Game Theory

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# Game Theory - PowerPoint PPT Presentation

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

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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:

Solving a Game Without Math

c. Not every game has a Nash Equilibrium

• Example:
Solving a Game Without Math

d. Some games have multiple Nash Equilibria

• 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!

Given a day, what can each person produce?

Fruit

• 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

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…
• 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?
• 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?