Hume's two neighbours have corn that ripens on different days, and each would harvest faster with the other's help. Played once, the exchange is the prisoner's dilemma and nobody helps; played season after season by farmers who answer help with help, it pays both of them. The farmers are no less selfish in the second case: what has changed is that there is a next season.
Core ideas
Hume's story is short, and the three ideas below are all that the lab needs from it.
Each farmer does best by being helped and not helping, next best by helping and being helped, worse by neither, and worst by helping alone. That is the ordering of the prisoner's dilemma, and the four payoffs here are the course's standard ones: 3, 0, 5 and 1.
Played once, refusing pays more than helping whatever the neighbour does. Both refuse, each collects 1, and the harvests are lost for want of the help that would have paid both.
Hume's farmers need no promise and no kindness. A farmer helps because the neighbour will stop helping otherwise, and the neighbour helps for the same reason. The habit is stable when each, keeping it, does best given that the other keeps it.
The material
Hume's story, the table behind it, and the one change that turns refusal into help.
Hume imagines two neighbours whose corn is ripe on different days. Your corn is ripe today and mine will be ripe tomorrow, and it would profit us both that I labour with you today and you aid me tomorrow. But I have no kindness for you and know you have as little for me, so I will not take pains on your account; and if I did, I should expect to be disappointed. You reason the same way. In his words, “both of us lose our harvests for want of mutual confidence and security.”
Write each season as a game in which both farmers choose, help or refuse, and the payoffs follow the order just described. Hume's farmers in fact move in turns, and the lab has them choose together; the dilemma is the same, and the turns are left out. The four numbers below are the course's standard ones and nothing in the lab knows about corn, so only their order should be read as Hume's.
Played once, this is the dilemma of “The Prisoner's Dilemma”. Refusing pays 5 against a farmer who helps and 1 against one who does not, so it beats helping either way. Both refuse and each collects 1. That dominance was computed in “The Prisoner's Dilemma”'s lab, which checks every cell; the lab here plays matches, and its first preset only plays the single harvest between two farmers who refuse and prints the 1 each. Hume's conclusion is right for one harvest, and it has nothing to do with the farmers being unkind: change them into saints with the same payoffs and the table is the same.
The tempted farmer is whichever one has just been helped. Refusing to return the help pays 5 this season in place of 3, a gain of 2. Hume's point is that the same farmer will want help again, and that the neighbour knows it. If the neighbour follows tit for tat, which helps first and afterwards does whatever the other did last, then a refusal is answered by a refusal, and the gain of 2 is followed by seasons at 1 where there would have been 3.
That trade is worth refusing exactly when the future counts for little. From “The Shadow of the Future”, with these payoffs tit for tat is stable against itself when the chance of another season is at least 2/3. The lab's second preset sets that chance at 9/10, and the tile says it sustains cooperation.
A convention is a regularity of behaviour that each member keeps because the others keep it, so that no one gains by departing from it alone. Hume's example is two men rowing a boat, who pull together without ever having promised to.
Twelve seasons of tit for tat against tit for tat are all help, and each farmer collects 12 times 3, which is 36. Nobody gave a word, and nobody is kind. Each is doing the best available given what the other will do, which is the definition of an equilibrium met in “Nash Equilibrium in Pure and Mixed Strategies”. A convention in Hume's sense is that equilibrium, reached by habit.
The word is wider here than in “Coordination, Conventions and the Stag Hunt”, where a convention picked one of several matches in a coordination game and another match would have served as well. Hume's rowers are a case of that kind. His farmers are not: helping is not an equilibrium of the one-harvest game at all, and becomes one only because the seasons repeat and a refusal can be answered. What the two uses share is the part that does the work in both lessons: a regularity that each keeps because the others keep it, with no promise behind it.
The third preset has one farmer suspicious: he refuses in the first season and afterwards copies his neighbour, who helps first. In the first season the suspicious farmer collects 5 and the neighbour 0. In the next each copies the other, and the roles reverse. The pair then alternate, helped and refused in turn, and over twelve seasons each collects 30, six short of what the two trusting farmers made. Neither can leave the pattern by following his own rule.
Three limits belong here. The twelve seasons are a stand-in for a game with no known end: farmers who knew which harvest was the last would, by the argument in the previous lesson, stop helping before it and unravel the whole run. Second, tit for tat is not the only way the pair could play, and two farmers who always refuse are an equilibrium too. Third, the payoffs are chosen, and a farmer who also enjoys the company is playing a different game.
Interactive
one match, a tournament of seven strategies, and the discount that sustains cooperation
Method
The same five moves for any pair of farmers.
Put the four outcomes in order of what each farmer would rather have, and give them the course's numbers. If help-and-be-helped is not below being helped for nothing, the game is not a dilemma.
Compare refusing with helping against each choice of the neighbour. Refusing wins both comparisons, so one harvest ends with nobody helping.
Tit for tat helps first and then copies. State what it plays in the opening season and what it plays after a refusal.
Sum the seasons for each farmer. Two rules that never refuse each other earn the reward every season.
The farmer who refuses gains 2 once and loses 2 in each later season the neighbour answers. Whether that is a loss depends on the continuation chance, not on the farmer's character.
Payoffs 3, 0, 5, 1 for help-and-help, help alone, refusal alone, neither.
The gap between 1 and 36 is the cost of playing once, and it is paid by two farmers who are exactly as self-interested in both cases. The gap between 36 and 30 is the cost of a single refusal to go first.
Given the standard payoffs, play one harvest and a twelve-season run between named rules, read both totals, and state the continuation chance above which repaying a favour is better than refusing it.
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