1 1 deep thought ba 210 lesson ii.8 beneficial grim punishment sometimes when i fell like killing...

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Deep ThoughtDeep Thought

BA 210 Lesson II.8 Beneficial Grim PunishmentBA 210 Lesson II.8 Beneficial Grim Punishment

Sometimes when I fell like killingSometimes when I fell like killing someone, I someone, I do a little trick to calm myself down. I’ll go do a little trick to calm myself down. I’ll go over to the person’s house and ring the over to the person’s house and ring the doorbell. When the person comes to the doorbell. When the person comes to the door, I’m gone, but you know what I’ve left door, I’m gone, but you know what I’ve left on the porch? A jack-o-lantern with a knife on the porch? A jack-o-lantern with a knife stuck in the side of its head with a note that stuck in the side of its head with a note that says “You”. After that I usually feel better, says “You”. After that I usually feel better, and no harm done. ~ Jack Handey.and no harm done. ~ Jack Handey.

(Translation: Today’s lesson shows credible threats do not have (Translation: Today’s lesson shows credible threats do not have to be executed.)to be executed.)

2 2BA 210 Lesson II.8 Beneficial Grim PunishmentBA 210 Lesson II.8 Beneficial Grim Punishment

OverviewOverview

OverviewOverview

3 3

Lesson OverviewLesson Overview

BA 210 Lesson II.8 Beneficial Grim Punishment

Lesson II.8 Beneficial Grim PunishmentLesson II.8 Beneficial Grim PunishmentExample 1: Grim Punishment with InterestExample 1: Grim Punishment with InterestSummarySummaryReview QuestionsReview Questions


Example 1: Grim Punishment with Example 1: Grim Punishment with InterestInterest

Example 1: Grim Punishment with InterestExample 1: Grim Punishment with Interest

5 5BA 210 Lesson II.8 Beneficial Grim Punishment

Comment:Comment: In any prisoners’ dilemma, there are mutual gains In any prisoners’ dilemma, there are mutual gains from Cooperating by choosing a particular action, but not from Cooperating by choosing a particular action, but not everyone can be trusted to cooperate because, for at least one everyone can be trusted to cooperate because, for at least one person, the cooperative action is not a best response to the other person, the cooperative action is not a best response to the other players selecting their cooperative actions. That is, cooperation players selecting their cooperative actions. That is, cooperation is not a Nash Equilibrium. is not a Nash Equilibrium.

We see that cooperation can become a Nash Equilibrium, and so We see that cooperation can become a Nash Equilibrium, and so players can be trusted to cooperate, if the dilemma game is players can be trusted to cooperate, if the dilemma game is repeated indefinitely, and players punish non-cooperation. The repeated indefinitely, and players punish non-cooperation. The most effective punishment is called the most effective punishment is called the Grim StrategyGrim Strategy. The . The punishment inflicts the maximum pain on non-cooperation, and it punishment inflicts the maximum pain on non-cooperation, and it lasts forever.lasts forever.



Question: R.J. Reynolds Tobacco Corp. and Philip Morris Corp. Question: R.J. Reynolds Tobacco Corp. and Philip Morris Corp. must decide how much money to spend on advertising each year, must decide how much money to spend on advertising each year, either $10,000 or zero. If one advertises and the other does not, either $10,000 or zero. If one advertises and the other does not, the advertiser pays $10,000, then takes $100,000 profit from the the advertiser pays $10,000, then takes $100,000 profit from the other. If each advertises, each pays $10,000 but the other. If each advertises, each pays $10,000 but the advertisements cancel out and neither player takes profit from the advertisements cancel out and neither player takes profit from the other. Suppose the yearly interest rate is 10%. other. Suppose the yearly interest rate is 10%.

What strategies (RWhat strategies (ROneOne,P,POneOne) should each player choose if they ) should each player choose if they

expect this game to last only one time period? Are there mutual expect this game to last only one time period? Are there mutual gains from cooperative strategies (Rgains from cooperative strategies (RCoopCoop,P,PCoopCoop)? If they expect this )? If they expect this

game to repeat indefinitely, would Reynolds cooperate each game to repeat indefinitely, would Reynolds cooperate each period and choose Rperiod and choose RCoop Coop if Philip followed the if Philip followed the Grim Strategy of Grim Strategy of

punishing non-cooperation? punishing non-cooperation? and would Philip cooperate each and would Philip cooperate each period and choose Pperiod and choose PCoop Coop if Reynolds followed the if Reynolds followed the Grim Strategy? Grim Strategy?


7 7

Ad No AdAd -10,-10 90,-100

No Ad -100,90 0,0

Philip

Reynolds


Answer: Answer: The essential data of the game are The essential data of the game are the interest rate the interest rate between periods, between periods, r = 10% or r = 0.10 (as a fraction), and the r = 10% or r = 0.10 (as a fraction), and the payoffs each period defined by the normal form. For example, payoffs each period defined by the normal form. For example, with payoffs in thousands of dollars, if Reynolds advertises and with payoffs in thousands of dollars, if Reynolds advertises and Philip does not, Reynolds pays $10,000, then takes $100,000 Philip does not, Reynolds pays $10,000, then takes $100,000 profit from Philip. Hence, Reynolds makes $90,000 and Philip profit from Philip. Hence, Reynolds makes $90,000 and Philip looses $100,000.looses $100,000.



What strategies (SWhat strategies (SOneOne,C,COneOne) )

should each player choose if should each player choose if they expect this game to last they expect this game to last only one time period? only one time period? In that In that one-shot game, one-shot game, each player should choose each player should choose Advertise Advertise (R(ROneOne=Ad,P=Ad,POneOne=Ad) =Ad) since it is the dominate strategy for each since it is the dominate strategy for each

player. Each player thus player. Each player thus earns -10earns -10. .

Are there mutual gains from cooperative strategies (SAre there mutual gains from cooperative strategies (SCoopCoop,C,CCoopCoop)? )?

Yes, if both players choose Yes, if both players choose (R(RCoopCoop=No Ad,P=No Ad,Pcoopcoop=No Ad)=No Ad), then , then

each player each player earns 0earns 0, rather than -10., rather than -10.


Ad No AdAd -10,-10 90,-100

No Ad -100,90 0,0

Philip

Reynolds


If players expect this game to If players expect this game to repeat indefinitely, each should repeat indefinitely, each should consider the Grim Strategy, consider the Grim Strategy, which has has two components. which has has two components. 1)1)The The CooperativeCooperative choice for choice for each player each player (R(RCoopCoop=No =No

Ad,PAd,Pcoopcoop=No Ad)=No Ad). .

2)2)The The PunishmentPunishment choice for choice for each player each player (R(RPunPun=Ad,P=Ad,PPunPun=Ad) =Ad) , ,

which gives the other player the which gives the other player the worst payoff after that player worst payoff after that player chooses his best response to his chooses his best response to his punishment.punishment.

Ad No AdAd -10,-10 90,-100

No Ad -100,90 0,0

Philip

Reynolds

Ad No AdAd -10,-10 90,-100

No Ad -100,90 0,0

Philip

Reynolds



The Grim Strategy is thus, in The Grim Strategy is thus, in each period, Cooperate and each period, Cooperate and choose choose (R(RCoopCoop=No Ad,P=No Ad,Pcoopcoop=No =No

Ad)Ad), as long as the other player , as long as the other player has Cooperated and chosen has Cooperated and chosen (R(Rcoopcoop=No Ad,P=No Ad,Pcoopcoop=No Ad)=No Ad) in in

every previous period. But every previous period. But otherwise (if the other player otherwise (if the other player has ever made a choice other has ever made a choice other than cooperation than cooperation (R(RCoopCoop=No =No

Ad,PAd,Pcoopcoop=No Ad)=No Ad)), then you ), then you

punish by choosing punish by choosing (R(RPunPun=Ad,P=Ad,PPunPun=Ad) =Ad) in the next in the next

period and in every period period and in every period thereafter --- forever.thereafter --- forever.

Ad No AdAd -10,-10 90,-100

No Ad -100,90 0,0

Philip

Reynolds

Ad No AdAd -10,-10 90,-100

No Ad -100,90 0,0

Philip

Reynolds



Suppose Philip Suppose Philip followed the followed the Grim Strategy. Grim Strategy. Would Would Reynolds cooperate each period Reynolds cooperate each period and choose and choose RRCoopCoop=No Ad=No Ad? ?

On the one hand, if Reynolds On the one hand, if Reynolds cooperated each period and cooperated each period and choose choose RRCoopCoop=No Ad=No Ad, then the , then the

Grim Strategy says Philip will Grim Strategy says Philip will also cooperate each period and also cooperate each period and choose choose PPcoopcoop=No Ad=No Ad, and so , and so

Reynolds Reynolds earns 0 earns 0 each period.each period.

Ad No AdAd -10,-10 90,-100

No Ad -100,90 0,0

Philip

Reynolds

Ad No AdAd -10,-10 90,-100

No Ad -100,90 0,0

Philip

Reynolds



On the other hand, if Reynolds On the other hand, if Reynolds cheated in any period, then cheated in any period, then consider the first period when consider the first period when he cheats. he cheats.

In that first period, the Grim In that first period, the Grim Strategy says Philip will still Strategy says Philip will still cooperate and choose cooperate and choose CCCoopCoop=No =No

AdAd. Reynolds’ best response to . Reynolds’ best response to PPcoopcoop=No Ad =No Ad is Ad, which earns is Ad, which earns

90 in that period, rather than 90 in that period, rather than earning 0 earning 0 if he had continued to if he had continued to cooperate. So cooperate. So the one period the one period gain from cheating is 90gain from cheating is 90..

Ad No AdAd -10,-10 90,-100

No Ad -100,90 0,0

Philip

Reynolds

Ad No AdAd -10,-10 90,-100

No Ad -100,90 0,0

Philip

Reynolds



But starting in the next period But starting in the next period and continuing forever, the and continuing forever, the Grim Strategy says Philip will Grim Strategy says Philip will punish by choosing punish by choosing PPPunPun=Ad=Ad. .

Reynolds’s best response to Reynolds’s best response to PPPunPun=Ad =Ad is Ad, which earns -10 is Ad, which earns -10

in each punishment period, in each punishment period, rather than rather than earning 0 earning 0 if he had if he had continued to cooperate. So continued to cooperate. So the the eventual loss from cheating is eventual loss from cheating is 1010..

Ad No AdAd -10,-10 90,-100

No Ad -100,90 0,0

Philip

Reynolds

Ad No AdAd -10,-10 90,-100

No Ad -100,90 0,0

Philip

Reynolds



Summing up, if Philip Summing up, if Philip followed the Grim Strategyfollowed the Grim Strategy, then Reynolds , then Reynolds would cooperate and choose Rwould cooperate and choose RCoop Coop each period exactly if the one each period exactly if the one

period gain of from cheating of period gain of from cheating of 90 90 does not compensate for the does not compensate for the eventual losses of eventual losses of 10 10 starting the next period. That answer starting the next period. That answer depends on the interest rate depends on the interest rate rr between periods. between periods.



Comment: Comment: To continue the answer, we need the formula that To continue the answer, we need the formula that $1 $1 starting next period and continuing for each subsequent period is starting next period and continuing for each subsequent period is worth $(1/r) today.worth $(1/r) today. To illustrate, consider earning 10 percent To illustrate, consider earning 10 percent interest each period, so r = 0.10. The formula thus becomes interest each period, so r = 0.10. The formula thus becomes $1 $1 starting next period and continuing for each subsequent period is starting next period and continuing for each subsequent period is worth $10 todayworth $10 today. To prove that formula, consider investing the . To prove that formula, consider investing the $10 today. At the end of the first period, you earn $1 interest. $10 today. At the end of the first period, you earn $1 interest. Suppose you withdraw that interest and reinvest the $10. Then at Suppose you withdraw that interest and reinvest the $10. Then at the end of the second period, you earn another $1 interest. If you the end of the second period, you earn another $1 interest. If you continue withdrawing interest each period, keeping the $10 continue withdrawing interest each period, keeping the $10 invested forever, you thus earn $1 interest each period.invested forever, you thus earn $1 interest each period.



Use the formula that $1 starting next month and continuing for Use the formula that $1 starting next month and continuing for each subsequent period is worth $(1/r) today. Since the interest each subsequent period is worth $(1/r) today. Since the interest rate r = 10% expressed as a fraction is r = 0.10, the eventual rate r = 10% expressed as a fraction is r = 0.10, the eventual losses of losses of 10 10 is the same as loosing 10/0.10 = 100 today. is the same as loosing 10/0.10 = 100 today.

Therefore, the one period gain of from cheating of Therefore, the one period gain of from cheating of 90 90 does not does not compensate for the eventual losses of compensate for the eventual losses of 10 10 starting the next period, starting the next period, and so Reynolds would cooperate and choose and so Reynolds would cooperate and choose RRCoopCoop = No Ad = No Ad each each

period. Since the game is symmetric, Philip would cooperate and period. Since the game is symmetric, Philip would cooperate and choose choose PPCoop Coop = No Ad= No Ad each period if Reynolds each period if Reynolds followed the Grim followed the Grim

Strategy.Strategy.


17 17

Review QuestionsReview Questions


Review QuestionsReview Questions You should try to answer some of the following questions You should try to answer some of the following questions before the next class. before the next class. You will not turn in your answers, but students may request You will not turn in your answers, but students may request to discuss their answers to begin the next class. to discuss their answers to begin the next class. Your upcoming Exam 2 and cumulative Final Exam will Your upcoming Exam 2 and cumulative Final Exam will contain some similar questions, so you should eventually contain some similar questions, so you should eventually consider every review question before taking your exams.consider every review question before taking your exams.


Review Question 1Review Question 1



Question 1. Sam’s Club and Costco Question 1. Sam’s Club and Costco both sell emergency food both sell emergency food supplies. The unit cost to both retailers is $75. The retailers supplies. The unit cost to both retailers is $75. The retailers compete on price: the low-price retailer gets all the market and compete on price: the low-price retailer gets all the market and they split the market if they have equal prices. Suppose, each they split the market if they have equal prices. Suppose, each month, they consider prices $85 and $95, and suppose monthly month, they consider prices $85 and $95, and suppose monthly market demands at those prices are 100 and 80. Suppose the market demands at those prices are 100 and 80. Suppose the monthly interest rate is 0.3%. monthly interest rate is 0.3%.

What strategies (SWhat strategies (SOneOne,C,COneOne) should each player choose if they ) should each player choose if they

expect this game to last only one time period? Are there mutual expect this game to last only one time period? Are there mutual gains from cooperative strategies (Sgains from cooperative strategies (SCoopCoop,C,CCoopCoop)? If they expect this )? If they expect this

game to repeat indefinitely, would Sam’s cooperate each period game to repeat indefinitely, would Sam’s cooperate each period and choose Sand choose SCoop Coop if Costco followed the if Costco followed the Grim Strategy of Grim Strategy of

punishing non-cooperation? punishing non-cooperation? and would Costco cooperate each and would Costco cooperate each period and choose Cperiod and choose CCoop Coop if Sam’s followed the if Sam’s followed the Grim Strategy? Grim Strategy?


20 20

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

Costco

Sam's


Answer: Answer: The essential data of the game are The essential data of the game are the interest rate the interest rate between periods, between periods, r = 0.3% or r = 0.003 (as a fraction), and the r = 0.3% or r = 0.003 (as a fraction), and the payoffs each period defined by the normal form. For example, at payoffs each period defined by the normal form. For example, at Sam’s Club price $95 and Costco price $85, Costco gets the Sam’s Club price $95 and Costco price $85, Costco gets the entire market demand of 100, and so makes $(85-75)entire market demand of 100, and so makes $(85-75)xx100 = 100 = $1,000.$1,000.



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

Costco

Sam's


What strategies (SWhat strategies (SOneOne,C,COneOne) )

should each player choose if should each player choose if they expect this game to last they expect this game to last only one time period? only one time period? In that In that one-shot game, one-shot game, each player should choose each player should choose $85 price $85 price (S(SOneOne=$85,C=$85,COneOne=$85) =$85) since it is the dominate strategy for each since it is the dominate strategy for each

player. Each player thus player. Each player thus earns 500earns 500. .

Are there mutual gains from cooperative strategies (SAre there mutual gains from cooperative strategies (SCoopCoop,C,CCoopCoop)? )?

Yes, if all players choose Yes, if all players choose (S(SCoopCoop=$95,C=$95,CCoopCoop=$95)=$95), then each player , then each player

earns 800earns 800, rather than 500., rather than 500.


If players expect this game to If players expect this game to repeat indefinitely, each should repeat indefinitely, each should consider the Grim Strategy, consider the Grim Strategy, which has has two components. which has has two components. 1)1)The The CooperativeCooperative choice for choice for each player each player (S(SCoopCoop=$95,C=$95,CCoopCoop=$95)=$95). .

2)2)The The PunishmentPunishment choice for choice for each player each player (S(SPunPun=$85,C=$85,CPunPun=$85) =$85) , which , which

gives the other player the worst gives the other player the worst payoff after that player chooses payoff after that player chooses his best response to his his best response to his punishment.punishment.

$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



The Grim Strategy is thus, in The Grim Strategy is thus, in each period, Cooperate and each period, Cooperate and choose choose (S(SCoopCoop=$95,C=$95,CCoopCoop=$95)=$95), ,

as long as the other player has as long as the other player has Cooperated and chosen Cooperated and chosen (S(SCoopCoop=$95,C=$95,CCoopCoop=$95)=$95) in every in every

previous period. But otherwise previous period. But otherwise (if the other player has ever (if the other player has ever made a choice other than made a choice other than cooperation cooperation (S(SCoopCoop=$95,C=$95,CCoopCoop=$95)=$95)), then ), then

you punish by choosing you punish by choosing (S(SPunPun=$85,C=$85,CPunPun=$85) =$85) in the next in the next


$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



Suppose Costco Suppose Costco followed the followed the Grim Strategy. Grim Strategy. Would Sam’s Would Sam’s cooperate each period and cooperate each period and choose choose SSCoopCoop=$95=$95? ?

On the one hand, if Sam’s On the one hand, if Sam’s cooperated each period and cooperated each period and choose choose SSCoopCoop=$95=$95, then the , then the

Grim Strategy says Costco will Grim Strategy says Costco will also cooperate each period and also cooperate each period and choose choose CCCoopCoop=$95=$95, and so Sam’s , and so Sam’s

earns 800 earns 800 each period.each period.

$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



On the other hand, if Sam’s On the other hand, if Sam’s cheated in any period, then cheated in any period, then consider the first period when consider the first period when he cheats. he cheats.

In that first period, the Grim In that first period, the Grim Strategy says Costco will still Strategy says Costco will still cooperate and choose cooperate and choose CCCoopCoop=$95=$95. Sam’s best response . Sam’s best response

to to CCCoopCoop=$95 =$95 is $85, which is $85, which

earns 1000 in that period, rather earns 1000 in that period, rather than than earning 800 earning 800 if he had if he had continued to cooperate. So continued to cooperate. So the the one period gain from cheating one period gain from cheating is 200is 200..

$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



But starting in the next period But starting in the next period and continuing forever, the and continuing forever, the Grim Strategy says Costco will Grim Strategy says Costco will punish by choosing punish by choosing CCPunPun=$85=$85. .

Sam’s best response to Sam’s best response to CCPunPun=$85 =$85 is $85, which earns is $85, which earns

500 in each punishment period, 500 in each punishment period, rather than rather than earning 800 earning 800 if he if he had continued to cooperate. So had continued to cooperate. So the eventual loss from cheating the eventual loss from cheating is 300is 300..

$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



Summing up, if Costco Summing up, if Costco followed the Grim Strategyfollowed the Grim Strategy, then Sam’s , then Sam’s would cooperate and choose Swould cooperate and choose SCoop Coop each period exactly if the one each period exactly if the one

period gain of from cheating of period gain of from cheating of 200 200 does not compensate for the does not compensate for the eventual losses of eventual losses of 300 300 starting the next period. starting the next period.

Use the formula that $1 starting next month and continuing for Use the formula that $1 starting next month and continuing for each subsequent period is worth $(1/r) today. Since the interest each subsequent period is worth $(1/r) today. Since the interest rate r = 0.3% expressed as a fraction is r = 0.003, the eventual rate r = 0.3% expressed as a fraction is r = 0.003, the eventual losses of losses of 300 300 is the same as loosing 300/0.003 = 300,000 today. is the same as loosing 300/0.003 = 300,000 today.

Therefore, the one period gain of from cheating of Therefore, the one period gain of from cheating of 200 200 does not does not compensate for the eventual losses of compensate for the eventual losses of 300 300 starting the next starting the next period, and Sam’s would cooperate and choose Speriod, and Sam’s would cooperate and choose SCoop Coop each period. each period.

Since the game is symmetric, Costco would cooperate and Since the game is symmetric, Costco would cooperate and choose Cchoose CCoop Coop each period if Sam’s each period if Sam’s followed the Grim Strategy.followed the Grim Strategy.



Review Question 2Review Question 2



Question 2. Question 2. Consider 2 bars (A and B) suffering from a serious Consider 2 bars (A and B) suffering from a serious drunkenness problem that detracts customers because of the drunkenness problem that detracts customers because of the violence and smell. It costs $98 weekly in foregone profit for violence and smell. It costs $98 weekly in foregone profit for each bar to enforce moderation by stopping service to customers each bar to enforce moderation by stopping service to customers before they become drunk. For each bar that enforces moderation before they become drunk. For each bar that enforces moderation during the week, both bars will have a $50 increase in profit. during the week, both bars will have a $50 increase in profit. Suppose the weekly interest rate is 5%. Suppose the weekly interest rate is 5%.

What strategies (AWhat strategies (AOneOne,B,BOneOne) should each player choose if they ) should each player choose if they

expect this game to last only one time period? Are there mutual expect this game to last only one time period? Are there mutual gains from cooperative strategies (Agains from cooperative strategies (ACoopCoop,B,BCoopCoop)? If they expect )? If they expect

this game to repeat indefinitely, would Player A cooperate each this game to repeat indefinitely, would Player A cooperate each period and choose Aperiod and choose ACoop Coop if Player B followed the if Player B followed the Grim Strategy Grim Strategy

of punishing non-cooperation? of punishing non-cooperation? and would B cooperate each and would B cooperate each period and choose Bperiod and choose BCoop Coop if A followed the if A followed the Grim Strategy? Grim Strategy?


30 30

Enforce Not En.Enforce 2,2 -48,50Not En. 50,-48 0,0

Bar B

Bar A


Answer: Answer: The essential data of the game are The essential data of the game are the interest rate the interest rate between periods, between periods, r = 5% or r = 0.05 (as a fraction), and the r = 5% or r = 0.05 (as a fraction), and the payoffs each period defined by the normal form. For example, if payoffs each period defined by the normal form. For example, if Bar A does not enforce moderation during the period but Bar B Bar A does not enforce moderation during the period but Bar B does enforce moderation, then both bars will have a $50 increase does enforce moderation, then both bars will have a $50 increase in profit but Bar B pays an extra $98 cost, and so nets profit $50-in profit but Bar B pays an extra $98 cost, and so nets profit $50-$98 = -$48. $98 = -$48.




What strategies (AWhat strategies (AOneOne,B,BOneOne) )

should each player choose if should each player choose if they expect this game to last they expect this game to last only one time period? only one time period? In that In that one-shot game, one-shot game, each player should choose each player should choose Not Enforce (ANot Enforce (AOneOne= =

Not Enforce,BNot Enforce,BOneOne= Not Enforce) = Not Enforce) since it is the dominate strategy since it is the dominate strategy

for each player. Each player thus for each player. Each player thus earns 0earns 0. .

Are there mutual gains from cooperative strategies (AAre there mutual gains from cooperative strategies (ACoopCoop,B,BCoopCoop)? )?

Yes, if both players choose Yes, if both players choose (A(ACoopCoop=Enforce,B=Enforce,BCoopCoop=Enforce)=Enforce), then , then

each player each player earns 2earns 2, rather than 0., rather than 0.


Bar B

Bar A


If players expect this game to If players expect this game to repeat indefinitely, each should repeat indefinitely, each should consider the Grim Strategy, consider the Grim Strategy, which has two components. which has two components. 1)1)The The CooperativeCooperative choice for choice for each player each player (A(ACoopCoop=Enforce, =Enforce,

BBCoopCoop=Enforce)=Enforce). .

2)2)The The PunishmentPunishment choice for choice for each player each player (A(APunPun=Not Enforce, =Not Enforce,

BBPunPun=Not Enforce)=Not Enforce), which gives , which gives

the other player the worst the other player the worst payoff after that player chooses payoff after that player chooses his best response to his his best response to his punishment.punishment.



Bar B

Bar A


Bar B

Bar A


The Grim Strategy is thus, in The Grim Strategy is thus, in each period, Cooperate and each period, Cooperate and choose choose (A(ACoopCoop=Enforce, =Enforce,

BBCoopCoop=Enforce)=Enforce), as long as the , as long as the

other player has Cooperated other player has Cooperated and chosen and chosen (A(ACoopCoop=Enforce, =Enforce,

BBCoopCoop=Enforce)=Enforce) in every in every

previous period. But otherwise previous period. But otherwise (if the other player has ever (if the other player has ever made a choice other than made a choice other than cooperation), then you punish cooperation), then you punish by choosing by choosing (A(APunPun=Not Enforce, =Not Enforce,

BBPunPun=Not Enforce) =Not Enforce) in the next in the next




Bar B

Bar A


Bar B

Bar A


Suppose Bar B Suppose Bar B followed the followed the Grim Strategy. Grim Strategy. Would Bar A Would Bar A cooperate each period and cooperate each period and choose choose AACoopCoop=Enforce=Enforce? ?

On the one hand, if Bar A On the one hand, if Bar A cooperated each period and cooperated each period and choose choose AACoopCoop=Enforce=Enforce, then the , then the

Grim Strategy says Bar B will Grim Strategy says Bar B will also cooperate each period and also cooperate each period and choose choose BBCoopCoop=Enforce=Enforce, and so , and so

Bar A Bar A earns 2 earns 2 each period.each period.



Bar B

Bar A


Bar B

Bar A


On the other hand, if Bar A On the other hand, if Bar A cheated in any period, then cheated in any period, then consider the first period when consider the first period when he cheats. he cheats.

In that first period, the Grim In that first period, the Grim Strategy says Bar B will still Strategy says Bar B will still cooperate and choose cooperate and choose BBCoopCoop=Enforce=Enforce. Bar A’s best . Bar A’s best

response to cooperation is Not response to cooperation is Not Enforce, which earns 50 in that Enforce, which earns 50 in that period, rather than period, rather than earning 2 earning 2 if if he had continued to cooperate. he had continued to cooperate. So So the one period gain from the one period gain from cheating is 48cheating is 48..



Bar B

Bar A


Bar B

Bar A


But starting in the next period But starting in the next period and continuing forever, the and continuing forever, the Grim Strategy says Bar B will Grim Strategy says Bar B will punish by choosing punish by choosing BBPunPun=Not =Not

EnforceEnforce. Bar A’s best response . Bar A’s best response to to BBPunPun=Not Enforce =Not Enforce is to Not is to Not

Enforce, which earns 0 in each Enforce, which earns 0 in each punishment period, rather than punishment period, rather than earning 2 earning 2 if he had continued to if he had continued to cooperate. So cooperate. So the eventual loss the eventual loss from cheating is 2from cheating is 2..



Bar B

Bar A


Bar B

Bar A


Summing up, if Player B Summing up, if Player B followed the Grim Strategyfollowed the Grim Strategy, then Player , then Player A would cooperate and choose AA would cooperate and choose ACoop Coop each period exactly if the each period exactly if the

one period gain of from cheating of one period gain of from cheating of 48 48 does not compensate for does not compensate for the eventual losses of the eventual losses of 2 2 starting the next period. starting the next period.

Use the formula that $1 starting next month and continuing for Use the formula that $1 starting next month and continuing for each subsequent period is worth $(1/r) today. Since the interest each subsequent period is worth $(1/r) today. Since the interest rate r = 5% expressed as a fraction is r = 0.05, the eventual losses rate r = 5% expressed as a fraction is r = 0.05, the eventual losses of of 2 2 is the same as loosing 2/0.05 = 40 today. is the same as loosing 2/0.05 = 40 today.

Therefore, the one period gain of from cheating of Therefore, the one period gain of from cheating of 48 48 compensates for the eventual losses of compensates for the eventual losses of 2 2 starting the next period, starting the next period, so Player A would not cooperate and, instead, choose Not so Player A would not cooperate and, instead, choose Not Enforce each period. Since the game is symmetric, Player B Enforce each period. Since the game is symmetric, Player B would not cooperate if Player A would not cooperate if Player A followed the Grim Strategy.followed the Grim Strategy.


38 38

End of Lesson II.8End of Lesson II.8


BA 210 Introduction to BA 210 Introduction to MicroeconomicsMicroeconomics

1 1 deep thought ba 210 lesson ii.8 beneficial grim punishment sometimes when i fell like killing...

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