Announcements• Pass Homework 2 to the aisle!• Lisa will check off your names, and you can pick up at end of class.• Expect Homework 3 to be posted by Wednesday.• Midterm isn’t until May 3, but make sure to catch up if ASAP if you find yourself falling behind• Definitely come to my office hours if you’re confused!
Last Class• Finished our discussion of market power• We spoke about how oligopolists, in particular, might compete with one another to set the lowest price. Segue…• Today, we will formalize our understanding of strategic interaction by introducing the fundamentals of game theory.
Learning Goals for Today• Be able to summarize the elements of a ``game.’’• Recognize the dominant strategies of a game.• Discern the Nash equilibria of a game.
How will we do this?• We will both: – define a games generally (abstractly). – work through explicit examples including two competing oligopolists.
Game Theory• Game theory helps us analyze situations in which the benefit of a given action depends on the actions of others.• Three elements of any game• (1) Players: The decision makers.• (2) Strategies: The actions players can take.• (3) Payoffs: The rewards for each possible combination of actions.
Advertising Game• Players: AT&T and Verizon• Strategies: Raise advertising spending or not.• Payoffs: – Both leave spending unchanged: 1.5 million each – Both increase spending (hurt each other): 1 million each – AT&T increases spending, Verizon does not (AT&T hurts Verizon): AT&T 2 million, Verizon 500k – Verizon increases spending, AT&T does not (Verizon hurts AT&T): Verizon 2 million, AT&T 500k.
Payoff Matrix for Advertising Game AT&T Increase Leave Spending Spending Unchanged Increase 1 million (Vz) 2 million (Vz) Spending 1 million (ATT) 0.5 million (ATT)Verizon Leave 0.5 million (Vz) 1.5 million (Vz) Spending 2 million (ATT) 1.5 million (ATT) Unchanged
Dominant Strategies• Dominant strategy: a strategy that yields a higher payoff not matter what the other player does.• Dominated strategy: any other strategy available to a player who has a dominant strategy.• In the advertising game did the players have a dominant strategy?• When players have dominant strategies, it’s easy to see what the outcome of the game will be.
Equilibrium• Nash equilibrium: a game is said to be in equilibrium if each player’s strategy is the best he or she can choose, given the other players choices.• A set of mutual best responses. – How are these defined?
Payoff Matrix for Advertising Game 2 AT&T Increase Leave Spending Spending Unchanged Increase 1 million (V) 1.5 million (V) Spending 1 million (A) 1.2 million (A)Verizon Leave 1.2 million (V) 0.5 million (V) Spending 1.8 million (A) 2 million (A) Unchanged
Payoff Matrix for Advertising Game 3 AT&T Increase Leave Spending Spending Unchanged Increase 1 million (V) 1.1 million (V) Spending 1.5 million (A) 1 million (A)Verizon Leave 1.8 million (V) 2 million (V) Spending 1 million (A) .5 million (A) Unchanged
The Prisoner’s Dilemma Criminal 1 Confess Deny -10 (S) -1 (S) Confess -10 (L) -25 (L)Criminal 2 -25 (S) 0 (S) Deny -1 (L) 0 (L)
Other Examples of the Prisoner’s Dilemma• Shouting at parties.• People crowding around the baggage claim area at airports.• Curbing CO2 emissions to slow climate change.• Global arms race.• Lance Armstrong.
How do two agents get out of this dilemma?• The dilemma: both agents have dominant, but suboptimal strategies.• Ideally, both agents’ dominant strategy would be the optimal contract.• Consider writing a contract to enforce the best outcome.
Recall the Original Advertising Game AT&T Increase Leave Spending Spending Unchanged Increase 1 million (Vz) 2 million (Vz) Spending 1 million (ATT) 0.5 million (ATT)Verizon Leave 0.5 million (Vz) 1.5 million (Vz) Spending 2 million (ATT) 1.5 million (ATT) UnchangedContract: If I cheat, I pay the other no less than X dollars.
What X makes (No Chg, No Chg) a Nash Equilibrium?A. 0 millionB. 0.5 millionC. 1 millionD. 1.5 millionE. 2 million
The Economics of Cartels• Cartel: any group of firms that agree to restrict output for the purpose of earning an economic profit.• But cartels are notoriously hard to maintain. Why?• Example:oligopolists Boeing and Airbus P $1 million Profit MC=ATC $600 million MR D 1000 Q (in thousands)
Payoff Matrix for a Cartel Agreement Boeing P=$1 million P=$999,999 (Cooperate) (Defect) P=$1 million $300 million (A) 0 (A) (Cooperate) $300 million (B) ≈$600 million (B)Airbus P=$999,999 ≈$600 million (A) ˂$300million (A) (Defect) 0 (B) ˂$300 million (B)
Next time• Games where timing matters• Extensive form games