Global stability in 𝑛-person games

Billera, Louis J.

doi:10.1090/S0002-9947-1972-0314469-1

Global stability in $n$-person games
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Trans. Amer. Math. Soc. 172 (1972), 45-56 Request permission

Abstract:

A class of bargaining sets, including the bargaining set $\mathfrak {M}_1^{(i)}$ and the kernel, is treated with regard to studying the tendency to reach stability from unstable points. A known discrete procedure is extended, and these results are applied to derive global stability properties for the solutions of certain differential equations. These differential equations are given in terms of the demand functions which define the bargaining sets, and the set of critical points is precisely the bargaining set in question.

References

A bargaining set for cooperative games without side payments

Louis J. Billera, Existence of general bargaining sets for cooperative games without side payments, Bull. Amer. Math. Soc. 76 (1970), 375–379. MR 258470, DOI 10.1090/S0002-9904-1970-12484-3
G. W. Brown and J. von Neumann, Solutions of games by differential equations, Contributions to the Theory of Games, Annals of Mathematics Studies, no. 24, Princeton University Press, Princeton, N.J., 1950, pp. 73–79. MR 0039220
Earl A. Coddington and Norman Levinson, Theory of ordinary differential equations, McGraw-Hill Book Co., Inc., New York-Toronto-London, 1955. MR 0069338
Morton Davis and Michael Maschler, The kernel of a cooperative game, Naval Res. Logist. Quart. 12 (1965), 223–259. MR 207404, DOI 10.1002/nav.3800120303
M. Maschler and B. Peleg, A characterization, existence proof and dimension bounds for the kernel of a game, Pacific J. Math. 18 (1966), 289–328. MR 205699
Michael Maschler and Bezalel Peleg, The structure of the kernel of a cooperative game, SIAM J. Appl. Math. 15 (1967), 569–604. MR 231630, DOI 10.1137/0115050

The kernel and the nucleolus of a cooperative game as locuses in the strong $\epsilon$-core

V. V. Nemyckiĭ and V. V. Stepanov, Kačestvennaya Teoriya Differencial′nyh Uravneniĭ, OGIZ, Moscow-Leningrad, 1947 (Russian). MR 0029483
Bezalel Peleg, Existence theorem for the bargaining set $M_{1}^{(i)}$, Bull. Amer. Math. Soc. 69 (1963), 109–110. MR 144792, DOI 10.1090/S0002-9904-1963-10881-2
Bezalel Peleg, Bargaining sets of cooperative games without side payments, Israel J. Math. 1 (1963), 197–200. MR 172725, DOI 10.1007/BF02759717
Bezalel Peleg, Equilibrium points for open acyclic relations, Canadian J. Math. 19 (1967), 366–369. MR 209017, DOI 10.4153/CJM-1967-028-4
David Schmeidler, The nucleolus of a characteristic function game, SIAM J. Appl. Math. 17 (1969), 1163–1170. MR 260432, DOI 10.1137/0117107
R. E. Stearns, Convergent transfer schemes for $N$-person games, Trans. Amer. Math. Soc. 134 (1968), 449–459. MR 230550, DOI 10.1090/S0002-9947-1968-0230550-9