3 Zero-Sum Games Alice and Bob are playing a zero-sum game whose payoff matrix is shown below. The jth entry of the matrix shows the payoff that Alice receives if she plays strategy 1' and Bob plays strategy 3'. Alice is the row player and is trying to maximize her payoff, and Bob is the column player trying to minimize Alice's payoff. Alice \\Bob 1 2 Now we will write a linear program to nd a strategy that maximizes Alice's payoff. Let the variables of the linear program be $1, $2 and p, where sci is the probability that Alice plays strategy '5' and p denotes Alice's payoff. (a) Write the linear program for maximizing Alice's payoff. (Hint : You should think of set- ting up the constraints of the program such that it nds the best worst case strategy. This would depend on the strategy Bob plays assuming he knows what Alice's (probabilistic) strategy is.) (b) Eliminate $2 from the linear program and write it in terms of p and $1 alone. (c) Draw the feasible region of the above linear program in p and 5161. You are encouraged to use a plotting software for this. ((1) Write a linear program from Bob's perspective trying to minimizing Alice's payoff. Let the variables of the linear program be y1, y2 and p, where y is the probability that Bob plays strategy 2' and p denotes Alice's payoff. (e) What is the optimal solution and What is the value of the game