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Section 2 – Ec1818 Jeremy Barofsky jbarofsk@hsph.harvard

Section 2 – Ec1818 Jeremy Barofsky jbarofsk@hsph.harvard.edu. February 10 and 11, 2010. Outline – Section 2. Prisoner’s Dilemma Iterated Prisoner’s Dilemma and Axelrod’s Computer Tournament Evolutionary Game Theory Evolutionary Stable Strategies Minimal Stabilizing Frequency.

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Section 2 – Ec1818 Jeremy Barofsky jbarofsk@hsph.harvard

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  1. Section 2 – Ec1818Jeremy Barofskyjbarofsk@hsph.harvard.edu February 10 and 11, 2010

  2. Outline – Section 2 • Prisoner’s Dilemma • Iterated Prisoner’s Dilemma and Axelrod’s Computer Tournament • Evolutionary Game Theory • Evolutionary Stable Strategies • Minimal Stabilizing Frequency

  3. Prisoner’s Dilemma Payoff Matrix • where CC > DC / CD > DD and CC > (DC + CD) / 2 ; Generally, T > R > P > S • Nash Equilibrium: Given each player knows the other player’s payoff matrix, no player • has an incentive to change their current strategy. • D is a dominant strategy for both players so that the only NE to the PD game is D,D and • the player’s receive payoffs (1, 1). • BUT THIS IS NOT PARETO OPTIMAL?!?!? Cooperation has productive value.

  4. Iterated Prisoner’s Dilemma • Nash equilibrium to any known finite, repeated game is D,D. Why? (Hint – all games reduce / unravel to a one-shot game). • When repeated games are of unknown length, (D, D) is not necessarily optimal. • No threats / promises allowed, only communication is through previous sequence of moves. • Only reason to cooperate today is because there might be a meeting in the future. • Empirically – we see much more cooperation in real world than PD would predict (WW I trench warfare, Senate reciprocity, duopolistic competition)

  5. The Evolution of Cooperation • Axelrod’s Fundamental Question: “Under what conditions will cooperation emerge in a world of egoists without central authority?” • Set-up a computer tournament: • Call for entries from game theorists • All entrants told of preliminary experiments • 15 strategies; 14 submitted and 1 random • Round-robin tournament with each strategy facing all others heads-up ; 200 iterations

  6. And the Winner Is…Tit-for-Tat • Tit-for-Tat: Cooperate on first move (nice) and reciprocate opponent’s previous movement afterward. • Nice rules did well against other nice rules (close to 600) and nice rules were separated by how well they did against the mean rules. • Downing rule: Kingmaker – used outcome maxmiziation (tries to respond optimally to other player’s strategy), starts with 2 D’s. • Loses, but helps the nice and forgiving rules win, Friedman / Grim Trigger does worst of nice rules because Downing.

  7. Results: Tournament 1 Nice guys finish first (top 8 strategies never defect first).

  8. Let’s run it back… • Axelrod ran another tournament, giving the results of the first tournament to all entrants • 63 entrants with a continuation probability of w = 0.99654 (discount rate). • Tit-for-Tat wins again!! • Of top 15 rules all but one were nice and of the bottom 15 all but one were mean.

  9. General Lessons for Axelrod’s PD Tournament (and for life?!?) • Be nice (Don’t defect first) • Retaliate swiftly (otherwise others will take advantage) • Forgive and forget (Feuds are costly) • Being too clever doesn’t work (too clever or complicated strategies look random and reduce cooperation) • Critiques: Simulations ignore theory and outcomes may depend on initial conditions

  10. Evolutionary Game Theory • Looks at how the “ecology” of players / strategies being played in a game changes over time. • Key difference with infinitely repeated games: the successful strategies “reproduce” and less successful ones die out. • Proportional Fitness Reproduction (PFR): grow proportional to score relative to average. P = proportion of pop. Dp / p = Wi / W. • Evolutionary Stable Strategy (ESS) is: • robust to invasions • a different equilibrium concept under evolutionary game theory instead of NE.

  11. Evolutionary Stable Strategies • Game to help show ESS is Hawk-Dove where two NE exist without dominant strategies. • With 2 strategies, ESS must exist but not with 3 and there can be more than one ESS.

  12. Rolling with the homeys • Minimal stabilizing frequency: the minimal proportion of the population that gives you protection against being wiped out by mutants. • The lower this number is, the better and the easier it is to achieve stability. The smaller it is the larger a strategies basin of attraction. • Must be > 50% for any strategy, but for nasty strategies MSF -> 1 as w -> 1 and for nice strategies MSF -> ½ as w -> 1.

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