1 1 Deep Thought BA 210 Lesson II.8 Beneficial Grim Punishment Sometimes when I fell like killing someone, I do a little trick to calm myself down. I’ll go over to the person’s house and ring the doorbell. When the person comes to the door, I’m gone, but you know what I’ve left on the porch? A jack-o-lantern with a knife stuck in the side of its head with a note that says “You”. After that I usually feel better, and no harm done. ~ Jack Handey. (Translation: Today’s lesson shows credible threats do not have to be executed.)

2 2 BA 210 Lesson II.8 Beneficial Grim PunishmentOverviewOverview

3 3 Lesson Overview BA 210 Lesson II.8 Beneficial Grim Punishment Lesson II.8 Beneficial Grim Punishment Example 1: Grim Punishment with Interest Summary Review Questions

4 4 BA 210 Lesson II.8 Beneficial Grim Punishment Example 1: Grim Punishment with Interest

5 5 BA 210 Lesson II.8 Beneficial Grim Punishment Comment: In any prisoners’ dilemma, there are mutual gains from Cooperating by choosing a particular action, but not everyone can be trusted to cooperate because, for at least one person, the cooperative action is not a best response to the other players selecting their cooperative actions. That is, cooperation is not a Nash Equilibrium. We see that cooperation can become a Nash Equilibrium, and so players can be trusted to cooperate, if the dilemma game is repeated indefinitely, and players punish non-cooperation. The most effective punishment is called the Grim Strategy. The punishment inflicts the maximum pain on non-cooperation, and it lasts forever. Example 1: Grim Punishment with Interest

6 6 BA 210 Lesson II.8 Beneficial Grim Punishment Question: R.J. Reynolds Tobacco Corp. and Philip Morris Corp. must decide how much money to spend on advertising each year, 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 other. If each advertises, each pays $10,000 but the advertisements cancel out and neither player takes profit from the other. Suppose the yearly interest rate is 10%. What strategies (R One,P One ) should each player choose if they expect this game to last only one time period? Are there mutual gains from cooperative strategies (R Coop,P Coop )? If they expect this game to repeat indefinitely, would Reynolds cooperate each period and choose R Coop if Philip followed the Grim Strategy of punishing non-cooperation? and would Philip cooperate each period and choose P Coop if Reynolds followed the Grim Strategy? Example 1: Grim Punishment with Interest

7 7 BA 210 Lesson II.8 Beneficial Grim Punishment Answer: The essential data of the game are the interest rate between periods, r = 10% or r = 0.10 (as a fraction), and the payoffs each period defined by the normal form. For example, with payoffs in thousands of dollars, if Reynolds advertises and Philip does not, Reynolds pays $10,000, then takes $100,000 profit from Philip. Hence, Reynolds makes $90,000 and Philip looses $100,000. Example 1: Grim Punishment with Interest

8 8 BA 210 Lesson II.8 Beneficial Grim Punishment What strategies (S One,C One ) should each player choose if they expect this game to last only one time period? In that one-shot game, each player should choose Advertise (R One =Ad,P One =Ad) since it is the dominate strategy for each player. Each player thus earns -10. Are there mutual gains from cooperative strategies (S Coop,C Coop )? Yes, if both players choose (R Coop =No Ad,P coop =No Ad), then each player earns 0, rather than -10. Example 1: Grim Punishment with Interest

9 9 BA 210 Lesson II.8 Beneficial Grim Punishment If players expect this game to repeat indefinitely, each should consider the Grim Strategy, which has has two components. 1)The Cooperative choice for each player (R Coop =No Ad,P coop =No Ad). 2)The Punishment choice for each player (R Pun =Ad,P Pun =Ad), which gives the other player the worst payoff after that player chooses his best response to his punishment. Example 1: Grim Punishment with Interest

10 BA 210 Lesson II.8 Beneficial Grim Punishment The Grim Strategy is thus, in each period, Cooperate and choose (R Coop =No Ad,P coop =No Ad), as long as the other player has Cooperated and chosen (R coop =No Ad,P coop =No Ad) in every previous period. But otherwise (if the other player has ever made a choice other than cooperation (R Coop =No Ad,P coop =No Ad)), then you punish by choosing (R Pun =Ad,P Pun =Ad) in the next period and in every period thereafter --- forever. Example 1: Grim Punishment with Interest

11 BA 210 Lesson II.8 Beneficial Grim Punishment Suppose Philip followed the Grim Strategy. Would Reynolds cooperate each period and choose R Coop =No Ad? On the one hand, if Reynolds cooperated each period and choose R Coop =No Ad, then the Grim Strategy says Philip will also cooperate each period and choose P coop =No Ad, and so Reynolds earns 0 each period. Example 1: Grim Punishment with Interest

12 BA 210 Lesson II.8 Beneficial Grim Punishment On the other hand, if Reynolds cheated in any period, then consider the first period when he cheats. In that first period, the Grim Strategy says Philip will still cooperate and choose C Coop =No Ad. Reynolds’ best response to P coop =No Ad is Ad, which earns 90 in that period, rather than earning 0 if he had continued to cooperate. So the one period gain from cheating is 90. Example 1: Grim Punishment with Interest

13 BA 210 Lesson II.8 Beneficial Grim Punishment But starting in the next period and continuing forever, the Grim Strategy says Philip will punish by choosing P Pun =Ad. Reynolds’s best response to P Pun =Ad is Ad, which earns -10 in each punishment period, rather than earning 0 if he had continued to cooperate. So the eventual loss from cheating is 10. Example 1: Grim Punishment with Interest

14 BA 210 Lesson II.8 Beneficial Grim Punishment Summing up, if Philip followed the Grim Strategy, then Reynolds would cooperate and choose R Coop each period exactly if the one period gain of from cheating of 90 does not compensate for the eventual losses of 10 starting the next period. That answer depends on the interest rate r between periods. Example 1: Grim Punishment with Interest

15 BA 210 Lesson II.8 Beneficial Grim Punishment Comment: To continue the answer, we need the formula that $1 starting next period and continuing for each subsequent period is worth $(1/r) today. To illustrate, consider earning 10 percent interest each period, so r = The formula thus becomes $1 starting next period and continuing for each subsequent period is worth $10 today. To prove that formula, consider investing the $10 today. At the end of the first period, you earn $1 interest. Suppose you withdraw that interest and reinvest the $10. Then at the end of the second period, you earn another $1 interest. If you continue withdrawing interest each period, keeping the $10 invested forever, you thus earn $1 interest each period. Example 1: Grim Punishment with Interest

16 BA 210 Lesson II.8 Beneficial Grim Punishment Use the formula that $1 starting next month and continuing for each subsequent period is worth $(1/r) today. Since the interest rate r = 10% expressed as a fraction is r = 0.10, the eventual losses of 10 is the same as loosing 10/0.10 = 100 today. Therefore, the one period gain of from cheating of 90 does not compensate for the eventual losses of 10 starting the next period, and so Reynolds would cooperate and choose R Coop = No Ad each period. Since the game is symmetric, Philip would cooperate and choose P Coop = No Ad each period if Reynolds followed the Grim Strategy. Example 1: Grim Punishment with Interest

17 Review Questions BA 210 Lesson II.8 Beneficial Grim Punishment 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.

18 BA 210 Lesson II.8 Beneficial Grim Punishment Review Question 1 Review Questions

19 BA 210 Lesson II.8 Beneficial Grim Punishment Question 1. Sam’s Club and Costco both sell emergency food supplies. 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, each month, they consider prices $85 and $95, and suppose monthly market demands at those prices are 100 and 80. Suppose the monthly interest rate is 0.3%. What strategies (S One,C One ) should each player choose if they expect this game to last only one time period? Are there mutual gains from cooperative strategies (S Coop,C Coop )? If they expect this game to repeat indefinitely, would Sam’s cooperate each period and choose S Coop if Costco followed the Grim Strategy of punishing non-cooperation? and would Costco cooperate each period and choose C Coop if Sam’s followed the Grim Strategy? Review Questions

20 BA 210 Lesson II.8 Beneficial Grim Punishment Answer: The essential data of the game are the interest rate between periods, r = 0.3% or r = (as a fraction), and the payoffs each period defined by the normal form. For example, at Sam’s Club price $95 and Costco price $85, Costco gets the entire market demand of 100, and so makes $(85-75) x 100 = $1,000. Review Questions

21 BA 210 Lesson II.8 Beneficial Grim Punishment Review Questions What strategies (S One,C One ) should each player choose if they expect this game to last only one time period? In that one-shot game, each player should choose $85 price (S One =$85,C One =$85) since it is the dominate strategy for each player. Each player thus earns 500. Are there mutual gains from cooperative strategies (S Coop,C Coop )? Yes, if all players choose (S Coop =$95,C Coop =$95), then each player earns 800, rather than 500.

22 BA 210 Lesson II.8 Beneficial Grim Punishment If players expect this game to repeat indefinitely, each should consider the Grim Strategy, which has has two components. 1)The Cooperative choice for each player (S Coop =$95,C Coop =$95). 2)The Punishment choice for each player (S Pun =$85,C Pun =$85), which gives the other player the worst payoff after that player chooses his best response to his punishment. Review Questions

23 BA 210 Lesson II.8 Beneficial Grim Punishment The Grim Strategy is thus, in each period, Cooperate and choose (S Coop =$95,C Coop =$95), as long as the other player has Cooperated and chosen (S Coop =$95,C Coop =$95) in every previous period. But otherwise (if the other player has ever made a choice other than cooperation (S Coop =$95,C Coop =$95)), then you punish by choosing (S Pun =$85,C Pun =$85) in the next period and in every period thereafter --- forever. Review Questions

24 BA 210 Lesson II.8 Beneficial Grim Punishment Suppose Costco followed the Grim Strategy. Would Sam’s cooperate each period and choose S Coop =$95? On the one hand, if Sam’s cooperated each period and choose S Coop =$95, then the Grim Strategy says Costco will also cooperate each period and choose C Coop =$95, and so Sam’s earns 800 each period. Review Questions

25 BA 210 Lesson II.8 Beneficial Grim Punishment On the other hand, if Sam’s cheated in any period, then consider the first period when he cheats. In that first period, the Grim Strategy says Costco will still cooperate and choose C Coop =$95. Sam’s best response to C Coop =$95 is $85, which earns 1000 in that period, rather than earning 800 if he had continued to cooperate. So the one period gain from cheating is 200. Review Questions

26 BA 210 Lesson II.8 Beneficial Grim Punishment But starting in the next period and continuing forever, the Grim Strategy says Costco will punish by choosing C Pun =$85. Sam’s best response to C Pun =$85 is $85, which earns 500 in each punishment period, rather than earning 800 if he had continued to cooperate. So the eventual loss from cheating is 300. Review Questions

27 BA 210 Lesson II.8 Beneficial Grim Punishment Summing up, if Costco followed the Grim Strategy, then Sam’s would cooperate and choose S Coop each period exactly if the one period gain of from cheating of 200 does not compensate for the eventual losses of 300 starting the next period. Use the formula that $1 starting next month and continuing for 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 losses of 300 is the same as loosing 300/0.003 = 300,000 today. Therefore, the one period gain of from cheating of 200 does not compensate for the eventual losses of 300 starting the next period, and Sam’s would cooperate and choose S Coop each period. Since the game is symmetric, Costco would cooperate and choose C Coop each period if Sam’s followed the Grim Strategy. Review Questions

28 BA 210 Lesson II.8 Beneficial Grim Punishment Review Question 2 Review Questions

29 BA 210 Lesson II.8 Beneficial Grim Punishment Question 2. Consider 2 bars (A and B) suffering from a serious drunkenness problem that detracts customers because of the violence and smell. It costs $98 weekly in foregone profit for each bar to enforce moderation by stopping service to customers before they become drunk. For each bar that enforces moderation during the week, both bars will have a $50 increase in profit. Suppose the weekly interest rate is 5%. What strategies (A One,B One ) should each player choose if they expect this game to last only one time period? Are there mutual gains from cooperative strategies (A Coop,B Coop )? If they expect this game to repeat indefinitely, would Player A cooperate each period and choose A Coop if Player B followed the Grim Strategy of punishing non-cooperation? and would B cooperate each period and choose B Coop if A followed the Grim Strategy? Review Questions

30 BA 210 Lesson II.8 Beneficial Grim Punishment Answer: The essential data of the game are the interest rate between periods, r = 5% or r = 0.05 (as a fraction), and the payoffs each period defined by the normal form. For example, if Bar A does not enforce moderation during the period but Bar B 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- $98 = -$48. Review Questions

31 BA 210 Lesson II.8 Beneficial Grim Punishment Review Questions What strategies (A One,B One ) should each player choose if they expect this game to last only one time period? In that one-shot game, each player should choose Not Enforce (A One = Not Enforce,B One = Not Enforce) since it is the dominate strategy for each player. Each player thus earns 0. Are there mutual gains from cooperative strategies (A Coop,B Coop )? Yes, if both players choose (A Coop =Enforce,B Coop =Enforce), then each player earns 2, rather than 0.

32 BA 210 Lesson II.8 Beneficial Grim Punishment If players expect this game to repeat indefinitely, each should consider the Grim Strategy, which has two components. 1)The Cooperative choice for each player (A Coop =Enforce, B Coop =Enforce). 2)The Punishment choice for each player (A Pun =Not Enforce, B Pun =Not Enforce), which gives the other player the worst payoff after that player chooses his best response to his punishment. Review Questions

33 BA 210 Lesson II.8 Beneficial Grim Punishment The Grim Strategy is thus, in each period, Cooperate and choose (A Coop =Enforce, B Coop =Enforce), as long as the other player has Cooperated and chosen (A Coop =Enforce, B Coop =Enforce) in every previous period. But otherwise (if the other player has ever made a choice other than cooperation), then you punish by choosing (A Pun =Not Enforce, B Pun =Not Enforce) in the next period and in every period thereafter --- forever. Review Questions

34 BA 210 Lesson II.8 Beneficial Grim Punishment Suppose Bar B followed the Grim Strategy. Would Bar A cooperate each period and choose A Coop =Enforce? On the one hand, if Bar A cooperated each period and choose A Coop =Enforce, then the Grim Strategy says Bar B will also cooperate each period and choose B Coop =Enforce, and so Bar A earns 2 each period. Review Questions

35 BA 210 Lesson II.8 Beneficial Grim Punishment On the other hand, if Bar A cheated in any period, then consider the first period when he cheats. In that first period, the Grim Strategy says Bar B will still cooperate and choose B Coop =Enforce. Bar A’s best response to cooperation is Not Enforce, which earns 50 in that period, rather than earning 2 if he had continued to cooperate. So the one period gain from cheating is 48. Review Questions

36 BA 210 Lesson II.8 Beneficial Grim Punishment But starting in the next period and continuing forever, the Grim Strategy says Bar B will punish by choosing B Pun =Not Enforce. Bar A’s best response to B Pun =Not Enforce is to Not Enforce, which earns 0 in each punishment period, rather than earning 2 if he had continued to cooperate. So the eventual loss from cheating is 2. Review Questions

37 BA 210 Lesson II.8 Beneficial Grim Punishment Summing up, if Player B followed the Grim Strategy, then Player A would cooperate and choose A Coop each period exactly if the one period gain of from cheating of 48 does not compensate for the eventual losses of 2 starting the next period. Use the formula that $1 starting next month and continuing for 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 of 2 is the same as loosing 2/0.05 = 40 today. Therefore, the one period gain of from cheating of 48 compensates for the eventual losses of 2 starting the next period, so Player A would not cooperate and, instead, choose Not Enforce each period. Since the game is symmetric, Player B would not cooperate if Player A followed the Grim Strategy. Review Questions

38 End of Lesson II.8 BA 210 Lesson II.8 Beneficial Grim Punishment BA 210 Introduction to Microeconomics