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Consistency and its converse: an introduction

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Abstract

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.

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References

  • Aumann R, Maschler M (1985) Game theoretic analysis of a bankruptcy problem from the Talmud. J Econ Theo 36: 195–213

    Article  Google Scholar 

  • Balinski M, Young P (1982) Fair representation. Yale University Press, New Haven

    Google Scholar 

  • Bevia C (1996) Identical preferences lower bound solution and consistency in economies with indivisible goods. Soc Choice Welfare 13: 113–126

    Article  Google Scholar 

  • Chambers C (2004) Consistency in the probabilistic assignment model. J Math Econ 40: 953–962

    Article  Google Scholar 

  • Chun Y (1999) Equivalence of axioms for bankruptcy problems. Int J Game Theory 28: 511–520

    Article  Google Scholar 

  • Chun Y (2002) The converse consistency principle in bargaining. Games Econ Behav 40: 25–43

    Article  Google Scholar 

  • Chun Y (2011) Consistency and monotonicity in sequencing problems. Int J Game Theory 40: 29–41

    Article  Google Scholar 

  • Dagan N (1996) A note on Thomson’s characterizations of the uniform rule. J Econ Theory 96: 255–261

    Article  Google Scholar 

  • 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

    Article  Google Scholar 

  • Davis M, Maschler M (1965) The Kernel of a cooperative game. Naval Res Logis Q 12: 223–259

    Article  Google Scholar 

  • Dutta B (1990) The egalitarian solution and reduced game properties in convex games. Int J Game Theory 19: 153–169

    Article  Google Scholar 

  • Dutta B, Kar A (2004) Cost monotonicity, consistency, and minimum cost spanning tree games. Games Econ Behav 48: 223–248

    Article  Google Scholar 

  • Ehlers L, Klaus B (2006) Efficient priority rules. Games Econ Behav 55: 372–384

    Article  Google Scholar 

  • Ergin H (2000) Consistency in house allocation problems. J Math Econ 34: 77–97

    Article  Google Scholar 

  • Fleurbaey M, Maniquet F (1996) Fair allocation with unequal production skills: the solidarity approach to compensation. Math Soc Sci 32: 71–93

    Article  Google Scholar 

  • Gale D, Shapley LS (1962) College admissions and the stability of marriage. Am Math Mon 69: 9–15

    Article  Google Scholar 

  • 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

    Google Scholar 

  • 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

  • Hokari T (2005) Consistency implies equal treatment in TU games. Games Econ Behav 51: 63–82

    Article  Google Scholar 

  • Hokari T, Kıbrıs O (2003) Consistency, converse consistency, and aspirations in coalitional games. Math Soc Sci 45: 249–297

    Article  Google Scholar 

  • Hokari T, Thomson W (2008) On properties of division rules lifted by bilateral consistency. J Math Econ 44: 1057–1071

    Article  Google Scholar 

  • Hwang Y-A, Yeh C-H, Ju B-G (2006) Reduction-consistency and the Condorcet principle in collective choice problems, mimeo

  • Ju B-G (2008) Efficiency and consistency for locating multiple public facilities. J Econ Theory 138: 165–183

    Article  Google Scholar 

  • Kalai E (1977) Proportional solution to bargaining problems: interpersonal utility comparisons. Econometrica 45: 1023–1030

    Article  Google Scholar 

  • Kalai E, Smorodinsky M (1975) Other solutions to Nash’s bargaining problem. Econometrica 43: 513–518

    Article  Google Scholar 

  • Klaus B, Nichifor A (2010) Consistency in one-sided assignment problems. Soc Choice Welfare 35: 415–433

    Article  Google Scholar 

  • Lensberg T (1987) Stability and collective rationality. Econometrica 55: 935–961

    Article  Google Scholar 

  • Lensberg T (1988) Stability and the Nash solution. J Econ Theory 45: 330–341

    Article  Google Scholar 

  • Maniquet F (1996) Horizontal equity and stability when the number of agents is variable in the fair division problem. Econ Lett 50: 85–90

    Article  Google Scholar 

  • Maschler M, Owen G (1989) The consistent shapley-value for hyperplane games. Int J Game Theory 18: 390–407

    Article  Google Scholar 

  • Moulin H (1987) Equal or proportional division of a surplus, and other methods. Int J Game Theory 16: 161–186

    Article  Google Scholar 

  • Moulin H (1988) Axioms of cooperative decision making. Cambridge University Press, Cambridge

    Google Scholar 

  • Nash J (1950) The bargaining problem. Econometrica 18: 155–162

    Article  Google Scholar 

  • O’Neill B (1982) A problem of rights arbitration from the Talmud. Math Soc Sci 2: 345–371

    Article  Google Scholar 

  • Özkal-Sanver I (2009) Minimal converse consistent extension of the men-optimal solution, mimeo

  • Peleg B (1985) An axiomatization of the core of cooperative games without side-payments. J Math Econ 14: 203–214

    Article  Google Scholar 

  • 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)

    Google Scholar 

  • Peleg B, Tijs S (1996) The consistency principle for games in strategic form. Int J Game Theory 25: 13–34

    Article  Google Scholar 

  • Potters JAM, Sudhölter P (1999) Airport problems and consistent solution rules. Math Soc Sci 38: 83–102

    Article  Google Scholar 

  • Roemer J (1988) Axiomatic bargaining on economic environments. J Econ Theory 45: 1–31

    Article  Google Scholar 

  • Sasaki H (1995) Consistency and monotonicity in assignment problems. Int J Game Theory 24: 373–397

    Article  Google Scholar 

  • Sasaki H, Toda M (1992) Consistency and characterization of the core of two-sided matching problems. J Econ Theory 56: 218–227

    Article  Google Scholar 

  • Schmeidler D (1969) The nucleolus of a characteristic function game. SIAM J Appl Math 17: 1163–1170

    Article  Google Scholar 

  • Serrano R (1995) Strategic bargaining, surplus sharing and the nucleolus. J Math Econ 24: 319–329

    Article  Google Scholar 

  • 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

    Google Scholar 

  • Shapley L, Shubik M (1972) The assignment game I: the core. Int J Game Theory 1: 111–130

    Article  Google Scholar 

  • Sobolev AI (1975) The characterization of optimality principles in cooperative games by functional equations. Math Methods Soc Sci 6: 150–165 (in Russian)

    Google Scholar 

  • Sprumont Y (1991) The division problem with single-peaked preferences. Econometrica 59: 509–519

    Article  Google Scholar 

  • Svensson L-G (1983) Large indivisibilities: an analysis with respect to price equilibrium and fairness. Econometrica 51: 939–954

    Article  Google Scholar 

  • Tadenuma K (1992) Reduced games, consistency, and the core. Int J Game Theory 20: 325–334

    Article  Google Scholar 

  • Tadenuma K, Thomson W (1991) No-envy and consistency in economies with indivisible goods. Econometrica 59: 1755–1767

    Article  Google Scholar 

  • Tadenuma K, Thomson W (1993) The fair allocation of an indivisible good when monetary compensations are possible. Math Soc Sci 25: 117–132

    Article  Google Scholar 

  • Thomson W (1983) The fair division of a fixed supply among a growing population. Math Oper Res 8: 319–326

    Article  Google Scholar 

  • Thomson W (1988) A study of choice correspondences in economies with a variable number of agents. J Econ Theory 46: 247–259

    Article  Google Scholar 

  • Thomson W (1994a) Consistent solutions to the problem of fair division when preferences are single-peaked. J Econ Theory 63: 219–245

    Article  Google Scholar 

  • Thomson W (1994b) Consistent extensions. Math Soc Sci 28: 35–49

    Article  Google Scholar 

  • Thomson W (2003) Axiomatic and game-theoretic analysis of bankruptcy and taxation problems: a survey. Math Soc Sci 45: 249–297

    Article  Google Scholar 

  • 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

    Google Scholar 

  • Thomson W (2010b) On the computational implications of converse consistency, University of Rochester mimeo

  • Thomson W (2011a) Lorenz rankings of rules for the adjudication of conflicting claims. Economic Theory (forthcoming)

  • Thomson W (2011b) Consistent allocation rules, monograph

  • Toda M (2003) Consistency and its converse in assignment games. Int J Math Game Theory Algebra 13: 1–14

    Google Scholar 

  • Toda M (2006) Monotonicity and consistency in matching markets. Int J Game Theory 34: 13–31

    Article  Google Scholar 

  • Yeh C-H (2006) Reduction-consistency in collective choice problems. J Math Econ 42: 637–652

    Article  Google Scholar 

  • Young P (1987) On dividing an amount according to individual claims or liabilities. Math Oper Res 12: 398–414

    Article  Google Scholar 

  • Young P (1988) Distributive justice in taxation. J Econ Theory 44: 321–335

    Article  Google Scholar 

Download references

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Authors and Affiliations

  1. University of Rochester, Rochester, NY, USA

    William Thomson

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  1. William Thomson
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Correspondence to William Thomson.

Additional information

I thank Julio Gonzalez, Bettina Klaus, Eiichi Miyagawa and Karol Szwagrzak for helpful comments and the NSF under grant 0721107 for its support.

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Thomson, W. Consistency and its converse: an introduction. Rev Econ Design 15, 257–291 (2011). https://doi.org/10.1007/s10058-011-0109-z

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