Fuzzy Mathematical Programming and Fuzzy Matrix Games
This book presents a systematic and focused study of the application of fuzzy sets to two basic areas of decision theory, namely Mathematical Programming and Matrix Game Theory. Apart from presenting most of the basic results available in the literature o
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1.3 Two person zero-sum matrix games
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1.3 Two person zero-sum matrix games In this section, we present certain basic definitions and preliminaries with regard to two person zero-sum matrix games. Let Rn denote the n-dimensional Euclidean space and R+n be its non-negative orthant. Let A ∈ Rm×n be an (m × n) real matrix and eT = (1, 1, . . . , 1) be a vector of ‘ones’ whose dimension is specified as per the specific context. By a (crisp) two person zero-sum matrix game G we mean the triplet G = (Sm , Sn , A) where Sm = {x ∈ R+m , eT x = 1} and Sn = {y ∈ R+n , eT y = 1}. In the terminology of the matrix game theory, Sm (respectively Sn ) is called the strategy space for Player I (respectively Player II ) and A is called the pay-off matrix. Then, the elements of Sm (respectively Sn ) which are of the form x = (0, 0, . . . , 1, . . . , 0)T = ei , where 1 is at the ith place (respectively y = (0, 0, . . . , 1, . . . , 0)T = e j , where 1 is at the jth place) are called pure strategies for Player I (respectively Player II). If Player I chooses ith pure strategy and Player II chooses jth pure strategy then aij is the amount paid by Player II to Player I. If the game is zero-sum then −aij is the amount paid by Player I to Player II i.e. the gain of one player is the loss of other player. The quantity E(x, y) = xT Ay is called the expected pay-off of Player I by Player II, as elements of Sm (respectively Sn ) can be thought of as a set of all probability distribution over I = {1, 2, . . . , m} (respectively J = {1, 2, . . . , n}). It is customary to assume tha
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