Intermediate Microeconomics 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.

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.

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 any observed state of the world the payoffs to each player for every possible choice of actions by that player and all the other players.

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?

Simultaneous Move Games “Prisoners’ Dilemma” Consider a game with 2 players, each player has two options: Keep quiet (Cooperate with each other) Blame partner (Defect against the other) Payoffs to Player-1; Cooperate utility of -3 if Player 2 also cooperates, utility of -8 if Player 2 defects. Defect utility of -5 if Player 2 also defects, utility of -1 if Player-2 cooperates. Payoffs to Player-2 are analogous. Players are in different rooms, so they can’t communicate nor know what their partner does when they make their decision.

Simultaneous Move Games 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) (-3,-3)(-8,-1) (-1,-8)(-5,-5) Player-2 CD C DC D Player-1

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 were to keep quiet (cooperate)? What is Player-1’s best action if Player-2 were to blame (defect)? How about for Player-2? So what do think each Player will do? Player-2 CD Player-1 C DC D (-3,-3)(-8,-1) (-1,-8)(-5,-5)

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?

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? (0,0)(-4,-1) (-1,-4)(-3,-3) Player-2 ignoreattack Player-1 ignore attack

Simultaneous Move Games “Battle of the Sexes” - Consider a game of the following form: What is Player-1’s best action to take if Player-2 chooses Italian? What is Player-1’s best action to take if Player-2 chooses Steak? How about for Player-2? (5,2)(1,1) (0,0)(2,5) Player-2 ItalianSteak Player-1 Italian Steak

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?”

Simultaneous Move Games Nash Equilibria of “Prisoner’s Dilemma”? Both Cooperate? One Cooperate, other Defect? Both Defect? Player-2 CD Player-1 C DC D (-3,-3)(-8,-1) (-1,-8)(-5,-5)

Simultaneous Move Games Nash Equilibria of Arms Race? Both Ignore? One Ignore, other Attack? Both Attack? (0,0)(-4,-1) (-1,-4)(-3,-3) Player-2 IA Player-1 I AI A

Simultaneous Move Games Nash Equilibria from “Battle of the Sexes” ? Both Italian? Both Steak? One Italian, one steak? (5,2)(1,1) (0,0)(2,5) Player-2 ItalianSteak Player-1 Italian Steak

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 can be Pareto ranked--- everyone is better off in one than the other---(e.g. Arms Race), or where they cannot be Pareto ranked---players may not agree on which equilibria is the “best”---(e.g. Battle of Sexes) Applications of these types of games?

Game Theory Application: Trade Suppose if Acme produces widgets domestically, its profits will only be $100K and China Corp. will have profits of $0. 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. Should trade happen? Will trade happen?

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

Game Theory: Sequential Games One particular way of committing is to act first. When there is an ordering to actions we call this a sequential game. Equilibria of interest are now what are referred to as Subgame Perfect Nash Equilibria (SPNE). Essentially, each player does not have any incentive to deviate from his SPNE strategy given other players play SPNE strategy in each subgame. What does this mean? Consider the following examples.

Sequential Move Games Consider the Prisoner’s Dilemma game But, suppose Player 1 can move first, and Player 2 can observe what he did. “Extensive form” Game Tree? Sub-game Perfect Nash Equilibria? A strategy is needed for each possible sub- game. (5,5)(2,8) (8,2)(3,3) Player-2 CD Player-1 C DC D

Sequential Move Games Consider the Arms Race game But, suppose Player 1 can move first, and Player 2 can observe what he did. “Extensive form” Game Tree? Sub-game Perfect Nash Equilibria? (0,0)(-4,-1) (-1,-4)(-3,-3) Player-2 IA Player-1 I AI A

Sequential Move Games Consider the “Battle of the Sexes” game But, suppose Player 1 can move first, and Player 2 can observe what he did. “Extensive form” Game Tree? Sub-game Perfect Nash Equilibria? (5,2)(1,1) (0,0)(2,5) Player-2 ItalianSteak Player-1 Italian Steak

Sequential Move Games Does Game theory work in the “Real-world”?