This input file gives examples of procedure MNEv regarding Mixed Nash equili\ bria --------------------- For Barry's Problem 1(a), the mixed equlibria are given by Our matrix game is [[4, 4] [0, 3]] [ ] [[3, 0] [2, 2]] Assume that player Row plays the Top strategy with probability p (and hence \ the Bottom strategy with prob. 1-p also assume that player Column plays the Left strategy with probability q (\ and hence the Right strategy with prob. 1-q The expected pay-off of Row is, 4 p q + 3 (1 - p) q + 2 (1 - p) (1 - q), which simplifies to, 3 p q - 2 p + q + 2 The expected pay-off of Column is, 4 p q + 3 p (1 - q) + 2 (1 - p) (1 - q), that simplifies to, 3 p q + p - 2 q + 2 Let's consider Row's best response to Column's paying Left with prob. , q Our function of , p, q, is , 3 p q - 2 p + q + 2 We have to find, for every, q, between 0 and 1 the value(s) of, p, that will make it largest The coefficient of p is the following expression linear function of q, 3 q - 2 The expression, 3 q - 2, has a zero when , q = 2/3 when q is striclty less than, 2/3, this expression is negative, hence the best response is p=0 when q equals, 2/3, this expression is zero, hence every possible p is a best response when q is strictly larger than , 2/3, this is positive, hence the best response is p=1 --------------------------------------------- Now Let's consider Column's best response to Row's paying Top with prob. , p Our function of , q, p, is , 3 p q + p - 2 q + 2 We have to find, for every, p, between 0 and 1 the value(s) of, q, that will make it largest The coefficient of p is the following expression linear function of q, 3 p - 2 The expression, 3 p - 2, has a zero when , p = 2/3 when q is striclty less than, 2/3, this expression is negative, hence the best response is p=0 when q equals, 2/3, this expression is zero, hence every possible p is a best response when q is strictly larger than , 2/3, this is positive, hence the best response is p=1 --------------------------------------------- Since when p equals , 2/3, every q is OK and when q equals , 2/3, every p is OK and The mixed Nash equilibrium is [2/3, 2/3] in other words Row should play Strategy Top with probability, 2/3 and hence Strategy Bottom with probability, 1/3 and Column should play Strategy left with probability, 2/3 and hence Strategy Right with probability, 1/3 With this mixed strategy the expected pay-off of player Row is, 8/3 and the expected pay-off of player Column is, 8/3 --------------------- --------------------- For Barry's Problem 1(b), the mixed equlibria are given by Our matrix game is [[1, 8] [7, 9]] [ ] [[3, 4] [5, 5]] Assume that player Row plays the Top strategy with probability p (and hence \ the Bottom strategy with prob. 1-p also assume that player Column plays the Left strategy with probability q (\ and hence the Right strategy with prob. 1-q The expected pay-off of Row is, p q + 7 p (1 - q) + 3 (1 - p) q + 5 (1 - p) (1 - q), which simplifies to, -4 p q + 2 p - 2 q + 5 The expected pay-off of Column is, 8 p q + 9 p (1 - q) + 4 (1 - p) q + 5 (1 - p) (1 - q), that simplifies to, 4 p - q + 5 Let's consider Row's best response to Column's paying Left with prob. , q Our function of , p, q, is , -4 p q + 2 p - 2 q + 5 We have to find, for every, q, between 0 and 1 the value(s) of, p, that will make it largest The coefficient of p is the following expression linear function of q, -4 q + 2 The expression, -4 q + 2, has a zero when , q = 1/2 when q is striclty less than, 1/2, this expression is positive, hence the best response is p=1 when q equals, 1/2, this expression is zero, hence every possible p is a best response when q is strictly larger than , 1/2, this expression is negative, hence the best response is p=0 --------------------------------------------- Now Let's consider Column's best response to Row's paying Top with prob. , p Our function of , q, p, is , 4 p - q + 5 We have to find, for every, p, between 0 and 1 the value(s) of, q, that will make it largest The coefficient of p is the following expression linear function of q, -1 Since this is always negative (for all q), the best respose is p=0, no matte\ r what q is --------------------------------------------- There are no mixed Nash Equilibrium, the pure ones are {[1, 2]} --------------------- --------------------- For Barry's Problem 1(c), the mixed equlibria are given by Our matrix game is [[2, 8] [3, 5]] [ ] [[8, 2] [1, 7]] Assume that player Row plays the Top strategy with probability p (and hence \ the Bottom strategy with prob. 1-p also assume that player Column plays the Left strategy with probability q (\ and hence the Right strategy with prob. 1-q The expected pay-off of Row is, 2 p q + 3 p (1 - q) + 8 (1 - p) q + (1 - p) (1 - q), which simplifies to, -8 p q + 2 p + 7 q + 1 The expected pay-off of Column is, 8 p q + 5 p (1 - q) + 2 (1 - p) q + 7 (1 - p) (1 - q), that simplifies to, 8 p q - 2 p - 5 q + 7 Let's consider Row's best response to Column's paying Left with prob. , q Our function of , p, q, is , -8 p q + 2 p + 7 q + 1 We have to find, for every, q, between 0 and 1 the value(s) of, p, that will make it largest The coefficient of p is the following expression linear function of q, -8 q + 2 The expression, -8 q + 2, has a zero when , q = 1/4 when q is striclty less than, 1/4, this expression is positive, hence the best response is p=1 when q equals, 1/4, this expression is zero, hence every possible p is a best response when q is strictly larger than , 1/4, this expression is negative, hence the best response is p=0 --------------------------------------------- Now Let's consider Column's best response to Row's paying Top with prob. , p Our function of , q, p, is , 8 p q - 2 p - 5 q + 7 We have to find, for every, p, between 0 and 1 the value(s) of, q, that will make it largest The coefficient of p is the following expression linear function of q, 8 p - 5 The expression, 8 p - 5, has a zero when , p = 5/8 when q is striclty less than, 5/8, this expression is negative, hence the best response is p=0 when q equals, 5/8, this expression is zero, hence every possible p is a best response when q is strictly larger than , 5/8, this is positive, hence the best response is p=1 --------------------------------------------- Since when p equals , 1/4, every q is OK and when q equals , 5/8, every p is OK and The mixed Nash equilibrium is [1/4, 5/8] in other words Row should play Strategy Top with probability, 1/4 and hence Strategy Bottom with probability, 3/4 and Column should play Strategy left with probability, 5/8 and hence Strategy Right with probability, 3/8 With this mixed strategy the expected pay-off of player Row is, 37/8 and the expected pay-off of player Column is, 37/8 ---------------------