On the Variational Inequality and Tykhonov Well-Posedness in Game Theory

C. A. Pensavalle; G. Pieri

Bollettino dell'Unione Matematica Italiana (2010)

  • Volume: 3, Issue: 2, page 337-348
  • ISSN: 0392-4041

Abstract

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Consider a M-player game in strategic form G = ( X 1 , ⋯ , X M , g 1 , ⋯ , 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 ∈ X i , for any i = 1 , ⋯ , 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.

How to cite

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Pensavalle, C. A., and Pieri, G.. "On the Variational Inequality and Tykhonov Well-Posedness in Game Theory." Bollettino dell'Unione Matematica Italiana 3.2 (2010): 337-348. <http://eudml.org/doc/290698>.

@article{Pensavalle2010,
abstract = {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.},
author = {Pensavalle, C. A., Pieri, G.},
journal = {Bollettino dell'Unione Matematica Italiana},
language = {eng},
month = {6},
number = {2},
pages = {337-348},
publisher = {Unione Matematica Italiana},
title = {On the Variational Inequality and Tykhonov Well-Posedness in Game Theory},
url = {http://eudml.org/doc/290698},
volume = {3},
year = {2010},
}

TY - JOUR
AU - Pensavalle, C. A.
AU - Pieri, G.
TI - On the Variational Inequality and Tykhonov Well-Posedness in Game Theory
JO - Bollettino dell'Unione Matematica Italiana
DA - 2010/6//
PB - Unione Matematica Italiana
VL - 3
IS - 2
SP - 337
EP - 348
AB - 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.
LA - eng
UR - http://eudml.org/doc/290698
ER -

References

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  2. BAIOCCHI, C. - CAPELO, A., Disequazioni variazionali e quasi variazionali. Applicazioni a problemi di frontiera libera, Pitagora Editrice, 1978. Zbl1308.49003
  3. CAVAZZUTI, E., Cobwebs and Something Else, in G. Ricci (ed.), Decision Processes in Economics, Springer-Verlag, 1991. MR1117912DOI10.1007/978-3-642-45686-2_5
  4. CAVAZZUTI, E. - MORGAN, J., Well-Posed Saddle Point Problems, in J. B. Hiriart-Urruty et al. (eds.), Proc. Conference in Confolant (France 1981), Springer-Verlag, 1983, 61-76. MR716357
  5. DONTCHEV, A. L. - ZOLEZZI, T., Well-Posed Optimization Problems, Spinger Verlag, 1993. Zbl0797.49001MR1239439DOI10.1007/BFb0084195
  6. FURI, M. - VIGNOLI, A., About Well-Posed Optimization Problems for Functionals in Metric spaces, Journal of Optimization Theory and Applications, 5 (1970), 225-229. Zbl0177.12904MR264482DOI10.1007/BF00927717
  7. GABAY, D. - MOULIN, H., 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. MR604934
  8. LUCCHETTI, R. - PATRONE, F., A Characterization of Tykhonov Well-Posedness for Minimum Problems, with Application to Variational Inequalities, Numerical Functional Analysis and Applications, 3 (1981), 461-476. Zbl0479.49025MR636739DOI10.1080/01630568108816100
  9. MOSCO, U., An Introduction to the Approximate Solution of Variational Inequalities, Cremonese, 1973. Zbl0266.49005
  10. MARGIOCCO, M. - PATRONE, F. - PUSILLO CHICCO, L., A new approach to Tykhonov Well-Posedness for Nash Equilibria, Journal of Optimization Theory and Applications, 40 (1997), 385-400. Zbl0881.90136MR1459911DOI10.1080/02331939708844321
  11. PENSAVALLE, C. A. - PIERI, G., Variational Inequalities in Cournot Oligopoly, International Game Theory Rewiew, 9 (2007), 1-16. Zbl1200.91024MR2388263DOI10.1142/S0219198907001618
  12. PIERI, G. - TORRE, A., Hadamard and Tykhonov Well-Posedness in two Player Games, International Game Theory Rewiew, 5 (2003), 375-384. Zbl1102.91007MR2034794DOI10.1142/S0219198903001124
  13. TYKHONOV, A. N., On The Stability of the Functional Optimization Problem, U.S.S.R. Computational Mathematics and Mathematical Physics, 6 (1966), 28-33. 

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