Game Theory

1
Game Theory

• Formalizing the Hobbesian Dilemma

2
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)

3
B. The Structure of a Simple Game
4
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

5
Solving a Game Without Math

• c. Not every game has a Nash Equilibrium
• Example

6
Solving a Game Without Math

• d. Some games have multiple Nash Equilibria
• Example

7
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.

8
2. Making a Game from Hobbes

• Outcomes What might come about from the
  combination of our choices?
• I cooperate but you defect Im 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 Youre dead (same
  logic as above)

9
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.

10
2. Making a Game from Hobbes

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

11
A Hobbesian World Less Than Optimal
12
C. Common Games Comparing Hobbes to Modern Games

• Prisoners 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)

13
C. Common Games

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

14
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!

15
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!

16
Absolute Advantage
Given a day, what can each person produce?
Fruit

• Production possibilities without trade
• Cain will buy Rabbits for buy Fruit for
• 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
10
5
Rabbits
17
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 at 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
10
5
Turkeys
18
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

19
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?

20
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 Hobbess 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?