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Rational actors, tit-for-tat types, and the evolution of cooperation

Rational actors, tit-for-tat types, and the evolution of cooperation. Summary by Mark Madrilejo January 17, 2006. (Finite) Prisoner’s Dilemma. In infinitely repeated PD, tit-for-tat (TFT) can be optimal When extended to finite cases

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Rational actors, tit-for-tat types, and the evolution of cooperation

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  1. Rational actors, tit-for-tat types, and the evolution of cooperation Summary by Mark Madrilejo January 17, 2006

  2. (Finite) Prisoner’s Dilemma • In infinitely repeated PD, tit-for-tat (TFT) can be optimal • When extended to finite cases • Mechanical strategies are not rational(under the game theory definition) • Rational actors use backward induction in finite cases • So TFT isn’t useful here

  3. What is a rational actor? • Considers everyone else’s strategy • Chooses a best response • Unlike mechanical strategies • All-D, All-C, tit-for-tat (TFT) • Possible scenarios • Everyone else is TFT • Everyone else is a rational actor

  4. But there is cooperation! • Kreps • In a population of rational actors and TFT robots, an actor may imitate a robot until near the end of the game • But can such a mixed population exist? • If choosing a best response is costly, then equilibrium can exist.

  5. Optimizers v. imitators • Consider costly optimizers and cheap imitators (who target average behavior) • Imitators are not the best, but don’t work hard. Hard workers attract free riders. • “Gossip, sexual recombination, and the El Farol bar” • In this paper, rational actors imitate TFT robots

  6. One-sided uncertainty • P1 is Rational, P2 is Maybe Rational • (p: probability that MR is rational) • Results (PBE: Perfect Bayesian Equilibrium) • MR should play as TFT robot and then preempt P1 (do them before they do you) • P1’s optimal choice depends on PD payoffs a, b, and probability p • P1 should cooperate until “some point” in the “endgame” (whose length is invariant to N)

  7. Two-sided uncertainty • Loops (C-to-D-to-C) can happen • Endgame length still invariant to N • Cooperate at least until endgame • p < 1/a: no pure PBE • p > 1/a: pure PBE says to switch simultaneously when t = N + [bp / (1 - p) ]

  8. Optimization costs • Fixed cost: just being a rational actor • Strategy selection cost: finding p • Implementation cost: counting turns until critical t is reached • Possible outcomes • Actors optimize • Actors always imitate robots • Robots win

  9. So what? • Mixed populations of actors and robots can exist • Thus, Kreps’ result stands: rational actors might cooperate to imitate TFT • Thus, cooperation can occur in a finitely repeated Prisoner’s Dilemma!

  10. Extensions • El Farol • Allow more than two types (add robots) • Multi-agent model • Rationals v. robots • Variables: a, b, N, p, c0, cP, ct • Implication: In some cases, it may be possible to let optimizers work, then invade later with cheap robots

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