Problem 3 (Adapted from a problem by Hoffman and Yoeli) Two players are fighting over a resource worth v > 0. The players choose how long to fight before giving up, and the player who chooses to fight for longer wins the resource. Players 1 and 2 devote times t1 2 0 and t2 2 0 to the fight, respectively. Fighting is costly and costs both players c > 0 per unit time. Hence the payoffs of the game are: -ct1 if t1 t2 -ct2 if t2 t1 (a) Compute the best response function for each player. Hint: To compute the best response function for player 1, you might find it useful to analyze by taking the cases (i) to v/c. Solution: For finding the best response for player 1, let us analyze the three cases for player 2. (i) Let us take the case to v/c, ul (t1, t2) = v - ct2. The utility for player 1 in this case is maximum when t1 > t2. Hence BRI(t2) = {t1 | t1 > t2}, when to v/c, u1 (t1, t2) = 0. The maximum utility for player 1 in this case is 0 which is achieved when ti > t2 or t1 = 0. Hence BRI(t2) = {0} U {ti | ti > t2}, when t2 = v/c. (iii) Let us take the case to > v/c. Then for t] v/c, ul (t1, t2) = v - ct2 v/c