Game Theory

1. Players
2. Strategies

   Annotations:

   • Complete plan of action
3. Outcomes

   Annotations:

   • Can be many things. Monetary, material, immaterial...

1. Players
2. Strategies
3. Outcomes for each strategy profile
4. Utility function for each player
   1. Captures preferences

      Annotations:

      • The preferences of another person creates an uncertainty for me. I don't know what they will do in the game. If I know their preferences, I know what they are likely to do
      1. Risk-neutral selfish materialist

         Annotations:

         • Risk-neutral implies a linear function. There are no diminishing returns to the utility of money
         1. u(i) = m(i)
      2. Risk-neutral egalitarian
         1. u(i) = min {m(j)}

            Annotations:

            • Prefers to minimise inequality, so my utility equals the outcome of the person who gets the least
      3. Altruistic
         1. u(i) = m(i) + γ m(j)

            Annotations:

            • Assign some weight to the outcome of the other person

Annotations:

• With outcomes expressed in terms other than utility, we can never know which strategy is dominant for a player. The highest monetary value may not yield the highest utility because of selfless preferences and so is not the dominant strategy
• Row player: Rowena (first values) Column player: Colin (second values)

1. Conditionally cooperative

   Annotations:

   • A normal argument against Rowena choosing T instead of B. If Colin chooses R, she would only lose 1 (from 1 in B to 0 in T), whereas Colin would earn 2. Conditionally cooperative means that she is unwilling to help someone who wouldn't help her back
   1. Differences between countries

      Annotations:

      • Is this related to why some countries are richer than others? Are they better at changing the situation so they end up in 2.2? Do they have more efficient contracts? Does the culture promote cooperation?

Annotations:

• There is a dominant strategy when the preferences are known. Players will always choose the highest utility

Annotations:

• Each player's strategy is a best reply to he opponent's strategies

1. Pure strategy equilibria

   Annotations:

   • All players pick a pure strategy
2. Mixed strategy equilibria

   Annotations:

   • At least one player chooses several pure strategies with positive probability
   • If you don't want to deviate in pure strategy, you TYPICALLY don't want to deviate in mixed
   1. Theorem: All games with a finite number of pure strategies have at least one equilibrium
      1. u(T) = PL*1 + (1-PL)*0 = u(B) = PL*0 + (1-PL)*2
3. Equilibrium vs efficiency

   Annotations:

   • Equilibrium is on (1,1). This is far below the Pareto frontier
   1. Outer line: Pareto frontier

      Annotations:

      • Possible to improve without making other person worse off
4. Interpretations
   1. Outcome of rational deliberations

      Annotations:

      • How rational is it to assume that everyone else is rational?
   2. Rest point of learning dynamics or evolutionary selection
   3. Self-enforcing agreement

      Annotations:

      • A player who believes that other players will stick to an agreement of the Nash equilibrium will not herself violate the agreement