Abstract
The reactive bargaining set (Granot [1994]) is the set of outcomes for which no justified objection exists. Here, in a justified objection the objector first watches how the target tries to act (if he has such an option), and then reacts by making a profit and ruining the target's attempt to maintain his share.
In this paper we explore properties of the reactive bargaining set, set up the system of inequalities that defines it, and construct a dynamic system in the sense of Stearns' transfer scheme that leads the players to this set. We also extend the definition of the reactive bargaining set toNTU games in a way that keeps it nonempty. To shed light on its nature and its relative ease of computation, we compute the reactive bargaining set for games that played important role in the game theory literature.
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References
Asscher N (1976) An ordinal bargaining set for games without side payments. Mathematics of Operations Research 1: 381–389
Asscher N (1977) A cardinal bargaining set for games without side payments. International Journal of Game Theory 6:87–114
Aumann RJ (1973) Disadvantageous monopolies. Journal of Economic Theory 6: 1–11
Billera LJ (1968) On cores and bargaining sets forn-person cooperative games without side payments. PhD Thesis. The City University of New York
Billera LJ (1970) Existence of general bargaining sets for cooperative games without side payments. Bulletin American Mathematical Society 76: 375–380
Billera LJ (1972) Global stability inn-person games. Transactions American Mathematical Society 172: 45–56
Davis M, Maschler M (1965) The kernel of a cooperative game. Naval Research Logistics Quarterly 12: 223–259
Davis M, Maschler M (1967) Existence of stable payoff configurations for cooperative games. In: Shubik M (ed) Essays in mathematical economics in honor of Oskar Morgenstern. Princeton University Press, Princeton, New Jersey 39–52
Granot D (1994) On a new bargaining set for cooperative games. Working Paper, Faculty of Commerce and Business Administration, University of British Columbia
Kahan JP, Rapoport A (1984) Theories of coalition formation. Lawrence Erlbaum Associates, Hillsdale, New Jersey-London
Kalai G, Maschler M, Owen G (1975) Asymptotic stability and other properties of trajectories and transfer sequences leading to the bargaining sets. International Journal of Game Theory 4: 193–213
Maschler M (1966) The inequalities that determine the bargaining setM (i) 1. Israel Journal of Mathematics 4: 127–134
Maschler M (1976) An advantage of the bargaining set over the core. Journal of Economic Theory 13: 184–192
Maschler M, Peleg B (1966) A characterization, existence proof and dimension bounds for the kernel of a game. Pacific journal of Mathematics 18: 289–328
Maschler M, Peleg B (1967) The structure of the kernel of a cooperative game. SIAM Journal of Applied Mathematics 15: 569–604
Maschler M, Peleg B (1976) Stable sets and stable points of set-valued dynamic systems with applications to game theory. SIAM Journal of Control and Optimization 14: 985–995
Maschler M, Peleg B, Shapley LS (1972) The kernel and bargaining set for convex games. International Journal of Game Theory 1: 73–93
Peleg B (1963) Bargaining sets of cooperative games without side payments. Israel Journal of Mathematics 1: 197–200
Postlewaite A, Rosenthal RW (1974) Disadvantageous syndicates. Journal of Economic Theory 9: 324–326
Potters JAM, Muto S, Tijs S (1990) Bargaining set and kernel for big boss games. Methods of Operations Research 60: 329–335
Selten R, Schuster KG (1968) Psychological variables and coalitions forming behavior. In: Borch K, Mossin J (eds) Risk and Uncertainty. Macmillan, London
Shapley LS, Shubik M (1972) The assignment game I: The core. International Journal of Game Theory 2: 111–130
Stearns RE (1968) Convergent transfer schemes forn-person games. Transactions American Mathematical Society 134: 449–459
Von Neumann J, Morgenstern O (1953) Theory of games and economic behavior, 1st edition: 1944. Princeton University Press, Princeton, New Jersey
Yarom M (1981) The lexicographic kernel of a cooperative game. Mathematics of Operations Research 6: 88–100
Yarom M (1985) Dynamic systems of differential inclusions for the bargaining sets. International Journal of Game Theory 14: 51–61
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This research was partially supported by Natural Sciences and Engineering Research Council of Canada, grant A4181.
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Granot, D., Maschler, M. The reactive bargaining set: Structure, dynamics and extension to NTU games. Int J Game Theory 26, 75–95 (1997). https://doi.org/10.1007/BF01262514
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DOI: https://doi.org/10.1007/BF01262514