Plate xii in print

The Hunt

the stag hunt and the assurance game; the great prize that needs every hand, and the mutual trust required before anyone dares reach for it.

The prize becomes reachable when each hand has reason to expect the others.

— the argument of this plate, after Rousseau, Hume and Skyrms

The great prize can be available to everyone and still remain untaken because each person must decide whether the others will stay. The earlier Collected Coordination ended in the forest, with a band whose stag requires all the hunters to hold their posts and whose hares can be caught alone. Rousseau’s passage supplies the departing hunter: a hare comes within reach, and he pursues it without concern for the companions whose deer he may cause them to lose. The later assurance game gives this scene a more exact structure. It supposes that each hunter prefers a successful joint hunt to the solitary alternative, yet prefers that alternative to waiting alone for a stag that will never come. Those rankings make cooperation an equilibrium. They also give retreat a rational place in the game, whenever the expectation of being abandoned makes the larger prize too doubtful.1

1. Rousseau, Discourse on Inequality, Part II, deer-and-hare passage, G. D. H. Cole translation in the 1920 collection; Skyrms, The Stag Hunt, March 2001 presidential-address manuscript, pp. 1–4. Skyrms distinguishes the story’s open questions from the formal game.

Confidence need only be sufficient for the gamble the hunter actually faces. Consider an invented two-person hunt with payoffs measured in units of satisfaction. Each receives six from a stag if both hunt it; a hunter who pursues stag alone receives zero; choosing hare gives four whatever the other does. Let q be the probability one hunter assigns to the other choosing stag. Stag then offers an expected payoff of 6q, against four from hare, so stag is strictly preferable when q exceeds two-thirds. At a three-quarter probability it yields four and a half; some doubt remains, but joining is worthwhile. At an even chance it yields three, and hare is the better response. The earlier forest image therefore needs a threshold, rather than the claim that the first trace of doubt sends every hunter away. Trust matters through the decision it makes reasonable.2

2. The six/four/zero example and all probabilities are invented. Skyrms, manuscript pp. 3–4, analyzes a different numerical example using expected utility and conditional best responses. Here q = 2/3 gives indifference; it does not make either action strictly preferable.

The equilibrium with the larger prize need not be the one supported by the broader range of beliefs. In this symmetric two-person example, stag with stag is better for both than hare with hare: six exceeds four. Yet hare is the best response for all beliefs below two-thirds, while stag requires the narrower interval above it. This is the sense in which hare is risk-dominant here, although stag is payoff-dominant. No special aversion to risk was needed to obtain that result; the calculation already used expected satisfaction. Nor does the name of the animal determine the conclusion. Raise the successful stag payoff to nine, keeping hare at four and failure at zero, and the threshold falls to four-ninths. Stag now wins even against an even chance. What appeared a contrast between courage and caution depends on the rewards, the exposure and the expectations built into the hunt.3

3. Skyrms, manuscript pp. 3–4 and n. 2 on p. 19. The belief-interval comparison is restricted to the symmetric two-action game here; neither risk dominance nor payoff dominance by itself predicts actual play. The nine-unit variation is invented.
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A larger band can make confidence in each companion insufficient for confidence in the whole undertaking. Extend the invented hunt to five people, all of whom must choose stag for anyone to receive six. Suppose one hunter assigns each of the other four a nine-in-ten chance of staying, independently. The probability that all four stay is 0.9⁴, or 0.6561, just below the two-thirds threshold; the expected payoff of stag is 3.9366, below hare’s four. If their actions instead move together, with all four staying nine times in ten, those same individual probabilities give a joint success probability of 0.9. The hunter should then join. Dependence between actions changes the answer. Hume’s contrast between two neighbours draining a meadow and a large undertaking put a related difficulty in social terms: people must know the plan, expect the others’ contributions and carry their own part through. Mere headcount cannot tell us whether they will.4

4. Hume, Treatise, 3.2.7.8, considers knowledge of others’ intentions, execution and attempts to shift burdens. The five-person calculation is invented and separates independent from perfectly correlated actions; it does not adopt his categorical claim about a thousand people.

Assurance can also be built by reducing what a willing hunter loses when the others fail to appear. Return to the two-person game and suppose a credible, externally funded arrangement raises the lone stag hunter’s payoff from zero to three, leaving the other payoffs unchanged. Stag now yields 6q + 3(1 − q), and beats hare when q exceeds one-third. The reassurance has changed the terms of the choice. A promise of compensation would accomplish less if nobody could pay it, if the promised payment arrived too late to be useful, or if the supposed protection concealed an offsetting charge. The model assumes these difficulties away so that its mechanism can be seen; an actual arrangement must solve them. Shared expectations may make an existing cooperative equilibrium reachable, while protection against failed cooperation may enlarge the circumstances in which reaching for it is sensible.5

5. The compensation and funding are invented assumptions, not a policy estimate. Compare Hume, Treatise, 3.2.7.5–8, on changing people’s situation and making observance their immediate interest, with Skyrms, manuscript pp. 3–4, on belief-dependent choice. Changing incentives and changing expectations are distinct possible routes.

The person who declines the hunt may therefore be answering a danger that the invitation has left with them. An imagined cooperative workshop might require every participant to give up a day’s paid employment before production can begin. Calling the common product valuable does not establish that each person can bear the loss if another fails to come, or that the finished product will be divided so that each prefers participation. A guarantee that protects only the organizer may leave the workers’ exposure untouched. Even where the formal assurance rankings do hold, a history of missed contributions can make retreat intelligible. If the rankings do not hold — if somebody is better off leaving even when everyone else stays — the undertaking contains a further incentive problem. The language of trust should help identify these circumstances, rather than turn the person who notices them into the explanation for failure.

We dare to reach together when the promised gain, the prospect of reciprocal action and the consequences of failure make participation worth choosing. Sometimes a familiar practice supplies the confidence; sometimes a public commitment, a workable sequence of contributions or a funded protection must give it substance. The stag hunt ends this volume with cooperation already possible and its realization still to be earned. An equilibrium describes a pattern in which no hunter wants to depart alone; it does not place the hunters at their posts, distribute the meat or reimburse the one who waited in vain. Those are tasks for the arrangements through which people come to rely on one another. The forest contains more than a prize and a temptation. It contains the distance between a benefit that could be shared and a shared undertaking that each person has reason to enter.