A study on linear and bi linear games

Abstract

newline ABSTRACT newlineKEYWORDS: Linear games; bi-linear games; Nash equilibrium; von-Neumann newlinesymmetrization; Z-transformation; semi-positive maps newlineNon-cooperative two-person non-zero-sum game, also known as bimatrix game, is a newlinegame played by two players say player I(row player) and player II(column player) newlinewhose pay-off matrices are A and B, respectively, in Mm×n(R). If player I plays with newlinei newlineth row(i newlineth pure strategy) and player II plays with j newlineth column(j newlineth pure strategy) then newlinethe expected pay-off for the player I and player II is aij and bij , respectively. In case newlineplayer I plays with the probability distribution x on rows(mixed strategy) and player newlineII plays with the probability distribution y on columns(mixed strategy), their expected newlinepay-off is hx, Ayi and hx, Byi, respectively. A pair of probability distribution vectors newline(strategies) (x newlineand#8727; newline, yand#8727; newline) is said to be an equilibrium pair if the following inequalities holds newlinefor all probability vectors x and#8712; R newlinem and y and#8712; R newlinen newline: newlinev1 := hx newlineand#8727; newline, Ayand#8727; newlinei and#8805; hx, Ayand#8727; newlinei, newlinev2 := hx newlineand#8727; newline, Byand#8727; newlinei and#8805; hx newlineand#8727; newline, Byi. newlineThat is, x newlineand#8727; newlineis the best response for player I against player II s y newlineand#8727; newline, likewise player II s best newlineresponse against x newlineand#8727; newlineis y newlineand#8727; newline. The expected pay-off v1 and v2 at equilibrium pair is called newlinethe equilibrium value of player I and player II, respectively. In the case of B = and#8722;A, the newlinegame is called as the zero-sum game. newlineThis thesis deals with a generalization of the non-cooperative two-person games. newlineFirst we consider the linear games - the generalized two-person zero-sum games; we newlinedefine a symmetric linear game corresponding to linear game whose solution yields a newlinesolution to the given linear game. Further, we list some sufficient conditions for the newlinesymmetric linear game to be completely mixed. Then we define the bi-linear game newlineas a generalization of the two-person non-zero-sum game and extend some classical newlinebimatrix game results to this general setting. In addition to that we prove some known newlineiii newlineclassical results from matrix theory and the theory of complementarity problems using newlinethe conce