Consider the following 3-players version of the war of attrition. All three players can stay and compete to get a reward that has a value V. Each player pays 1 for each unit of time he/she stays. The winner that gets the reward is the last to quit. If several agents are the last to stay and they quit at exactly the same time, the reward is given to one of them following a fair lottery (each of them has the same probability to get the reward). In the first round, each player chooses her/his first exit time, 0 ≤ xi < ∞, and the player with the lowest value, xmin, pays xmin and quits. The other two players also pay xmin, but they continue to the second round. If several players try to quit together at xmin, only one player actually quits, and that player is determined in a lottery in which each of the players that try to quit at xmin have the same chance. The second round is the usual 2-players war of attrition, where each of the remaining two players chooses her/his exit time 0 ≤ yi < ∞. Both players pay ymin (in addition to the xmin paid in the first round), and the player with the higher exit time gets a reward V . (If both players quit the second round at exactly the same time, the reward is given to one of them following a fair lottery.) Fing the symmetric Nash equilibrium, as well as one non-symmetric Nash equilibrium. What is the expected payoff for each player in each equilibrium? Use backward induction.
Consider the following 3-players version of the war of attrition. All three players can stay and compete to get a reward that has a value V. Each player pays 1 for each unit of time he/she stays. The winner that gets the reward is the last to quit. If several agents are the last to stay and they quit at exactly the same time, the reward is given to one of them following a fair lottery (each of them has the same probability to get the reward). In the first round, each player chooses her/his first exit time, 0 ≤ xi < ∞, and the player with the lowest value, xmin, pays xmin and quits. The other two players also pay xmin, but they continue to the second round. If several players try to quit together at xmin, only one player actually quits, and that player is determined in a lottery in which each of the players that try to quit at xmin have the same chance. The second round is the usual 2-players war of attrition, where each of the remaining two players chooses her/his exit time 0 ≤ yi < ∞. Both players pay ymin (in addition to the xmin paid in the first round), and the player with the higher exit time gets a reward V . (If both players quit the second round at exactly the same time, the reward is given to one of them following a fair lottery.) Fing the symmetric Nash equilibrium, as well as one non-symmetric Nash equilibrium. What is the expected payoff for each player in each equilibrium? Use backward induction.
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