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CHAPTER 14 Markov Chains Chapter 15 Theory of Games • Deals with decision-making in situations where There are two or more rational players Who all have a set of strategies each Who are involved in conditions of competition and conflicting interests, and They are all aware of the pay-offs resulting from the play of various combinations of strategies by different players • The solution to a game calls for determining optimal strategies for the players to play Quantitative Techniques in Management by N.D.Vohra 2 Two-person zero-sum Games There are two players Each player has a finite number of strategies to play Conditional pay-offs resulting from play of various combinations of strategies are known Each pay-off is a gain for one player and loss for the other The solution calls for determining optimal strategies for each of the players, whether pure or mixed, and the resulting value of game Quantitative Techniques in Management by N.D.Vohra 3 Solution to Games To check if a saddle point exists: Find minimum pay-off in each row Select the largest of the minimum pay-offs This is maximin strategy of the maximising player Find maximum pay-off in each column Select the smallest of these pay-offs This is minimax strategy of the minimising player If the maximin and minimax strategies have same pay-offs, the game has a saddle point Quantitative Techniques in Management by N.D.Vohra 4 Solution to Games If a game has a saddle point, the maximin and minimax strategies involved are optimal strategies for the players and these are called pure strategies A game can have more than one saddle point, resulting in multiple optimal strategies If the game has no saddle point, the players have to play mixed strategies If value of the game, v= 0, it is called a fair game If v > 0, the game favours maximising player and if v < 0, it favours minimising player Quantitative Techniques in Management by N.D.Vohra 5 Solution to Games with no Saddle Point • If it is a 2 × 2 game Player B b1 b2 Player a1 a11 a12 A a2 a21 a22 If A plays a1 with probability x and a2 with probability 1-x, and B plays b1 with probability y and b2 with probability 1-y, then a22 – a21 a22 – a12 x = ----------------------------- y = --------------------------(a11 + a22) – (a21 + a12) (a11 + a22) – (a21 + a12) and (a11 × a22) – (a21 × a12) v = ----------------------------(a11 + a22) – (a21 + a12) Quantitative Techniques in Management by N.D.Vohra 6 Solution to Games with no Saddle Point • If it is a 2×n game or an m×2 game Plot expected pay-off of each strategy on a graph Locate the highest point in the lower envelop (in case of a 2×n game) and the lowest point in the upper envelop (in case of an m×2 game) Consider the pair of lines whose intersection yields the highest/lowest point and use the strategies represented by it This reduces the game to a 2×2 game and it is solved accordingly Quantitative Techniques in Management by N.D.Vohra 7 Solution to Games with no Saddle Point • If a game is of the order m × n: Attempt to reduce the order of the problem by applying dominance rule If it can be reduced to a 2×2 game, solve it accordingly If it can not be reduced to a 2×2 game, solve it as an LPP Quantitative Techniques in Management by N.D.Vohra 8 Rule of Dominance If a strategy is inferior to another, it is said to be dominated A dominated strategy can be deleted If each value in a row (say R1) is greater than, or equal to, the corresponding value in another row (say R2), then R1 dominates R2 If each value in a column (say C1) is smaller than, or equal to, the corresponding value in another column (say C2), then C1 dominates C2 A linear combination of two strategies [for example, αR1 + (1 – α) R2 ] may also dominate a strategy Quantitative Techniques in Management by N.D.Vohra 9 LP Solution to Games Games can be formulated and solved as LPPs The solution is obtained by simplex and uses the concept of duality Before formulation, make sure that all pay-offs are non-negative In case of negative values, add such a constant to all values that leaves all values non-negative Quantitative Techniques in Management by N.D.Vohra 10 LP Formulation of Games • Sample formulation of a 3×3 game from maximising player’s point of view Minimise 1/U = X1 + X2 +X3 Subject to a11X1 + a21X2 + a31X3 ≥ 1 a12X1 + a22X2 + a32X3 ≥ 1 a13X1 + a23X2 + a33X3 ≥ 1 X1, X2, X3 ≥ 0 where Xi = xi/U; xi is the probability that player plays ith strategy and U is the value of game Quantitative Techniques in Management by N.D.Vohra 11