Finance 30210: Managerial Economics The Basics of Game Theory

What is a Game?

Prisoner’s Dilemma…A Classic! Jake Two prisoners (Jake & Clyde) have been arrested. The DA has enough evidence to convict them both for 1 year, but would like to convict them of a more serious crime. Clyde The DA puts Jake & Clyde in separate rooms and makes each the following offer:  Keep your mouth shut and you both get one year in jail  If you rat on your partner, you get off free while your partner does 8 years  If you both rat, you each get 4 years.

Strategic (Normal) Form Jake Clyde ConfessDon’t Confess Confess Don’t Confess -8 0 Jake is choosing rows Clyde is choosing columns

Jake Clyde ConfessDon’t Confess Confess Don’t Confess -8 0 Suppose that Jake believes that Clyde will confess. What is Jake’s best response? If Clyde confesses, then Jake’s best strategy is also to confess

Jake Clyde ConfessDon’t Confess Confess Don’t Confess -8 0 Suppose that Jake believes that Clyde will not confess. What is Jake’s best response? If Clyde doesn’t confesses, then Jake’s best strategy is still to confess

Jake Clyde ConfessDon’t Confess Confess Don’t Confess -8 0 Dominant Strategies Jake’s optimal strategy REGARDLESS OF CLYDE’S DECISION is to confess. Therefore, confess is a dominant strategy for Jake Note that Clyde’s dominant strategy is also to confess

Nash Equilibrium Jake Clyde ConfessDon’t Confess Confess Don’t Confess -8 0 The Nash equilibrium is the outcome (or set of outcomes) where each player is following his/her best response to their opponent’s moves Here, the Nash equilibrium is both Jake and Clyde confessing

The Prisoner’s Dilemma Jake Clyde ConfessDon’t Confess Confess Don’t Confess -8 0 The prisoner’s dilemma game is used to describe circumstances where competition forces sub-optimal outcomes Note that if Jake and Clyde can collude, they would never confess!

“Winston tastes good like a cigarette should!” “Us Tareyton smokers would rather fight than switch!” AdvertiseDon’t Advertise Advertise Don’t Advertise

Repeated Games JakeClyde The previous example was a “one shot” game. Would it matter if the game were played over and over? Suppose that Jake and Clyde were habitual (and very lousy) thieves. After their stay in prison, they immediately commit the same crime and get arrested. Is it possible for them to learn to cooperate? Time Play PD Game

Repeated Games JakeClyde Time Play PD Game We can use backward induction to solve this. At time 5 (the last period), this is a one shot game (there is no future). Therefore, we know the equilibrium is for both to confess. Confess However, once the equilibrium for period 5 is known, there is no advantage to cooperating in period 4 ConfessConfessConfessConfessConfess Similar arguments take us back to period 0

Infinitely Repeated Games JakeClyde 012 Play PD Game …………… Suppose that Jake knows Clyde is planning on NOT CONFESSING at time 0. If Jake confesses, Clyde never trusts him again and they stay in the non- cooperative equilibrium forever Lifetime Reward from confessing Lifetime Reward from not confessing Not confessing is an equilibrium as long as i < 3 (300%)!!

Infinitely Repeated Games JakeClyde 012 Play PD Game …………… Suppose that Jake knows Clyde is planning on NOT CONFESSING at time 0. If Jake confesses, Clyde never trusts him again and they stay in the non- cooperative equilibrium forever The Folk Theorem basically states that if we can “escape” from the prisoner’s dilemma as long as we play the game “enough” times (infinite times) and our discount rate is low enough

Suppose that McDonald’s is currently the only restaurant in town, but Burger King is considering opening a location. Should McDonald's fight for it’s territory? IN Out Fight Cooperate

Now, suppose that this game is played repeatedly. That is, suppose that McDonald's faces possible entry by burger King in 20 different locations. Can entry deterrence be a credible strategy? Enter Don’t Fight 2 EnterDon’t Enter FightDon’t Enter 0 OR Total =2*20 = 40 Total =19*5 = 95 Cooperate Fight

Enter Don’t Enter Enter Don’t Enter Enter Don’t Enter Fight Don’t Fight Fight Don’t Fight Fight Don’t Fight 20 th location Does McDonald’s have an incentive to fight here? What will Burger King do here? If there is an “end date” then McDonald's threat loses its credibility!! Now, suppose that this game is played repeatedly. That is, suppose that McDonald's faces possible entry by burger King is 20 different locations. Can entry deterrence be a credible strategy?

How about this game? $.95$1.30$1.95 $ $ $ Allied Acme Acme and Allied are introducing a new product to the market and need to set a price. Below are the payoffs for each price combination. What is the Nash Equilibrium?

Iterative Dominance $.95$1.30$1.95 $ $ $ Allied Acme Note that Allied would never charge $1 regardless of what Acme charges ($1 is a dominated strategy). Therefore, we can eliminate it from consideration. With the $1 Allied Strategy eliminated, Acme’s strategies of both $.95 and $1.30 become dominated. With Acme’s strategies reduced to $1.95, Allied will respond with $1.35

Choosing Classes! Suppose that you and a friend are choosing classes for the semester. You want to be in the same class. However, you prefer Microeconomics while your friend prefers Macroeconomics. You both have the same registration time and, therefore, must register simultaneously MicroMacro Micro Macro0 1 2 Player A Player B What is the equilibrium to this game?

MicroMacro Micro Macro0 1 2 Player A Player B Choosing Classes! If Player B chooses Micro, then the best response for Player A is Micro If Player B chooses Macro, then the best response for Player A is Macro There are two types of equilibria for this game: Pure strategies and mixed strategies!

A quick detour: Expected Value Suppose that I offer you a lottery ticket: This ticket has a 2/3 chance of winning $100 and a 1/3 chance of losing $100. How much is this ticket worth to you? Suppose you played this ticket 6 times: AttemptOutcome 1$ $100 4$100 5-$100 6$100 Total Winnings: $200 Attempts: 6 Average Winnings: $200/6 = $33.33

A quick detour: Expected Value Given a set of probabilities, Expected Value measures the average outcome Expected Value = A weighted average of the possible outcomes where the weights are the probabilities assigned to each outcome Suppose that I offer you a lottery ticket: This ticket has a 2/3 chance of winning $100 and a 1/3 chance of losing $100. How much is this ticket worth to you?

Choosing Classes! Suppose that player B chooses Micro 20% of the time. What should Player A do? Micro: Macro: If player B chooses Micro 20% of the time, you are better off choosing Macro. MicroMacro Micro Macro0 1 2 Player A Player B

Micro: Macro: Suppose Player B chooses Micro with probability Chooses Macro with probability If you are indifferent… Choosing Classes! MicroMacro Micro Macro0 1 2 Player B Player A

There are three possible Nash Equilibrium for this game Both always choose Micro Both always choose Macro Both Randomize between Micro and Macro Note that the strategies are known with certainty, but the outcome is random!

Don’t Audit Audit Cheat Don’t Cheat 0 What is the equilibrium to this game? Ever Cheat on your taxes? In this game you get to decide whether or not to cheat on your taxes while the IRS decides whether or not to audit you

If the IRS never audited, your best strategy is to cheat (this would only make sense for the IRS if you never cheated) The Equilibrium for this game will involve only mixed strategies! Don’t Audit Audit Cheat Don’t Cheat 0 If the IRS always audited, your best strategy is to never cheat (this would only make sense for the IRS if you always cheated)

Cheating on your taxes! Don’t Audit Audit Cheat Don’t Cheat 0 Suppose that the IRS Audits 25% of all returns. What should you do? Cheat: Don’t Cheat: If the IRS audits 25% of all returns, you are better off not cheating. But if you never cheat, they will never audit, …

Don’t AuditAudit Cheat Don’t Cheat0 The only way this game can work is for you to cheat sometime, but not all the time. That can only happen if you are indifferent between the two! Cheat: Don’t Cheat: Suppose the government audits with probability Doesn’t audit with probability If you are indifferent… (83%)(17%)

Don’t AuditAudit Cheat Don’t Cheat0 We also need for the government to audit sometime, but not all the time. For this to be the case, they have to be indifferent! Audit: Don’t Audit: Suppose you cheat with probability Don’t cheat with probability If they are indifferent… (91%)(9%)

Don’t AuditAudit Cheat Don’t Cheat0 Now we have an equilibrium for this game that is sustainable! The government audits with probability Doesn’t audit with probability Suppose you cheat with probability Don’t cheat with probability We can find the odds of any particular event happening…. You Cheat and get audited:(1.5%) (7.5%) (15%)(75%)

The Airline Price Wars $500 $ Suppose that American and Delta face the given demand for flights to NYC and that the unit cost for the trip is $200. If they charge the same fare, they split the market P = $500P = $220 P = $500$9,000 $3,600 $0 P = $220$0 $3,600 $1,800 American Delta What will the equilibrium be?

The Airline Price Wars P = $500P = $220 P = $500$9,000 $3,600 $0 P = $220$0 $3,600 $1,800 American Delta If American follows a strategy of charging $500 all the time, Delta’s best response is to also charge $500 all the time If American follows a strategy of charging $220 all the time, Delta’s best response is to also charge $220 all the time This game has multiple equilibria and the result depends critically on each company’s beliefs about the other company’s strategy

The Airline Price Wars: Mixed Strategy Equilibria P = $500P = $220 P = $500$9,000 $3,600 $0 P = $220$0 $3,600 $1,800 American Delta Charge $500: Charge $220: Suppose American charges $500 with probability Charges $220 with probability (75%) (25%) (56%)(19%) (6%)

Suppose that we make the game sequential. That is, one side makes its decision (and that decision is public) before the other Cheat Audit Don’t Cheat Don’t Audit (-25, 5)(5, -5)(-1, -1)(0, 0) Don’t AuditAudit Cheat Don’t Cheat0

If the IRS observes you cheating, their best choice is to Audit Cheat Audit Don’t Cheat Don’t Audit (-25, 5)(5, -5)(-1, -1)(0, 0) Don’t AuditAudit Cheat Don’t Cheat0 vs

If the IRS observes you not cheating, their best choice is to not audit Cheat Audit Don’t Cheat Don’t Audit (-25, 5)(5, -5)(-1, -1)(0, 0) Don’t AuditAudit Cheat Don’t Cheat0 vs

Knowing how the IRS will respond, you never cheat and they never audit!! Cheat Audit Don’t Cheat Don’t Audit (-25, 5)(5, -5)(-1, -1)(0, 0) Don’t AuditAudit Cheat Don’t Cheat0 vs (0%) (100%)

Now, lets switch positions…suppose the IRS chooses first Audit Cheat Don’t Audit Don’t Cheat (-25, 5)(-1, -1)(5, -5)(0, 0) Don’t AuditAudit Cheat Don’t Cheat0 (0%) (100%)(0%)

Again, we could play this game sequentially $500 $220 (9,000, 9,000)(3,600, 0)(0, 3,600)(1,800, 1,800) Delta’s reward is on the left P = $500P = $220 P = $500$9,000 $3,600 $0 P = $220$0 $3,600 $1,800 (0%)(100%) (0%)

Terrorists President Take Hostages Negotiate Kill Don’t Take Hostages Don’t Kill Don’t Negotiate (1, -.5) (-.5, -1)(-1, 1) (0, 1) In the Movie Air Force One, Terrorists hijack Air Force One and take the president hostage. Can we write this as a game? (Terrorists payouts on left) In the third stage, the best response is to kill the hostages Given the terrorist response, it is optimal for the president to negotiate in stage 2 Given Stage two, it is optimal for the terrorists to take hostages

Terrorists President Take Hostages Negotiate Kill Don’t Take Hostages Don’t Kill Don’t Negotiate (1, -.5) (-.5, -1)(-1, 1) (0, 1) The equilibrium is always (Take Hostages/Negotiate). How could we change this outcome? Suppose that a constitutional amendment is passed ruling out hostage negotiation (a commitment device) Without the possibility of negotiation, the new equilibrium becomes (No Hostages)

A bargaining example…How do you divide $20? Two players have $20 to divide up between them. On day one, Player A makes an offer, on day two player B makes a counteroffer, and on day three player A gets to make a final offer. If no agreement has been made after three days, both players get $0. Player A Player B Offer AcceptReject Player B Offer Player A AcceptReject Player A Offer Player B AcceptReject (0,0) Day 1 Day 2 Day 3

Player A Player B Offer AcceptReject Player B Offer Player A AcceptReject Player A Offer Player B AcceptReject (0,0) Day 1 Day 2 Day 3 If day 3 arrives, player B should accept any offer – a rejection pays out $0! Player A: $19.99 Player B: $.01 Player B knows what happens in day 3 and wants to avoid that! Player A: $19.99 Player B: $.01 Player A knows what happens in day 2 and knows that player B wants to avoid that! Player A: $19.99 Player B: $.01

Player A Player B Offer AcceptReject Player B Offer Player A AcceptReject Player A Offer Player B AcceptReject (0,0) Year 1 Year 2 Year 3 Lets consider a variation… Variation : Negotiations take a lot of time and each player has an opportunity cost of waiting: Player A has an investment opportunity that pays 20% per year. Player B has an investment strategy that pays 10% per year

Player A Player B Offer AcceptReject Player B Offer Player A AcceptReject Player A Offer Player B AcceptReject (0,0) Year 1 Year 2 Year 3 If year 3 arrives, player B should accept any offer – a rejection pays out $0! Player A: $19.99 Player B: $.01 If player A rejects, she gets $19.99 in one year. That’s worth $19.99/1.20 today Player A: $16.65 Player B: $3.35 If player B rejects, she gets $3.35 in one year. That’s worth $3.35/1.10 today Player A: $16.95 Player B: $3.05