Volunteer's dilemma
The n-person version of chicken. The good is provided if at least one person pays the cost; each would rather someone else did, and the chance that nobody acts rises with group size.
Volunteer's dilemma
A good is provided if at least one person volunteers. Everyone gets the benefit U; the volunteer also pays a cost K, smaller than U. If nobody volunteers, everyone gets nothing. The name and model are Diekmann's (Journal of Conflict Resolution, 1985 [?]; original not read). The same model and group-size result are in Dixit, Skeath and Reiley (Games of Strategy, 4th ed., 2015, pp. 455–457), who base it on Palfrey and Rosenthal (1984) and present it as chicken with mixed strategies.
Two players, U = 4, K = 1, payoffs (row, column):
| Column volunteers | Column waits | |
|---|---|---|
| Row volunteers | 3, 3 | 3, 4 |
| Row waits | 4, 3 | 0, 0 |
This is Chicken (game) with a tie: volunteering costs the same whether or not the other also volunteers. Greed, no fear (Greed and fear).
Equilibria
- Pure: exactly one volunteer. Nobody volunteering is not an equilibrium: if nobody else will, I should.
- Symmetric mixed: each waits with probability q = (K/U)^(1/(n−1)). The probability that nobody acts is (K/U)^(n/(n−1)), which rises with n towards K/U. With U = 4 and K = 1, nobody acts 1 time in 16 with two people, and about 1 time in 5 with ten.
Each person is less likely to act in a bigger group, and so is the group as a whole, though the chance of help levels off rather than reaching zero:
the larger the group, the less likely is help to be given at all.
— Dixit, Skeath and Reiley, Games of Strategy (4th ed., 2015), p. 457
This is the formal match to the bystander effect (Darley and Latané 1968 [?]).
Names
Bystander problem, "someone should do something". The course teaches it as chicken, the many-player form of Chicken (game). More aliases: Names of collective action problems.
Typical failure and fix
- Failure: waiting, diffusion of responsibility, the wrong person paying. In big groups, nobody acts.
- Fix: allocation. Point at one person ("you, call an ambulance"), rotate the duty, take turns, or pay the volunteer. Assurance backfires: if I know you'll act, I wait. A big player who gains enough to act alone often resolves it.
each prefers the other to do all the work.
— Michael Taylor, The Possibility of Cooperation (1987), p. 36, on two cultivators maintaining dykes
Taylor adds that the consequences of nobody doing the work are bad enough that either would do it if the other didn't (p. 36).
Cases
- A person collapses in a crowded street.
- The Alanya fishing-site rotation, a taking-side case: Ostrom (Governing the Commons, 1990, p. 47) says it "cannot be represented as a PD game" and calls it an "assignment" game. The rotation and annual lottery are the fix (pp. 18–20).
- Heartbleed and the under-funded open source maintainer fit here and as a Prisoner's Dilemma: one big funder is enough, but many small users each gain little.
Related
- Aggregation technologies: best-shot goods have the volunteer's payoff order, though Sandler calls the game a coordination game.
- Chicken vs prisoner's dilemma
- Patterns of collective action
