Problem 1 We consider a twoplayer strategic form game where Alice and Bob are the two players. Alice's pure strategies are U and D and Bob's pure strategies are L and R. The game is depicted in Table 1. Bob L R U a, e 0, Alice f D b, g d, h Table 1: Game for Problem 1 Denote by q the probability with which Bob plays L (and thus the probability to play R is 1 q). We will assume further in the problem that before playing the game Bob can commit not to play certain mixed strategies. That is, Bob can commit to play a mixed strategy such that o: S q S 8. Hence, for instance, if 0: > 0 then Bob commits not to play the pure strategy L. This commitment yields a new, \"constrained\" game where Bob's pure strategies are L' and R' such that o L' is playing L with probability 0: and R with probability 1 oz. 0 R' is playing L with probability ,8 and R with probability 1 ,8. For instance, if Alice and Bob play the game depicted in Table 2 and Alice commits to Bob L R C 2, 3 -1, 4 Alice D 1, 1 2, 0 Table 2: Numerical example play a mixed strategy such that the probability that she plays U is between } and }, then the new strategic form game is the one depicted in Table 3 Bob L R U' 5, 1 Alice D' 0, Table 3: Numerical example The payoff in Table 3 are obtained as follows: UAlice (U', L) = =uAlice (U, I) + ( U Alice ( D, L) = = x 2+~ x 1=> U Alice ( D', L) = = uAlice(U, L) + U Alice ( D, L) U Alice (U', R) = -uAlice (U, R) + UAlice (D', R) = ; UAlice (U, R) + (1- 2 UAlice ( D, R) For the following questions you need to work with the game depicted in Table 1. Do NOT answer using the game depicted in Table 2 or Table 3 (those two tables were just here to show you a numerical example of a game with constrained strategy sets). 1. Suppose that Bob is playing a mixed strategy where he plays L with probability q. Give the formula for q*, the probability for Bob playing L such that Alice is indifferent between playing U and D from the game depicted in Table 1.. Calculate Alice's expected payoff if they play a mixed strategy equilibrium where Bob plays L with probability {3* (we assume that a, I), c, and d are such that such an equi- librium exists). Here we assume that Bob plays a non-trivial mixed strategy equilibrium, i.e., 0