1 1 Deep Thought BA 445 Lesson B.7 Repeated Dilemmas The face of a child can say it all, especially...

1 1

Deep Thought

BA 445 Lesson B.7 Repeated Dilemmas

The face of a child can say it all, especially the mouth part of the face. ~ Jack Handey.

(Translation: Today’s lesson considers whether people can cooperate or collude when they can communicate but not punish.)

2 2BA 445 Lesson B.7 Repeated Dilemmas

Readings

Readings

Baye “Repeated Games” (see the index)Dixit Chapter 11


Overview

Overview

4 4

Overview


Bonuses and Tipping can be prisoner dilemmas. Prompt service and tipping together can be profitable, but prompt service without tipping hurts the server, and tipping without prompt service hurts the served.

Stackelberg Duopoly is a prisoners’ dilemma. Lower output by both leader and follower is profitable, but lower output by the follower is unprofitable given the output chosen by the leader.

Buying Online is a prisoners’ dilemma because traders risk sending money or goods and receiving nothing. Trading a little at a time reduces risk, but it is still most profitable for you to send nothing.

Solving a Dilemma with Pre-Commitment works when, given a commitment to cooperate by one of the players, the other players’ best response is cooperation.

Solving a Dilemma with Immediate Punishment works when each player can immediately switch from cooperation if the other player does not cooperate, as with a price guarantee.

Repeating Dilemmas does not enforce cooperation if the number of repetitions is finite and fixed. Rollback has players working backward and choosing non-cooperation in each period.


Example 1: Bonuses and Tipping



Overview

Bonuses and Tipping can be prisoner dilemmas. Prompt service and tipping together can be profitable, but prompt service without tipping hurts the server, and tipping without prompt service hurts the served.



Question: Consider ways for Professor Burke To Insure Promptness (TIP) when out to lunch at Islands Fine Burgers and Drinks. Suppose Maria, his usual server, chooses whether to offer Prompt service or Slow Service, and after being served Prof. B chooses whether to tip $2 or 0. If Maria chooses Prompt, she loses $1 worth of energy because of the extra effort but Prof. B gains $3 worth of time.

Should Maria be Prompt? Should Prof. B Tip? Are there mutual gains from cooperation? Can Prof. B trust Maria to cooperate? Can Maria trust Prof. B to cooperate?


8 8

Prompt SlowTip $2 1,1 -2,2No Tip 3,-1 0,0

Maria

Professor


Answer: To begin, fill out the normal form for this game of sequential moves. On the one hand, if Prof. B chooses to Tip $2 (first row), then Prof. B and Maria gain 1=3-2 and 1=2-1 if Maria chooses Prompt; and gain -2=0-2 and 2=2-0 if Maria chooses Slow.

On the other hand, if Prof. B chooses to No Tip (second row), then Prof. B and Maria gain 3=3-0 and -1=0-1 if Maria chooses Prompt; and gain 0=0-0 and 0=0-0 if chooses Slow.



Prof. B should choose No Tip since he moves second and No Tip is the dominate strategy. Maria should then choose Slow as the best response to Prof. B choosing No Tip. {No Tip, Slow} is thus the rollback solution.

There are mutual gains if both Prof. B and Maria cooperate with Prof. B choosing Tip $2 and Maria X choosing Prompt. But Maria cannot trust Prof. B to cooperate because Prof. B cooperating and choosing Tip $2 is not a best response to Maria cooperating and choosing Prompt.

The question of whether Prof. B can trust Maria to cooperate is irrelevant because Maria chooses before Prof. B.


Maria

Professor



Comment: Since No Tip is the dominate strategy for Prof. B and Slow is the dominate strategy for Maria, the solution to the game remains {No Tip, Slow} regardless of whether Maria continues to serve first, or whether Prof. B tips first (like he does at the Bellagio Gourmet Buffet in Vegas), or whether they (somehow) serve and tip simultaneously.


Maria

Professor



Example 2: Stackelberg Duopoly



Overview

Stackelberg Duopoly is a prisoners’ dilemma. Lower output by both leader and follower is profitable, but lower output by the follower is unprofitable given the output chosen by the leader.



Comment: A Prisoners’ Dilemma demonstrates why people might not cooperate or collude even if it is in their best interests to do so. While the strongest form of a prisoners’ dilemma is when non-cooperation is a dominate strategy for each person (as in Example 1), there are weaker forms, like when non-cooperation is the unique dominance solution, or when non-cooperation is the unique rollback solution.



Question: Airbus and Boeing control a large share of the commercial large jet aircraft market. Airlines find the two products to be indistinguishable. The inverse market demand for jets is P = 50-Q (in hundreds of millions of dollars). The Airbus unit costs of production are 10 (in hundreds of millions of dollars), and the unit costs of Boeing are 10 (in hundreds of millions of dollars). Suppose Airbus chooses its output of jets before Boeing, and Boeing knows the Airbus output before they decide their own output. Suppose Airbus and Boeing each consider producing quantities 10 or 11 or 18 or 20 jets.

How many jets should each produce? Are there mutual gains from cooperation? Can Airbus trust Boeing to cooperate? Can Boeing trust Airbus to cooperate?



Answer: To begin, fill out the normal form for this game of sequential moves. For example, at Airbus quantity 18 and Boeing quantity 20, price = 50-38 = 12, so Airbus profits = (12-10)18 = 36 and Boeing profits = (12-10)20 = 40

10 11 18 2010 200,200 190,209 120,216 100,20011 209,190 198,198 121,198 99,18018 216,120 198,121 72,72 36,4020 200,100 180,99 40,36 0,0

Boeing

Airbus



Airbus is the leader in a Stackelberg Duopoly Game.Start from the end of the game:Given Airbus’s choice 10, Boeing’s best response is 18; given Airbus’s choice 11, Boeing’s best response is either 11 or 18; given Airbus’s choice 18, Boeing’s best response is 11; and given Airbus’s choice 20, Boeing’s best response is 10. Of those four choices, the best payoffs for Airbus are for choosing 20 and generating payoff 200.


10 11 18 2010 200,200 190,209 120,216 100,20011 209,190 198,198 121,198 99,18018 216,120 198,121 72,72 36,4020 200,100 180,99 40,36 0,0

Boeing

Airbus


There are mutual gains if both Airbus and Boeing cooperate and Airbus produces either 11 or 18 and Boeing produces 10.

But Airbus cannot trust Boeing to cooperate because Boeing cooperating and choosing 10 is not a best response to Airbus cooperating and choosing either 11 or 18.

The question of whether Boeing can trust Airbus to cooperate is irrelevant because Airbus chooses before Boeing.


10 11 18 2010 200,200 190,209 120,216 100,20011 209,190 198,198 121,198 99,18018 216,120 198,121 72,72 36,4020 200,100 180,99 40,36 0,0

Boeing

Airbus


Example 3: Buying Online



Overview

Buying Online is a prisoners’ dilemma because traders risk sending money or goods and receiving nothing. Trading a little at a time reduces risk, but it is still most profitable for you to send nothing.



Comment: Buyers and Sellers trading over the internet face a prisoners’ dilemma because they risk sending money or goods and not getting what was agreed upon. One attempted solution that reduces their exposure to risk is to trade a little at a time.

Does trading a little at a time solve that prisoners’ dilemma?



Question: Suppose Charlie values 4 disposable DVDs (Gladiator, …) at $3 each, suppose it costs Blockbuster $1 to provide each DVD, and suppose Blockbuster sells DVDs for $2 each. Should Blockbuster send the first DVD to Charlie? • If the first DVD is sent, Charlie (C) faces a decision: steal the DVD

and terminate the relationship; or, send $2 for the first DVD.• If the first $2 is sent, Blockbuster (B) faces a decision: take the $2

and terminate the relationship; or, send the second DVD to C.• If the second DVD is sent, C faces a decision: steal the DVD and

terminate the relationship; or, send $2 for the second DVD.• If the second $2 is sent, B faces a decision: take the $2 and

terminate the relationship; or, send the third DVD to C. • And so on.• If the fourth DVD is sent, C faces a decision: steal the DVD and

terminate the relationship; or, send $2 for the fourth DVD.

Are there mutual gains from cooperation? Can Charlie trust Blockbuster to cooperate? Can Blockbuster trust Charlie to cooperate?



Answer: Here is a Game Tree where payoffs list the gains from trade, with payoffs to the first player to act (Blockbuster) listed first. (This game is sometimes called the Centipede Game since the game tree looks like a Centipede.)

0 ,0

D o N o t S en d 1

-1 ,3

D o N o t P a y 1

1 ,1

D o N o t S en d 2

0 ,4

D o N o t P a y 2

2 ,2

D o N o t S en d 2

1 ,5

D o N o t P a y 3

3 ,3

D o N o t S en d 4

2 ,6

D o N o t P a y 4

4 ,4

P a y 4

C

S e nd 4

B

P a y 3

C

S e nd 3

B

P a y 2

C

S e nd 2

B

P a y 1

C

S e nd 1

B



Starting at the end of the game, Blockbuster predicts rational responses to alternative strategies. Charlie chooses Do Not Pay for the fourth DVD if it is sent.

0 ,0

D o N o t S en d 1

-1 ,3

D o N o t P a y 1

1 ,1

D o N o t S en d 2

0 ,4

D o N o t P a y 2

2 ,2

D o N o t S en d 2

1 ,5

D o N o t P a y 3

3 ,3

D o N o t S en d 4

2 ,6

D o N o t P a y 4

4 ,4

P a y 4

C

S e nd 4

B

P a y 3

C

S e nd 3

B

P a y 2

C

S e nd 2

B

P a y 1

C

S e nd 1

B



Rolling back one period from the end of game, Charlie predicts rational responses to alternative strategies. Blockbuster chooses Do Not Send the fourth DVD since Blockbuster predicts Charlie will not pay for that fourth DVD.

0 ,0

D o N o t S en d 1

-1 ,3

D o N o t P a y 1

1 ,1

D o N o t S en d 2

0 ,4

D o N o t P a y 2

2 ,2

D o N o t S en d 2

1 ,5

D o N o t P a y 3

3 ,3

D o N o t S en d 4

2 ,6

D o N o t P a y 4

4 ,4

P a y 4

C

S e nd 4

B

P a y 3

C

S e nd 3

B

P a y 2

C

S e nd 2

B

P a y 1

C

S e nd 1

B



Rolling back one more period from the end of game, Blockbuster predicts rational responses to alternative strategies. Charlie chooses Do Not Pay for the third DVD if it is sent since Charlie predicts Blockbuster will not send the fourth DVD.

0 ,0

D o N o t S en d 1

-1 ,3

D o N o t P a y 1

1 ,1

D o N o t S en d 2

0 ,4

D o N o t P a y 2

2 ,2

D o N o t S en d 2

1 ,5

D o N o t P a y 3

3 ,3

D o N o t S en d 4

2 ,6

D o N o t P a y 4

4 ,4

P a y 4

C

S e nd 4

B

P a y 3

C

S e nd 3

B

P a y 2

C

S e nd 2

B

P a y 1

C

S e nd 1

B



Rolling back one more period from the end of game, Charlie predicts rational responses to alternative strategies. Blockbuster chooses Do Not Send the third DVD since Blockbuster predicts Charlie will not pay for that third DVD.

0 ,0

D o N o t S en d 1

-1 ,3

D o N o t P a y 1

1 ,1

D o N o t S en d 2

0 ,4

D o N o t P a y 2

2 ,2

D o N o t S en d 2

1 ,5

D o N o t P a y 3

3 ,3

D o N o t S en d 4

2 ,6

D o N o t P a y 4

4 ,4

P a y 4

C

S e nd 4

B

P a y 3

C

S e nd 3

B

P a y 2

C

S e nd 2

B

P a y 1

C

S e nd 1

B



Rolling back one more period from the end of game, Blockbuster predicts rational responses to alternative strategies. Charlie chooses Do Not Pay for the second DVD if it is sent since Charlie predicts Blockbuster will not send the third DVD.

0 ,0

D o N o t S en d 1

-1 ,3

D o N o t P a y 1

1 ,1

D o N o t S en d 2

0 ,4

D o N o t P a y 2

2 ,2

D o N o t S en d 2

1 ,5

D o N o t P a y 3

3 ,3

D o N o t S en d 4

2 ,6

D o N o t P a y 4

4 ,4

P a y 4

C

S e nd 4

B

P a y 3

C

S e nd 3

B

P a y 2

C

S e nd 2

B

P a y 1

C

S e nd 1

B



Rolling back one more period from the end of game, Charlie predicts rational responses to alternative strategies. Blockbuster chooses Do Not Send the second DVD since Blockbuster predicts Charlie will not pay for that second DVD.

0 ,0

D o N o t S en d 1

-1 ,3

D o N o t P a y 1

1 ,1

D o N o t S en d 2

0 ,4

D o N o t P a y 2

2 ,2

D o N o t S en d 2

1 ,5

D o N o t P a y 3

3 ,3

D o N o t S en d 4

2 ,6

D o N o t P a y 4

4 ,4

P a y 4

C

S e nd 4

B

P a y 3

C

S e nd 3

B

P a y 2

C

S e nd 2

B

P a y 1

C

S e nd 1

B



Rolling back one more period from the end of game, Blockbuster predicts rational responses to alternative strategies. Charlie chooses Do Not Pay for the first DVD if it is sent since Charlie predicts Blockbuster will not send the second DVD.

0 ,0

D o N o t S en d 1

-1 ,3

D o N o t P a y 1

1 ,1

D o N o t S en d 2

0 ,4

D o N o t P a y 2

2 ,2

D o N o t S en d 2

1 ,5

D o N o t P a y 3

3 ,3

D o N o t S en d 4

2 ,6

D o N o t P a y 4

4 ,4

P a y 4

C

S e nd 4

B

P a y 3

C

S e nd 3

B

P a y 2

C

S e nd 2

B

P a y 1

C

S e nd 1

B



Rolling back to the beginning of game, Blockbuster chooses Do Not Send the first DVD since Blockbuster predicts Charlie will not pay for that first DVD.

0 ,0

D o N o t S en d 1

-1 ,3

D o N o t P a y 1

1 ,1

D o N o t S en d 2

0 ,4

D o N o t P a y 2

2 ,2

D o N o t S en d 2

1 ,5

D o N o t P a y 3

3 ,3

D o N o t S en d 4

2 ,6

D o N o t P a y 4

4 ,4

P a y 4

C

S e nd 4

B

P a y 3

C

S e nd 3

B

P a y 2

C

S e nd 2

B

P a y 1

C

S e nd 1

B



Centipede Game: Should Blockbuster send the first DVD to Charlie? In the rollback solution, Charlie will steal the first DVD and terminate the relationship. So Blockbuster should not send the first DVD.

There are mutual gains if, say, Charlie always pays and Blockbuster sends every DVD.

But Blockbuster cannot trust Charlie to cooperate because Charlie paying for the fourth DVD is not a best response to Blockbuster sending every DVD.



Example 4: Solving a Dilemma with Pre-Commitment



Overview

Solving a Dilemma with Pre-Commitment works when, given a commitment to cooperate by one of the players, the other players’ best response is cooperation.



Question: Consider ways Pepperdine University can get better Teaching or Research from its faculty. Suppose Pepperdine chooses whether to spend $1,000 to send Prof. X to a conference in Las Vegas over Christmas break, and Prof. X chooses whether to spend $2,000 of effort to Work over Christmas break. If Pepperdine chooses the conference, then both Pepperdine and Prof. X gain $3,000 of benefit if Prof. X chooses to work (since he is working more effectively) but Prof. X alone gains $3,000 if Prof. X chooses to not work (since he is on a free holiday).

Should Pepperdine send Prof. X to the conference? Are there mutual gains from cooperation? Can Pepperdine trust Prof. X to cooperate? Can Prof. X trust Pepperdine to cooperate? Can the two cooperate by having Pepperdine pre-commit to its choice before Prof. X chooses? Or by having Prof. X pre-commit (say, by agreeing to present his results after Christmas break)?


35 35

Work No WorkConf. 2,1 -1,3

No Conf. 0,-2 0,0

Prof. X

Pepperdine


Answer: To begin, fill out the normal form for this game of simultaneous moves. On the one hand, if Pepperdine chooses to send Prof. X to the conference (first row), then both Pepperdine and Prof. X gain $3,000 of benefit if Prof. X chooses to work, resulting in net payoffs of $3,000-$1,000 = 2 (thousand) to Pepperdine and $3,000-$2,000 = 1 (thousand) to Prof. X, but Prof. X alone gains $3,000 if Prof. X chooses to not work, resulting in net payoffs of $0-$1,000 = -1 (thousand) to Pepperdine and $3,000 = 3 (thousand) to Prof. X. On the other hand, if Pepperdine chooses to send Prof. X (second row), then Pepperdine has no gains or costs, and Prof. X has no gains but $2,000 cost if he chooses Work.



Prof. X should choose No Worksince it is the dominate strategy. Pepperdine should then chooseNo Conference as the best response to Prof. X choosing No Work. {No Conference, No Work} is thus the dominance solution.

There are mutual gains if both Pepperdine and Prof. X cooperate with Pepperdine choosing Conference and Prof. X choosing Work. But Pepperdine cannot trust Prof. X to cooperate becauseProf. X cooperating and choosing Work is not a best response to Pepperdine cooperating and choosing Conference. However, Prof. X can trust Pepperdine to cooperate because Pepperdine cooperating and choosing conference is a best response to Prof. X cooperating and choosing work.


No Conf. 0,-2 0,0

Prof. X

Pepperdine



Can the two cooperate by having Pepperdine pre-commit to its choice before Prof. X chooses? If Pepperdine pre-commits to Conference, then the two do not both cooperate since Prof. X cooperating and choosing Work is not a best response to Pepperdine cooperating and choosing Conference.

Can the two cooperate by having Prof. X pre-commit to its choice before Pepperdine chooses? If Prof. X pre-commits to Work, then the two do both cooperate since Pepperdine cooperating and choosing Conference is a best response to Prof. X cooperating and choosing Work.


No Conf. 0,-2 0,0

Prof. X

Pepperdine



Example 5: Solving a Dilemma with Immediate Punishment



Overview

Solving a Dilemma with Immediate Punishment works when each player can immediately switch from cooperation if the other player does not cooperate, as with a price guarantee.



Question: Sam’s Club and Costco consider modifying the price competition described in a previous lesson. They both sell emergency food supplies in a weather-proof bucket that provides 275 delicious easy-to-prepare meals, including potato soup and corn chowder. The unit cost to both retailers is $75. The retailers compete on price: the low-price retailer gets all the market and they split the market if they have equal prices. Suppose they consider prices $85 and $95, and suppose market demands at those prices are 100 and 80.

What price should Costco choose in this Price Competition Game? Are there mutual gains from cooperation? Can Costco trust Sam’s to cooperate? Can Sam’s trust Costco to cooperate?

Reconsider your answers if there is a third strategy of price matching --- charging $95 but matching a competitors $85 price.



Answer: To begin, fill out the normal form for this game of simultaneous moves. For example, at Sam's Club price $95 and Costco price $85, Costco gets the entire market demand of 100. Hence, Sam's makes $0 and Costco makes $(85-75)x100 = $1,000.

$85 $95$85 500,500 1000,0$95 0,1000 800,800

Costco

Sam's



$85 $95$85 500,500 1000,0$95 0,1000 800,800

Costco

Sam's

Each player should choose $85 since it is the dominate strategy for each player: $85 it gives better payoffs for that player compared with $95, no matter whether the other player chooses $85 or $95.

There are mutual gains if both Sam’s and Costco cooperate and charge $95. But Costco cannot trust Sam’s to cooperate becauseSam’s cooperating and choosing $95 is not a best response to Costco cooperating and choosing $95. Likewise, Sam’s cannottrust Costco to cooperate because Costco cooperating and choosing $95 is not a best response to Sam’s cooperating and choosing $95.


43 43

$85 $95 Match$85 500,500 1000,0 500,500$95 0,1000 800,800 800,800

Match 500,500 800,800 800,800

Costco

Sam's


Reconsider those answers if there is a third strategy of price matching --- charging $95 but matching a competitors $85 price. To begin, fill out the normal form for this game of simultaneous moves. For example, at Sam's Club Match and Costco price $85, profits are the same as at Sam's Club price $85 and Costco price $85. And at Sam's Club Match and Costco price $95, profits are the same as at Sam's Club price $95 and Costco price $95.



$85 $95 Match$85 500,500 1000,0 500,500$95 0,1000 800,800 800,800

Match 500,500 800,800 800,800

Costco

Sam's

There are no dominate or weakly-dominate strategies in the original game, but $95 Is weakly-dominated by Match for each player. After $95 is eliminated for each player, Match is the weakly-dominate strategy for each player, resulting in Payoffs of 800 for each player.

There are no mutual gains if both Sam’s and Costco select alternative strategies.



$85 $95 Match$85 500,500 1000,0 500,500$95 0,1000 800,800 800,800

Match 500,500 800,800 800,800

Costco

Sam's

Comment: The original gamewithout price matching was aprisoners’ dilemma, with a dominance solution of low prices that was worse for both players than a cooperative solution of high prices.

The Match strategy solved the dilemma by immediately punishing a player that did not cooperate. The punishment against the non-cooperative player charging low prices was for the cooperative player to also charge low prices. And since that punishment was immediate, there was no period of time that the non-cooperative player benefited from his charging low prices.



Example 6: Repeating Dilemmas



Overview

Repeating Dilemmas does not enforce cooperation if the number of repetitions is finite and fixed. Rollback has players working backward and choosing non-cooperation in each period.



Question: Wii video game consoles are made by Nintendo, and some games are produced by third parties, including Sega. The unit cost of a console to Nintendo is $50, and of a game to Sega is $10. Suppose, each month, Nintendo considers prices $250 and $350 for consoles, and Sega considers $40 and $50 for games. If they choose prices $250 and $40 for consoles and games, then demands are 1 and 2 (in millions); if $250 and $50, then .8 and 1.6 (in millions); if $350 and $40, then .7 and 1.4 (in millions); and if $350 and $50, then .6 and 1.2 (in millions).

What price should Nintendo choose in this pricing game if Nintendo and Sega can set different prices in January and in February and in March? Are there mutual gains from cooperation? Can Nintendo trust Sega to cooperate? Can Sega trust Nintendo to cooperate?


49 49

$40 $50$250 200,60 160,64$350 210,42 180,48

Sega

Nintendo


Answer: To begin, fill out the normal form for this three-times-repeated game of simultaneous moves. For example, at Nintendo price $350 and Sega price $40, Nintendo’s demand is .7 and Sega’s is 1.4, so Nintendo profits $(350-50)x.7 = $210 and Sega profits $(40-10)x1.4 = $42.

(Goods are gross complements because a higher price for one means lower demand for the other.)



Since Nintendo and Sega can set different prices in January and in February and in March, their strategies for prices chosen in February and in March can potentially be a function of prices chosen in earlier months. For example, Nintendo could choose the strategy that its price in February or March is high if the price Sega chose in the preceding month is low and vice versa.

$40 $50$250 200,60 160,64$350 210,42 180,48

Sega

Nintendo

$40 $50$250 200,60 160,64$350 210,42 180,48

Sega

Nintendo

$40 $50$250 200,60 160,64$350 210,42 180,48

Sega

Nintendo



To compute the rollback solution, start at the end of the game (March). Regardless of prices in previous months, Nintendo should choose $350 since it is the dominate strategy, and Sega should choose $50 since it is the dominate strategy.

$40 $50$250 200,60 160,64$350 210,42 180,48

Sega

Nintendo

$40 $50$250 200,60 160,64$350 210,42 180,48

Sega

Nintendo

$40 $50$250 200,60 160,64$350 210,42 180,48

Sega

Nintendo



Rolling back to the middle of the game (February). Each player knows that prices at the end of the game (March) do not actually depend on prices chosen in the middle of the game (February). Thus each should choose February prices without regard to the future. So, Nintendo should choose $350 in February since it is the dominate strategy, and Sega should choose $50 in February since it is the dominate strategy.

$40 $50$250 200,60 160,64$350 210,42 180,48

Sega

Nintendo

$40 $50$250 200,60 160,64$350 210,42 180,48

Sega

Nintendo

$40 $50$250 200,60 160,64$350 210,42 180,48

Sega

Nintendo



Rolling back to the beginning of the game (January). Each player knows that prices at the middle and end of the game do not actually depend on prices chosen should at the beginning (January). Thus each should choose January prices without regard to the future. So, Nintendo should choose $350 in January since it is the dominate strategy, and Sega should choose $50 in January since it is the dominate strategy.

$40 $50$250 200,60 160,64$350 210,42 180,48

Sega

Nintendo

$40 $50$250 200,60 160,64$350 210,42 180,48

Sega

Nintendo

$40 $50$250 200,60 160,64$350 210,42 180,48

Sega

Nintendo



There are mutual gains if both Nintendo and Sega cooperate and charge their lower price. But Nintendo cannot trust Sega to cooperate because, in the third period, Sega cooperating and choosing $40 is not a best response to Nintendo cooperating and choosing $250. Likewise, Sega cannot trust Nintendo to cooperate because in the third period, Nintendo cooperating and choosing $250 is not a best response to Sega cooperating and choosing $40.

$40 $50$250 200,60 160,64$350 210,42 180,48

Sega

Nintendo

$40 $50$250 200,60 160,64$350 210,42 180,48

Sega

Nintendo

$40 $50$250 200,60 160,64$350 210,42 180,48

Sega

Nintendo



Comment: Repeating the prisoners’ dilemma pricing game with gross complements thus generates another prisoners’ dilemma --- meaning, there are gains from cooperation, but no one cooperates. That is a general result: It turns out that repeating any prisoners’ dilemma any finite number of times results in another prisoners’ dilemma.

$40 $50$250 200,60 160,64$350 210,42 180,48

Sega

Nintendo

$40 $50$250 200,60 160,64$350 210,42 180,48

Sega

Nintendo

$40 $50$250 200,60 160,64$350 210,42 180,48

Sega

Nintendo


56 56

Review Questions


Review Questions You should try to answer some of the following

questions before the next class. You will not turn in your answers, but students may

request to discuss their answers to begin the next class. Your upcoming Exam 2 and cumulative Final Exam will

contain some similar questions, so you should eventually consider every review question before taking your exams.


Review 1: Repeating Dilemmas



Question: Sam’s Club and Costco control a large share of the US retail wholesale market. Both sell emergency food supplies in a weather-proof bucket that provides 275 delicious easy-to-prepare meals, including potato soup and corn chowder. The unit cost to both retailers is $75. Each month, the retailers compete on price but consumers do not find the goods to be perfect substitutes. Suppose Sam’s Costco consider prices $85 and $95. If both choose price $85, each has demand 50; if both $95, each has 40; and if one chooses $85 and the other $95, the lower price has demand 85 and the higher price 5.

What price should Costco choose in this Price Competition Game if Sam’s Club and Costco can set different prices in January and in February?

Are there mutual gains from cooperation? Can Costco trust Sam’s to cooperate? Can Sam’s trust Costco to cooperate?


59 59

$85 $95$85 500,500 850,100$95 100,850 800,800

Costco

Sam's


Answer: To begin, fill out the normal form for this twice-repeated game of simultaneous moves. For example, at Sam’s Club price $95 and Costco price $85, Sam’s demand is 5 and Costco’s is 85, so Sam’s profits $(95-75)x5 = $100 and Costco profits $(85-75)x85 = $850.

(Goods are gross substitutes because a higher price for one means higher demand for the other.)



$85 $95$85 500,500 850,100$95 100,850 800,800

Costco

Sam's

Since Sam’s Club and Costco can set different prices in January and in February, their strategies for prices chosen in February can potentially be a function of prices chosen in January. For example, Costco could choose the strategy that its price in February is whatever price Sam’s chose in January.

$85 $95$85 500,500 850,100$95 100,850 800,800

Costco

Sam's



$85 $95$85 500,500 850,100$95 100,850 800,800

Costco

Sam's

To compute the rollback solution, start at the end of the game (February). Regardless of prices in the previous month, Each player should choose $85 since it is the dominate strategy for each player: $85 it gives better payoffs for that player compared with $95, no matter whether the other player chooses $85 or $95.

$85 $95$85 500,500 850,100$95 100,850 800,800

Costco

Sam's



$85 $95$85 500,500 850,100$95 100,850 800,800

Costco

Sam's

Rolling back to the beginning of the game (January). Each player knows that prices at the end of the game (February) do not actually depend on prices chosen should at the beginning (January). Thus each should choose January prices without regard to the future. So choose $85 in the first since it is the dominate strategy for each player.

$85 $95$85 500,500 850,100$95 100,850 800,800

Costco

Sam's



$85 $95$85 500,500 850,100$95 100,850 800,800

Costco

Sam's

There are mutual gains if both Sam’s and Costco cooperate and charge $95 each period. But Costco cannot trust Sam’s to cooperate because, in the second period, Sam’s cooperating and choosing $95 is not a best response to Costco cooperating and choosing $95. Likewise, Sam’s cannottrust Costco to cooperate because, in the second period, Costco choosing $95 is not a best response to Sam’s cooperating and choosing $95.

$85 $95$85 500,500 850,100$95 100,850 800,800

Costco

Sam's



$85 $95$85 500,500 850,100$95 100,850 800,800

Costco

Sam's

Comment: Repeating the prisoners’ dilemma price competition game with gross substitutes thus generates another prisoners’ dilemma --- meaning, there are gains from cooperation, but no one cooperates. That is a general result: It turns out that repeating any prisoners’ dilemma two times results in another prisoners’ dilemma.

$85 $95$85 500,500 850,100$95 100,850 800,800

Costco

Sam's


65 65

End of Lesson B.7

BA 445 Managerial Economics


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