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On the core of dynamic cooperative games. (English) Zbl 1281.91015

The paper concerns dynamic cooperative games where a sequence of allocations uniquely induces a sequence of stage games. After considering in Section 2 a motivating example based on the classical majority game, in Section 3 the authors introduce a Markovian dynamic game and the notion of an allocation plan. In Section 4 the intertemporal and \(\varepsilon\)-intertemporal core are defined. “A sequence of allocations is in the intertemporal core if no coalition can deviate and get on its own a greater share than the one proposed by the sequence.” When a coalition deviates, then the game is no longer played with the original set of players but the deviating coalition becomes the new grand coalition which induces a new dynamic game. In Section 5 the main theorem that characterizes the nonemptyness of the \(\varepsilon\)-intertemporal core is proved. While the authors assume a common discount factor, in the final remarks presented in Section 6 they mention different discount rates as one of the issues for further studies. Also investigating a different core definition, the fair core, and considering a stochastic dynamic game where the stage games are endogenously determined are listed as possible further research.

MSC:

91A12 Cooperative games
91A15 Stochastic games, stochastic differential games
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[1] Allouch N, Wooders M (2008) Price taking equilibrium in economies with multiple memberships in clubs and unbounded club sizes. J Econ Theory 140:246-278. http://www.sciencedirect.com/science/article/pii/S0022053107001123 · Zbl 1136.91390 · doi:10.1016/j.jet.2007.07.006
[2] Becker RA, Chakrabarti SK (1995) The recursive core. Econometrica 63:401-423 · Zbl 0835.90014 · doi:10.2307/2951631
[3] Chander P, Wooders M (2010) Subgame perfect cooperation in an extensive game. Working Papers 1008, Department of Economics, Vanderbilt University. http://ideas.repec.org/p/van/wpaper/1008.html · Zbl 1437.91063
[4] Chwe MS-Y (1994) Farsighted coalitional stability. J Econ Theory 63:299-325. doi:10.1006/jeth.1994.1044 · Zbl 0841.90131 · doi:10.1006/jeth.1994.1044
[5] Diamantoudi E, Xue L (2007) Coalitions, agreements and efficiency. J Econ Theory 136:105-125 · Zbl 1256.91009 · doi:10.1016/j.jet.2006.02.012
[6] Gale D (1978) The core of a monetary economy without trust. J Econ Theory 19:456-491 · Zbl 0413.90012 · doi:10.1016/0022-0531(78)90104-7
[7] Habis H, Herings PJ-J (2010) A note on the weak sequential core of dynamic TU games. Research Memoranda 022, Maastricht: METEOR, Maastricht Research School of Economics of Technology and Organization. http://econpapers.repec.org/RePEc:dgr:umamet:2010022 · Zbl 1215.91008
[8] Hellman Z (2008) Bargaining set solution concepts in dynamic cooperative games. Technical report, Munich Personal RePEc Archive. http://mpra.ub.uni-muenchen.de/8798/ · Zbl 0719.90099
[9] Konishi H, Ray D (2003) Coalition formation as a dynamic process. J Econ Theory 110:1-41 · Zbl 1052.91017 · doi:10.1016/S0022-0531(03)00004-8
[10] Koutsougeras LC (1998) A two-stage core with applications to asset market and differential information economies. Econom Theory 11:563-584 · Zbl 0903.90018 · doi:10.1007/s001990050202
[11] Kovalenkov A, Wooders M (2003) Approximate cores of games and economies with clubs. J Econ Theory 110:87-120 · Zbl 1056.91005 · doi:10.1016/S0022-0531(03)00003-6
[12] Kovalenkov A, Wooders M (2005) Laws of scarcity for a finite game—exact bounds on estimations. Econom Theory 26:383-396. doi:10.1007/s00199-003-0443-7 · Zbl 1132.91336 · doi:10.1007/s00199-003-0443-7
[13] Kovalenkov A, Wooders MH (2001) Epsilon cores of games with limited side payments: nonemptiness and equal treatment. Games Econ Behav 36:193-218. doi:10.1006/game.2000.0815 · Zbl 1011.91007 · doi:10.1006/game.2000.0815
[14] Kovalenkov A, Wooders MH (2001) An exact bound on epsilon for nonemptiness of epsilon cores of games. Math Oper Res 26:654-678. doi:10.1287/moor.26.4.654.10001 · Zbl 1082.91505 · doi:10.1287/moor.26.4.654.10001
[15] Kranich L, Perea A, Peters H (2005) Core concepts for dynamic TU games. Int Game Theory Rev 7:43-61 · Zbl 1134.91319 · doi:10.1142/S0219198905000417
[16] Lehrer E, Pauzner A (1999) Repeated games with differential time preferences. Econometrica 67:393-412. doi:10.1111/1468-0262.00024 · Zbl 1020.91009 · doi:10.1111/1468-0262.00024
[17] Lehrer E, Scarsini M (2012) On the core of dynamic cooperative games. arXiv:1203.2832 · Zbl 1281.91015
[18] Oviedo J (2000) The core of a repeated n-person cooperative game. Eur J Oper Res 127:519-524 · Zbl 0982.91011 · doi:10.1016/S0377-2217(99)00335-5
[19] Petrosjan L, Zaccour G (2003) Time-consistent Shapley value allocation of pollution cost reduction. J Econ Dyn Control 27:381-398 · Zbl 1027.91005 · doi:10.1016/S0165-1889(01)00053-7
[20] Petrosjan LA (1977) Stability of the solutions in differential games with several players. Vestn Leningr Univ 19:46-52 · Zbl 0397.90116
[21] Petrosjan LA (1993) Differential games of pursuit. World Scientific, River Edge · doi:10.1142/1670
[22] Predtetchinski A (2007) The strong sequential core for stationary cooperative games. Games Econ Behav 61:50-66 · Zbl 1271.91021 · doi:10.1016/j.geb.2006.10.013
[23] Predtetchinski A, Herings PJ-J, Peters H (2002) The strong sequential core for two-period economies. J Math Econ 38:465-482 · Zbl 1024.91018
[24] Predtetchinski A, Herings PJ-J, Peters H (2004) The strong sequential core in a dynamic exchange economy. Econom Theory 24:147-162 · Zbl 1093.91043 · doi:10.1007/s00199-003-0404-1
[25] Predtetchinski A, Herings PJJ, Perea A (2006) The weak sequential core for two-period economies. Int J Game Theory 34:55-65 · Zbl 1154.91325 · doi:10.1007/s00182-005-0007-0
[26] Ray D (1989) Credible coalitions and the core. Int J Game Theory 18:185-187. doi:10.1007/BF01268157 · Zbl 0719.90099 · doi:10.1007/BF01268157
[27] Ray D (2007) A game-theoretic perspective on coalition formation. Lipsey lectures. Oxford University Press, Oxford · Zbl 1188.91005 · doi:10.1093/acprof:oso/9780199207954.001.0001
[28] Ray D, Vohra R (1999) A theory of endogenous coalition structures. Games Econ Behav 26:286-336. doi:10.1006/game.1998.0648 · Zbl 0918.90146 · doi:10.1006/game.1998.0648
[29] Shapley LS, Shubik M (1966) Quasi-cores in a monetary economy with nonconvex preferences. Econometrica 34:805-827. http://www.jstor.org/stable/1910101 · Zbl 0154.45303 · doi:10.2307/1910101
[30] Wooders MH (1983) The epsilon core of a large replica game. J Math Econ 11:277-300. doi:10.1016/0304-4068(83)90005-8 · Zbl 0518.90101 · doi:10.1016/0304-4068(83)90005-8
[31] Wooders MH, Zame WR (1984) Approximate cores of large games. Econometrica 52:1327-1350. doi:10.2307/1913508 · Zbl 0616.90105 · doi:10.2307/1913508
[32] Xue L (1998) Coalitional stability under perfect foresight. Econom Theory 11:603-627. doi:10.1007/s001990050204 · Zbl 0903.90005 · doi:10.1007/s001990050204
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