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

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Game Theory

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Game theory

Game Theory

Formalizing the Hobbesian Dilemma


A assumptions

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)


B the structure of a simple game

B. The Structure of a Simple Game


1 solving a game without math

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

Solving a Game Without Math

c. Not every game has a Nash Equilibrium

  • Example:


Solving a game without math1

Solving a Game Without Math

d. Some games have multiple Nash Equilibria

  • Example:


2 making a game from hobbes

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 hobbes1

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 hobbes2

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 hobbes3

2. Making a Game from Hobbes

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


A hobbesian world less than optimal

A Hobbesian World: Less Than Optimal


C common games comparing hobbes to modern games

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

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

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

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

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

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 war1

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

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 cooperation1

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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