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Keywords:
bimatrix game; nash equilibrium; ${\bf Z}$-transformation; semi positive map
bimatrix game; nash equilibrium; ${\bf Z}$-transformation; semi positive map
Summary:
In this paper, we define bi-linear games as a generalization of the bimatrix games. In particular, we generalize concepts like the value and equilibrium of a bimatrix game to the general linear transformations defined on a finite dimensional space. For a special type of ${\bf Z}$-transformation we observe relationship between the values of the linear and bi-linear games. Using this relationship, we prove some known classical results in the theory of linear complementarity problems for this type of ${\bf Z}$-transformations.
References:
[1] Berman, A., Plemmons, R. J.: Nonnegative Matrices in the Mathematical Sciences. Classics in Applied Mathematics 9. SIAM, Philadelphia (1994). DOI 10.1137/1.9781611971262 | MR 1298430 | Zbl 0815.15016
[2] Cottle, R. W., Pang, J.-S., Stone, R. E.: The Linear Complementarity Problem. Computer Science and Scientific Computing. Academic Press, Boston (1992). DOI 10.1137/1.9780898719000 | MR 1150683 | Zbl 0757.90078
[3] Ferris, M. C., Pang, J. S.: Engineering and economic applications of complementarity problems. SIAM Rev. 39 (1997), 669-713. DOI 10.1137/S0036144595285963 | MR 1491052 | Zbl 0891.90158
[4] Fiedler, M., Pták, V.: On matrices with non-positive off-diagonal elements and positive principal minors. Czech. Math. J. 12 (1962), 382-400. DOI 10.21136/CMJ.1962.100526 | MR 0142565 | Zbl 0131.24806
[5] Gale, D., Nikaidô, H.: The Jacobian matrix and global univalence of mappings. Math. Ann. 159 (1965), 81-93. DOI 10.1007/BF01360282 | MR 0204592 | Zbl 0158.04903
[6] Gowda, M. S., Ravindran, G.: On the game-theoretic value of a linear transformation relative to a self-dual cone. Linear Algebra Appl. 469 (2015), 440-463. DOI 10.1016/j.laa.2014.11.032 | MR 3299071 | Zbl 1309.91008
[7] Gowda, M. S., Tao, J.: Z-transformations on proper and symmetric cones. Math. Program. 117 (2009), 195-221. DOI 10.1007/s10107-007-0159-8 | MR 2421305 | Zbl 1167.90022
[8] Isac, G.: Complementarity Problems. Lecture Notes in Mathematics 1528. Springer, Berlin (1992). DOI 10.1007/BFb0084653 | MR 1222647 | Zbl 0795.90072
[9] Kaneko, I.: A linear complementarity problem with an $n$ by $2n$ ``$P$''-matrix. Math. Program. Study 7 (1978), 120-141. DOI 10.1007/BFb0120786 | MR 0483316 | Zbl 0378.90054
[10] Kaneko, I.: Linear complementarity problems and characterizations of Minkowski matrices. Linear Algebra Appl. 20 (1978), 111-129. DOI 10.1016/0024-3795(78)90045-9 | MR 0480554 | Zbl 0382.15011
[11] Karamardian, S.: The complementarity problem. Math. Program. 2 (1972), 107-129. DOI 10.1007/BF01584538 | MR 295762 | Zbl 0247.90058
[12] Murty, K. G.: Linear Complementarity, Linear and Nonlinear Programming. Sigma Series in Applied Mathematics 3. Heldermann Verlag, Berlin (1988). MR 0949214 | Zbl 0634.90037
[13] Nikaidô, H.: Convex Structures and Economic Theory. Mathematics in Science and Engineering 51. Academic Press, New York (1968). DOI 10.1016/S0076-5392(08)63326-3 | MR 0277233 | Zbl 0172.44502
[14] Orlitzky, M. J.: Positive Operators, Z-Operators, Lyapunov Rank, and Linear Games on Closed Convex Cones: Ph.D. Thesis. University of Maryland, Baltimore County (2017). MR 3697664
[15] Parthasarathy, T., Raghavan, T. E. S.: Some Topics in Two-Person Games. American Elsevier, New York (1971). MR 0277260 | Zbl 0225.90049
[16] Schneider, H., Vidyasagar, M.: Cross-positive matrices. SIAM J. Numer. Anal. 7 (1970), 508-519. DOI 10.1137/0707041 | MR 0277550 | Zbl 0245.15008
[17] Varga, R. S.: Matrix Iterative Analysis. Springer Series in Computational Mathematics 27. Springer, Berlin (2000). DOI 10.1007/978-3-642-05156-2 | MR 1753713 | Zbl 0998.65505
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