Overview Economics 3030 I. Introduction to Chapter 10 II. Simultaneous-Move, One-Shot Games Game Theory: III. Infinitely Repeated Games Inside Oligopoly IV. Finitely Repeated Games V. Multistage Games

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Normal Form Game A Normal Form Game

• A Normal Form Game consists of:

n Players Player 2 n Strategies or feasible actions n Payoffs A B C a 12,11 11,12 14,13 b 11,10 10,11 12,12 Player 1 c 10,15 10,13 13,14

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Normal Form Game: Normal Form Game: Scenario Analysis Scenario Analysis • Suppose 1 thinks 2 will choose “A”. • Then 1 should choose “a”. n Player 1’s best response to “A” is “a”. Player 2 Player 2 Strategy A B C a 12,11 11,12 14,13 Strategy A B C b 11,10 10,11 12,12 a 12,11 11,12 14,13

Player 1 11,10 10,11 12,12 c 10,15 10,13 13,14 b Player 1 c 10,15 10,13 13,14

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1 Normal Form Game: Normal Form Game: Scenario Analysis Scenario Analysis • Suppose 1 thinks 2 will choose “B”. • Then 1 should choose “a”. n Player 1’s best response to “B” is “a”. Player 2 Player 2 Strategy A B C Strategy A B C a 12,11 11,12 14,13 a 12,11 11,12 14,13 b 11,10 10,11 12,12 11,10 10,11 12,12

Player 1 b

c 10,15 10,13 13,14 Player 1 c 10,15 10,13 13,14

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Normal Form Game Dominant Strategy • Regardless of whether Player 2 chooses A, B, or C, Scenario Analysis Player 1 is better off choosing “a”! • “a” is Player 1’s Dominant Strategy (i.e., the • Similarly, if 1 thinks 2 will choose C… strategy that results in the highest payoff regardless n Player 1’s best response to “C” is “a”. of the opponent’s action). Player 2 Player 2

Strategy A B C Strategy A B C a 12,11 11,12 14,13 a 12,11 11,12 14,13 b 11,10 10,11 12,12 b 11,10 10,11 12,12 Player 1 c 10,15 10,13 13,14 Player 1 c 10,15 10,13 13,14

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Putting Yourself in your Rival’s The Outcome Shoes Player 2 • What should player 2 do? Strategy A B C n 2 has no dominant strategy! a 12,11 11,12 14,13 n But 2 should reason that 1 will play “a”. b 11,10 10,11 12,12 n Therefore 2 should choose “C”. Player 1 10,15 10,13 13,14 Player 2 c Strategy A B C • This outcome is called a (i.e., a 12,11 11,12 14,13 no way a player can unilaterally change strategies b 11,10 10,11 12,12 and be better off). Player 1 c 10,15 10,13 13,14 n “a” is player 1’s best response to “C”. n “C” is player 2’s best response to “a”. 11 12

2 Key Insights E.g., A Market Share Game

• Look for dominant strategies • Two managers want to maximize market • Put yourself in your rival’s shoes share • Strategies are pricing decisions • Simultaneous moves • One-shot game

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Market-Share Game The Market-Share Game Equilibrium in Normal Form

Manager 2 Manager 2 Strategy P=$10 P=$5 P = $1 Strategy P=$10 P=$5 P = $1 P=$10 .5, .5 .2, .8 .1, .9 P=$10 .5, .5 .2, .8 .1, .9 P=$5 .8, .2 .5, .5 .2, .8 P=$5 .8, .2 .5, .5 .2, .8 Manager 1

Manager 1 P=$1 .9, .1 .8, .2 .5, .5 P=$1 .9, .1 .8, .2 .5, .5

Note: P = $1 is the dominant strategy for both managers Nash Equilibrium

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Examples of Coordination Key Insight: Games • Game theory can also be used to analyze • Product standards situations where “payoffs” are non n size of floppy disks monetary! n size of CDs n VHS vs. Betamax • National standards n electric current

n traffic laws

n etc. • It may be beneficial for all to cooperate and

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3 A in A Coordination Problem: Normal Form Three Nash Equilibria!

Player 2 Player 2 Strategy A B C Strategy A B C 1 0,0 0,0 $10,$10 1 0,0 0,0 $10,$10 2 $10,$10 0,0 0,0 Player 1 2 $10,$10 0,0 0,0 Player 1 3 0,0 $10,$10 0,0 3 0,0 $10, $10 0,0

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Key Insights: An Advertising Game • Not all games are games of conflict. • Two firms (Kellogg’s & General Mills) • Communication can help solve coordination managers want to maximize profits problems. • Strategies consist of advertising campaigns • Sequential moves can help solve coordination • Simultaneous moves problems (i.e., let one player move first) • One-shot interaction • Repeated interaction

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Equilibrium to the One-Shot A One-Shot Advertising Game Advertising Game

General Mills General Mills Strategy None Moderate High Strategy None Moderate High None 12,12 1, 20 -1, 15 None 12,12 1, 20 -1, 15 Moderate 20, 1 6, 6 0, 9 Kellogg’s Moderate 20, 1 6, 6 0, 9 High 15, -1 9, 0 2, 2 Kellogg’s High 15, -1 9, 0 2, 2 Nash Equilibrium

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4 Can work if the game No (by backwards induction). is repeated 2 times? • In period 2, the game is a one-shot game, so equilibrium entails High Advertising in the last period. General Mills • This means period 1 is “really” the last Strategy None Moderate High period, since everyone knows what will None 12,12 1, 20 -1, 15 happen in period 2. Moderate 20, 1 6, 6 0, 9 Kellogg’s High 15, -1 9, 0 2, 2 • Equilibrium entails High Advertising by each firm in both periods. • The same holds true if we repeat the game

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Suppose General Mills adopts this Can collusion work if firms play the trigger strategy. Kellogg’s profits? game each year, forever? 2 3 PCooperate = 12 +12/(1+i) + 12/(1+i) + 12/(1+i) + … • Consider the following “trigger strategy” Value of a perpetuity of $12 paid = 12 + 12/i at the end of every year by each firm: 2 3 PCheat = 20 +2/(1+i) + 2/(1+i) + 2/(1+i) + … n “Don’t advertise, provided the rival has not advertised in the past. If the rival ever advertises, “punish” it by = 20 + 2/i engaging in a high level of advertising forever after.” General Mills • In effect, each firm agrees to “cooperate” so long as the rival hasn’t “cheated” in the Strategy None Moderate High past. “Cheating” triggers punishment in all None 12,12 1, 20 -1, 15 future periods. Moderate 20, 1 6, 6 0, 9

Kellogg’s High 15, -1 9, 0 2, 2 27 28

Kellogg’s Gain to Cheating: Benefits & Costs of Cheating

• PCheat - PCooperate = 20 + 2/i - (12 + 12/i) = 8 - 10/i • PCheat - PCooperate = 8 - 10/i n Suppose i = .05 n 8 = Immediate Benefit (20 - 12 today) • PCheat - PCooperate = 8 - 10/.05 = 8 - 200 = -192 n 10/i = PV of Future Cost (12 - 2 forever after) • It doesn’t pay to deviate. • If Immediate Benefit > PV of Future Cost

n Collusion is a Nash equilibrium in the infinitely repeated n Pays to “cheat”. game! General Mills • If Immediate Benefit £ PV of Future Cost n Doesn’t pay to “cheat”. Strategy None Moderate High General Mills None 12,12 1, 20 -1, 15 Strategy None Moderate High None 12,12 1, 20 -1, 15 Moderate 20, 1 6, 6 0, 9 Kellogg’s Moderate 20, 1 6, 6 0, 9 High 15, -1 9, 0 2, 2 Kellogg’s High 15, -1 9, 0 2, 2 29 30

5 Real World Examples of Key Insight Collusion • Collusion can be sustained as a Nash 1. Garbage Collection Industry equilibrium when there is no certain “end” 2. OPEC to a game. • Doing so requires:

n Ability to monitor actions of rivals

n Ability (and reputation for) punishing defectors

n Low interest rate n High probability of future interaction

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1. Garbage Collection Industry Normal Form Bertrand Game

• Homogeneous products • Bertrand oligopoly Firm 2 • Identity of customers is known Strategy Low Price High Price • Identity of competitors is known Firm 1 Low Price 0,0 20,-1 High Price -1, 20 15, 15

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One-Shot Bertrand Potential (Nash) Equilibrium Equilibrium Outcome

Firm 2 Firm 2 Strategy Low Price High Price Strategy Low Price High Price Firm 1 Low Price 0,0 20,-1 Firm 1 Low Price 0,0 20,-1 High Price -1, 20 15, 15 High Price -1, 20 15, 15

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6 2. OPEC Current OPEC Members • Cartel founded in 1960 by Iran, Iraq, Kuwait, Saudi Arabia, and Venezuela • Currently has 11 members • “OPEC’s objective is to co-ordinate and unify petroleum policies among Member Countries, in order to secure fair and stable prices for petroleum producers…” (www.opec.com) • Cournot oligopoly

• With no collusion: P Competition < PCournot < PMonopoly 37 38

Cournot Game in Normal One-Shot Cournot Form (Nash) Equilibrium Venezuela Venezuela Strategy High Q Med Q Low Q High Q 5, 3 9,4 3, 6 Strategy High Q Med Q Low Q Med Q 6, 7 12,10 20, 8 High Q 5, 3 9,4 3, 6

Saudi Arabia Low Q 8, 1 10, 18 18, 15 Med Q 6, 7 12,10 20, 8 Saudi Arabia Low Q 8, 1 10, 18 18, 15

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Effect of Collusion on Oil Repeated Game Equilibrium* Prices Price Venezuela $30 Strategy High Q Med Q Low Q High Q 5, 3 9,4 3, 6 Med Q 6, 7 12,10 20, 8 $15 Saudi Arabia Low Q 8, 1 10, 18 18, 15 World Demand for Oil * (Assuming a Low Interest Rate)

41 Low Medium Quantity of Oil42

7 OPEC’s Demise Caveat

40 Low Interest High Interest 35 • Collusion is illegal in most countries Rates Rates 30 • Firms are constantly been investigated by the 25 Competition Bureau in Canada and brought to 20 trial in Federal Court 15 • OPEC isn’t illegal; North American laws don’t 10 apply 5

0 1970 1972 1974 1976 1978 1980 1982 1984 1986 -5

Real Interest Rate Price of Oil 43 44

Simultaneous-Move Bargaining The Bargaining Game • Management and a union are negotiating a wage increase. in Normal Form • Strategies are wage offers & wage demands • Successful negotiations lead to $600 million in surplus, Union which must be split among the parties • Failure to reach an agreement results in a loss to the firm Strategy W = $10 W = $5 W = $1 of $100 million and a union loss of $3 million W = $10 100, 500 -100, -3 -100, -3 • Simultaneous moves, and time permits only one-shot at W=$5 -100, -3 300, 300 -100, -3 W=$1 -100, -3 -100, -3 500, 100

making a deal. Management

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Three Nash Equilibria! Fairness: The “Natural” Focal Point

Union Union

Strategy W = $10 W = $5 W = $1 Strategy W = $10 W = $5 W = $1 W = $10 100, 500 -100, -3 -100, -3 W = $10 100, 500 -100, -3 -100, -3 W=$5 -100, -3 300, 300 -100, -3 W=$5 -100, -3 300, 300 -100, -3 W=$1 -100, -3 -100, -3 500, 100 W=$1 -100, -3 -100, -3 500, 100 Management Management

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8 Lessons in Single Offer Bargaining Simultaneous Bargaining • Now suppose the game is sequential in nature, and • Simultaneous-move bargaining results in a management gets to make the union a “take-it-or- coordination problem leave-it” offer. • Experiments suggests that, in the absence of • This is a Multistage Game any “history,” real players typically • Analysis Tool: Write the game in extensive form coordinate on the “fair outcome” n Summarize the players • When there is a “bargaining history,” other n Their potential actions outcomes may prevail n Their information at each decision point n The sequence of moves and

n 49 Each player’s payoff 50

Step 2: Add the Union’s Step 1: Management’s Move Move Accept Union Reject 10 10 Firm 5 Accept Firm 5 Union

1 1 Reject Accept Union Reject

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Step 3: Add the Payoffs The Game in Extensive Form

Accept 100, 500 Accept 100, 500 Union -100, -3 Union Reject Reject -100, -3 10 10 Accept 300, 300 Firm 5 Accept 300, 300 Union Firm 5 Union Reject -100, -3 -100, -3 1 1 Reject Accept 500, 100 Accept 500, 100 Union -100, -3 Union Reject Reject -100, -3

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9 Step 4: Identify the Firm’s Step 5: Identify the Union’s Feasible Strategies Feasible Strategies

• Management has one information set and n Accept $10, Accept $5, Accept $1 thus three feasible strategies: n Accept $10, Accept $5, Reject $1 n Accept $10, Reject $5, Accept $1 n Offer $10 n Reject $10, Accept $5, Accept $1 Nine feasible n Offer $5 n Accept $10, Reject $5, Reject $1 strategies n Offer $1 n Reject $10, Accept $5, Reject $1

n Reject $10, Reject $5, Accept $1

n Reject $10, Reject $5, Reject $1

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Step 6: Identify Nash There are 3 Nash Equilibrium Outcomes: Equilibrium Outcomes! Accept • Outcomes such that neither the firm nor the 100, 500 Union union has an incentive to change its strategy, Reject -100, -3 given the strategy of the other 10 Accept 300, 300 Firm 5 Union -100, -3 1 Reject Accept 500, 100 Union Reject -100, -3 57 58

Step 7: Find the Subgame The “Credible” Union Strategy Perfect Nash Equilibrium Outcomes Are all Union's Strategy Actions • Outcomes where no player has an incentive Credible? to change its strategy, given the strategy of Accept $10, Accept $5, Accept $1 Yes Accept $10, Accept $5, Reject $1 No the rival, and Accept $10, Reject $5, Accept $1 No • The outcomes are based on “credible Reject $10, Accept $5, Accept $1 No Accept $10, Reject $5, Reject $1 No actions;” that is, they are not the result of Reject $10, Accept $5, Reject $1 No “empty threats” by the rival. Reject $10, Reject $5, Accept $1 No Reject $10, Reject $5, Reject $1 No

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10 To Summarize: Only 1 Subgame-Perfect Nash Equilibrium Outcome! Accept • We have identified many combinations of 100, 500 Union Nash equilibrium strategies Reject -100, -3 • In all but one the union does something that 10 isn’t in its self interest (and thus entail Accept 300, 300 Firm 5 Union threats that are not credible) Reject -100, -3 • Graphically: 1 Accept 500, 100 Union Reject -100, -3 61 62

Re-Cap Pricing to Prevent Entry: An Application of Game Theory • In take-it-or-leave-it bargaining, there is a first-mover advantage. • Management can gain by making a take-it • Two firms: an incumbent and potential or leave-it offer to the union. But... entrant • Management should be careful. Real world • The game in extensive form: evidence suggests that people sometimes reject offers on the the basis of “principle” instead of cash considerations.

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The Entry Game in Extensive Identify Nash and Subgame Form Perfect Equilibria -1, 1 -1, 1 Hard Hard Incumbent Incumbent Enter Enter Soft Soft Entrant 5, 5 Entrant 5, 5

Out 0, 10 Out 0, 10

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11 Two Nash Equilibria One Subgame Perfect Equilibrium

-1, 1 -1, 1 Hard Hard Incumbent Incumbent Enter Enter Soft Soft Entrant 5, 5 Entrant 5, 5

Out 0, 10 Out 0, 10

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Insights Summary

• Establishing a reputation for being unkind • Game theory is a useful way of analyzing to entrants can enhance long-term profits the potential outcomes of decision making. • It is costly to do so in the short-term, so • Outcomes can depend on the number of much so that it isn’t optimal to do so in a times a game is played (one-shot vs. one-shot game. repeated), the interest rate, trigger strategies and the credibility of an opponent’s threat, etc.

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