Finance 30210 Managerial Economics The Basics of Game

Finance 30210: Managerial Economics The Basics of Game Theory

What is a Game?

Prisoner’s Dilemma…A Classic! 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. Jake Clyde The DA puts Jake & Clyde in separate rooms and makes each the following offer: ØKeep your mouth shut and you both get one year in jail ØIf you rat on your partner, you get off free while your partner does 8 years ØIf you both rat, you each get 4 years.

Jake is choosing rows Clyde is choosing columns Clyde Jake Confess Don’t Confess -4 -4 0 -8 Don’t Confess -8 0 -1 -1

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 Confess Don’t Confess -4 -4 0 -8 Don’t Confess -8 0 -1 -1

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 Confess Don’t Confess -4 -4 0 -8 Don’t Confess -8 0 -1 -1

Dominant Strategies Jake’s optimal strategy REGARDLESS OF CLYDE’S DECISION is to confess. Therefore, confess is a dominant strategy for Jake Clyde Jake Note that Clyde’s dominant strategy is also to confess Confess Don’t Confess -4 -4 0 -8 Don’t Confess -8 0 -1 -1

The Nash equilibrium is the outcome (or set of outcomes) where each player is following his/her best response to their opponent’s moves Clyde Jake Here, the Nash equilibrium is both Jake and Clyde confessing Confess Don’t Confess -4 -4 0 -8 Don’t Confess -8 0 -1 -1

The prisoner’s dilemma game is used to describe circumstances where competition forces sub-optimal outcomes Clyde Jake Note that if Jake and Clyde can collude, they would never confess! Confess Don’t Confess -4 -4 0 -8 Don’t Confess -8 0 -1 -1

“Winston tastes good like a cigarette should!” “Us Tareyton smokers would rather fight than switch!” Advertise Don’t Advertise 10 10 30 Don’t Advertise 5 20 20 30 5

Jake The previous example was a “one shot” game. Would it matter if the game were played over and over? Clyde 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? 0 1 2 Play PD Game Time 3 Play PD Game 4 Play PD Game 5 Play PD Game

Perhaps the dynamics of the game changes because the members are interacting over time – this brings is a possible punishment for cheating. Clyde Confess Jake -4 -4 Don’t Confess -8 0 0 -8 -1 -1 “I plan on not confessing today…if you don’t confess today, I will not confess tomorrow, but if you confess today, I will confess forever!” 0 1 Play Prisoner’s Dilemma Game Time Confess Don’t Confess 2 Play Prisoner’s Dilemma Game 3 Play Prisoner’s Dilemma Game 4 5 Play Prisoner’s Dilemma Game

Jake Don’t Confess: -1 Clyde -1 0 1 Play Prisoner’s Dilemma Game Time Confess: Confess “I plan on not confessing today…if you don’t confess today, I will not confess tomorrow, but if you confess today, I will confess forever!” 0 Don’t Confess: -6 Cheat: -20 -4 -1 -4 -4 -4 Don’t Confess -8 -1 2 Play Prisoner’s Dilemma Game Confess -1 3 Play Prisoner’s Dilemma Game -4 0 Don’t Confess 0 -8 -1 -1 -1 4 5 Play Prisoner’s Dilemma Game -4 Clyde shouldn’t confess, right? -4

We need to use backward induction to solve this. Jake Clyde 0 1 Play Prisoner’s Dilemma Game Time 2 Play Prisoner’s Dilemma Game 3 Play Prisoner’s Dilemma Game Clyde 4 5 Play Prisoner’s Dilemma Game Confess Regardless of what took place the first four time periods, what will happen in period 5? Confess -4 -4 Don’t Confess -8 Don’t Confess 0 What should Clyde do here? 0 -8 -1 -1

We need to use backward induction to solve this. Clyde 0 1 Play Prisoner’s Dilemma Game Time Jake 2 Play Prisoner’s Dilemma Game 3 Play Prisoner’s Dilemma Game Clyde 4 5 Play Prisoner’s Dilemma Game Confess Given what happens in period 5, what should happen in period 4? Confess -4 -4 Don’t Confess -8 Don’t Confess 0 What should Clyde do here? 0 -8 -1 -1

We need to use backward induction to solve this. Clyde Jake 0 1 Play Prisoner’s Dilemma Game Confess Time Confess -4 -4 Don’t Confess -8 0 Knowing the future prevents credible promises/threats! 2 3 Clyde 4 5 Play Prisoner’s Dilemma Game Confess Don’t Confess 0 -8 -1 -1 What we would need is for this game to never end!

Infinitely Repeated Games 0 1 2 Play PD Game Jake Clyde …………… 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 noncooperative 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 we value the future enough.

Suppose that Mc. Donald’s is currently the only restaurant in town, but Burger King is considering opening a location. Should Mc. Donald's fight for it’s territory? Fight 0 0 IN 2 Cooperate 2 The equilibrium will be that Mc. Donalds cooperates and Burger King Enters 5 Out 0

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

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

How about this game? 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. Acme What is the Nash Equilibrium? Allied $. 95 $1. 30 $1. 95 $1. 00 3 6 7 3 10 4 $1. 35 5 1 8 2 14 7 $1. 65 6 0 6 2 8 5

Note that Allied would never charge $1 regardless of what Acme charges ($1 is a dominated strategy). Therefore, we can eliminate it from consideration. Acme With the $1 Allied Strategy eliminated, Acme’s strategies of both $. 95 and $1. 30 become dominated. Allied With Acme’s strategies reduced to $1. 95, Allied will respond with $1. 35 $. 95 $1. 30 $1. 95 $1. 00 3 6 7 3 10 4 $1. 35 5 1 8 2 14 7 $1. 65 6 0 6 2 8 5

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 Player B What is the equilibrium to this game? Player A Micro Macro Micro 2 1 0 0 Macro 0 1 2 0

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 Player B Player A Micro Macro Micro 2 1 0 0 Macro 0 1 2 0

If both choose Micro, neither side has an incentive to deviate – an equilibrium! Player B Player A Micro A pure strategy equilibrium refers to an outcome where both sides make a move with certainty. An equilibrium can be maintained because neither side has any incentive to deviate from their choice Macro Micro 2 1 0 0 Macro 0 1 2 0 If both choose Macro, neither has an incentive to deviate – an equilibrium!

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: Attempt Outcome 1 $100 2 $100 3 -$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?

Mixed strategy equilibria involve players making moves with various probabilities Suppose that player B chooses Micro 20% of the time. What should Player A do? Micro: Player B Macro: If player B chooses Micro 20% of the time, you are better off choosing Macro. Player A Micro Macro Micro 2 1 0 0 Macro 0 1 2 0

To maintain a mixed strategy equilibrium, neither player can have a superior choice Suppose Player B chooses Micro with probability Chooses Macro with probability Player B Micro Macro: If you are indifferent… Player A Micro: Macro Micro 2 1 0 0 Macro 0 1 2 0

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

The Stag Hunt: Two individuals are out on a hunt. Each must make a decision on what to hunt without knowledge of the other individual’s choice ØOnly one hunter is required to catch a rabbit – a small, sure reward ØTwo hunters are required to take down a stag – a bigger but riskier reward Stag What’s the equilibrium here? Rabbit Stag 4 4 0 1 Rabbit 1 0 1 1

The Stag Hunt: Two individuals are out on a hunt. Each must make a decision on what to hunt without knowledge of the other individual’s choice If both hunt the stag, neither has an incentive to deviate – an equilibrium! Stag Rabbit Stag 4 4 0 1 Rabbit 1 0 1 1 If both hunt the rabbit, neither has an incentive to deviate – an equilibrium!

Suppose that you believed that your fellow hunter was equally likely to hunt the stag or the rabbit what would you do? 50% Stag Rabbit Stag 4 4 0 1 Rabbit 1 0 1 1 If you hunt the rabbit: You are guaranteed a reward of 1 with certainty If you hunt the stag: 50% of the time you get 4, 50% of the time you get 0 In this example, hunting the stag is reward dominant (better average payout), while hunting the rabbit is risk dominant (lower risk)

What if we change the odds…? 10% 90% Stag Rabbit Stag 4 4 0 1 Rabbit 1 0 1 1 If you hunt the rabbit: You are guaranteed a reward of 1 with certainty If you hunt the stag: 10% of the time you get 2, 90% of the time you get 0 Now, hunting the rabbit is both reward dominant and risk dominant!! Choosing the stag would never be a good idea here.

Let’s find the odds that make the stag and rabbit equally attractive on average… X% Y% Stag Rabbit Stag 4 4 0 1 Rabbit 1 0 1 1 If you hunt the rabbit: You are guaranteed a reward of 1 with certainty If you hunt the stag: For them to be equal on average: X = 25%, Y = 75%

Therefore, in this example, you will only hunt the stag if your fellow hunter hunts the stag at least 25% of the time. Similarly, your fellow hunter will only hunt the stag if you hunt the stag at least 25% of the time. 25% 75% Stag 4 Rabbit 4 6. 25% Rabbit 1 0 18. 75% 0 1 18. 75% 1 1 56. 25%

Therefore, in this case, the stag hunt has three possible equilibria: 50% Stag Rabbit Stag 4 4 0 1 Rabbit 1 0 1 1 Equilibrium #1: Both players always hunt the stag Equilibrium #2: Both players sometimes hunt the stag (each player must hunt the stag at least 25% of the time) Equilibrium #3: Both players never hunt the stag

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 Cheat Don’t Cheat What is the equilibrium to this game? Don’t Audit 5 -5 -25 5 0 0 -1 -1

If the IRS never audited, your best strategy is to cheat (this would only make sense for the IRS if you never cheated) If the IRS always audited, your best strategy is to never cheat (this would only make sense for the IRS if you always cheated) The Equilibrium for this game will involve only mixed strategies! Cheat Don’t Audit 5 -5 -25 5 0 0 -1 -1

Cheating on your taxes! 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, … Cheat Don’t Audit 5 -5 -25 5 0 0 -1 -1

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! Suppose the government audits with probability Doesn’t audit with probability Don’t Audit Cheat: Don’t Cheat: If you are indifferent… Audit Cheat 5 -5 -25 Don’t Cheat 0 0 -1 (83%) 5 -1 (17%)

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! Suppose you cheat with probability Don’t Audit: If they are indifferent… Audit Cheat 5 -5 -25 Don’t Cheat 0 0 -1 (91%) 5 -1 (9%)

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 Audit Don’t cheat with probability Cheat 5 -5 (7. 5%) We can find the odds of any particular event happening…. You Cheat and get audited: Don’t Cheat 0 0 (75%) (1. 5%) Audit -25 5 (1. 5%) -1 -1 (15%)

Suppose that we make the game sequential. That is, one side makes its decision (and that decision is public) before the other Don’t Audit Au Au (5, -5)(-1, -1) t n’ Do dit Au dit Ch ea t t n’ Do eat Ch t n’ Do dit Au (-25, 5) (0, 0) Audit Cheat 5 -5 -25 Don’t Cheat 0 0 -1 5 -1

If the IRS observes you cheating, their best choice is to Audit Don’t Audit dit Au (5, -5)(-1, -1) vs t n’ Do dit Au Au Ch ea t t n’ Do eat Ch t n’ Do dit Au (-25, 5) (0, 0) Audit Cheat 5 -5 -25 Don’t Cheat 0 0 -1 5 -1

If the IRS observes you not cheating, their best choice is to not audit Don’t Audit Au Au t n’ Do dit Au dit Ch ea t t n’ Do eat Ch t n’ Do dit Au (-25, 5) (5, -5)(-1, -1) (0, 0) vs Audit Cheat 5 -5 -25 Don’t Cheat 0 0 -1 5 -1

Knowing how the IRS will respond, you never cheat and they never audit!! Don’t Audit Cheat t ea -5 (0%) t n’ Do eat Ch Ch 5 Don’t Cheat 0 0 dit Au Au (5, -5)(-1, -1) vs t n’ Do dit Au (-25, 5) t n’ Do dit Au dit (100%) (0, 0) Audit -25 5 (0%) -1 -1 (0%)

Now, lets switch positions…suppose the IRS chooses first Don’t Audit Cheat dit -5 (0%) t n’ Do dit Au Au 5 Don’t Cheat 0 0 t ea ea Ch Ch (-1, -1)(5, -5) t n’ Do eat Ch (-25, 5) t n’ Do eat Ch t (0%) (0, 0) Audit -25 5 (0%) -1 -1 (100%)

e ak t. T s n’ ge Do sta Ho Ta Ho ke sta ge s Terrorists President (0, 1) Ne go tia te e t n’ tiat Do go Ne In the third stage, the best response is to kill the hostages Terrorists ill ll (-. 5, -1) t. K n’ Do Ki (1, -. 5) 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) (-1, 1) 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

e ak t. T s n’ ge Do sta Ho Ta Ho ke sta ge s Terrorists President (0, 1) Ne go tia te e t n’ tiat Do go Ne Suppose that a constitutional amendment is passed ruling out hostage negotiation (a commitment device) Terrorists ill ll (-. 5, -1) t. K n’ Do Ki (1, -. 5) The equilibrium is always (Take Hostages/Negotiate). How could we change this outcome? (-1, 1) Without the possibility of negotiation, the new equilibrium becomes (No Hostages)

A bargaining example…How do you divide $20? Player A Offer Player B Accept 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. Day 1 Reject Player B Offer Player A Accept Day 2 Reject Player A Offer Player B Accept Day 3 Reject (0, 0)

Player A Offer Player A knows Day 1 what happens in day 2 and knows that player B wants to avoid that! Player B Accept Reject Player A: $19. 99 Player B: $. 01 Player B Offer Player B knows what happens in Day 2 day 3 and wants to avoid that! Player A Accept Reject Player A: $19. 99 Player B: $. 01 Player A Offer Player B Accept Reject (0, 0) If day 3 arrives, Day 3 player B should accept any offer – a rejection pays out $0! Player A: $19. 99 Player B: $. 01

Lets consider a variation… Player A Offer Player B 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 Accept Year 1 Reject Player B Offer Player A Accept Year 2 Reject Player A Offer Player B Accept Year 3 Reject (0, 0)

Player A If player B rejects, she gets $3. 35 in Year 1 one year. That’s worth $3. 35/1. 10 today Offer Player B Accept Reject Player A: $16. 95 Player B: $3. 05 Player B Offer If player A rejects, she gets $19. 99 in Year 2 one year. That’s worth $19. 99/1. 20 today Player A Accept Reject Player A: $16. 65 Player B: $3. 35 Player A Offer Player B Accept Reject (0, 0) If year 3 arrives, Year 3 player B should accept any offer – a rejection pays out $0! Player A: $19. 99 Player B: $. 01

Lets consider a variation… Player A Offer Player B The Shrinking Pie Game: Negotiations are costly. After each round, the pot gets reduced by 50%: Accept Day 1 $20 Reject Player B Offer Player A Accept Day 2 $10 Reject Player A Offer Player B Accept Day 3 Reject (0, 0) $5

Player A If player B rejects, she gets $5 Day 1 tomorrow. She will accept anything $20 better than $5 Offer Player B Accept Reject Player A: $14. 99 Player B: $5. 01 Player B Offer Player A Accept Day 2 Reject $10 If player A rejects, she gets $4. 99 in one year. She will accept anything better than $4. 99 Player A: $5. 00 Player B: $5. 00 Player A Offer Player B Accept Reject (0, 0) If day 3 arrives, Day 3 player B should accept any offer – a rejection pays $5 out $0! Player A: $4. 99 Player B: $. 01