Solution
The Volunteer's Dilemma
Show the problem again
Three people witness a problem. Each simultaneously decides whether to volunteer to fix it. If at least one person volunteers, everyone receives a benefit of 1. A volunteer additionally pays a private cost of 1/9. If nobody volunteers, everyone receives 0. In the symmetric Nash equilibrium in which each person volunteers independently with the same probability p strictly between 0 and 1, what is the probability that exactly one person volunteers? Give a reduced fraction.
Worked solution
The answer is 2/9. A volunteer's payoff is 1 − 1/9 = 8/9 for certain. A non-volunteer gets 1 only if at least one of the other two volunteers, so her payoff is 1 − (1 − p)². Indifference requires (1 − p)² = 1/9, so 1 − p = 1/3 and p = 2/3. The probability that exactly one of the three volunteers is 3·p·(1 − p)² = 3·(2/3)·(1/9) = 2/9. The chance that nobody volunteers is (1/3)³ = 1/27. With n people it equals (1/9)^(n/(n−1)), which rises with n toward 1/9, the bystander effect in miniature: each person's p falls as the group grows.
Source: Volunteer's dilemma, introduced by Diekmann (1985). Numbers written for AxiomIQ.