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Scenario Analysis Normal Form Game Overview Economics 3030 I. Introduction to Game Theory Chapter 10 II. Simultaneous-Move, One-Shot Games Game Theory: III. Infinitely Repeated Games Inside Oligopoly IV. Finitely Repeated Games V. Multistage Games 1 2 Normal Form Game A Normal Form Game • A Normal Form Game consists of: n Players Player 2 n Strategies or feasible actions n Payoffs Strategy 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 3 4 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 5 6 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 7 8 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 9 10 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 Nash equilibrium (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 13 14 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 15 16 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 17 have “standards” 18 3 A Coordination Game 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 19 20 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 21 22 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 23 24 4 Can collusion 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 25 any known, finite number of times. 26 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 31 32 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 33 34 One-Shot Bertrand Potential Repeated Game (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 35 36 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 39 40 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.
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