Measured Games · Paper 09 of 07 · Social psychology

The volunteer's dilemma

After A. Diekmann (1985), formalising the bystander effect reported by J. M. Darley and B. Latané (1968)

Abstract

A group benefits if at least one member acts, but acting is costly and one volunteer suffices. The symmetric mixed equilibrium has each member volunteering with a probability that falls as the group grows, and the probability that nobody at all volunteers rises with group size. Diffusion of responsibility is therefore a prediction of rational play rather than a failure of it, and the effect is present in the arithmetic before any psychology is added.

Keywordsbystander effect · diffusion of responsibility · N-person game · mixed equilibrium

1Setup

Setup

A group of N people faces a situation that will resolve well if at least one of them acts. Acting costs the actor something. Everyone receives the benefit whether or not they were the one who acted, and a second volunteer adds nothing. Calling an ambulance, reporting a fault, staying late to finish a shared task.

Nobody has an incentive to act if they expect someone else to. Nobody can rely on that, because everyone is reasoning the same way. The interesting question is not whether a particular person volunteers but how the probability that nobody does changes as the group grows.

2The model

The model

# N players, benefit b if at least one volunteers, cost c to the volunteer volunteer = b - c abstain, someone else volunteers = b abstain, nobody volunteers = 0 # no pure symmetric equilibrium: if nobody else volunteers you should, # and if somebody else does you should not # symmetric mixed equilibrium, each volunteers with probability p indifference: c/b = (1 - p)N-1 p* = 1 - (c/b)^(1/(N-1)) # and the quantity that matters for the group P(nobody volunteers) = (c/b)^(N/(N-1))

Both consequences follow immediately. Individual willingness p* falls as N rises, which is unsurprising. The probability that the act never happens at all also rises with N, which is not: adding people to a group makes the group less likely to respond.

Figure 1Symmetric mixed play across group sizes

This figure needs JavaScript. The panels below are live simulations rather than images, so with scripting turned off they stay empty and the controls do nothing. Everything the figure demonstrates is also stated in the surrounding text: the model is given in full in section 2 and the result in section 3.

One group per round · filled markers volunteered

Failure rate against group size · measured against the closed form

Individual willingness and group failure, as predicted

Volunteered Nobody volunteered Predicted

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Group size N

Cost as a fraction of benefit, c/b

Rounds per batch

Predicted p*—

Observed p—

Predicted failure—

Observed failure—

Volunteers per round—

Rounds0

Figure 1. Every member plays the symmetric equilibrium mixed strategy for the current N and cost ratio, so the failure rate is an emergent count rather than a formula being plotted back at you. Sweep N runs every group size from two to forty and traces the measured failure rate against the closed form. The lower panel shows the two curves that make the result uncomfortable: individual willingness falling while group failure rises.

3Results

Results

Diekmann (1985) set out the game and its symmetric mixed equilibrium. The equilibrium exists and is unique among symmetric strategies, and both comparative statics follow from it directly. As N grows, each person volunteers less often and the group fails more often, with failure tending to the cost ratio c/b in the limit of a large group.

The result is worth separating from the psychology it resembles. No diffusion of responsibility, no pluralistic ignorance and no failure of nerve is required to produce it. Perfectly rational players who care about the outcome, know the payoffs and reason correctly will still leave the act undone more often in a large group than in a small one. Any psychological mechanism operating on top of that is additional to it, not a substitute for it.

The asymmetric versions matter for the same reason. If one member has a lower cost or a higher benefit, the equilibrium loads the responsibility onto them, and everyone else volunteers less. Being visibly the most capable person present makes it more likely that you will have to act and less likely that anyone will help.

4Evidence

Evidence

Darley and Latané (1968) ran the founding experiment. Subjects who believed they were the only witness to an apparent seizure reported it in the large majority of cases and quickly. Subjects who believed four others were also present reported it substantially less often and more slowly. The effect was in group size, not in whether the subject believed an emergency was occurring.

Diekmann (1986) tested the game-theoretic version directly, varying group size in a laboratory volunteer's dilemma with monetary payoffs, and found the predicted decline in individual volunteering. Franzen (1995) and subsequent experimental work found the same qualitative pattern, with the usual caveat that observed volunteering rates are often higher than the equilibrium predicts.

Fischer and colleagues (2011) meta-analysed roughly fifty years of bystander research across more than 100 studies. The effect is real and robust, and they also report a moderator worth knowing: in situations that are clearly dangerous, the bystander effect weakens and can reverse, because bystanders become a source of physical support rather than only of diffused responsibility.

5Limitations

Limitations

The model assumes everyone knows N, that costs and benefits are common knowledge, and that one volunteer genuinely suffices. Real situations violate all three, and the third is often the important one: where a second helper adds real value, the game is not a volunteer's dilemma at all.

The bigger caution is about what the model licenses you to say. It shows that group size alone is enough to produce inaction, which means observing inaction in a crowd does not license any inference about the character of the people in it. It does not show that psychological diffusion is absent, and the experimental literature suggests it is not. Treat the model as establishing a floor under the effect rather than as an account of the whole of it.

Observed volunteering usually exceeds the equilibrium rate. Whatever produces that, altruism, social norms or inability to compute a mixed strategy, is outside the model.

References

Darley, J. M., & Latané, B. (1968). Bystander intervention in emergencies: Diffusion of responsibility. Journal of Personality and Social Psychology, 8(4, Pt.1), 377–383. https://doi.org/10.1037/h0025589
Diekmann, A. (1985). Volunteer's dilemma. Journal of Conflict Resolution, 29(4), 605–610. https://doi.org/10.1177/0022002785029004003
Diekmann, A. (1986). Volunteer's dilemma: A social trap without a dominant strategy and some empirical results. In A. Diekmann & P. Mitter (Eds.), Paradoxical effects of social behavior (pp. 187–197). Physica-Verlag.
Fischer, P., Krueger, J. I., Greitemeyer, T., Vogrincic, C., Kastenmüller, A., Frey, D., Heene, M., Wicher, M., & Kainbacher, M. (2011). The bystander-effect: A meta-analytic review on bystander intervention in dangerous and non-dangerous emergencies. Psychological Bulletin, 137(4), 517–537. https://doi.org/10.1037/a0023304
Franzen, A. (1995). Group size and one-shot collective action. Rationality and Society, 7(2), 183–200. https://doi.org/10.1177/1043463195007002006
Latané, B., & Darley, J. M. (1970). The unresponsive bystander: Why doesn't he help? Appleton-Century-Crofts.

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