Game Theory Formalizing the Hobbesian Dilemma

A. Assumptions 1. Assumptions a. Rational choice – People act according to their preferences (desires, for Hobbes) b. Strategic interaction – What one person does affects what others should do 2. Elements a. Players – Two or more b. Strategies – The choices players have (Means) c. Outcomes – The results of the players’ choices (Ends) d. Payoffs – How much each player values each Outcome (Desires)

B. The Structure of a Simple Game Player 2 Player 1 Strategy AStrategy B Strategy A Outcome 1 Player 1 Payoff, Player 2 Payoff Outcome 2 Player 1 Payoff, Player 2 Payoff Strategy B Outcome 3 Player 1 Payoff, Player 2 Payoff Outcome 4 Player 1 Payoff, Player 2 Payoff

1. Solving a Game Without Math a. Nash Equilibrium  Neither player could do any better by unilaterally changing its strategy choice b. 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: Player 2 Player 1 Strategy AStrategy B Strategy A 2,33,4 Strategy B 0,04,2

Solving a Game Without Math Player 2 Player 1 Strategy AStrategy B Strategy A 2,33,4 Strategy B 0,54,2 c. Not every game has a Nash Equilibrium Example:

Solving a Game Without Math Player 2 Player 1 Strategy AStrategy B Strategy A 2,53,4 Strategy B 0,04,1 d. Some games have multiple Nash Equilibria Example:

2. Making a Game from Hobbes a. Players – Limit to two for simplicity (result holds with more than two) b. 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.

2. Making a Game from Hobbes c. Outcomes – What might come about from the combination of our choices? i. 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”) ii. You cooperate but I defect – You’re dead (same logic as above)

2. Making a Game from Hobbes iii. We both defect – Life is nasty, brutish, and short – but since we each know the other is prepared, death is less likely iv. 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.

2. Making a Game from Hobbes d. Preferences (Desires) – Which outcome is best for each of us? PAYOFFSFor MeFor Thee Best (4)I kill you. All for me!You kill me! 2 nd Best (3)We compromise 2 nd Worst (2)We both fight Worst (1)You kill me!I kill you!

A Hobbesian World: Less Than Optimal Player 2 Player 1 CooperateDefect Cooperate 3,31,4 Defect 4,12,2

C. Common Games: Comparing Hobbes to Modern Games 1. Prisoner’s Dilemma a. Both players end up worse, even though each plays rationally!  Hobbesian Dilemma b. Used to model the “Security Dilemma” by Realists (Efforts to increase own security make others less secure) Player 2 Player 1 Remain SilentConfess Remain Silent Misdemeanor, Misdemeanor Life, Walk Free Confess Walk Free, LifeFelony, Felony

C. Common Games 2. Chicken – Another Possibility a. Equilibria: Someone swerves – but who? b. Used to model nuclear crises c. Could this be the state of nature? Player 2 Player 1 SwerveDrive Straight Swerve Status Quo, Status Quo Wimp, Cool Drive Straight Cool, WimpDEAD, DEAD

D. Liberal Alternatives to Hobbes 1.“Stag Hunt”, aka the Assurance Game, aka Mixed-Motive PD a.Used to model non-predatory security dilemma, driven by fear instead of aggression (Rousseau) b.Equilibria: depends on trust – Nobody wants to be the only one looking for a stag! Player 2 Player 1 DeerRabbit Deer Deer, Deer Nothing, Rabbit Rabbit Rabbit, Nothing Rabbit, Rabbit

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!

Absolute Advantage CainAbel RabbitsOR 410 Fruit 107 Given a day, what can each person produce? 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 Fruit Rabbits 5 5

Comparative Advantage BartLisa TurkeysOR 510 Potatoes 1012 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 Turkeys 5

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…

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?

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?