Section 1: Preference, Utility, and Games in Normal Form

Key Learning Points

• Explain the dominant approach to modelling an agent’s interests, utility theory
• Explain non-cooperative game theory
• Describe two of the most fundamental solution concepts: Pareto optimality and Nash equilibrium
• Discuss a list of useful solution concepts:
  • Maxmin and minmax strategies
  • Minimax regret
  • Dominant strategy
  • Dominated strategy
  • Rationalizable strategy
  • Correlated equilibrium
  • Trembling hand perfect equilibrium
  • ε-Nash equilibrium

Activities

1. Read Chapter 3 of the textbook and the tutorial on mixed strategies.
2. Watch the following short videos from YouTube:
   1. Introduction to Game Theory
   2. Self-Interest and Utility Theory
   3. Defining Games
   4. Examples of Games from Game Theory
   5. Nash Equilibrium Introduction and the Keynes Beauty Contest
   6. Strategic Reasoning and the Keynes Beauty Contest Game
   7. Best Response and Nash Equilibrium
   8. Examples
   9. Maxmin Strategies (Basic)
   10. Maxmin Strategies (Advanced)
   11. Dominant Strategies
   12. Pareto Optimality
   13. Computing Mixed-Strategy Nash Equilibria
3. Do the following exercises:
   1. Two players decide to play the following game. They start driving toward each other, and a collision is unavoidable unless one (or both) of the drivers decides to change its driving course (chicken out). For each player, the best outcome is that it keeps driving straight while the other player chickens out. The next outcome is that both chicken out. The third-best option is that the player itself chickens out while the other player drives straight. Finally, the worst outcome is that they both keep driving straight till the collision occurs. Write down a normal-form game that represents this situation, and find its Nash equilibria.
   2. Consider a two-player game defined by the following payoff matrix:

      	W	X	Y	Z
      A	15, 42	13, 23	9, 43	0, 23
      B	2, 19	2, 14	2, 23	1, 0
      C	20, 2	20, 21	19, 4	3, 1
      D	70, 45	3, 11	0, 45	1, 2

      Decide whether the following statements are true or false. Explain your answer.

      1. A strictly dominates B.
      2. Z strictly dominates W.
      3. C weakly dominates D.
      4. X weakly dominates W.
      5. C is a best response to X.
      6. Z is a best response to A.
   3. Two business partners are working on a joint project. For the project to be successfully implemented, it is necessary that both partners engage in the project and exert the same amount of effort. The payoff from the project is one unit to each partner, whereas the cost of the effort required is given by some constant c, with 0 < c < 1. This can be modelled by the following game (where W stands for work and S stands for slack):

      	S	W
      S	0, 0	0, −c
      W	−c, 0	1 − c, 1 − c

   Find all pure and mixed Nash equilibria of this game. How do the mixed strategy equilibria change as a function of the effort c?

4. Discuss the following scenarios in the discussion forum:

   1. Consider the following game in matrix form with two players. Payoffs for the row player Shelia are indicated first in each cell, and payoffs for the column player Thomas are second.

      	C	D
      A	10, 16	14, 24
      B	15, 20	6, 12

      1. Does either player have a dominant strategy? Explain your answer.
      2. What are the pure-strategy Nash equilibria (if any) of this game? Justify your answer. If there is more than one pure equilibrium, which would Thomas prefer?
      3. This game has a fully mixed strategy Nash equilibrium in which both Shelia and Thomas play each of their actions with positive probability. What are the mixed strategies for each player in this equilibrium? Show how you would compute such a mixed equilibrium and verify that your mixed strategies are indeed in equilibrium.
   2. In the following game, find:

      Player 1/Player 2	L2	R2
      L1	1, 0	3, 1
      R1	2, 2	2, 2

      1. All pure Nash equilibria;
      2. A trembling-hand perfect equilibrium.