Applications of Mathematics, Vol. 65, No. 5, pp. 665-675, 2020
Linear complementarity problems and bi-linear games
Gokulraj Sengodan, Chandrashekaran Arumugasamy
Received December 26, 2019. Published online June 25, 2020.
Abstract: 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 $Z$-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 $Z$-transformations.
References: [1] A. Berman, R. J. Plemmons: 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] R. W. Cottle, J.-S. Pang, R. E. Stone: The Linear Complementarity Problem. Computer Science and Scientific Computing. Academic Press, Boston (1992). DOI 10.1137/1.9780898719000 | MR 1150683 | Zbl 0757.90078
[3] M. C. Ferris, J. S. Pang: Engineering and economic applications of complementarity problems. SIAM Rev. 39 (1997), 669-713. DOI 10.1137/S0036144595285963 | MR 1491052 | Zbl 0891.90158
[4] M. Fiedler, V. Pták: On matrices with non-positive off-diagonal elements and positive principal minors. Czech. Math. J. 12 (1962), 382-400. MR 0142565 | Zbl 0131.24806
[5] D. Gale, H. Nikaidô: The Jacobian matrix and global univalence of mappings. Math. Ann. 159 (1965), 81-93. DOI 10.1007/BF01360282 | MR 0204592 | Zbl 0158.04903
[6] M. S. Gowda, G. Ravindran: 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] M. S. Gowda, J. Tao: 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] G. Isac: Complementarity Problems. Lecture Notes in Mathematics 1528. Springer, Berlin (1992). DOI 10.1007/BFb0084653 | MR 1222647 | Zbl 0795.90072
[9] I. Kaneko: 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] I. Kaneko: 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] S. Karamardian: The complementarity problem. Math. Program. 2 (1972), 107-129. DOI 10.1007/BF01584538 | MR 295762 | Zbl 0247.90058
[12] K. G. Murty: Linear Complementarity, Linear and Nonlinear Programming. Sigma Series in Applied Mathematics 3. Heldermann Verlag, Berlin (1988). MR 0949214 | Zbl 0634.90037
[13] H. Nikaidô: 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] M. J. Orlitzky: 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] T. Parthasarathy, T. E. S. Raghavan: Some Topics in Two-Person Games. American Elsevier, New York (1971). MR 0277260 | Zbl 0225.90049
[16] H. Schneider, M. Vidyasagar: Cross-positive matrices. SIAM J. Numer. Anal. 7 (1970), 508-519. DOI 10.1137/0707041 | MR 0277550 | Zbl 0245.15008
[17] R. S. Varga: 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
Affiliations: Gokulraj Sengodan (corresponding author), Chandrashekaran Arumugasamy, Department of Mathematics, Central University of Tamil Nadu, Neelakudi Campus, Thiruvarur, Tamil Nadu, 610 005, India, e-mail: gokulrajs93@gmail.com, chandru1782@gmail.com
