The payoff matrix (a ij ) for a two-person zero-sum game is said to be skew symmetric if the…
The payoff matrix (aij) for a two-person zero-sum game is said to be skew symmetric if the matrix has as many rows as columns and aij = −aji for each choice of i and j. The payoff matrix for the game Rock, Paper and Scissors discussed in Exercise 20 has this property.
a) Show that if players 1 and 2 use the same strategy,
b) Given any strategy x1, x2, . . . , xn for player 2, i.e., satisfying:
multiply the first n constraints by y1 = x1, y2 = x2, . . . , and yn = xn, respectively, and add. Use this manipulation, together with part (a), to show that w ≤ 0. Similarly, show that the value to the player 1 problem satisfies v ≥ 0. Use linear-programming duality to conclude that v = w = 0 is the value of the game.