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A new approach to Tikhonov well-posedness for Nash equilibria. (English) Zbl 0881.90136
Summary: It is suggested a new approach to Tikhonov well-posedness for Nash equilibria. Loosely speaking, Tikhonov well-posedness of a problem means that approximate solutions converge to the true solution when the degree of approximation goes to zero.
The novelty of our approach consists in a suitable definition of what could be considered an approximate solution of a Nash equilibrium problem. We add to the requirement of being an $$\varepsilon$$-equilibrium also that of being $$\varepsilon$$ close in value to some Nash equilibrium. In this way, we can get rid of some problems which affect Tikhonov well-posedness when the last condition is not taken into account, like the usual lack of uniqueness for Nash equilibria. Furthermore, it can be proved that this property of well-posedness is preserved under monotonic transformations of the payoffs: a result which is relevant in view of economic interpretation.

##### MSC:
 91A10 Noncooperative games 49J45 Methods involving semicontinuity and convergence; relaxation
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##### References:
 [1] Bednarczuk E., Control and Cybern 23 pp 107– (1994) [2] DOI: 10.1287/moor.17.3.715 · Zbl 0767.49011 · doi:10.1287/moor.17.3.715 [3] Cavazzuti, E. and Morgan, J. Optimization, theory and algorithms. Proc. Conf. Confolant. 1981, France. Edited by: Hiriart-Urruty, J.B., Oettli, W. and Stoer, J. pp.61–76. Well-Posed Saddle Point Problems [4] Dontchev, A. and Zolezzi, T. 1993. ”Well-Posed Optimization Problems”. Berlin: Springer. · Zbl 0797.49001 [5] DOI: 10.1006/game.1995.1012 · Zbl 0835.90122 · doi:10.1006/game.1995.1012 [6] DOI: 10.1007/BF00927717 · Zbl 0177.12904 · doi:10.1007/BF00927717 [7] Levitin E.S., Societ Matiz. Dokl 7 pp 764– (1966) [8] Loridan P., Recent Decelopmeizts in Well-Posed Variational Problenzs pp 171– (1995) [9] DOI: 10.1080/01630568108816100 · Zbl 0479.49025 · doi:10.1080/01630568108816100 [10] DOI: 10.1080/01630568308816145 · Zbl 0517.49007 · doi:10.1080/01630568308816145 [11] Lucchetti, R. and Revalski, J. 1995. ”Recent Developments in Well-Posed Variational Problems”. Edited by: Lucchetti, R. and Revalski, J. Kluwer: Dordrecht. · Zbl 0823.00006 [12] Morgan J., Non-Smooth Optinzization and Related Topics (1989) [13] Myerson, R.B. 1991. ”Game Theory: Analysis of Conflict”. Cambridge, MA: Harvard University Press. · Zbl 0729.90092 [14] Patrone F., Riv. Mat. Pura Appl 1 pp 95– (1987) [15] Patrone F., Recent Deceloptnents in Well-Posed Variational Problerns pp 211– (1995) [16] Patrone F. Pusillo Chicco L. Antagonism for two-person games: taxsonomy and applications to Tikhonov well-posedness 1995 preprint · Zbl 0881.90136 [17] Revalski, I.P. Mathematics and Edzicatioil $$si;in$$ei; Mathematics. Proc. 14th Spring Confer. of the Union of Bulgarian Mathematicians. Sofia. Variational inequalities with unique solution [18] Revalski J.P., Acta Univ. Carolinae Math,et Phys 28 pp 117– (1987) [19] Tikhonov A.N., USSR J. Comp. Math. Math. Phys 6 pp 631– (1966) [20] DOI: 10.1137/0121011 · doi:10.1137/0121011
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