1. Section 2 – Ec1818 Jeremy Barofsky jbarofsk@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
Cooperate
Defect
3,3
0, 5
5, 0
1,1
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 = (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.

12. 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.
Hawk
Dove
-1, -1
0, 4**
2,2

13. 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.