Again consider an extension of the Stag Hunt game. There are N = 3 hunters. This time each hunter has multiple effort levels in hunting for stag. Specifically, assume each hunter i can choose effort level ei = 0, 1, 2, 3, 4. Given a strategy profile (e1, e2, e3), each player i 0 s payoff = 2 min{e1, e2, e3} − ei . In words, payoff equals twice of the least effort level of everyone subtract player i’s own effort level. For example, if player 3 chooses e3 = 1, players 1 and 2 both choose effort level 4 then the minimum effort level min{e1, e2, e3} = 1, so player 3’s payoff equals 2 × 1 − e3 = 1 and players 1 and 2 each gets payoff 2 × 1 − 4 = −2. a. Is a strategy profile in which all players choose the same effort level a Nash equilibrium? Explain. (Consider what a player’s payoff is given this profile and what payoff he could get by increasing or decreasing his effort.) b. Is any profile in which not all effort levels are equal a Nash equilibrium? Explain.
Again consider an extension of the Stag Hunt game. There are N = 3 hunters. This time each hunter has multiple effort levels in hunting for stag. Specifically, assume each hunter i can choose effort level ei = 0, 1, 2, 3, 4. Given a strategy profile (e1, e2, e3), each player i 0 s payoff = 2 min{e1, e2, e3} − ei . In words, payoff equals twice of the least effort level of everyone subtract player i’s own effort level. For example, if player 3 chooses e3 = 1, players 1 and 2 both choose effort level 4 then the minimum effort level min{e1, e2, e3} = 1, so player 3’s payoff equals 2 × 1 − e3 = 1 and players 1 and 2 each gets payoff 2 × 1 − 4 = −2.
a. Is a strategy profile in which all players choose the same effort level a Nash equilibrium? Explain. (Consider what a player’s payoff is given this profile and what payoff he could get by increasing or decreasing his effort.)
b. Is any profile in which not all effort levels are equal a Nash equilibrium? Explain.
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