Consider a M-player game in strategic form where the set is a closed interval of real numbers and the payoff function is concave and differentiable with respect to the variable , for any . 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.
@article{BUMI_2010_9_3_2_337_0, author = {C. A. Pensavalle and G. Pieri}, title = {On the Variational Inequality and Tykhonov Well-Posedness in Game Theory}, journal = {Bollettino dell'Unione Matematica Italiana}, volume = {3}, year = {2010}, pages = {337-348}, zbl = {1195.49031}, mrnumber = {2666362}, language = {en}, url = {http://dml.mathdoc.fr/item/BUMI_2010_9_3_2_337_0} }
Pensavalle, C. A.; Pieri, G. On the Variational Inequality and Tykhonov Well-Posedness in Game Theory. Bollettino dell'Unione Matematica Italiana, Tome 3 (2010) pp. 337-348. http://gdmltest.u-ga.fr/item/BUMI_2010_9_3_2_337_0/
[1] | MR 556865 | Zbl 0452.90093
, Mathematical Methods of Games and Economic Theory, North Holland, 1979.[2] | Zbl 1308.49003
- , Disequazioni variazionali e quasi variazionali. Applicazioni a problemi di frontiera libera, Pitagora Editrice, 1978.[3] Cobwebs and Something Else, in (ed.), Decision Processes in Economics, Springer-Verlag, 1991. | MR 1117912
,[4] Well-Posed Saddle Point Problems, in et al. (eds.), Proc. Conference in Confolant (France 1981), Springer-Verlag, 1983, 61-76. | MR 716357
- ,[5] | MR 1239439 | Zbl 0797.49001
- , Well-Posed Optimization Problems, Spinger Verlag, 1993.[6] About Well-Posed Optimization Problems for Functionals in Metric spaces, Journal of Optimization Theory and Applications, 5 (1970), 225-229. | MR 264482 | Zbl 0177.12904
- ,[7] On the Uniqueness a Stability of Nash-Equilibria in Non Co-operative Games, in A. et al. (eds.), Applied Stochastic Control in Econometrics and Management Science (North-Holland, 1980), 271-293. | MR 604934
- ,[8] A Characterization of Tykhonov Well-Posedness for Minimum Problems, with Application to Variational Inequalities, Numerical Functional Analysis and Applications, 3 (1981), 461-476. | MR 636739 | Zbl 0479.49025
- ,[9] | Zbl 0266.49005
, An Introduction to the Approximate Solution of Variational Inequalities, Cremonese, 1973.[10] A new approach to Tykhonov Well-Posedness for Nash Equilibria, Journal of Optimization Theory and Applications, 40 (1997), 385-400. | MR 1459911 | Zbl 0881.90136
- - ,[11] Variational Inequalities in Cournot Oligopoly, International Game Theory Rewiew, 9 (2007), 1-16. | MR 2388263 | Zbl 1200.91024
- ,[12] Hadamard and Tykhonov Well-Posedness in two Player Games, International Game Theory Rewiew, 5 (2003), 375-384. | MR 2034794 | Zbl 1102.91007
- ,[13] On The Stability of the Functional Optimization Problem, U.S.S.R. Computational Mathematics and Mathematical Physics, 6 (1966), 28-33.
,