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The Basics of Game Theory Finance 510: Microeconomic Analysis

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What is a Game?

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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.

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

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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

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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

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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

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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

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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!

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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

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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

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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%)!!

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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

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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? Present Value of Entry Deterrence Cost of Entry Deterrence

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

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

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

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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?

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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 The Equilibrium for this game will involve mixed strategies!

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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

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

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First Order Necessary Conditions

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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

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It’s straightforward to show that 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!

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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. MicroMacro Micro Macro0 1 2 Player A Player B Consider the previous game, (with three possible equilibria), but now, let Player A choose first.

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

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

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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 Player B Micro Macro (2, 1)(0, 0) (1, 2) In this game, player A has a first mover advantage

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Player A Player B Micro Macro (2, 1)(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. But always choosing Macro is not a solution in the subgame. Therefore, Macro/Macro is not subgame perfect

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

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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? 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

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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)

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

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Backward Induction…the Centipede game! ABABAB $5.00 $3.00 $6.00$2.50 $0.00 $3.00 $1.50 $4.50$3.50$ 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)

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Backward Induction…the Centipede game! ABABAB $5.00 $3.00 $6.00$2.50 $0.00 $3.00 $1.50 $4.50$3.50$ 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)

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Backward Induction…the Centipede game! ABABAB $5.00 $3.00 $6.00$2.50 $0.00 $3.00 $1.50 $4.50$3.50$ Through backward induction, we find that the equilibrium of this game is A choosing down in the first stage and ending the game!

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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

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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 What should Player A offer in Day 3? If player A offers $0, Player B is indifferent Player A = $1, Player B = $0

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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 What should Player B offer in Day 2? We know that Player A is indifferent between $1 tomorrow and $ today Player A = $Player B = $

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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 What should Player A offer in Day 1? We know that Player B is indifferent between $ today and $ tomorrow Player A = $ Player B = $

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

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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?

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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

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