TEMPTATIONgtREWARDgtPUNISHMENTgtSUCKER Repeated Prisoners Dilemma Consider a prisoners dilemma game played many times A strategy specifies what you do in each stage game Ex cooperate in every stage game ID: 248885
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Slide1
Prisoner’s dilemma
TEMPTATION>REWARD>PUNISHMENT>SUCKERSlide2Slide3
Repeated Prisoner’s Dilemma
Consider a prisoner’s dilemma game played many times A strategy specifies what you do in each stage gameEx: cooperate in every stage gameEx: cooperate in every odd-numbered stage game, defect in every even-numbered stage gameEtc…Slide4
Axelrod’s tournament Slide5
Axelrod’s tournament
The game above repeated 200 times15 strategies submitted Random strategyAlways defectAlways cooperate
Etc.
Each strategy played against all other strategies including itself
15x15=225 games in total
After all games played, earnings added and strategy with the most points declared winnerSlide6
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 strategySlide7
Tit-for-tat
The winning strategy was called tit-for-tatThis strategy starts off by cooperating and then mimics what the other player doesExample: imagine tit-for-tat playing against naïve proberNaïve 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?Slide8
Tit-for-two-tats
Same at Tit-for-tat but allows two defections in a rowAxelrod found that if tit-for-two-tats participated in his tournament, it would have won Slide9
Second tournament
More strategies (63)John Maynard Smith submitted tit-for-two-tatsRandom 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 submittedSlide10
Third tournament (Evolution)
Started with the same 63 strategies in equal proportionAfter the first round of repeated games was played, winnings paid out in “offspring”New round with different proportions of strategiesAfter 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)Slide11
Collectively stable strategies
If there are lots of nasty strategies, always defect does bestIf there are lots of nice strategies, tit-for-tat does bestConsider a world where only these two strategies are playedCan we argue that the system will tend toward tit-for-tat?Kinship: related individuals live close together
Small clusters grow into large clustersSlide12
Examples of repeated games
“Live and let live” in WWIVampire batsSlide13