= The 2. Contest where players are asymmetric (8 points). Two players participate in a contest: player 1 and player 2. Both players decide how much effort to spend (any positive number) in order to win the contest. The effort expenditures are denoted by x and X2. The player l's probability of winning the contest is axi X2 P1(x1, x2) = - and the player 2's probability of winning the contest is p2(x1, x2) axy + x2 axy + x2 player who wins the contest receives a prize V. Regardless of who wins the contest both players have to pay their effort expenditures. The cost to player i of expending effort x; is C}(x;) = x;. (Note: the difference between this problem and the problem considered in the class is that in the class we assumed that both players affect the X probabilities of winning in the same way p:(x1, x2) -, while here player 1 is a times more skilled than X1 + X2 ax , player 2 and thus he has a greater effect on the probability of winning p.(x1, x2) -). axi + x2 = a) Write down the normal form representation of this game (2 points). b) Find the best response functions of player 1 and player 2 (2 points). c) Find the Nash equilibrium efforts x* and x* of this game using the FOCs (2 points). d) Calculate the equilibrium effort of player 2 when a = 1 and V = 100. Then calculate the equilibrium effort of player 2 when a = 4 and V = 100. (Note: a = 1 means that players have the same abilities, while a = 4 means that one player is a superstar). Provide intuition as to why these efforts are different. (2 points). = The 2. Contest where players are asymmetric (8 points). Two players participate in a contest: player 1 and player 2. Both players decide how much effort to spend (any positive number) in order to win the contest. The effort expenditures are denoted by x and X2. The player l's probability of winning the contest is axi X2 P1(x1, x2) = - and the player 2's probability of winning the contest is p2(x1, x2) axy + x2 axy + x2 player who wins the contest receives a prize V. Regardless of who wins the contest both players have to pay their effort expenditures. The cost to player i of expending effort x; is C}(x;) = x;. (Note: the difference between this problem and the problem considered in the class is that in the class we assumed that both players affect the X probabilities of winning in the same way p:(x1, x2) -, while here player 1 is a times more skilled than X1 + X2 ax , player 2 and thus he has a greater effect on the probability of winning p.(x1, x2) -). axi + x2 = a) Write down the normal form representation of this game (2 points). b) Find the best response functions of player 1 and player 2 (2 points). c) Find the Nash equilibrium efforts x* and x* of this game using the FOCs (2 points). d) Calculate the equilibrium effort of player 2 when a = 1 and V = 100. Then calculate the equilibrium effort of player 2 when a = 4 and V = 100. (Note: a = 1 means that players have the same abilities, while a = 4 means that one player is a superstar). Provide intuition as to why these efforts are different. (2 points)
