. Consider a game between two players where each player can make up to n different moves

Chapter 10, Problem T2

(choose chapter or problem)

. Consider a game between two players where each player can make up to n different moves (n > 1). If both players make the same move, then player C pays player R $(n 1). However, if both players make different moves, then player R pays player C $1. Assume that both players have the same strategythat is, pn = [i]1n and qn = [i]n1, where 1 + 2 + 3 ++ n = 1. Use a computer to show that E(p2, q2) = 1 2 (1 1)2 + 1 2 (1 2)2 + 1 2 (2 1)2 + 1 2 (2 2)2 E(p3, q3) = 1 2 (1 1)2 + 1 2 (1 2)2 + 1 2 (1 3)2 + 1 2 (2 1)2 + 1 2 (2 2)2 + 1 2 (2 3)2 + 1 2 (3 1)2 + 1 2 (3 2)2 + 1 2 (3 3)2 E(p4, q4) = 1 2 (1 1)2 + 1 2 (1 2)2 + 1 2 (1 3)2 + 1 2 (1 4)2 + 1 2 (2 1)2 + 1 2 (2 2)2 + 1 2 (2 3)2 + 1 2 (2 4)2 + 1 2 (3 1)2 + 1 2 (3 2)2 + 1 2 (3 3)2 + 1 2 (3 4)2 + 1 2 (4 1)2 + 1 2 (4 2)2 + 1 2 (4 3)2 + 1 2 (4 4)2 Using these results as a guide, prove in general that the expected payoff to player R is E(pn, qn) = 1 2 -n i=1 -n j=1 (i j ) 2 0 which shows that in the long run, player R will not lose in this game.