Prisoners dilemma TEMPTATIONREWARDPUNISHMENTSUCKER Repeated Prisoners Dilemma Consider a

Prisoner’s dilemma • TEMPTATION>REWARD>PUNISHMENT>SUCKER

Repeated Prisoner’s Dilemma • Consider a prisoner’s dilemma game played many times • A strategy specifies what you do in each stage game • Ex: cooperate in every odd-numbered stage game, defect in every even-numbered stage game • Etc…

Axelrod’s tournament

Axelrod’s tournament • The game above repeated 200 times • 15 strategies submitted • • Random strategy Always defect Always cooperate Etc. • Each strategy played against all other strategies including itself • 15 x 15=225 games in total • After all games played, earnings added and strategy with the most points declared winner

Tournament results • On average, no strategy scored above 600 points per game (what you would get if everyone mutually cooperated 200 rounds) • The best scoring strategies were nice (never first to defect) • 8 top scoring strategies were nice • The worst scoring strategies were nasty (first to defect) • Forgiving strategies did better than unforgiving ones • A forgiving strategy has a short memory. For example, it doesn’t punish forever • Of the 8 nice strategies, one of the strategies punished a defection by defecting forever in response. This was the worst scoring nice strategy

Tit-for-tat • The winning strategy was called tit-for-tat • This strategy starts off by cooperating and then mimics what the other player does • Example: imagine tit-for-tat playing against naïve prober • Naïve prober is the same as tit for tat, except it defects 1 in 10 rounds chosen at random • U(TFT, TFT)>U(NP, TFT) >U(NP, NP) • Example: imagine tit-for-tat playing against remorseful prober • Remorseful prober is the same as naïve prober but allows “one free hit” • U(TFT, TFT)>U(RP, TFT)>U(NP, TFT) • But is tit-for-tat an equilibrium?

Tit-for-two-tats • Same at Tit-for-tat but allows two defections in a row • Axelrod found that if tit-for-two-tats participated in his tournament, it would have won

Second tournament • More strategies (63) • John Maynard Smith submitted tit-for-two-tats • Random termination times for each game (“infinitely” repeated game) • Tit-for-tat won again! • One problem with these tournaments is that the winner depends on the strategies that were submitted

Third tournament (Evolution) • Started with the same 63 strategies in equal proportion • After the first round of repeated games was played, winnings paid out in “offspring” • New round with different proportions of strategies • After 1000 rounds, no changes in the population • Nasty strategies driven out, tit-for-tat and some other nice strategies survived • Note tit-for-tat is not ESS • Can be invaded by always cooperate • Can be invaded by a mixture of tit-for-two-tats and suspicious tit-for-tat (who defects on the first move, otherwise behaves like tit-for-tat)

Collectively stable strategies • If there are lots of nasty strategies, always defect does best • If there are lots of nice strategies, tit-for-tat does best • Consider a world where only these two strategies are played • Can we argue that the system will tend toward tit-for-tat? • Kinship: related individuals live close together • Small clusters grow into large clusters

Examples of repeated games • “Live and let live” in WWI • Vampire bats