. J. Roy. Statist. Soc. 109(1):53–57.Crossref, Google Scholar
• [24] Roth AE (1977) Bargaining ability, the utility of playing a game, and models of coalition formation. J. Math. Psych. 16(2):153–160.Crossref, Google Scholar
• [25] Roth AE (1977) The Shapley value as a von Neumann-Morgenstern utility. Econometrica 45(3):657–664.Crossref, Google Scholar
• [26] Roth AE (1988) The expected utility of playing a game. Roth AE, ed. The Shapley Value: Essays in Honor of Lloyd S. Shapley (Cambridge University Press, Cambridge, UK), 51–70.Crossref, Google Scholar
• [27] Shapley LS (1953) A value for n-person games. Kuhn HW, Tucker AW, eds. Contributions to the Theory of Games II, Annals of Mathematical Studies 28 (Princeton University Press, Princeton, NJ), 307–317.Crossref, Google Scholar
• [28] Shapley LS (1964) Solutions of compound simple games. Dresher M, Shapley LS, Tucker AW, eds. Annals of Mathematics, vol. 52 (Princeton University Press, Princeton, NJ), 267–306.Crossref, Google Scholar
• [29] Shapley LS, Shubik M (1954) A method for evaluating the distribution of power in a committee system. Amer. Political Sci. Rev. 48(3):787–792.Crossref, Google Scholar