Source
Axelrod & Hamilton (1981), LIT-498 (Science 211: 1390–1396), read in full in NOTE-411. Two inequalities lost their right-hand sides in extraction and are reconstructed from the payoff sums the text states; they agree with condition (1), which survived.
What was actually shown
The game. A two-player Prisoner’s Dilemma with T > R > P > S and R > (S + T)/2, payoffs in fitness. After each move the pair meets again with probability w. Players recognize a previous partner and remember at least the last move.
Stability. Against TIT FOR TAT only two replies need checking. TIT FOR TAT against itself earns R/(1 − w). Always-defect earns T + wP/(1 − w). Alternating defection and cooperation earns (T + wS)/(1 − w²). So no strategy does better than TIT FOR TAT iff w ≥ (T − R)/(T − P) and w ≥ (T − R)/(R − S) (condition 1).
Always-defect is evolutionarily stable for every w: a single reciprocator among defectors loses.
Initial viability. A cluster of TIT FOR TAT whose members spend a fraction p of their interactions with one another earns p[R/(1 − w)] + (1 − p)[S + wP/(1 − w)]. That beats always-defect’s P/(1 − w) for large enough p and w. Kinship is the other route: reckoned in inclusive fitness, the dilemma can disappear between relatives. That route is argued verbally.
The ratchet. A nice strategy that is evolutionarily stable cannot be invaded by a cluster of any other strategy.
Robustness. TIT FOR TAT won two computer tournaments (14 entries and then 62) and went to fixation in an ecological replay. NOTE-411 rates this moderate: the environment was whatever strategies people submitted.
What could have come out otherwise. The best reply to TIT FOR TAT could have been some cleverer exploiter than the two checked, and clusters could have failed to beat always-defect at any p. Neither holds.
What this does not say
- That TIT FOR TAT is an ESS in the strict sense. Always cooperate does exactly as well against it, so by the two-part definition in Maynard Smith & Price (LIT-505) it is only neutrally stable. Boyd & Lorberbaum (1987) proved that no pure strategy is evolutionarily stable in the repeated game. That paper was checked on Crossref and not read.
- That TIT FOR TAT is best with noise. Strategies here are error-free. Nowak (LIT-501) reports that TIT FOR TAT declines under errors and that generous variants and win-stay, lose-shift replace it.
- That reciprocity is common in nature. The applications (fig wasps, cleaner fish, territorial birds) are illustrations. Those on cancer onset and nondisjunction are labelled “pure speculation” by the authors.
- That the formal proofs are given. They are deferred to Axelrod’s forthcoming paper (note 19).
- Anything about morality. TIT FOR TAT’s properties (nice, provocable, forgiving) are behaviours, not judgements.
Connections
- Trivers (LIT-510, THEORY-663) gave the verbal model this formalizes. Hamilton had given him the Prisoner’s Dilemma framing.
- Hamilton’s Part I (LIT-497, THEORY-408) supplies the kinship route.
- Nowak (LIT-501, THEORY-567) restates direct reciprocity for the donation game as w > c/b.
- The continuation probability w is the “shadow of the future” that moral disciplining (LIT-101) needs. Its reading (NOTE-132, A1) cites Axelrod for the claim that reciprocal cooperation requires delayed gratification.