The volunteer's dilemma is a

game A game is a structured type of play usually undertaken for entertainment or fun, and sometimes used as an educational tool. Many games are also considered to be work (such as professional players of spectator sports or video games) or art ...

that models a situation in which each player can either make a small sacrifice that benefits everybody, or instead wait in hope of benefiting from someone else's sacrifice. One example is a scenario in which the electricity supply has failed for an entire neighborhood. All inhabitants know that the electricity company will fix the problem as long as at least one person calls to notify them, at some cost. If no one volunteers, the worst possible outcome is obtained for all participants. If any one person elects to volunteer, the rest benefit by not doing so. A

public good In economics, a public good (also referred to as a social good or collective good)Oakland, W. H. (1987). Theory of public goods. In Handbook of public economics (Vol. 2, pp. 485–535). Elsevier. is a commodity, product or service that is bo ...

is only produced if at least one person volunteers to pay an arbitrary cost. In this game, bystanders decide independently on whether to sacrifice themselves for the benefit of the group. Because the volunteer receives no benefit, there is a greater incentive for freeriding than to sacrifice oneself for the group. If no one volunteers, everyone loses. The social phenomena of the

bystander effect The bystander effect, or bystander apathy, is a social psychological theory that states that individuals are less likely to offer help to a victim in the presence of other people. The theory was first proposed in 1964 after the murder of Kitty ...

and

diffusion of responsibility Diffusion of responsibility is a sociopsychological phenomenon whereby a person is less likely to take responsibility for action or inaction when other bystanders or witnesses are present. Considered a form of attribution, the individual assume ...

heavily relate to the volunteer's dilemma.

Payoff matrix

The

payoff matrix In game theory, normal form is a description of a ''game''. Unlike extensive-form game, extensive form, normal-form representations are not Graph (discrete mathematics), graphical ''per se'', but rather represent the game by way of a matrix (mathe ...

for the game is shown below: When the volunteer's dilemma takes place between only two players, the game gets the character of the game "

chicken The chicken (''Gallus gallus domesticus'') is a domesticated subspecies of the red junglefowl (''Gallus gallus''), originally native to Southeast Asia. It was first domesticated around 8,000 years ago and is now one of the most common and w ...

". As seen by the payoff matrix, there is no dominant strategy in the volunteer's dilemma. In a mixed-strategy

Nash equilibrium In game theory, the Nash equilibrium is the most commonly used solution concept for non-cooperative games. A Nash equilibrium is a situation where no player could gain by changing their own strategy (holding all other players' strategies fixed) ...

, an increase in N players will decrease the likelihood that at least one person volunteers, which is consistent with the

Examples in real life

The murder of Kitty Genovese

The story of

Kitty Genovese In the early hours of March 13, 1964, Kitty Genovese, a 28-year-old bartender, was raped and stabbed outside the apartment building where she lived in the Kew Gardens, Queens, Kew Gardens neighborhood of the Queens borough of New York City, Uni ...

is often cited as an example of the volunteer's dilemma. Genovese was stabbed to death outside her apartment building in Queens, New York, in 1964. According to a highly influential ''New York Times'' account, dozens of people witnessed the assault but did not get involved because they thought others would contact the police anyway and did not want to incur the personal cost of getting involved. Subsequent investigations have shown the original account to have been unfounded, and although it inspired sound scientific research, its use as a simplistic parable in psychology textbooks has been criticized.

The meerkat

The

meerkat The meerkat (''Suricata suricatta'') or suricate is a small mongoose found in southern Africa. It is characterised by a broad head, large eyes, a pointed snout, long legs, a thin tapering tail, and a brindled coat pattern. The head-and-body ...

exhibits the volunteer's dilemma in nature. One or more meerkats act as sentries while the others forage for food. If a predator approaches, the sentry meerkat lets out a warning call so the others can burrow to safety. However, the altruism of this meerkat puts it at risk of being discovered by the predator.

Quantum volunteer's dilemma

One significant volunteer's dilemma variant was introduced by Weesie and Franzen in 1998 and involves cost-sharing among volunteers. In this variant of the volunteer's dilemma, if there is no volunteer, all players receive a payoff of 0. If there is at least one volunteer, the reward of b units is distributed to all players. In contrast, the total cost of c units incurred by volunteering is divided equally among all the volunteers. It is shown that for classical mixed strategies setting, there is a unique symmetric Nash equilibrium and it is obtained by setting the probability of volunteering for each player to be the unique root in the open interval (0,1) of the degree-n polynomial

g_n

given by

g_n(\alpha) = (1-\alpha)^ (2n \alpha + 1 - \alpha) - 1

In 2024, a quantum variant of the classical volunteer’s dilemma was introduced with b=2 and c=1. This generalizes the classical setting by allowing players to utilize quantum strategies. This is achieved by employing the Eisert–Wilkens–Lewenstein quantization framework. In this setting, the players receive an entangled n-qubit state with each player controlling one qubit. The decision of each player can be viewed as determining two angles. Symmetric Nash equilibria that attain a payoff value of

2-1/n

for each player is shown, and each player volunteers at this Nash equilibrium. Furthermore, these Nash equilibria are

Pareto optimal In welfare economics, a Pareto improvement formalizes the idea of an outcome being "better in every possible way". A change is called a Pareto improvement if it leaves at least one person in society better off without leaving anyone else worse ...

. It is shown that the payoff function of Nash equilibria in the quantum setting is higher than the payoff of Nash equilibria in the classical setting.

References