Modification of Shapley Value and its Implementation in Decision Making

[1] Aumann, R., Dreze, J., 1975. Solutions of cooperative games with coalition structures. International Journal of Game Theory, (4), pp.180-192. 10.1007/BF01766876Search in Google Scholar

[2] Aumann, R., Hart, S., 1994. Handbook of Game Theory with Economic Applications, Vol. 2. North-Holland: Elsevier. Search in Google Scholar

[3] Aumann, R., Hart, S., 2002. Handbook of Game Theory with Economic Applications, Vol. 3. North-Holland: Elsevier. Search in Google Scholar

[4] Aumann, R., Heifetz, A., Incomplete Information. In: R. Aumann and S. Hart, eds., 2002. Handbook of Game Theory with Economic Applications, Vol. 3. North-Holland: Elsevier, pp.1665-1685. 10.1016/S1574-0005(02)03006-0Search in Google Scholar

[5] Aumann, R., Myerson, R., 2005. Endogenous Formation of Links Between Players and of Coalitions: An Application of the Shapley Value. In: Roth A. Shapley Value - Essays in Honor of L. Shapley. Cambridge: Cambridge Univ. Press, pp.175-194. 10.1017/CBO9780511528446.013Search in Google Scholar

[6] Avenhaus, R., von Stengel, B., and Zamir, S., Inspection Games. In: R., Aumann and S., Hart, eds., 2002. Handbook of Game Theory with Economic Applications, Vol. 3. North-Holland: Elsevier, pp.1947-1984. 10.1016/S1574-0005(02)03014-XSearch in Google Scholar

[7] Banzhaf, J.F., III, 1965. Weighted Voting Doesn’t Work: A Mathematical Analysis. Rutgers Law Review, 19, pp.317-343. Search in Google Scholar

[8] Banzhaf, J.F., III, 1968. One Man, 3312 Votes: A Mathematical Analysis of the Electoral College. Villanova Law Review, 13, pp.303-346. Search in Google Scholar

[9] Billera, L., Heath, D., and Raanan, J., 1978. Internal Telephone Billing, Rates: A Novel Application of Non-Atomic Games Theory. Operations Research, 26, pp.956-965. 10.1287/opre.26.6.956Search in Google Scholar

[10] Blackwell, D., and Ferguson, T., 1968. The Big Match. Annals of Mathematical Statistics, 39, pp.159-163. 10.1214/aoms/1177698513Search in Google Scholar

[11] Bondareva, O., 1963. Some Applications of Linear Programming Methods to the Theory of Cooperative Games. Problemy Kibernetiki, 10, pp.119-139 (in Russian). Search in Google Scholar

[12] Brams, S., Lake, M., 1977. Power and Satisfaction in a Representative Democracy. In: Game Theory and Political Science Conference. Hyannis MA., USA. Search in Google Scholar

[13] Carreras F. Owen G. 1988. Evaluation in Catalonian Parliament 1980-1984. Mathematical Social Science, 15, pp.87-92. Search in Google Scholar

[14] Casajus, A., Huettner, F., 2013. Null Players, Solidarity and the Egalitarian Shapely Values. Journal of Mathematical Economics, 49(1), pp.58-61. 10.1016/j.jmateco.2012.09.008Search in Google Scholar

[15] Colleman, J., 1971. Control of Collectivities and Power of a Collectivity to Act. In: Social Choice, B. Lieberman, ed., Gordon and Breach, pp.269-300. Search in Google Scholar

[16] Deegal, J., Packel, E., 1979. A New Index of Power for Simple n-Person Games. International Journal of Game Theory, 7, pp.113-123. 10.1007/BF01753239Search in Google Scholar

[17] Dubey, P., Shapley, L., 1979. Mathematical Properties of the Banzhaf Power Index. Mathematics of Operations Research, 4, pp.99-131. 10.1287/moor.4.2.99Search in Google Scholar

[18] Dutta, P., Radner R., Moral Hazard. In: R. Aumann and S. Hart, eds., 1994. Handbook of Game Theory with Economic Applications, Vol.2. North-Holland: Elsevier, pp.870-900. Search in Google Scholar

[19] Felsenthal, D., Machover, M., 1977. The Weighted Rule in the EU’s Council of Ministers, 1958-1995; Intentions and Outcomes. Electoral Studies, (16), pp.33-47. 10.1016/S0261-3794(96)00055-8Search in Google Scholar

[20] Friedman, A., 1994. Differential Games. In: R., Aumann and S., Hart, eds. Handbook of Game Theory with Economic Applications, Vol. 2. North-Holland: Elsevier, pp.782-798. Search in Google Scholar

[21] Gale, D., Shapley, L., 1962. College Admissions and Stability of Marriage. American Mathematical Monthly, 69, pp.9-15. 10.1080/00029890.1962.11989827Search in Google Scholar

[22] Geanakoplos, J., 1994. Common Knowledge. In: R., Aumann and S., Hart, eds. Handbook of Game Theory with Economic Applications, Vol. 2. North-Holland: Elsevier, pp.1438-1495. Search in Google Scholar

[23] Gillette, D., 1957. Stochastic Games with Zero Stop Probabilities. In: Contribution to the Theory of Games., M., Dresher, A., Tucker and P., Wolfe, eds., vol. III Ann. Math. Studies, 39. Princeton: Princeton University Press, pp.179-187. 10.1515/9781400882151-011Search in Google Scholar

[24] Gillies, D., 1959. Solutions to General non-zerosum Games. In: Contributions to the Theory of Games, Vol. 4, Ann. Math. Studies, (40), eds. A. Tucker and D. Luce. Princeton: Princeton University Press, pp.47-85. 10.1515/9781400882168-005Search in Google Scholar

[25] Gillies, D., 1953a. Locations of Solutions. In: Report of an Informal Conference on the theory of N-person games, ed. H.W. Kuhn, Princeton University. Search in Google Scholar

[26] Gillies, D., 1953b. Some Theorems on N-person Games, Ph. D. dissertation. Department of Mathematics, Princeton University. Search in Google Scholar

[27] Holler, M., 1982. Forming Coalitions and Measuring Voting Power. Political Studies, 30(2), pp.266-231.10.1111/j.1467-9248.1982.tb00537.xSearch in Google Scholar

[28] Holler, M., Packel, E., 1983. Power, Luck and the Right Index. Journal of Economics (Zeitschrift fur Nationalekonomie), 43(1), pp.21-29. 10.1007/BF01283881Search in Google Scholar

[29] Holler, M., 1984. A Public Good Power Index. In: Coalitions and Collective Actions, ed., M. Holler. Wurzburg and Wein: Physica-Verlag, pp.51-59. Search in Google Scholar

[30] Holler, M., Owen, G., 2013. Power Indices and Coalition Formation. Springer Science & Business Media (386 pp.). Search in Google Scholar

[31] Holler, M., Li, X., 1995. From Public Good Index to Public Value: An Axiomatic Approach and Generalization. Control & Cybernetics, 24(2), pp.257-270. Search in Google Scholar

[32] Holler, M., 1997. Power, Monotocity and Expectations. Control & Cybernetics, 26(4), pp.605-608.Search in Google Scholar

[33] Homenda, W., 2009. Decision Making in Voting Games: An Insight into Theory and Practice. In: MDAI’09 Proceedings of the 6th International Conference on Modelling Decisions for Artificial Intelligence. Awaji Island, Japan, Nov. 30 - Dec. 02, pp.60-71. 10.1007/978-3-642-04820-3_6Search in Google Scholar

[34] Johnston, R., 1978. On the Measurement of Power: Some Reactions to Laver. Environment and Planning A, 10, pp.907-914. 10.1068/a100907Search in Google Scholar

[35] Lucas, W., 1983. Measuring Power in Weighted Voting Systems. In: Political and Related Models, S. Brams, W. Lucas, P. Straffin, eds., Berlin: Springer, pp.183-238. 10.1007/978-1-4612-5430-0_9Search in Google Scholar

[36] Malawski, M., Wieczorek, A., and Sosnowska, H., 2006. Konkurencja i kooperacja - teoria gier w ekonomii i naukach społecznych (Competition and Cooperation - Theory of Games in Economics and Social Science). Warszawa: Wydawnictwo Naukowe PWN. Search in Google Scholar

[37] Malawski, M., 2013. Procedural Values for Cooperative Games. International Journal of Game Theory, 42(1), pp.305-324. 10.1007/s00182-012-0361-7Search in Google Scholar

[38] Mann, I., Shapley, L., 1960. Values of Large Games IV: Evaluating the Electoral College by Monte Carlo Techniques. RM-2651, Rand Corporation, Santa Monica. Search in Google Scholar

[39] Mann, I., Shapley, L., 1962. Values of Large Games VI: Evaluating the Electoral College Exactly. RM-3158, Rand Corporation, Santa Monica. Search in Google Scholar

[40] Mercik, J., Kolodziejczyk, W., 1986. Taxonomy Approach to a Cabinet Formation. Mathematical Social Science, 12, pp.159-167. 10.1016/0165-4896(86)90034-XSearch in Google Scholar

[41] Mercik, J., 1997. Power and Expectations. Control & Cybernetics, 26(4), pp.617-622. Search in Google Scholar

[42] Mercik, J., 1999. Siła i oczekiwania; Decyzje grupowe (Power and Expectations; Group Deci sions). Warszawa-Wrocław: Wydawnictwo Naukowe PWN. Search in Google Scholar

[43] Mercik, J., 2013. Index of Power for Cabinet. In: Holler and Owen, eds., Power Indices and Coalition Formation, pp.371-380. New York: Springer Science+ Business Media. Search in Google Scholar

[44] Mertens, J-F., Neyman, A., 1981. Stochastic Games. International Journal of game Theory, 10, pp.53-66.10.1007/BF01769259Search in Google Scholar

[45] Mertens, J-F., 2002. Stochastic Games. In: R. Aumann and S. Hart, eds., 2002. Handbook of Game Theory with Economic Applications, Vol.3. North-Holland: Elsevier, pp.1809-1827. 10.1016/S1574-0005(02)03010-2Search in Google Scholar

[46] Milnor, J., Shapley, L., 1978. Values of Large Games II: Oceanic Games. Mathematics of Operations Research, 3, pp. 290-307; initially in: Rand Corporation, RM-2649, Santa Monica, 1961. Search in Google Scholar

[47] Monderer, D., Samet, D., and Shapely, L., 1992. Weighted Values and the Core. Journal of Political Economy, 21, pp.27-39. 10.1007/BF01240250Search in Google Scholar

[48] Monderer, D., Samet, D., 2002. Variations on the Shapley Value. In: R. Aumann and S. Hart, eds., 2002. Handbook of Game Theory with Economic Applications, Vol. 3. North-Holland: Elsevier, pp.2055-2075. 10.1016/S1574-0005(02)03017-5Search in Google Scholar

[49] Moriariti, S. ed., 1983. Joint Cost Allocation. Tulsa: Univ. of Oklahoma Press. Search in Google Scholar

[50] Nowak, A., Radzik, T., 1994. A Solidarity Value for n-Person Transferable Utility Games. International Journal of Game Theory, 23(1), pp.43-48. 10.1007/BF01242845Search in Google Scholar

[51] Ott, U., 2006. International Joint Ventures: An Interplay of Cooperative and Noncooperative Games under Incomplete Information. Houndmills Basingstoke: Palgrave Macmillan. Search in Google Scholar

[52] Owen, G., 1977. Values of Games with a Priori Unions. In: R. Henn and O. Moeschlin. Mathematical Economics and Game Theory. Berlin: Springer Verlag, pp.76-88. 10.1007/978-3-642-45494-3_7Search in Google Scholar

[53] Peleq, B., 2005. Introduction to the Theory of Cooperative Games, 2 ed. Berlin: Springer. Search in Google Scholar

[54] Raghavan, T., Non-sero-sum Two-Person Games. In: R. Aumann and S. Hart, eds., 2002. Handbook of Game Theory with Economic Applications, Vol. 3. North-Holland: Elsevier, pp.1687-1718.10.1016/S1574-0005(02)03007-2Search in Google Scholar

[55] Roth, A., 1984. The Evolution of the Labor Market for Medical Interns and Residents: A Case Study in Game Theory, Journal of Political Economy, 92, pp.991-1016. 10.1086/261272Search in Google Scholar

[56] Roth, A., Sotomayor, M., 1990. Two-Sided Matching: A Study in Game-Theoretic Modelling and Analysis. Econometric Society Monograph. Cambridge: Cambridge University Press. 10.1017/CCOL052139015XSearch in Google Scholar

[57] Roth, A., 2005. The Shapley Value: Essays in Honor of Lloyd S. Shapley. Cambridge: Cambridge Univ. Press. Search in Google Scholar

[58] Shapley, L., 1953a. A Value for n-Person Games. In: eds., H.W. Kuhn and A.W. Tucker. Contributions to the Theory of Games, Vol. 2. Ann. Math. Studies 28. Princeton: Princeton Univ. Press, pp.307-317. 10.1515/9781400881970-018Search in Google Scholar

[59] Shapley, L., 1953b. Adaptive and Non-Adaptive Set Functions. Ph.D. thesis. Department of Mathematics, Princeton Univ. Search in Google Scholar

[60] Shapley, L., 1953c. Open Questions. In: ed., H.W. Kuhn. Report of an Informal Conference on the Theory of n-Person Games. Princeton University. Search in Google Scholar

[61] Shapley, L., 1953d. Quota Solutions of n-Person Games. In: eds., H.W. Kuhn and A.W. Tucker. Contributions to the theory of Games, Vol. 2, Ann. Math. Studies 28, Princeton Univ. Press, pp.343-359. 10.1515/9781400881970-021Search in Google Scholar

[62] Shapley, L., 1953. Stochastic Games. Proceedings of the National Academy of Sciences, 39, pp.1095-1100. 10.1073/pnas.39.10.1953Search in Google Scholar

[63] Shapley, L., Shubik M., 1954. A Method for Evaluating a Distribution of Power in a Committee System. American Political Science Review, 48, pp.787-792. 10.2307/1951053Search in Google Scholar

[64] Shapley, L., 1959. The Solution of a Symmetric Market Game. Annals of Mathematics Studies, 40, pp.145-162. 10.1515/9781400882168-010Search in Google Scholar

[65] Shapley, L., 1961a. Values of Large Games III: A Cooperation with 2 Large Stockholders. RM- 26-50 Rand Corporation, Santa Monica. Search in Google Scholar

[66] Shapley, L., 1961b. Values of Large Games IV: An 18-Person Market Game. RM-2860 Rand Corporation, Santa Monica. Search in Google Scholar

[67] Shapley, L., 1962a. Compound Simple Games, I: Solutions of Sums and Products. RM -3192-PR, The Rand Corporation, Santa Monica. Search in Google Scholar

[68] Shapley, L., 1963). Compound Simple Games, II: Some General Composition Theorems. RM - 3643-PR, The Rand Corporation, Santa Monica. Search in Google Scholar

[69] Shapley, L., 1964a. Solutions of Compound Simple Games, I: Solutions of Sums and Products. Annals of Mathematics Studies, 52, pp.267-305. 10.1515/9781400882014-016Search in Google Scholar

[70] Shapley, L. 1964b. Values of Large Games VII: A General Exchange Economy with Money. RM-4248, Rand Corporation, Santa Monica. Search in Google Scholar

[71] Shapley, L., 1967a. On Balanced Sets and Cores. Naval Research Logistics Quarterly, 14, pp.453-460. 10.1002/nav.3800140404Search in Google Scholar

[72] Shapley, L., 1967b. On Committees. In: eds., F. Zwicky and A. Wilson. New Methods of Thought and Procedure. New York: Springer, pp.246-270. 10.1007/978-3-642-87617-2_13Search in Google Scholar

[73] Shapley, L., Shubik, M., 1969. On Market Games. Journal of Economic Theory, 1, pp.9-25. 10.1016/0022-0531(69)90008-8Search in Google Scholar

[74] Shapley, L., 1971. Cores of Convex Games. International Journal of Game Theory, 1, pp.11-26. 10.1007/BF01753431Search in Google Scholar

[75] Shapley, L., Shubik, M., 1972. The assignment Game I: The Core. International Journal of Game Theory, 1, pp.111-130. 10.1007/BF01753437Search in Google Scholar

[76] Shapley, L., 1977. A Comparison of Power Indices and a Nonsymmetric Generalization. Paper P-5872. The Rand Corporation, Santa Monica. Search in Google Scholar

[77] Shapley, L., Shapiro, N., 1978. Values of Large Games I: A Limit Theorem. Mathematics of Operations Research, 3, pp.1-9; initially in Rand Corporation, RM-2648, Santa Monica (1960). Search in Google Scholar

[78] Shubik, M., 1962. Incentives, Decentralized Control, the Assignment of Joint Costs and Internal Pricing. Management Science, 8(3), pp.325-343. 10.1287/mnsc.8.3.325Search in Google Scholar

[79] Shubik, M., 1982. Games Theory in the Social Science. Cambridge: MIT Press. Search in Google Scholar

[80] Sosnowska, H., 1997. Coalitions and Rationality, Control & Cybernetics 26(4), pp.623-624. Search in Google Scholar

[81] Sosnowska, H., 2014. Banzhaf Value for Games Analyzing Voting with Rotation. Operations Research & Decisions, 24(4), pp.75-88. Search in Google Scholar

[82] Sosnowska, H., 2016. Approval Voting as a Method of Prediction in Political Votings. Case of Polish Elections. LNCS Transactions on Computational Collective Intelligence, 23, pp.17-28. 10.1007/978-3-662-52886-0_2Search in Google Scholar

[83] Steunenberg, B., Schmidtchen, D., and Kobolt, C., 1999. Strategic Power in the European Union: Evaluating the Distribution of Power in Policy Games. Journal of Theoretical Politics, 11(3), pp.339-366. 10.1177/0951692899011003005Search in Google Scholar

[84] Thomas L., 1986. Games, Theory & Applications. Chichester: Ellis Horwood. Search in Google Scholar

[85] Weirich, P., 2012. Collective Rationality: Equilibrium in Cooperative Games. Oxford: Oxford Univ. Press. Search in Google Scholar

[86] Young, H., 1994. Cost Allocation. In: Aumann and Hart, eds., Handbook of Game Theory with Economic Applications, Vol.2. North-Holland: Elsevier, pp.1194-1230. Search in Google Scholar

[87] Zaremba, L., 1979. On the Existence of Value in the Varaiya-Lin Sense in Differential Games of Pursuit. Journal of Optimization Theory & Applications, 29, pp.135-145. 10.1007/BF00932640Search in Google Scholar

[88] Zaremba, L., 1980. On the Existence of Value in Pursuit-Evasion Games with Restricted Coordinates, Journal of Optimization Theory & Applications, Vol. 30, pp.451-470. 10.1007/BF00935497Search in Google Scholar

[89] Zaremba, L., 1982. Existence of Value in Differential Games with Fixed Time Duration. Journal of Optimization Theory & Applications, Vol. 38, pp.581-598. 10.1007/BF00934489Search in Google Scholar

[90] Zaremba, L., 1984a. Existence of Value in Games Governed by Generalized Differential Equations. Systems & Control Letters, Vol. 4, pp.237-341 10.1016/S0167-6911(84)80032-8Search in Google Scholar

[91] Zaremba, L., 1984b. Existence of Value in Generalized Pursuit-Evasion Games. SIAM Journal on Control & Optimization Vol. 22, pp.894-901. 10.1137/0322057Search in Google Scholar

[92] Zaremba, L., 1986. Existence Theorems for Games of Survival. Journal of Optimization Theory & Applications, Vol. 48, pp.431-446. 10.1007/BF00940678Search in Google Scholar

[93] Zaremba, L., 1989. Optimality Principles of Dynamic Programming in Differential Games. Journal of Mathematical Analysis and Applications, Vol. 138, pp.43-51. 10.1016/0022-247X(89)90317-XSearch in Google Scholar

[94] Zaremba, L., Zaremba, C., 2010. Rozwiązanie uogólnionego problemu optymalnej alokacji zasobów (Solution to Generalized Optimal Resource Allocation Problem). Zarządzanie Zmianami - Biuletyn Naukowy i Informacyjny Wyższej Szkoły Zarządzanie - the Polish Open University, 6, pp.1-19. Search in Google Scholar