Geodesic
On the Variational Inequality and Tykhonov Well-Posedness in Game Theory
Bollettino della Unione matematica italiana, Série 9, Tome 3 (2010) no. 2, pp. 337-348.

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Consider a M-player game in strategic form $G = (X_{1},\cdots,X_{M},g_{1},\cdots,g_{M})$ where the set $X_{i}$ is a closed interval of real numbers and the payoff function $g_{i}$ is concave and differentiable with respect to the variable $x_{i} \in X_{i}$, for any $i = 1,\cdots,M$. The aim of this paper is to find appropriate conditions on the payoff functions under the well-posedness with respect to the related variational inequality is equivalent to the formulation of the Tykhonov well-posedness in a game context. The idea of the proof is to appeal to a third equivalence, which is the well-posedness of an appropriate minimum problem. 
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Pensavalle, C. A.; Pieri, G. On the Variational Inequality and Tykhonov Well-Posedness in Game Theory. Bollettino della Unione matematica italiana, Série 9, Tome 3 (2010) no. 2, pp. 337-348. http://geodesic.mathdoc.fr/item/BUMI_2010_9_3_2_a5/

[1] J. P. Aubin, Mathematical Methods of Games and Economic Theory, North Holland, 1979. | MR | Zbl

[2] C. Baiocchi - A. Capelo, Disequazioni variazionali e quasi variazionali. Applicazioni a problemi di frontiera libera, Pitagora Editrice, 1978. | Zbl

[3] E. Cavazzuti, Cobwebs and Something Else, in G. Ricci (ed.), Decision Processes in Economics, Springer-Verlag, 1991. | DOI | MR

[4] E. Cavazzuti - J. Morgan, Well-Posed Saddle Point Problems, in J. B. Hiriart-Urruty et al. (eds.), Proc. Conference in Confolant (France 1981), Springer-Verlag, 1983, 61-76. | MR

[5] A. L. Dontchev - T. Zolezzi, Well-Posed Optimization Problems, Spinger Verlag, 1993. | DOI | MR | Zbl

[6] M. Furi - A. Vignoli, About Well-Posed Optimization Problems for Functionals in Metric spaces, Journal of Optimization Theory and Applications, 5 (1970), 225-229. | DOI | MR | Zbl

[7] D. Gabay - H. Moulin, On the Uniqueness a Stability of Nash-Equilibria in Non Co-operative Games, in A. Bensoussan et al. (eds.), Applied Stochastic Control in Econometrics and Management Science (North-Holland, 1980), 271-293. | MR

[8] R. Lucchetti - F. Patrone, A Characterization of Tykhonov Well-Posedness for Minimum Problems, with Application to Variational Inequalities, Numerical Functional Analysis and Applications, 3 (1981), 461-476. | DOI | MR | Zbl

[9] U. Mosco, An Introduction to the Approximate Solution of Variational Inequalities, Cremonese, 1973. | Zbl

[10] M. Margiocco - F. Patrone - L. Pusillo Chicco, A new approach to Tykhonov Well-Posedness for Nash Equilibria, Journal of Optimization Theory and Applications, 40 (1997), 385-400. | DOI | MR | Zbl

[11] C. A. Pensavalle - G. Pieri, Variational Inequalities in Cournot Oligopoly, International Game Theory Rewiew, 9 (2007), 1-16. | DOI | MR | Zbl

[12] G. Pieri - A. Torre, Hadamard and Tykhonov Well-Posedness in two Player Games, International Game Theory Rewiew, 5 (2003), 375-384. | DOI | MR | Zbl

[13] A. N. Tykhonov, On The Stability of the Functional Optimization Problem, U.S.S.R. Computational Mathematics and Mathematical Physics, 6 (1966), 28-33.