The game "Rock-Paper-Scissors" is a two-person game in which at the count of three John and Jane form his or her hand into a rock (a balled fist), a sheet of paper (a flat hand) or scissors (two fingers in a V shape). The outcome of the game is determined as follows: if both players form the same object, then the round is a draw; otherwise Rock beats Scissors (since a rock can dull the scissors); Scissors beats Paper (since scissors can cut paper); and Paper beats Rock (since the paper can cover up a rock). You may denote the choice of actions by either Rock or (1), Paper or (2) and Scissors or (3). (a) Suppose the payoff for each game is $1 from the loser to the winner and $0 if the outcome of the game is a draw. i. Write out the game matrix for John's payoffs. John uses the mixed strategy p = (p1, p2, p3) and Jane adopts q = (q1, q2, q3). ii. Give an expression in terms of p and q for John's expected reward, E[R(p, q)]. iii. Write out the linear program to determine John's maximin mixed strategy