THE FOLD / CHEAT / THE EXPLOIT / THE FOLK THEOREM
THE FOLK THEOREM
why tomorrow makes today honest
1 WHAT IT IS · WHAT IT DOES · FACT OR FICTION
In a one-shot prisoner’s dilemma, defection is strictly dominant and cooperation is irrational. Repeat the game indefinitely and cooperation becomes an equilibrium — not because anyone became decent, but because the threat of never being trusted again outweighs one round of gain. The folk theorem says more than that: above a threshold discount factor, almost any average payoff above the punishment level can be sustained.
LIT verified live with T = 5, R = 3, P = 1: the grim-trigger threshold is (T−R)/(T−P) = 0.500000000000, exactly. At δ = 0.49 cooperating forever pays 5.8824 against 5.9608 for defecting once and being punished — defection wins. At δ = 0.51 it is 6.1224 against 6.0408 and cooperation wins. At δ = 0.50 both come to 6.000000000, so the switch is exactly at the threshold rather than near it. At δ = 0.9, 91 of 101 target average payoffs between P and T are sustainable.
LIT verified live with T = 5, R = 3, P = 1: the grim-trigger threshold is (T−R)/(T−P) = 0.500000000000, exactly. At δ = 0.49 cooperating forever pays 5.8824 against 5.9608 for defecting once and being punished — defection wins. At δ = 0.51 it is 6.1224 against 6.0408 and cooperation wins. At δ = 0.50 both come to 6.000000000, so the switch is exactly at the threshold rather than near it. At δ = 0.9, 91 of 101 target average payoffs between P and T are sustainable.
2 HOW IT WAS WEAVED · AI + HUMAN
David (human) seated this at THE EXPLOIT: repetition is the hole in the dilemma, and cooperation is what climbs through it.
AVAN (AI) tested the threshold by checking the payoffs on both sides and exactly at it. A comparison that only samples 0.2 and 0.8 would confirm the direction while saying nothing about the value; showing that the two payoffs are equal to nine decimal places at δ = 0.5 is what makes it a threshold rather than a trend. The second half is the folk theorem proper and it is the part usually skipped: cooperation is not the only thing repetition sustains. Above the threshold a continuum of outcomes becomes equilibrium behaviour, including thoroughly unpleasant ones, which is why the theorem is a statement about how little repetition determines rather than a proof that repeated interaction produces good behaviour.
AVAN (AI) tested the threshold by checking the payoffs on both sides and exactly at it. A comparison that only samples 0.2 and 0.8 would confirm the direction while saying nothing about the value; showing that the two payoffs are equal to nine decimal places at δ = 0.5 is what makes it a threshold rather than a trend. The second half is the folk theorem proper and it is the part usually skipped: cooperation is not the only thing repetition sustains. Above the threshold a continuum of outcomes becomes equilibrium behaviour, including thoroughly unpleasant ones, which is why the theorem is a statement about how little repetition determines rather than a proof that repeated interaction produces good behaviour.
3 ONE DIMENSION
Two payoff curves against the discount factor. They cross once, exactly at 0.5.
4 TWO DIMENSIONS · INTERACTIVE
Turn the dial through the threshold and watch the incentive flip.
5 THREE DIMENSIONS + AVAN’S INVERSE
The green forward object: the set of sustainable payoffs, widening as patience grows.
AVAN’s addition (the inverse-companion): the forward reading is “repetition makes cooperation rational.” The inverse is that repetition makes almost everything rational, and that is a weakness dressed as a result. The same argument that sustains mutual cooperation sustains extortion, collusion and elaborate punishment schedules — the theorem produces a set, and the set is nearly everything. Read backwards, the folk theorem does not explain why cooperation happens; it removes the explanation that it could not, and leaves the actual question — which of the sustainable outcomes people land on — entirely open.
LIT with T = 5, R = 3, P = 1 the grim-trigger threshold is (T-R)/(T-P) = 0.500000000000 exactly; at d = 0.49 cooperating forever pays 5.8824 against 5.9608 for defecting once and being punished, so defection wins; at d = 0.51 it is 6.1224 against 6.0408 and cooperation wins; at d = 0.50 both come to 6.000000000, so the switch is exactly at the threshold rather than near it; and at d = 0.9, 91 of 101 target average payoffs between P and T are sustainable
FIG The threshold was tested on both sides AND exactly at it. A comparison sampling only 0.2 and 0.8 would confirm the direction while saying nothing about the value; showing the two payoffs equal to nine decimal places at d = 0.5 is what makes it a threshold rather than a trend. The second half is the folk theorem proper and is usually skipped: cooperation is not the only thing repetition sustains. Above the threshold a continuum of outcomes becomes equilibrium behaviour, including thoroughly unpleasant ones - which is why the theorem is a statement about how LITTLE repetition determines.
FIG The threshold was tested on both sides AND exactly at it. A comparison sampling only 0.2 and 0.8 would confirm the direction while saying nothing about the value; showing the two payoffs equal to nine decimal places at d = 0.5 is what makes it a threshold rather than a trend. The second half is the folk theorem proper and is usually skipped: cooperation is not the only thing repetition sustains. Above the threshold a continuum of outcomes becomes equilibrium behaviour, including thoroughly unpleasant ones - which is why the theorem is a statement about how LITTLE repetition determines.
◆ sealed .dlw.fold → folded to ROOT_0 · a sphere of THE EXPLOIT · David Lee Wise (ROOT0), with AVAN