Finance 510: Microeconomic Analysis 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.

Confess Don’t Confess -4 -4 0 -8 -8 0 -1 -1 Strategic (Normal) Form Jake is choosing rows Clyde is choosing columns Clyde Confess Don’t Confess -4 -4 0 -8 -8 0 -1 -1 Jake

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

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

Confess Don’t Confess -4 -4 0 -8 -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 Confess Don’t Confess -4 -4 0 -8 -8 0 -1 -1 Jake Note that Clyde’s dominant strategy is also to confess

Confess Don’t Confess -4 -4 0 -8 -8 0 -1 -1 Nash Equilibrium 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 Confess Don’t Confess -4 -4 0 -8 -8 0 -1 -1 Jake Here, the Nash equilibrium is both Jake and Clyde confessing

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

Repeated Games 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? 1 2 3 4 5 Time Play PD Game Play PD Game Play PD Game Play PD Game Play PD Game Play PD Game

Repeated Games We can use backward induction to solve this. 1 2 3 4 5 Jake Clyde We can use backward induction to solve this. 1 2 3 4 5 Time Play PD Game Play PD Game Play PD Game Play PD Game Play PD Game Play PD Game Confess Confess Confess Confess Confess Confess Confess Confess Confess Confess Confess Confess Similar arguments take us back to period 0 However, once the equilibrium for period 5 is known, there is no advantage to cooperating in period 4 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.

Infinitely Repeated Games Jake Clyde 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 Lifetime Reward from confessing Lifetime Reward from not confessing Not confessing is an equilibrium as long as i < 3 (300%)!!

Infinitely Repeated Games Jake Clyde 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 our discount rate is low enough

The Chain Store Paradox Suppose that McDonalds has an exclusive territory where is earns $100,000 per year, but faces the constant threat of Burger King moving in. If Burger King enters, McDonald's profits fall to $80,000. If it fights, it loses $10,000 today, but creates a reputation that deters future entry. Should McDonalds fight? Cost of Entry Deterrence Present Value of Entry Deterrence

The Chain Store Paradox Accommodate ($80,$10) Enter Fight ($-10,-$10) Accommodate Accommodate ($100,$0) ($80,$10) Stay Out ($80,$10) A Enter Fight ($60,$0) Fight F ($-10,-$10) ($-10,-$10) ($100,$0) A Stay Out A ($60,$0) At the end game, it is always optimal for McDonalds to Accommodate.

The Chain Store Paradox Accommodate ($80,$10) Enter Fight ($-10,-$10) Accommodate Accommodate ($100,$0) ($80,$10) Stay Out ($80,$10) A Enter Fight ($60,$0) Fight F ($-10,-$10) ($-10,-$10) ($100,$0) A Stay Out A ($60,$0) However, given McDonald's accommodation, Burger King always enters!

The Chain Store Paradox Accommodate ($80,$10) Enter Fight ($-10,-$10) Accommodate Accommodate ($100,$0) ($80,$10) Stay Out ($80,$10) A Enter Fight ($60,$0) Fight F ($-10,-$10) ($-10,-$10) ($100,$0) A Stay Out A ($60,$0) However, if entry always occurs, then fighting is not optimal in the prior period!

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

Micro Macro 2 1 0 0 1 2 Choosing Classes! Player B 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 Micro Macro 2 1 0 0 1 2 The Equilibrium for this game will involve mixed strategies! Player A

Choosing Classes! Suppose that Player A has the following beliefs about Player B’s Strategy Probabilities of choosing Micro or Macro Player A’s best response will be his own set of probabilities to maximize expected utility

Subject to Probabilities always have to sum to one Both classes have a chance of being chosen

First Order Necessary Conditions

Best Responses What this says is that if Player A believes that Player B will select Macro with a 2/3 probability, then Player A is willing to randomize between Micro and Macro Notice that if we 1/3 and 2/3 for the above probabilities, we get If Player B is following a 1/3, 2/3 strategy, then any strategy yields the same expected utility for player B

It’s straightforward to show that 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!

Micro Macro 2 1 0 0 1 2 Player B Player A Sequential Games In many games of interest, some of the choices are made sequentially. That is, one player may know the opponents choice before she makes her decision. Player B Micro Macro 2 1 0 0 1 2 Consider the previous game, (with three possible equilibria), but now, let Player A choose first. Player A

We can use a decision tree to write out the extensive form of the game Player A Player A moves first in stage one. Micro Macro Player B Player B The second stage (after the first decision is made) is known as the subgame. Micro Macro Micro Macro (2, 1) (0, 0) (0, 0) (1, 2)

We can use a decision tree to write out the extensive form of the game Now, if Player A chooses Micro Suppose that Player A chooses Macro. Player A Micro Player B should choose Micro Macro Player B should choose Macro Player B Player B Micro Macro Micro Macro (2, 1) (0, 0) (0, 0) (1, 2)

In this game, player A has a first mover advantage Player A knows how player B will respond, and therefore will always choose Micro (and a utility level of 2) over Macro (and a utility level of 1) Player A In this game, player A has a first mover advantage Micro Macro Player B Player B Micro Macro Micro Macro (2, 1) (0, 0) (0, 0) (1, 2)

What about the Macro/Macro equilibrium? If player A know that Player B was following a pure strategy of always choosing Macro, then we could get a Macro/Macro result. Player A Micro Macro But always choosing Macro is not a solution in the subgame. Therefore, Macro/Macro is not subgame perfect Player B Player B Micro Macro Micro Macro (2, 1) (0, 0) (0, 0) (1, 2)

Note: Simultaneous Move Games Suppose that we assume Player A moves first, but Player B can’t observe Player A’s choice? Player A Micro Macro Player B Player B We are back to the original mixed strategy equilibrium! Micro Macro Micro Macro (2, 1) (0, 0) (0, 0) (1, 2)

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

Backward Induction…the Centipede game! 1 2 3 4 5 6 A B A B A B $5.00 $5.00 $1.00 $1.00 $0.00 $3.00 $2.50 $2.50 $1.50 $4.50 $3.50 $3.50 $3.00 $6.00 Two players (A and B) make alternating decisions (Right or Down). Note that at each stage in the game, the total reward increases.

Backward Induction…the Centipede game! 1 2 3 4 5 6 A B A B A B $5.00 $5.00 $1.00 $1.00 $0.00 $3.00 $2.50 $2.50 $1.50 $4.50 $3.50 $3.50 $3.00 $6.00 In stage 6, B’s best move is down In stage 5, Given B’s expected move in stage 6, A will choose down ($3.50 vs. $3) In stage 4, Given A’s move in stage 5, B will choose down ($4.50 vs. $3.50)

Backward Induction…the Centipede game! 1 2 3 4 5 6 A B A B A B $5.00 $5.00 $1.00 $1.00 $0.00 $3.00 $2.50 $2.50 $1.50 $4.50 $3.50 $3.50 $3.00 $6.00 In stage 3, Given B’s move in stage 4, A will choose down ($2.50 vs. $1.50) In stage 2, Given A’s move in stage 3, B will choose down ($3.00 vs. $2.50) In stage 1, Given B’s move in stage 2, A will choose down ($1.00 vs. $0)

Backward Induction…the Centipede game! 1 2 3 4 5 6 A B A B A B $5.00 $5.00 $1.00 $1.00 $0.00 $3.00 $2.50 $2.50 $1.50 $4.50 $3.50 $3.50 $3.00 $6.00 Through backward induction, we find that the equilibrium of this game is A choosing down in the first stage and ending the game!

A bargaining example… Two players have $1 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 discounts future payments at rate Player B discounts future payments at rate Player A is more impatient

What should Player A offer in Day 3? Player B Day 1 What should Player A offer in Day 3? Accept Reject If player A offers $0, Player B is indifferent Player B Offer Player A Day 2 Accept Reject Player A Offer Player B Day 3 Player A = $1, Player B = $0 Accept Reject (0,0)

What should Player B offer in Day 2? Player A Offer Player B Day 1 What should Player B offer in Day 2? Accept Reject We know that Player A is indifferent between $1 tomorrow and $ Player B today Offer Player A Day 2 Player A = $ Player B = $ Accept Reject Player A Offer Player B Day 3 Accept Reject (0,0)

What should Player A offer in Day 1? Player A Player B Day 1 Player B = $ Accept Reject Player B Offer Player A Day 2 Accept Reject What should Player A offer in Day 1? Player A Offer We know that Player B is indifferent between $ Player B Day 3 today and $ tomorrow Accept Reject (0,0)

In this game, player A has a last mover advantage Player B = $0.09 Offer Player A = $ Player B Day 1 Player B = $ Accept Reject The Nash Equilibrium is Player B accepting Player A’s offer on Day one. A couple points… Player A = $0.91 In this game, player A has a last mover advantage Player B = $0.09 This advantage grows as either A becomes more patient or B becomes less patient Player A = $1 or Player B = $0

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 $.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 Allied What is the Nash Equilibrium?

Iterative Dominance 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. $.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 Allied With Acme’s strategies reduced to $1.95, Allied will respond with $1.35