# zbMATH — the first resource for mathematics

Consistency and its converse: an introduction. (English) Zbl 1233.91163
Summary: This essay is a didactic introduction to the literature on the “consistency principle” and its “converse”. An allocation rule is consistent if for each problem in its domain of definition and each alternative that it chooses for it, then for the “reduced problem” obtained by imagining the departure of an arbitrary subgroup of the agents with their “components of the alternative” and reassessing the options open to the remaining agents, it chooses the restriction of the alternative to that subgroup. Converse consistency pertains to the opposite operation. It allows us to deduce that a rule chooses an alternative for a problem from the knowledge that for each two-agent subgroup, it chooses its restriction to the subgroup for the associated reduced problem this subgroup faces. We present two lemmas that have played a critical role in helping understand the implications of these properties in a great variety of models, the elevator lemma and the bracing lemma. We describe several applications. Finally, we illustrate the versatility of consistency and of its converse by means of a sample of characterizations based on them.

##### MSC:
 91B32 Resource and cost allocation (including fair division, apportionment, etc.)
##### Keywords:
consistency; converse consistency; elevator lemma; bracing lemma
Full Text:
##### References:
 [1] Aumann R, Maschler M (1985) Game theoretic analysis of a bankruptcy problem from the Talmud. J Econ Theo 36: 195–213 · Zbl 0578.90100 · doi:10.1016/0022-0531(85)90102-4 [2] Balinski M, Young P (1982) Fair representation. Yale University Press, New Haven [3] Bevia C (1996) Identical preferences lower bound solution and consistency in economies with indivisible goods. Soc Choice Welfare 13: 113–126 · Zbl 0841.90003 · doi:10.1007/BF00179102 [4] Chambers C (2004) Consistency in the probabilistic assignment model. J Math Econ 40: 953–962 · Zbl 1117.90319 · doi:10.1016/j.jmateco.2003.10.004 [5] Chun Y (1999) Equivalence of axioms for bankruptcy problems. Int J Game Theory 28: 511–520 · Zbl 0940.91022 · doi:10.1007/s001820050122 [6] Chun Y (2002) The converse consistency principle in bargaining. Games Econ Behav 40: 25–43 · Zbl 1036.91012 · doi:10.1006/game.2001.0907 [7] Chun Y (2011) Consistency and monotonicity in sequencing problems. Int J Game Theory 40: 29–41 · Zbl 1214.91031 · doi:10.1007/s00182-010-0225-y [8] Dagan N (1996) A note on Thomson’s characterizations of the uniform rule. J Econ Theory 96: 255–261 · Zbl 0893.90007 · doi:10.1006/jeth.1996.0051 [9] Dagan N, Volij O (1997) Bilateral comparisons and consistent fair division rules in the context of bankruptcy problems. Int J Game Theory 26: 11–25 · Zbl 0872.90119 · doi:10.1007/BF01262509 [10] Davis M, Maschler M (1965) The Kernel of a cooperative game. Naval Res Logis Q 12: 223–259 · Zbl 0204.20202 · doi:10.1002/nav.3800120303 [11] Dutta B (1990) The egalitarian solution and reduced game properties in convex games. Int J Game Theory 19: 153–169 · Zbl 0716.90106 · doi:10.1007/BF01761074 [12] Dutta B, Kar A (2004) Cost monotonicity, consistency, and minimum cost spanning tree games. Games Econ Behav 48: 223–248 · Zbl 1117.91308 · doi:10.1016/j.geb.2003.09.008 [13] Ehlers L, Klaus B (2006) Efficient priority rules. Games Econ Behav 55: 372–384 · Zbl 1125.91033 · doi:10.1016/j.geb.2005.02.009 [14] Ergin H (2000) Consistency in house allocation problems. J Math Econ 34: 77–97 · Zbl 1145.91360 · doi:10.1016/S0304-4068(99)00038-5 [15] Fleurbaey M, Maniquet F (1996) Fair allocation with unequal production skills: the solidarity approach to compensation. Math Soc Sci 32: 71–93 · Zbl 0917.90026 · doi:10.1016/0165-4896(96)00811-6 [16] Gale D, Shapley LS (1962) College admissions and the stability of marriage. Am Math Mon 69: 9–15 · Zbl 0109.24403 · doi:10.2307/2312726 [17] Gillies DB (1959) Solutions to general non-zero sum games. In: Tucker AW, Luce RD (eds) Contributions to the theory of games IV, annals of mathematical studies 40. Princeton University Press, Princeton, pp 595–614 [18] Hart S, Mas-Colell A (1988) The potential of the Shapley-value, Chapter 9 in the shapley-value: essays in honor of Lloyd S. Shapley Roth AE (ed), Cambridge University Press, pp 127–137 [19] Hokari T (2005) Consistency implies equal treatment in TU games. Games Econ Behav 51: 63–82 · Zbl 1115.91005 · doi:10.1016/j.geb.2004.04.006 [20] Hokari T, Kıbrıs O (2003) Consistency, converse consistency, and aspirations in coalitional games. Math Soc Sci 45: 249–297 · Zbl 1042.91014 · doi:10.1016/S0165-4896(02)00070-7 [21] Hokari T, Thomson W (2008) On properties of division rules lifted by bilateral consistency. J Math Econ 44: 1057–1071 · Zbl 1152.91022 · doi:10.1016/j.jmateco.2008.01.001 [22] Hwang Y-A, Yeh C-H, Ju B-G (2006) Reduction-consistency and the Condorcet principle in collective choice problems, mimeo [23] Ju B-G (2008) Efficiency and consistency for locating multiple public facilities. J Econ Theory 138: 165–183 · Zbl 1140.91350 · doi:10.1016/j.jet.2007.03.006 [24] Kalai E (1977) Proportional solution to bargaining problems: interpersonal utility comparisons. Econometrica 45: 1023–1030 · Zbl 0382.62057 · doi:10.2307/1912690 [25] Kalai E, Smorodinsky M (1975) Other solutions to Nash’s bargaining problem. Econometrica 43: 513–518 · Zbl 0308.90053 · doi:10.2307/1914280 [26] Klaus B, Nichifor A (2010) Consistency in one-sided assignment problems. Soc Choice Welfare 35: 415–433 · Zbl 1232.91532 · doi:10.1007/s00355-010-0447-8 [27] Lensberg T (1987) Stability and collective rationality. Econometrica 55: 935–961 · Zbl 0622.90007 · doi:10.2307/1911037 [28] Lensberg T (1988) Stability and the Nash solution. J Econ Theory 45: 330–341 · Zbl 0657.90106 · doi:10.1016/0022-0531(88)90273-6 [29] Maniquet F (1996) Horizontal equity and stability when the number of agents is variable in the fair division problem. Econ Lett 50: 85–90 · Zbl 0900.90261 · doi:10.1016/0165-1765(95)00718-0 [30] Maschler M, Owen G (1989) The consistent shapley-value for hyperplane games. Int J Game Theory 18: 390–407 · Zbl 0682.90105 [31] Moulin H (1987) Equal or proportional division of a surplus, and other methods. Int J Game Theory 16: 161–186 · Zbl 0631.90093 · doi:10.1007/BF01756289 [32] Moulin H (1988) Axioms of cooperative decision making. Cambridge University Press, Cambridge · Zbl 0699.90001 [33] Nash J (1950) The bargaining problem. Econometrica 18: 155–162 · Zbl 1202.91122 · doi:10.2307/1907266 [34] O’Neill B (1982) A problem of rights arbitration from the Talmud. Math Soc Sci 2: 345–371 · Zbl 0489.90090 · doi:10.1016/0165-4896(82)90029-4 [35] Özkal-Sanver I (2009) Minimal converse consistent extension of the men-optimal solution, mimeo [36] Peleg B (1985) An axiomatization of the core of cooperative games without side-payments. J Math Econ 14: 203–214 · Zbl 0581.90102 · doi:10.1016/0304-4068(85)90020-5 [37] Peleg B (1986) On the reduced game property and its converse. Int J Game Theory 15 , 187–200. A correction. International Journal of Game Theory 16 (1987) · Zbl 0629.90099 [38] Peleg B, Tijs S (1996) The consistency principle for games in strategic form. Int J Game Theory 25: 13–34 · Zbl 0856.90147 · doi:10.1007/BF01254381 [39] Potters JAM, Sudhölter P (1999) Airport problems and consistent solution rules. Math Soc Sci 38: 83–102 · Zbl 1111.91310 · doi:10.1016/S0165-4896(99)00004-9 [40] Roemer J (1988) Axiomatic bargaining on economic environments. J Econ Theory 45: 1–31 · Zbl 0637.90107 · doi:10.1016/0022-0531(88)90251-7 [41] Sasaki H (1995) Consistency and monotonicity in assignment problems. Int J Game Theory 24: 373–397 · Zbl 0843.90139 · doi:10.1007/BF01243039 [42] Sasaki H, Toda M (1992) Consistency and characterization of the core of two-sided matching problems. J Econ Theory 56: 218–227 · Zbl 0763.90040 · doi:10.1016/0022-0531(92)90078-V [43] Schmeidler D (1969) The nucleolus of a characteristic function game. SIAM J Appl Math 17: 1163–1170 · Zbl 0191.49502 · doi:10.1137/0117107 [44] Serrano R (1995) Strategic bargaining, surplus sharing and the nucleolus. J Math Econ 24: 319–329 · Zbl 0834.90143 · doi:10.1016/0304-4068(94)00696-8 [45] Shapley L (1953) A value for n-person games. In: Kuhn HW, Tucker AW (eds) Contributions to the theory of games II (annals of matheLLLmatics studies 28). Princeton University Press, Princeton, pp 307–317 [46] Shapley L, Shubik M (1972) The assignment game I: the core. Int J Game Theory 1: 111–130 · Zbl 0236.90078 · doi:10.1007/BF01753437 [47] Sobolev AI (1975) The characterization of optimality principles in cooperative games by functional equations. Math Methods Soc Sci 6: 150–165 (in Russian) [48] Sprumont Y (1991) The division problem with single-peaked preferences. Econometrica 59: 509–519 · Zbl 0721.90012 · doi:10.2307/2938268 [49] Svensson L-G (1983) Large indivisibilities: an analysis with respect to price equilibrium and fairness. Econometrica 51: 939–954 · Zbl 0526.90017 · doi:10.2307/1912044 [50] Tadenuma K (1992) Reduced games, consistency, and the core. Int J Game Theory 20: 325–334 · Zbl 0751.90104 · doi:10.1007/BF01271129 [51] Tadenuma K, Thomson W (1991) No-envy and consistency in economies with indivisible goods. Econometrica 59: 1755–1767 · Zbl 0744.90007 · doi:10.2307/2938288 [52] Tadenuma K, Thomson W (1993) The fair allocation of an indivisible good when monetary compensations are possible. Math Soc Sci 25: 117–132 · Zbl 0780.90027 · doi:10.1016/0165-4896(93)90047-M [53] Thomson W (1983) The fair division of a fixed supply among a growing population. Math Oper Res 8: 319–326 · Zbl 0524.90102 · doi:10.1287/moor.8.3.319 [54] Thomson W (1988) A study of choice correspondences in economies with a variable number of agents. J Econ Theory 46: 247–259 · Zbl 0657.90019 [55] Thomson W (1994a) Consistent solutions to the problem of fair division when preferences are single-peaked. J Econ Theory 63: 219–245 · Zbl 0864.90008 · doi:10.1006/jeth.1994.1041 [56] Thomson W (1994b) Consistent extensions. Math Soc Sci 28: 35–49 · Zbl 0874.90071 · doi:10.1016/0165-4896(93)00744-F [57] Thomson W (2003) Axiomatic and game-theoretic analysis of bankruptcy and taxation problems: a survey. Math Soc Sci 45: 249–297 · Zbl 1042.91014 · doi:10.1016/S0165-4896(02)00070-7 [58] Thomson W (2010a) Fair allocation rules. In: Arrow K, Sen A, Suzumura K (eds) Handbook of social choice and welfare. North-Holland, Amsterdam, pp 393–506 [59] Thomson W (2010b) On the computational implications of converse consistency, University of Rochester mimeo [60] Thomson W (2011a) Lorenz rankings of rules for the adjudication of conflicting claims. Economic Theory (forthcoming) [61] Thomson W (2011b) Consistent allocation rules, monograph [62] Toda M (2003) Consistency and its converse in assignment games. Int J Math Game Theory Algebra 13: 1–14 · Zbl 1078.91002 [63] Toda M (2006) Monotonicity and consistency in matching markets. Int J Game Theory 34: 13–31 · Zbl 1151.91664 · doi:10.1007/s00182-005-0002-5 [64] Yeh C-H (2006) Reduction-consistency in collective choice problems. J Math Econ 42: 637–652 · Zbl 1201.91058 · doi:10.1016/j.jmateco.2005.12.002 [65] Young P (1987) On dividing an amount according to individual claims or liabilities. Math Oper Res 12: 398–414 · Zbl 0629.90003 · doi:10.1287/moor.12.3.398 [66] Young P (1988) Distributive justice in taxation. J Econ Theory 44: 321–335 · Zbl 0637.90027 · doi:10.1016/0022-0531(88)90007-5
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.