Nontransferable utility bankruptcy games. (English) Zbl 1435.91014
Summary: In this paper, we analyze bankruptcy problems with nontransferable utility (NTU) from a game theoretical perspective by redefining corresponding NTU-bankruptcy games in a tailor-made way. It is shown that NTU-bankruptcy games are both coalition-merge convex and ordinally convex. Generalizing the notions of core cover and compromise stability for transferable utility (TU) games to NTU-games, we also show that each NTU-bankruptcy game is compromise stable. Thus, NTU-bankruptcy games are shown to retain the two characterizing properties of TU-bankruptcy games: convexity and compromise stability. As a first example of a game theoretical NTU-bankruptcy rule, we analyze the adjusted proportional rule and show that this rule corresponds to the compromise value of NTU-bankruptcy games.
MSC:
| 91A12 | Cooperative games |
| 91B32 | Resource and cost allocation (including fair division, apportionment, etc.) |
| 91A80 | Applications of game theory |
Keywords:
NTU-bankruptcy problem; NTU-bankruptcy game; coalition-merge convexity; ordinal convexity; compromise stability; adjusted proportional ruleReferences:
| [1] | Aumann, RJ, An axiomatization of the non-transferable utility value, Econometrica, 53, 599-612 (1985) · Zbl 0583.90110 · doi:10.2307/1911657 |
| [2] | Aumann, R.; Maschler, M., Game theoretic analysis of a bankruptcy problem from the talmud, J Econ Theory, 36, 195-213 (1985) · Zbl 0578.90100 · doi:10.1016/0022-0531(85)90102-4 |
| [3] | Borm, P.; Keiding, H.; McLean, RP; Oortwijn, S.; Tijs, S., The compromise value for NTU-games, Int J Game Theory, 21, 175-189 (1992) · Zbl 0766.90095 · doi:10.1007/BF01245460 |
| [4] | Chun, Y.; Thomson, W., Bargaining problems with claims, Math Soc Sci, 24, 19-33 (1992) · Zbl 0769.90082 · doi:10.1016/0165-4896(92)90003-N |
| [5] | Csóka, P.; Herings, P.; Kóczy, L.; Pintér, M., Convex and exact games with non-transferable utility, Eur J Oper Res, 209, 57-62 (2011) · Zbl 1205.91021 · doi:10.1016/j.ejor.2010.08.004 |
| [6] | Curiel, IJ; Maschler, M.; Tijs, S., Bankruptcy games, Z Oper Res, 31, A 143-A 159 (1987) · Zbl 0636.90100 |
| [7] | Dietzenbacher, BJ, Bankruptcy games with nontransferable utility, Math Soc Sci, 92, 16-21 (2018) · Zbl 1397.91032 · doi:10.1016/j.mathsocsci.2017.12.003 |
| [8] | Estévez-Fernández, A.; Fiestras-Janeiro, MG; Mosquera, MA; Sánchez, E., A bankruptcy approach to the core cover, Math Methods Oper Res, 76, 343-359 (2012) · Zbl 1262.91022 · doi:10.1007/s00186-012-0409-2 |
| [9] | Fragnelli, V.; Gastaldi, F., Remarks on the integer Talmud solution for integer bankruptcy problems, TOP, 25, 127-163 (2017) · Zbl 1414.91250 · doi:10.1007/s11750-016-0426-z |
| [10] | Fragnelli, V.; Gagliardo, S.; Gastaldi, F., Integer solutions to bankruptcy problems with non-integer claims, TOP, 22, 892-933 (2014) · Zbl 1336.91049 · doi:10.1007/s11750-013-0304-x |
| [11] | Fragnelli, V.; Gagliardo, S.; Gastaldi, F., Bankruptcy problems with non-integer claims: definition and characterizations of the ICEA solution, TOP, 24, 88-130 (2016) · Zbl 1341.91095 · doi:10.1007/s11750-015-0376-x |
| [12] | Greenberg, J., Cores of convex games without side payments, Math Oper Res, 10, 523-525 (1985) · Zbl 0581.90099 · doi:10.1287/moor.10.3.523 |
| [13] | Hendrickx, R.; Borm, P.; Timmer, J., A note on NTU convexity, Int J Game Theory, 31, 29-37 (2002) · Zbl 1083.91018 · doi:10.1007/s001820200105 |
| [14] | Herrero, C.; Martínez, R., Ballanced allocation methods for claims problems with indivisibilities, Soc Choice Welf, 30, 603-617 (2008) · Zbl 1135.91370 · doi:10.1007/s00355-007-0262-z |
| [15] | O’Neill, B., A problem of rights arbitration from the talmud, Math Soc Sci, 2, 345-371 (1982) · Zbl 0489.90090 · doi:10.1016/0165-4896(82)90029-4 |
| [16] | Orshan, G.; Valenciano, F.; Zarzuelo, JM, The bilateral consistent prekernel, the core, and NTU bankruptcy problems, Math Oper Res, 28, 268-282 (2003) · Zbl 1082.91020 · doi:10.1287/moor.28.2.268.14476 |
| [17] | Peleg, B.; Sudhölter, P., Introduction to the theory of cooperative games (2007), Berlin: Springer, Berlin · Zbl 1193.91002 |
| [18] | Predtetchinski, A.; Herings, P., A necessary and sufficient condition for the non-emptiness of the core of a non-transferable utility game, J Econ Theory, 116, 84-92 (2004) · Zbl 1070.91009 · doi:10.1016/S0022-0531(03)00261-8 |
| [19] | Quant, M.; Borm, P.; Reijnierse, H.; van Velzen, B., The core cover in relation to the nucleolus and the Weber set, Int J Game Theory, 33, 491-503 (2005) · Zbl 1086.91008 · doi:10.1007/s00182-005-0210-z |
| [20] | Shapley, LS, Cores of convex games, Int J Game Theory, 1, 11-26 (1971) · Zbl 0222.90054 · doi:10.1007/BF01753431 |
| [21] | Sharkey, WW, Convex games without side payments, Int J Game Theory, 10, 101-106 (1981) · Zbl 0461.90088 · doi:10.1007/BF01769263 |
| [22] | Thomson, W., Axiomatic and game-theoretic analysis of bankruptcy and taxation problems: a survey, Math Soc Sci, 45, 249-297 (2003) · Zbl 1042.91014 · doi:10.1016/S0165-4896(02)00070-7 |
| [23] | Thomson, W., Axiomatic and game-theoretic analysis of bankruptcy and taxation problems: an update, Math Soc Sci, 74, 867-880 (2015) · Zbl 1310.91084 · doi:10.1016/j.mathsocsci.2014.09.002 |
| [24] | Tijs, SH; Moeschlin, O.; Pallaschke, D., Bounds for the core and the \(\tau \)-value, Game Theory and Mathematical Economics, 123-132 (1981), Amsterdam: North-Holland, Amsterdam · Zbl 0467.90087 |
| [25] | Tijs, SH; Lipperts, F., The hypercube and the core cover of \(n\)-person cooperative games, Cahiers du Centre d’Études de Researche Opérationelle, 24, 27-37 (1982) · Zbl 0479.90093 |
| [26] | Vidal-Puga, J., Forming coalitions and the Shapley NTU value, Eur J Oper Res, 190, 659-671 (2008) · Zbl 1161.91330 · doi:10.1016/j.ejor.2007.07.006 |
| [27] | Vilkov, V., Convex games without side payments, Vestnik Leningradskiva Universitata, 7, 21-24 (1977) · Zbl 0363.90116 |
| [28] | Weber, RJ; Roth, AE, Probabilistic values for games, The Shapley value: essays in honor of Lloyd S Shapley, 101-119 (1988), Cambridge: Cambridge University Press, Cambridge · Zbl 0707.90100 |
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