Intermediate Microeconomics

1. Intermediate Microeconomics Game Theory and Oligopoly

2. Game Theory • So far we have only studied situations that were not “strategic”. • The optimal behavior of any given individual or firm did not depend on what other individuals or firms did. • E.g. • An individual buys something if its price is less than his willingness to pay. • A firm enters a market if there are positive economic profits to be made at going prices. • Obviously, we might want to expand this. • For example, what happens when firms recognize how price will be affected by their behavior (i.e., not price takers)? • Or when one firm or person’s optimal behavior depends on what another firm or person does.

3. Game Theory • Game theory helps to model strategic behavior --- or interactions where what is optimal for a given agent depends on what actions are taken by another agent and vice versa. • Applications: • The study of oligopolies (industries containing only a few firms) • The study of externalities and public goods; e.g. using a common resource such as a fishery. • The study of military strategies. • Bargaining. • How markets work. • Behavior in the courts. • Behavior of news media. • Crime.

4. What is a Game? • A game consists of: • a set of players • a set of strategies for each player • i.e. actions to be performed given anyobserved state of the world • the payoffs to each player for every possible choice of actions by that player and all the other players.

5. Simultaneous Move Games • Consider games where players must choose an action without knowing what the other players have chosen. • Does a defendant agree to testify against his co-defendants when he doesn’t know whether or not his co-defendants are going to do the same? • How much should a firm bid for a given item in a silent auction? • Should I act friendly or defensively when I encounter a stranger on an empty street late at night? • How do we model the outcomes in these types of games?

6. Simultaneous Move Games • “Prisoners’ Dilemma” • Consider a game with 2 players, each player has two options: • Keep quiet (“cooperate” with each other), • or talk to Police (“defect”) • Payoffs to Player-1; • Cooperate utility of 5 if Player-2 cooperates, utility of 2 if Player-2 defects. • Defect utility of 8 if Player-2 cooperates, utility of 3 if Player-2 defects. • Payoffs to Player-2 are analogous.

7. One way to summarize the payoffs associated with each action is to use a payoff matrix. Payoff for Player-1 (row player) shown first, followed by payoff for Player-2 (column player) Simultaneous Move Games Player-2 C D C D Player-1

8. Simultaneous Move Games • So how should we think of how to model the outcome of such games? • What is Player-1’s best action to take if Player-2 Cooperates? • What is Player-1’s best action if Player-2 Defects? • How about for Player-2? • So what do think each Player will do? Player-2 C D C D Player-1

9. Simultaneous Move Games • Dominant Strategy - A strategy that gives higher utility than all other strategies given any actions taken by other players. • Does a dominant strategy always exist? • Meeting time for dinner? • Wearing a costume to a Halloween party?

10. Simultaneous Move Games • “Arms Race” - Consider a game of the following form: • What is Player-1’s best action to take if Player-2 chooses ignore? • What is Player-1’s best action to take if Player-2 chooses attack? • How about for Player-2? Player-2 ignore attack ignore attack Player-1

11. Nash Equilibrium • Nash Equilibrium – A set of actions such that each person’s action is (privately) optimal given the actions of others. • Key to a Nash Equilibrium: • No person has an incentive to deviate from his Nash equilibrium action given everyone else behaves according to their Nash equilibrium action. • Nash Equilibrium in “A Beautiful Mind?”

12. Simultaneous Move Games • Nash Equilibria of “Prisoner’s Dilemma”? • Both Cooperate? • One Cooperate, other Defect? • Both Defect? Player-2 C D C D Player-1

13. Simultaneous Move Games • Nash Equilibria of Arms Race? • Both Ignore? • One Ignore, other Attack? • Both Attack? Player-2 I A I A Player-1

14. Nash Equilibria • Three things to notice: • Playing Dominant Strategies are always a Nash Equilibrium (e.g. Prisoner’s Dilemma). • Nash Equilibria do not have to be Pareto Efficient (e.g. Prisoner’s Dilemma and Arms Race). • There can be multiple equilibria that often that can be Pareto ranked (e.g. Arms Race). • Applications of these types of games?

15. Game Theory Application: Trade • Suppose Acme Corp. could make a deal with China Corp. to produce widgets abroad. • If both stick with the deal (i.e. China Corp. produces quality widgets and Acme Corp. pays China Corp. the agreed upon fee), Acme’s profits will be $200K while China Corp’s profits will be $50K. • If Acme cheats and pays less than the agreed upon rate after delivery, Acme has profits of $250K and China Corp. ends up losing $50K. • Alternatively, if Acme acts honestly, but China Corp. cheats and produces sub-standard widgets, Acme Corp.’s profits will only be $50K, but China Corp.’s profits will be $90K. • If both act dishonestly, Acme will make only $75K while China Corp. will lose $20K. • If Acme produces widgets domestically, its profits will only be $100K and China Corp. will have profits of $0. • Should trade happen? Will trade happen?

16. Game Theory Application: Trade • What is the key problem that leads to inefficiency? • How could this problem be overcome?

17. Continuous Actions • What if players could choose among a continuum of actions? • The standard way to handle such situation is to use Reaction Functions. • Reaction function – a function that maps any possible action by Player b into optimal action for player a. • A Nash Equilibrium will arise at the point where Reaction functions intersect.

18. Continuous Actions: Cournot Equilibrium • To see the role played by reaction functions we can look at an Oligopoly setting. • So far, we have examined 2 types of market structure. • Markets where each supplier was small enough that its decision regarding how much to supply had no effect on price (competition) • Markets where there was only one supplier, so its decision regarding how much to supply fully determined price (Monopoly) • What happens with “a few” suppliers who don’t collude (i.e. and oligopoly)?

19. Continuous Actions: Cournot Equilibrium • Cournot Equilibrium • Consider a market with two firms, each with cost function equal to C(q) = 4q • Suppose the market (inverse) demand function is p(Q) = 84 – 8(Q) where Q = q1 + q2 • How do we find optimal quantity supplied by each firm?

20. So each firm’s reaction function will be: So what will be Nash Equilibrium? Will it be efficient? Continuous Actions: Cournot Equilibrium q1 10 Firm 2’s reaction fn 5 Firm 1’s reaction fn q2 5 10

21. Continuous Actions: Cournot Equilibrium • What happens as the number of firms increases? • Firm 1 : max p(Q)q1 – C(q1) • FOC: • So what happens when s1 = 1 (i.e. firm 1 is a monopolist)? How about when then number of firms becomes larger, so s1 gets smaller? What about when s1 goes to zero (i.e. firm 1 is an extremely small part of the overall market)? divide both terms in brackets by p(Q) multiply second term in brackets by Q/Q (s1 = q1/Q or firm 1’s market share)

22. Continuous Actions: Bertrand Equilibrium • Is Cournot Equilibrium always a realistic model? • How else might competition work? What would happen in this case?

23. Continuous Actions: Bertrand Equilibrium • Bertrand Equilibrium • Instead of choosing quantity, firms choose price to sell at. • Consider again two firms with C(q) = 4q who face an inverse market demand function of p(Q) = 84 – 8(Q) or equivalently a market demand curve Q(p) = 80.5 – p/8 • What is the optimal strategy for firm 1 for any given action by firm 2?