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### Game Theory

Formalizing the Hobbesian Dilemma

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)

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:

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.

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)

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.

2. Making a Game from Hobbes

- Preferences (Desires) – Which outcome is best for each of us?

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)

C. Common Games

- Chicken – Another Possibility
- Equilibria: Someone swerves – but who?
- Used to model nuclear crises
- Could this be the state of nature?

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!

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

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

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

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?

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