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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

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)
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?
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?