Axioms of invariance for TU-games

Béal, Sylvain and Rémila, Eric and Solal, Philippe (2012): Axioms of invariance for TU-games.

Abstract

We introduce new axioms for the class of all TU-games with a fixed but arbitrary player set, which require either invariance of an allocation rule or invariance of the payoff assigned by an allocation rule to a specified subset of players in two related TU-games. Comparisons with other axioms are provided. These new axioms are used to characterize the Shapley value, the equal division rule, the equal surplus division rule and the Banzhaf value. The classical axioms of efficiency, anonymity, symmetry and additivity are not used.

• Item Type: MPRA Paper
• Language: English
• Keywords: uniform addition invariance ; uniform transfer invariance ; Shapley value ; equal division rule ; equal surplus division rule ; Banzhaf value
• Subjects: C - Mathematical and Quantitative Methods > C7 - Game Theory and Bargaining Theory > C71 - Cooperative Games

References

K. J. Arrow, "A Difficulty in the Concept of Social Welfare", Journal of Political Economy 58 (1950), p. 326–346.

J. F. Banzhaf, "Weighted Voting doesn't Work: a Mathematical Analysis", Rutgers Law Review 19 (1965), p. 317–343.

A. Casajus, "Differential Marginality, van den Brink Fairness, and the Shapley Value", Theory and Decision 71 (2011), p. 163–174.

Y. Chun, "A new Axiomatization of the Shapley Value", Games and Economic Behavior 1 (1989), p. 119–130.

T. S. H. Driessen, "Associated Consistency and Values for TU Games", International Journal of Game Theory 39 (2010), p. 467–482.

C. A. Futia, "Invariant Distributions and the Limiting Behavior of Markovian Economic Models", Econometrica 50 (1982), p. 377–408.

G. Hamiache, "Associated Consistency and Shapley Value", International Journal of Game Theory 30 (2001), p. 279–289.

Y. Ju, P. Borm and P. Ruys, "The Consensus Value: A New Solution Concept for Cooperative Games", Social Choice and Welfare 28 (2007), p. 685–703.

Y. Kamijo and T. Kongo, "Whose Deletion does not affect your Payoff? The Difference between the Shapley Value, the Egalitarian Value, the Solidarity Value, and the Banzhaf Value", European Journal of Operational Research 216 2012), p. 638–646.

K. Kamiya and D. Talman, "Matching Models with a Conservation Law: The Existence and Global Structure of the Set of Stationary Equilibria", Journal of Mathematical Economics 45 (2009), p. 397–413.

G. C. Kemp, "Invariance and theWald Test", Journal of Econometrics 104 (2001), p. 209–217.

N. L. Kleinberg and J. H. Weiss, "Equivalent N-Person Games and the Null Space of the Shapley Value", Mathematics of Operations Research 10 (1985), p. 233–243.

E. Kohlberg and J. Mertens, "On Strategic Stability of Equilibria", Econometrica 54 (1986), p. 1003–1037.

J. F. Nash, "Two-Person Cooperative Games", Econometrica 21 (1953), p. 128–140.

M. Pintér, "Young's Axiomatization of the Shapley Value – A New Proof", arXiv paper number 0805.2797v3, 2012.

M. K. Richter, "Invariance Axioms and Economic Indexes", Econometrica 34 (1966), p. 739– 755.

P. A. Samuelson, "Law of Conservation of the Capital-Output Ratio", Proceedings of the National Academy of Sciences 67 (1970), p. 1477–1479.

P. A. Samuleson and S. Swamy, "Invariant Economic Index Numbers and Canonical Duality: Survey and Synthesis", American Economic Review 64 (1974), p. 566–593.

R. Sato, Theory of Technical Change and Economic Invariance: Application of Lie Groups, Edward Elgar, 1981.

L. S. Shapley, "A Value for n-person Games", in Contribution to the Theory of Games vol. II (H.W. Kuhn and A.W. Tucker eds). Annals of Mathematics Studies 28, Princeton University Press, Princeton, 1953.

R. van den Brink, "An Axiomatization of the Shapley Value using a Fairness Property", International Journal of Game Theory 30 (2001), p. 309–319.

R. van den Brink, "Null Players or Nullifying Players: the Difference between the Shapley Value and Equal Division Solutions", Journal of Economic Theory 136 (2007), p. 767–775.

R. D. Willig, "Risk Invariance and Ordinally Additive Utility Functions", Econometrica 45 (1977), p. 621–640.

H. Young, "Monotonic Solutions of Cooperative Games", International Journal of Game Theory 14 (1985), p. 65–72.