# Game Theory - PowerPoint PPT Presentation

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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

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Game Theory

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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…

### 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?