Thought Toys · Games & choices · Exhibit 129
Something needs doing and several people can see it. Whoever acts pays a small price; everybody benefits either way. Nobody can arrange who goes. Each of them ends up exactly as willing to act as to wait — that indifference is the only way a room of equals can settle — and yet the more people are watching, the likelier it becomes that nobody does. Not because anyone stops caring. Because each of them is right that somebody else probably will.
One round at a time — who acts, and how often nobody does ● acted this round ● waited — if willingness never changed
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Each round, every person decides independently whether to act. Nobody signals, nobody takes turns. The only thing they have in common is that they are all reasoning the same way about the same situation, and each of them knows the others are too.
That fixes how willing each one is. If you were sure to act, everyone else would happily leave it to you — so being sure cannot be right. If you were sure to wait, so would they, and the thing would never get done — so that cannot be right either. The only stable answer is the willingness that leaves you exactly indifferent: the chance that somebody else handles it has to make waiting worth precisely what acting is worth. Nothing more mysterious than that determines the number in the first box.
Now add people. Each newcomer makes it likelier that somebody will act, which makes waiting more attractive, which makes each individual less willing. Those two effects pull against each other — and the second one wins, slightly, every time. The chance nobody acts climbs. Drag the crowd from 2 up to 60 and watch the amber curve rise.
Be careful what that does not say. Each person is not becoming more likely to act; each becomes far less likely to. At the settings this page opens with, one of two people acts 60% of the time and one of sixty acts 1.5% of the time. What holds steady is the indifference, not the willingness. And the climb has a ceiling: the chance nobody acts rises toward the cost of acting and never past it, so a crowd makes things worse by a bounded amount, not an unbounded one.
Freeze the two-person habit is the control that shows this is about the response and not the crowd. Hold each person's willingness at whatever it was when there were only two of them, then add people: now the cyan curve climbs toward certainty, exactly as intuition says it should. The crowd was never the problem. Reacting sensibly to the crowd is.
One more thing worth noticing in the third box. As you add people the expected number of volunteers does not grow — it settles. Past a certain size, a bigger crowd contributes no more helpers than a small one; it just spreads the same expected effort more thinly, and misses more often.
This is not a model of panic, apathy or moral failure, and it does not need one. Every person here is attentive, willing, and correct. It is also not a claim about what any particular real bystander does — real situations add noise, visibility, cost differences and the ability to catch someone's eye, all of which this leaves out on purpose. What it does show is that you do not need any of those to get the result. The arithmetic of several people independently reasoning well is enough.
The rule, exactly. N people; benefit 1 to everyone if at least one acts; the volunteer pays
c, with 0 < c < 1. Playing probability p against everyone else's
q, a person expects p(1−c) + (1−p)[1 −
(1−q)N−1], so the symmetric equilibrium sits where the bracket makes
them indifferent:
p* = 1 − c1/(N−1),
P(nobody acts) = cN/(N−1)
which is c2 at N = 2 and rises toward c as the crowd grows
(Diekmann, 1985). The expected number of volunteers, N·p*, converges to
ln(1/c) — a constant. The rounds you see are drawn independently at p*, so the
tally in the fourth box is a genuine sample, not the formula redrawn.
Checked before this page was written (improve/verify/129-volunteers-dilemma.js):
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