Classic Games: Prisoner’s Dilemma, Battle of the Sexes and Volunteer’s Dilemma

Three classic games illustrate the main strategic tensions: in the prisoner’s dilemma individual incentives lead to a collectively worse outcome, in the battle of the sexes players want to coordinate but disagree about where, and in the volunteer’s dilemma one person’s costly action benefits everyone, so each hopes that somebody else will act. In brief: in the prisoner’s dilemma the only equilibrium (defect, defect) pays (1, 1) instead of (3, 3). In the battle of the sexes there are two pure equilibria and a mixed one with the probabilities 2/3 and 1/3. In the volunteer’s dilemma with benefit 10 and cost 2, a group of 3 persons each volunteer with the probability of about 0.55 and the chance that nobody acts is about 9%.

These games are the standard models for cooperation, coordination and responsibility. This guide gives the payoff structure of each, the equilibria, and the real-world interpretation, with worked calculations and a comparison table.

What is the prisoner’s dilemma?

Two players each choose to cooperate (C) or defect (D). The payoffs satisfy T > R > P > S: the temptation to defect T, the reward for mutual cooperation R, the punishment for mutual defection P, and the sucker’s payoff S. Typically 2R > T + S, so that alternating exploitation is not better than steady cooperation. With T = 5, R = 3, P = 1, S = 0: D strictly dominates C for both players, so (D, D) is the unique equilibrium with the payoff (1, 1), although (C, C) gives (3, 3).

Example 1: the dilemma in real situations

Two firms agreeing to keep prices high each gain by undercutting; two countries agreeing on a mutual arms reduction each gain by cheating; several fishing fleets sharing a lake each gain by overfishing. Whenever individual gain from defecting exceeds the loss from the other’s defection spread over everyone, a dilemma arises, with the common resource (the tragedy of the commons) as the many-player version.

Example 2: checking the structure

Payoffs (row, column): (C, C) = (3, 3), (C, D) = (0, 5), (D, C) = (5, 0), (D, D) = (1, 1). T = 5 > R = 3 > P = 1 > S = 0 ✓ and 2R = 6 > T + S = 5 ✓. The game is a dilemma, since the sum of payoffs is highest (6) at (C, C) but each player’s best reply is D.

What is the battle of the sexes?

A couple wants to spend the evening together, but one prefers the opera (O) and the other football (F). Payoffs (Alex, Blair): (O, O) = (2, 1), (F, F) = (1, 2), and (O, F) = (F, O) = (0, 0), since going separately is worst. There are two pure equilibria, (O, O) and (F, F), which favor different players, and a mixed equilibrium. Let Alex choose O with the probability p and Blair choose O with the probability q.

Example 3: the mixed equilibrium

Blair is indifferent between O and F when Alex’s mix gives equal expected payoffs: for O, 1 · p; for F, 2 · (1 − p). So p = 2(1 − p), p = 2/3. Alex is indifferent when 2q = 1 · (1 − q), q = 1/3. In the mixed equilibrium, Alex goes to the opera with probability 2/3 and Blair with the probability 1/3. Each player’s expected payoff is 2/3 (Alex: 2q = 2/3; Blair: p = 2/3), lower than any pure equilibrium: they meet only with probability pq + (1 − p)(1 − q) = 2/9 + 2/9 = 4/9. This shows the cost of miscoordination.

Example 4: resolving the conflict

Communication, turn-taking (opera this week, football the next), or a public randomizing device (a coin flip) lead to coordination. With a fair coin, the expected payoff is (2 + 1)/2 = 1.5 for each player, better than 2/3.

What is the volunteer’s dilemma?

In a group of n players, if at least one volunteers, everybody receives the benefit b, but the volunteer pays the cost c (0 < c < b). If nobody volunteers, everybody gets 0. A volunteer therefore receives b − c, a nonvolunteer receives b if someone else volunteers, and 0 otherwise. Examples: a bystander calling for help, a coworker fixing a shared problem, one neighbor reporting a hazard.

In the pure equilibria exactly one player volunteers (nobody else gains by joining). It is not clear who should do so, and symmetric players cannot coordinate; so one looks for the symmetric mixed equilibrium. If every other player volunteers independently with the probability p, then a given player who does not volunteer gets b with the probability 1 − (1 − p)ⁿ⁻¹. Indifference with b − c gives (1 − p)ⁿ⁻¹ = c/b.

Example 5: the equilibrium probabilities

With b = 10 and c = 2, c/b = 0.2. For n = 2: 1 − p = 0.2, p = 0.8. For n = 3: (1 − p)² = 0.2, 1 − p = √0.2 ≈ 0.447, p ≈ 0.553. For n = 10: 1 − p = 0.2^1/9 ≈ 0.836, p ≈ 0.164.

Example 6: the probability that nobody acts

The probability that nobody volunteers is (1 − p)ⁿ = (c/b)^n/(n−1). For n = 2: 0.2² = 0.04. For n = 3: 0.2^1.5 ≈ 0.089. For n = 10: 0.2^10/9 ≈ 0.167. Paradoxically, the larger the group, the more likely it is that nobody helps, the formal analogue of the bystander effect: each person’s incentive to act falls faster than the number of potential helpers rises.

Can cooperation survive repetition?

If the prisoner’s dilemma is repeated indefinitely, players can cooperate by threatening punishment. With the trigger strategy “cooperate until the other defects, then defect forever” and the discount factor δ per round (how much the future counts), cooperating forever yields R/(1 − δ), while deviating yields T + δP/(1 − δ). Cooperation is an equilibrium when R/(1 − δ) ≥ T + δP/(1 − δ), that is δ ≥ (T − R)/(T − P).

Example 7: how patient must players be?

With T = 5, R = 3, P = 1: δ ≥ (5 − 3)/(5 − 1) = 1/2. Check at δ = 0.5: cooperating forever gives 3/(0.5) = 6; deviating gives 5 + 0.5 · 1/0.5 = 6: equal, as expected at the threshold. A well-known tournament by Robert Axelrod (1980) found that the simple strategy tit-for-tat (cooperate first, then copy the opponent’s last move) did best in repeated play.

Table of the three games

Where do these games come from?

The prisoner’s dilemma was formulated by Merrill Flood and Melvin Dresher in 1950 and named by Albert Tucker. The battle of the sexes appears in the textbook of R. Duncan Luce and Howard Raiffa (1957). The volunteer’s dilemma was named and analyzed by Andreas Diekmann in 1985, building on the work on the bystander effect by the psychologists Bibb Latané and John Darley in the 1960s. The collective version of the first game, the tragedy of the commons, was popularized by Garrett Hardin in 1968.

Which mistakes are common?

Calling any conflict a prisoner’s dilemma

The defining feature is the payoff ordering T > R > P > S with a dominant defect strategy. Many conflicts have other structures, such as coordination games.

Expecting the mixed equilibrium to be a good outcome

In the battle of the sexes the mixed equilibrium is worse for both than either pure one, because of the chance of miscoordination.

Mixing up the probabilities

In a mixed equilibrium, each player’s probability makes the other player indifferent, not themselves. Alex’s p = 2/3 comes from Blair’s payoffs.

Assuming that more bystanders mean more help

In the volunteer’s dilemma, the chance that nobody helps increases with the group size, for a fixed cost-benefit ratio.