Game Theory

1. Game Theory Formalizing the Hobbesian Dilemma

2. A. Assumptions • Assumptions • Rational choice – People act according to their preferences (desires, for Hobbes) • Strategic interaction – What one person does affects what others should do • Elements • Players – Two or more • Strategies – The choices players have (Means) • Outcomes – The results of the players’ choices (Ends) • Payoffs – How much each player values each Outcome (Desires)

3. B. The Structure of a Simple Game

4. 1. Solving a Game Without Math • Nash Equilibrium  Neither player could do any better by unilaterally changing its strategy choice • To Solve: Examine each cell to see if either player could do better by unilaterally choosing a different Strategy, given that its opponent does nothing different. Example:

5. Solving a Game Without Math c. Not every game has a Nash Equilibrium • Example:

6. Solving a Game Without Math d. Some games have multiple Nash Equilibria • Example:

7. 2. Making a Game from Hobbes • Players – Limit to two for simplicity (result holds with more than two) • Strategies (Means) – We can be nice (help others or at least not harm them) or nasty (use violence to get what we want). Usual termnology is Cooperate vs Defect.

8. 2. Making a Game from Hobbes • Outcomes – What might come about from the combination of our choices? • I cooperate but you defect – I’m dead. May not be able to defect later if I cooperate now (“there is no way for any man to secure himself so reasonable as anticipation”) • You cooperate but I defect – You’re dead (same logic as above)

9. 2. Making a Game from Hobbes • We both defect – Life is nasty, brutish, and short – but since we each know the other is prepared, death is less likely • We both cooperate – We get along fine, but this means we have to each give up some things we desire. “Diffidence” = we both want the same thing.

10. 2. Making a Game from Hobbes • Preferences (Desires) – Which outcome is best for each of us?

11. A Hobbesian World: Less Than Optimal

12. C. Common Games: Comparing Hobbes to Modern Games • Prisoner’s Dilemma • Both players end up worse, even though each plays rationally!  Hobbesian Dilemma • Used to model the “Security Dilemma” by Realists (Efforts to increase own security make others less secure)

13. C. Common Games • Chicken – Another Possibility • Equilibria: Someone swerves – but who? • Used to model nuclear crises • Could this be the state of nature?

14. D. Liberal Alternatives to Hobbes • “Stag Hunt”, aka the Assurance Game, aka Mixed-Motive PD • Used to model non-predatory security dilemma, driven by fear instead of aggression (Rousseau) • Equilibria: depends on trust – Nobody wants to be the only one looking for a stag!

15. 2. Does trade provide a rational alternative to war? • Hobbes assumes life is zero-sum in state of nature, because we want the same things • Liberals assume we have different tastes AND that we have different talents/interests • If you and I are each better at making/gathering something, we can both do better by trade than predation!

16. Absolute Advantage Given a day, what can each person produce? Fruit • Production possibilities without trade • Cain will buy Rabbits for < 2.5 fruit. Abel will buy Fruit for < 10/7 Rabbits. • Exchange rate must be between 2.5 fruit/rabbit and .7 fruit/rabbit • Example: Abel hunts 10 rabbits, trades 3 to Cain for 5 fruits. (1.67 fruits/rabbit = good deal for Cain, .6 rabbits/fruit = good deal for Abel!). • Result: Both sides achieve consumption beyond original production possibilities! 10 5 5 10 Rabbits

17. Comparative Advantage Given a day, what can each person produce? • Lisa has absolute advantage in both goods! • Lisa has comparative advantage in… • 2 to 1 in turkey, 1.2 to 1 in taters  turkey • Bart has comparative advantage in taters (5/6 as productive rather than only 1/2) • Bart buys turkey at < 2 taters, Lisa buys taters at < 5/6 turkey. Exchange rate must be between 2 and 1.2 taters/turkey • Example: Bart grows 10 taters, Lisa catches 10 turkeys. Bart trades 6 taters for 4 turkeys (1.5 taters/turkey) Taters 20 10 5 10 Turkeys

18. 2. Does trade provide a rational alternative to war? • Is trade possible in the state of nature? • Does it matter whether there are two people or thousands? Does this change incentives for predation vs. trade? • Could some type of money evolve in a state of nature? Locke argues yes…

19. 3. A Surprising Twist: Can a Hobbesian World Evolve Cooperation? • Hobbesian tournament: Each player must play each other player in a series of Prisoners’ Dilemma (Hobbesian Dilemma) games. • Best strategy in a single-shot game is always Defect, but… • Which strategies produce the highest total payoff over many games against different players?

20. 3. A Surprising Twist: Can a Hobbesian World Evolve Cooperation? • Best strategy is almost always Tit-for-Tat • Start by cooperating • Then do what opponent did last time • Matches some of Hobbes’s advice: • Cooperate at first, but retain ability to defect (Law of Complacence) • Match cooperation with cooperation (Law of Gratitude) • Respond to renewed good behavior (Law of Pardon) • Implication: People playing the best strategy will get along. If poor strategy = earlier death, only TFT players will survive. • Did Hobbes miss this implication? Is the state of Nature a repeated game? What happens if I fail to defect when I should have defected?