On Public Values and Power Indices
DOI:
https://doi.org/10.7494/dmms.2015.9.1.9Keywords:
cooperative game theory, simple game, values, public values, power indices, public power indicesAbstract
In this paper, we analyze some values and power indices from a different point of view that are well-defined in the social context where the goods are public. In particular, we consider the Public Good index (Holler, 1982), the Public Good value (Holler and Li, 1995), the Public Help index (Bertini et al., 2008), the König and Bräuninger index (1998) also called the Zipke index (Nevison et al., 1978), and the Rae index (1969). The aims of this paper are: to propose an extension of the Public Help index to cooperative games; to introduce a new power index with its extension to a game value; and to provide some characterizations of the new index and values.
References
Banzhaf J.F. III (1965). Weighted voting doesn’t work: A mathematical analysis. Rutgers Law Review, 19(2), pp. 317–343.
Bertini C., Freixas J., Gambarelli G. & Stach I. (2013a). Comparing power indices. International Game Theory Review, 15(2), art. no. 1340004, pp. 1–19. DOI: http://doi.org/10.1142/S0219198913400045.
Bertini C., Freixas J., Gambarelli G. & Stach I. (2013b). Some open problems in simple games. International Game Theory Review, 15(2), art. no. 1340005, pp. 1–18. DOI: http://doi.org/10.1142/S0219198913400057.
Bertini C., Gambarelli G. & Stach I. (2008). A public help index. In: Braham M. & Steffen F. (Eds.), Power, Freedom, and Voting, pp. 83–98. Berlin–Heidelberg: Springer. DOI: http://doi.org/10.1007/978-3-540-73382-9_5.
Brams S.J. & Lake M. (1978). Power and satisfaction in a representative democracy. In: Ordeshook P.C. (Ed.), Game Theory and Political Science, pp. 529–562. New York: New York University Press.
Coleman J.S. (1971). Control of collectivities and the power of a collectivity to act. In: Lieberman B. (Ed.), Social Choice, pp. 269–300. New York: Gordon and Breach.
Deegan J. Jr. & Packel E.W. (1978). A new index of power for simple n-person games. International Journal of Game Theory, 7(2), pp. 113–123. DOI: http://doi.org/10.1007/BF01753239.
Dubey P. & Shapley L.S. (1979). Mathematical properties of the Banzhaf power index. Mathematics of Operations Research, 4(2), pp. 99–131. DOI: http://doi.org/10.1287/moor.4.2.99.
Felsenthal D.S. & Machover M. (1995). Postulates and paradoxes of relative voting power: A critical re-appraisal. Theory and Decision, 38(2), pp. 195–229. DOI: http://doi.org/10.1007/BF01079500.
Felsenthal D.S. & Machover M. (1998). The Measurement of Voting Power: Theory and Practice, Problems and Paradoxes. Cheltenham: Edward Elgar Publishing. DOI: http://doi.org/10.4337/9781840647761.
Fragnelli V., Ottone S. & Sattanino R. (2009). A new family of power indices for voting games. Homo Oeconomicus, 26, pp. 381–394.
Holler M.J. (1982). Forming coalitions and measuring voting power. Political Studies, 30(2), pp. 262–271. DOI: http://doi.org/10.1111/j.1467-9248.1982.tb00537.x.
Holler M.J. & Li X. (1995). From public good index to public value: An axiomatic approach and generalization. Control and Cybernetics, 24(3), pp. 257–270.
Holler M.J. & Packel E.W. (1983). Power, luck and the right index. Zeitschrift für Nationalökonomie – Journal of Economics, 43(1), pp. 21–29. DOI: http://doi.org/10.1007/BF01283881.
Jasiński M. (2013). The terms of cooperation’s stability: What is the reason of flow between coalitions? Roczniki Kolegium Analiz Ekonomicznych, 32, pp. 55–76.
Johnston R.J. (1978). On the measurement of power: Some reactions to Laver. Environment and Planning A, 10(8), pp. 907–914. DOI: http://doi.org/10.1068/a100907.
König T. & Bräuninger T. (1998). The inclusiveness of European decision rules. Journal of Theoretical Politics, 10(1), pp. 125–142. DOI: http://doi.org/10.1177/0951692898010001006.
Kurz S. (2016). The inverse problem for power distributions in committees. Social Choice and Welfare, 47(1), pp. 65–88. DOI: http://doi.org/10.1007/s00355-015-0946-8.
Laruelle A. & Valenciano F. (2008). Voting and Collective Decision-Making: Bargaining and Power. Cambridge: Cambridge University Press. DOI: http://doi.org/10.1017/CBO9780511493553.
Mercik J.W. (1997). Power and expectations. Control and Cybernetics, 26(4), pp. 617–621.
Napel S. (1999). The Holler–Packel axiomatization of the public good index completed. Homo Oeconomicus, 15, pp. 513–520.
Napel S. (2001). A note on the Holler–Packel axiomatization of the public good index (PGI). In: Holler M.J. & Owen G. (Eds.), Power Indices and Coalition Formation, pp. 143–151. Boston, MA: Springer. DOI: http://doi.org/10.1007/978-1-4757-6221-1_9.
Nevison C.H., Zicht B. & Schoepke S. (1978). A naive approach to the Banzhaf index of power. Behavioral Science, 23(2), pp. 130–131. DOI: http://doi.org/10.1002/bs.3830230209.
Nevison C.H. (1979). Structural power and satisfaction in simple games. In: Brams S.J., Schotter A. & Schwödiauer G. (Eds.), Applied Game Theory, pp. 39–57. Heidelberg: Physica-Verlag. DOI: http://doi.org/10.1007/978-3-662-41501-6_3.
Rae D.W. (1969). Decision-rules and individual values in constitutional choice. American Political Science Review, 63(1), pp. 40–56. DOI: http://doi.org/10.2307/1954283.
Shapley L.S. (1953). A value for n-person games. In: Kuhn H.W. & Tucker A.W. (Eds.), Contributions to the Theory of Games II. Annals of Mathematics Studies, vol. 28, pp. 307–317. Princeton: Princeton University Press. DOI: http://doi.org/10.1515/9781400881970-018.
Shapley L.S. & Shubik M. (1954). A method for evaluating the distribution of power in a committee system. American Political Science Review, 48(3), pp. 787–792. DOI: http://doi.org/10.2307/1951053.
