Finance 30210: Managerial Economics

Finance 30210: Managerial Economics The Basics of Game Theory

What is a Game? Players Rules Payoffs

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: • 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.

Jake is choosing rows Clyde is choosing columns Clyde Jake

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 Clyde Jake

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 Clyde Jake

Dominant Strategies Jake’s optimal strategy REGARDLESS OF CLYDE’S DECISION is to confess. Therefore, confess is a dominant strategyfor Jake Clyde Jake Note that Clyde’s dominant strategy is also to confess

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

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!

“Winston tastes good like a cigarette should!” “Us Tareyton smokers would rather fight than switch!”

Jake Clyde 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? 0 1 2 3 4 5 Time Play PD Game Play PD Game Play PD Game Play PD Game Play PD Game 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 Jake Jake “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 2 3 4 5 Time Play Prisoner’s Dilemma Game Play Prisoner’s Dilemma Game Play Prisoner’s Dilemma Game Play Prisoner’s Dilemma Game Play Prisoner’s Dilemma Game Play Prisoner’s Dilemma Game

“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!” Jake Don’t Confess: -1 -1 -1 -1 -1 -1 Clyde 0 1 2 3 4 5 Time Play Prisoner’s Dilemma Game Play Prisoner’s Dilemma Game Play Prisoner’s Dilemma Game Play Prisoner’s Dilemma Game Play Prisoner’s Dilemma Game Play Prisoner’s Dilemma Game -4 Confess: 0 -4 -4 -4 -4 Don’t Confess: -6 Cheat: -20 Clyde shouldn’t confess, right?

We need to use backward induction to solve this. Jake Clyde Clyde 0 1 2 3 4 5 Time Play Prisoner’s Dilemma Game Play Prisoner’s Dilemma Game Play Prisoner’s Dilemma Game Play Prisoner’s Dilemma Game Play Prisoner’s Dilemma Game Play Prisoner’s Dilemma Game Regardless of what took place the first four time periods, what will happen in period 5? What should Clyde do here?

We need to use backward induction to solve this. Jake Clyde Clyde 0 1 2 3 4 5 Time Play Prisoner’s Dilemma Game Play Prisoner’s Dilemma Game Play Prisoner’s Dilemma Game Play Prisoner’s Dilemma Game Play Prisoner’s Dilemma Game Play Prisoner’s Dilemma Game Confess Given what happens in period 5, what should happen in period 4? What should Clyde do here?

We need to use backward induction to solve this. Jake Clyde Clyde 0 1 2 3 4 5 Time Play Prisoner’s Dilemma Game Play Prisoner’s Dilemma Game Play Prisoner’s Dilemma Game Play Prisoner’s Dilemma Game Play Prisoner’s Dilemma Game Play Prisoner’s Dilemma Game Confess Confess Confess Confess Confess Knowing the future prevents credible promises/threats! What we would need is for this game to never end!

Infinitely Repeated Games Jake Clyde 0 1 2 Play PD Game Play PD Game 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 we value the future 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? 0 Fight 0 IN 2 Cooperate 2 The equilibrium will be that McDonalds cooperates and Burger King Enters 5 Out 0

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 Enter Don’t Fight Enter Don’t Fight 2 2 2 Cooperate Total =2*20 = 40 OR Enter Fight Don’t Enter Don’t Enter Don’t Enter Don’t Enter 0 5 5 5 5 Fight Total =19*5 = 95

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? Enter Don’t Enter Fight Don’t Fight Enter Don’t Enter Fight Don’t Fight Enter Don’t Enter Fight Don’t Fight 20th 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!!

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 Allied What is the Nash Equilibrium?

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

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

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

If both choose Micro, neither side has an incentive to deviate – an equilibrium! Player B Player A 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 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: 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

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: Player A Macro: If you are indifferent…

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 • Only one hunter is required to catch a rabbit – a small, sure reward • Two hunters are required to take down a stag – a bigger but riskier reward What’s the equilibrium here?

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! 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% 50% 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% 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% 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% 25% 6.25% 18.75% 75% 18.75% 56.25%

Therefore, in this case, the stag hunt has three possible equilibria: 50% 50% 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 What is the equilibrium to this game?

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!

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, …

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 Cheat: Don’t Cheat: If you are indifferent… (83%) (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 cheat with probability Audit: Don’t Audit: If they are indifferent… (91%) (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 cheat with probability (7.5%) (1.5%) We can find the odds of any particular event happening…. (75%) (15%) You Cheat and get audited: (1.5%)

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

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

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

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

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

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) Terrorists Take Hostages Don’t Take Hostages President (0, 1) Don’t Negotiate Negotiate In the third stage, the best response is to kill the hostages (1, -.5) Terrorists Given the terrorist response, it is optimal for the president to negotiate in stage 2 Kill Don’t Kill Given Stage two, it is optimal for the terrorists to take hostages (-.5, -1) (-1, 1)

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

Finance 30210: Managerial Economics