| Title:
|
Mixed complementarity problems for robust optimization equilibrium in bimatrix game (English) |
| Author:
|
Luo, Guimei |
| Language:
|
English |
| Journal:
|
Applications of Mathematics |
| ISSN:
|
0862-7940 (print) |
| ISSN:
|
1572-9109 (online) |
| Volume:
|
57 |
| Issue:
|
5 |
| Year:
|
2012 |
| Pages:
|
503-520 |
| Summary lang:
|
English |
| . |
| Category:
|
math |
| . |
| Summary:
|
In this paper, we investigate the bimatrix game using the robust optimization approach, in which each player may neither exactly estimate his opponent's strategies nor evaluate his own cost matrix accurately while he may estimate a bounded uncertain set. We obtain computationally tractable robust formulations which turn to be linear programming problems and then solving a robust optimization equilibrium can be converted to solving a mixed complementarity problem under the $l_1\cap l_\infty $-norm. Some numerical results are presented to illustrate the behavior of the robust optimization equilibrium. (English) |
| Keyword:
|
robust optimization equilibrium |
| Keyword:
|
bimatrix game |
| Keyword:
|
$l_1\cap l_\infty $-norm |
| Keyword:
|
mixed complementarity problem |
| MSC:
|
90C05 |
| MSC:
|
90C33 |
| MSC:
|
90C46 |
| MSC:
|
91A05 |
| idZBL:
|
Zbl 1265.91003 |
| idMR:
|
MR2984616 |
| DOI:
|
10.1007/s10492-012-0029-4 |
| . |
| Date available:
|
2012-08-19T22:04:56Z |
| Last updated:
|
2020-07-02 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/142913 |
| . |
| Reference:
|
[1] Aghassi, M., Bertsimas, D.: Robust game theory.Math. Program. 107 (2006), 231-273. Zbl 1134.91309, MR 2218128, 10.1007/s10107-005-0686-0 |
| Reference:
|
[2] Ben-Tal, A., Nemirovski, A.: Robust convex optimization.Math. Oper. Res. 23 (1998), 769-805. Zbl 0977.90052, MR 1662410, 10.1287/moor.23.4.769 |
| Reference:
|
[3] Ben-Tal, A., Nemirovski, A.: Robust solutions of uncertain linear programs.Oper. Res. Lett. 25 (1999), 1-13. Zbl 0941.90053, MR 1702364, 10.1016/S0167-6377(99)00016-4 |
| Reference:
|
[4] Ben-Tal, A., Nemirovski, A.: Robust solutions of linear programming problems contaminated with uncertain data.Math. Program. 88 (2000), 411-424. Zbl 0964.90025, MR 1782149, 10.1007/PL00011380 |
| Reference:
|
[5] Bertsimas, D., Pachamanova, D., Sim, M.: Robust linear optimization under general norms.Oper. Res. Lett. 32 (2004), 510-516. Zbl 1054.90046, MR 2077451, 10.1016/j.orl.2003.12.007 |
| Reference:
|
[6] Bertsimas, D., Sim, M.: The price of robustness.Oper. Res. 52 (2004), 35-53. Zbl 1165.90565, MR 2066239, 10.1287/opre.1030.0065 |
| Reference:
|
[7] Bertsimas, D., Sim, M.: Tractable approximations to robust conic optimization problems.Math. Program. 107 (2006), 5-36. Zbl 1134.90026, MR 2216799, 10.1007/s10107-005-0677-1 |
| Reference:
|
[8] Chen, X., Sim, M., Sun, P.: A robust optimization perspective of stochastic programming.Oper. Res. 55 (2007), 1058-1071. MR 2372277, 10.1287/opre.1070.0441 |
| Reference:
|
[9] Ghaoui, L. El, Oustry, F., Lebret, H.: Robust solutions to least-squares problems with uncertain data.SIAM J. Matrix Anal. Appl. 18 (1997), 1035-1064. MR 1472008, 10.1137/S0895479896298130 |
| Reference:
|
[10] Ghaoui, L. El, Oustry, F., Lebret, H.: Robust solutions to uncertain semidefinite programs.SIAM J. Optim. 9 (1998), 33-52. Zbl 0960.93007, MR 1660106, 10.1137/S1052623496305717 |
| Reference:
|
[11] Facchinei, F., Pang, J. S.: Finite-Dimensional Variational Inequalities and Complementarity Problems, Vol. I.Springer New York (2003). Zbl 1062.90001, MR 1955648 |
| Reference:
|
[12] Hayashi, S., Yamashita, N., Fukushima, M.: A combined smoothing and regularization method for monotone second-order cone complementarity problems.SIAM J. Optim. 15 (2005), 593-615. Zbl 1114.90139, MR 2144183, 10.1137/S1052623403421516 |
| Reference:
|
[13] Hayashi, S., Yamashita, N., Fukushima, M.: Robust Nash equilibria and second-order cone complementarity problems.J. Nonlinear. Convex Anal. 6 (2005), 283-296. Zbl 1137.91310, MR 2159841 |
| Reference:
|
[14] Harsanyi, J. C.: Games with incomplete information played by ``Bayesian'' playes, Part II.Manage. Sci. 14 (1968), 320-334. MR 0246650, 10.1287/mnsc.14.5.320 |
| Reference:
|
[15] Holmström, B., Myerson, R.: Efficient and durable decision rules with incomplete information.Econometrica 51 (1983), 1799-1820. Zbl 0521.90008, 10.2307/1912117 |
| Reference:
|
[16] Luo, G. M., Li, D. H.: Robust optimization equilibrium with deviation measures.Pac. J. Optim. 5 (2009), 427-441. Zbl 1175.91017, MR 2567016 |
| Reference:
|
[17] Mertens, J., Zamir, S.: Formualation of Bayesian analysis for games with incomplete information.Int. J. Game Theory 14 (1985), 1-29. MR 0784702, 10.1007/BF01770224 |
| Reference:
|
[18] jun., J. F. Nash: Equilibrium points in $n$-person games.Proc. Natl. Acad. Sci. USA 36 (1950), 48-49. Zbl 0036.01104, MR 0031701, 10.1073/pnas.36.1.48 |
| Reference:
|
[19] Nash, J.: Non-cooperative games.Ann. Math. 54 (1951), 286-295. Zbl 0045.08202, MR 0043432, 10.2307/1969529 |
| Reference:
|
[20] Soyster, A. L.: Convex programming with set-inclusive constraints and applications to inexact linear programming.Oper. Res. 21 (1973), 1154-1157. Zbl 0266.90046, 10.1287/opre.21.5.1154 |
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