Solving two-person zero-sum game by Matlab

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1 Appled Mechancs and Materals Onlne: ISSN: , Vols , pp do: / Trans Tech Publcatons, Swtzerland Solvng two-person zero-sum game by Matlab Yanme Yang 1, a, Yan Guo 2, b, Lchao Feng 1,c, and Janyong D 3,d 1 College of Scence, Hebe Polytechnc Unversty, Tangshan, , Chna; 2 Department of Mathematcs & Physcs, North Chna Electrc Power Unversty, Baodng, , Chna 3 College of lght ndustry, Hebe Polytechnc Unversty, Tangshan, , Chna ; a b c d Keywords: game theory; two-person zero-sum games; lnear programmng; Matlab; Abstract: In ths artcle we present an overvew on two-person zero-sum games, whch play a central role n the development of the theory of games. Two-person zero-sum games s a specal class of game theory n whch one player wns what the other player loses wth only two players. It s dffcult to solve 2-person matrx game wth the order m n(m 3,n 3). The am of the artcle s to determne the method on how to solve a 2-person matrx game by lnear programmng functon lnprog() n matlab. Wth lnear programmng technques n the Matlab software, we present effectve method for solvng large zero-sum games problems. Introducton Game theory s a branch of appled mathematcs that s used n the socal scences, most notably n economcs [1]. Two-person zero-sum games refer to games of pure conflct, whch s the smplest case of game theory. In zero-sum games the total beneft to all players n the game, for every combnaton of strateges, always adds to zero, the payoff of one player s the negatve of the payoff of the other player [2]. Two-person zero-sum games play a central role n the development of the theory of games, but t s dffcult to solve 2-person matrx game wth the order m n (m 3,n 3). The connecton wth lnear programmng was dscovered by von Neumann n Mxed strateges must exst for two-player zero-sum games had been proven by John von Neumann and Oskar Morgenstern. The Mnmax Theorem s a smple consequence of the Dualty Theorem of lnear programmng [2,3]. So usng lnear programmng technques can solve two-person zero-sum games. In ths paper we dscuss how to convert a two-person zero-sum game nto a lnear programmng problem Lnprog command n Matlab Lnear programmng s one of the most wdespread methods used to solve management and economc problems. The command lnprog from the optmzaton toolbox mplements the smplex algorthm to solve a lnear programmng problem. Matlab uses the followng format for lnear programs: All rghts reserved. No part of contents of ths paper may be reproduced or transmtted n any form or by any means wthout the wrtten permsson of Trans Tech Publcatons, (ID: , Pennsylvana State Unversty, Unversty Park, USA-09/05/16,18:08:00)

2 Appled Mechancs and Materals Vols mn z= T f x A x b s. t. Aeq x= beq lb x ub Where f, x, b, beq, lb, and ub are vectors, and A and Aeq are matrces. x = lnprog(f, A, b, Aeq, beq, lb, ub) solves the above lnear programmng mn T f x such that A x b. Whle addtonally satsfyng the equalty constrants Aeq x= beq, set A = [] and b = [] f no nequaltes exst. Defnes a set of lower and upper bounds on the desgn varables, x, so that the soluton s always n the range lb x ub, set Aeq = [] and beq = [] f no equaltes exst [4]. For example, our orgnal problem s translated nto the format: mn Z = 4x + 3x x1+ x2 5 6 x1 0 1 x Solve the problem by Matlab, frst we set up the vectors and matrces: c=[4,3]; a=[1,1]; b=[5]; lb=[-6;-1]; ub=[10;4]; Fnally, nput the command x=lnprog(c,a,b,[],[],lb,ub). Applyng wth the command lnprog n optmzaton toolbox can be solved the above problem. Solvng two-person zero-sum game by Matlab It s stll dffcult to solve the matrx game for hgh order, although there are many solvng ways. The Mnmax Theorem s the fundamental theorem n the theory of two-person zero-sum game. The Mnmax Theorem as follow: For every two-person, zero-sum game wth fnte strateges, there exsts a value V and a mxed strategy for each player, such that (a) Gven player 2's strategy, the best payoff possble for player 1 s V, and (b) Gven player 1's strategy, the best payoff possble for player 2 s V. It can be proved by technques from lnear programmng [5-7]. Seeng the relatonshp of the Mnmax Theorem and Dualty Theorem, lnear programmng can be used to solve two-person zero-sum game. We have an m n payoff matrx A=(a ) m n, The entry a represents the payoff to the row player when she pcks the th strategy and column player pcks hs th strategy. The two-person zero-sum game can be solved by solvng thus two lnear programmng:

3 264 Intellgent Structure and Vbraton Control mn Z = x a x 1 ( = 1,2,, n) x 0 ( = 1, 2,, m) max w= y a y ( = 1, 2,, m) y 0 ( = 1,2,, n) Suppose we want to solve the two-person zero-sum game, whch payoff matrx as follow, Step 1 Reduce the payoff matrx by domnance. The above payoff matrx s a smplest form. Step 2 Convert to a payoff matrx wth no negatve entres by addng a sutable fxed number to all the entres. Addng 3 to every element of the above payoff matrx, then the new matrx s Step 3 Solve the assocated standard lnear programmng problem. Solvng thus two lnear programmng: mn x + x + x max y + y + y 6x1 + 2x2+ 5x3 x 1 + 7x2 + 5x3 1 4x1+ 5x2+ 6x3 x Step 4 Calculate the optmal strateges by Matlab mn x + x + x 2y1+ 7y2+ 5y3 5y1+ 5y2+ 6y3 y mn x + x + x 6x1 + 2x2+ 5x3 6x1 2x2 5x3 1 Translated LP x 1 + 7x2 + 5x3 1 nto the format x 1 7x 2 5x 3 1 4x1+ 5x2+ 6x3 4x1 5x2 6x3 1 x x set up the vectors and matrces:: c=[1,1,1]; a=[-6,-2,-5;-1,-7,-5;-4,-5,-6]; b=[-1,-1,-1]; lb=[0;0;0]; To fnd the optmal soluton, the optmzaton toolbox has the command lnprog: x=lnprog(c,a,b,[],[],lb)

4 Appled Mechancs and Materals Vols max y + y + y Translated LP 2y1+ 7y2+ 5y3 5y1+ 5y2+ 6y3 y Solved command n Matlab s nto the format mn y y y 2y1+ 7y2+ 5y3 5y1+ 5y2+ 6y3 y clear c=[-1,-1,-1]; a=[6,1,4;2,7,5;5,5,6]; b=[1,1,1]; lb=[0;0;0]; y=lnprog(c,a,b,[],[],lb) The soluton of above two LP s x = 0,0,, y =,,0, v = The fnal result of the two-person zero-sum game s * * 4 1 x = ( 0,0,1 ), y =,,0, v= 5 3= Step by step, we show how to use ths lnear programmng method solve the two- person zero-sum game n Matlab. The help of computer tools, we can greatly reduce the solvng work tme. Concluson Matlab s a hgh-level techncal computng language. In optmzaton toolbox of Matlab, lnear programmng problem can be solved smply by the command lnprog. Many algorthms have been proposed for solvng two-player zero-sum games, but t s not easy to solve for large matrx game. Usng lnear programmng s effcent method. Especally, t uses the lnear programmng functon lnprog() n Matlab, calculates the two-person zero-sum game. Ths method s effcent and effectve for large matrx game. References [1] Fernandez, L F.; Berman, H S., Game theory wth economc applcatons, Addson-Wesley, (1998) [2] Dutta. Prat K, Strateges and games: theory and practce, MIT Press, (1999) [3] Osborne, Martn J. (2004), An ntroducton to game theory, Oxford Unversty Press [4] B.R. Hunt, R.L. Lpsman, and J.M. Rosenberg. A Gude to MATLAB, for begnners and experenced users. Cambrdge Unversty Press, 2001 [5] Phlp D.Straffn. Game theory and strategy, Mathematcal assocaton of Amerca, [6] Andrey Garnaev, Search games and other applcatons of game theory, Lecture notes n economcs and mathematcal systems 485, Sprnger, [7] Mehrotra, S., On the Implementaton of a Prmal-Dual Interor Pont Method, SIAM Journal on Optmzaton, Vol. 2, pp , 1992.

5 Intellgent Structure and Vbraton Control / Solvng Two-Person Zero-Sum Game by Matlab / DOI References [7] Mehrotra, S., On the Implementaton of a Prmal-Dual Interor Pont Method, SIAM Journal on Optmzaton, Vol. 2, pp , /

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