1 1 Deep Thought BA 445 Lesson B.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 445 Lesson B.8 Beneficial Grim Punishment ReadingsReadings Baye “Repeated Games” (see the index) Dixit Chapter 11

3 3 BA 445 Lesson B.8 Beneficial Grim Punishment OverviewOverview

4 4Overview Grim Punishment solves some prisoners’ dilemmas with threats. Cooperation becomes a best response to other players’ threats to punish non-cooperation, so the punishment never happens. Uncertainty and the Effective Interest Rate affect which prisoners’ dilemmas are solved by grim punishment. The interest rate discounts the effect of future punishment on current non-cooperation. Non-Symmetric Games complicate solving dilemmas. Player A must threaten enough to cause Player B to cooperate, and Player B must threaten enough to cause Player A to cooperate. Multiple Actions complicate solving dilemmas. The most effective punishment to non-cooperation is selected, from multiple alternative actions, to minimize the offending player’s payoff. Multiple Players complicate solving dilemmas. The most effective punishment to non-cooperation has every other player punishing the offending player.

5 5 BA 445 Lesson B.8 Beneficial Grim Punishment Example 1: Grim Punishment

6 6 BA 445 Lesson B.8 Beneficial Grim Punishment Overview Grim Punishment solves some prisoners’ dilemmas with threats. Cooperation becomes a best response to other players’ threats to punish non-cooperation, so the punishment never happens. Example 1: Grim Punishment

7 7 BA 445 Lesson B.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

8 8 BA 445 Lesson B.8 Beneficial Grim Punishment Question: R.J. Reynolds Tobacco Corp. and Philip Morris Corp. must decide how much money to spend on advertising each month. They consider spending either $10,000 or zero each month. 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. Are there mutual gains from both players following the Grim Strategy for the repeated game (the game repeated month after month) rather than repeating the solution to the one-shot game? And is it a Nash Equilibrium for both players to follow the Grim Strategy? That is, can R.J. Reynolds trust Philip Morris to cooperate and follow the Grim Strategy? and can Philip Morris trust R.J. Reynolds to cooperate and follow the Grim Strategy?

9 9 Example 1: Grim Punishment BA 445 Lesson B.8 Beneficial Grim Punishment Answer: To begin, fill out the normal form for each month for this game of simultaneous moves, with payoffs in thousands of dollars. For example, 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.

10 Example 1: Grim Punishment BA 445 Lesson B.8 Beneficial Grim Punishment The one-shot version of the game each month is defined by the normal form. On the one hand, in the hypothetical one-shot game, each player should choose to advertise since it is the dominate strategy. Each thus earns -10. If players repeat the solution to the one-shot game, then each player earns 10 each period.

11 Example 1: Grim Punishment BA 445 Lesson B.8 Beneficial Grim Punishment On the other hand, if the game continues indefinitely, then each player should consider the Grim Strategy. The Grim Strategy has two components. 1) The Cooperative action of No Advertise, which is mutually-better than the non-cooperative strategy of Advertise. 2) The Punishment action of Advertise, which gives the other player the worst payoff after that player chooses his best response to (makes the best of) his punishment.

12 Example 1: Grim Punishment BA 445 Lesson B.8 Beneficial Grim Punishment The Grim Strategy is thus, in each month, Cooperate and choose No Advertise, as long as the other player has Cooperated and chosen No Advertise in every previous month. But if the other player ever chooses to Not Cooperate and so to Advertise, then you punish by choosing Advertise in the next month and in every month thereafter --- forever.

13 Example 1: Grim Punishment BA 445 Lesson B.8 Beneficial Grim Punishment Can R.J. Reynolds trust Philip Morris to cooperate and follow the Grim Strategy? To answer, consider the benefits of Philip Not Cooperating and choosing Advertise in Month X while Reynolds chooses to Cooperate and choose No Advertise. In Month X, Philip gains  Cheat = 90 from Advertise rather than the  Cooperate = 0 he would have had from No Advertise.

14 Example 1: Grim Punishment BA 445 Lesson B.8 Beneficial Grim Punishment But starting in Month X+1 and continuing forever, Reynolds punishes Philip by choosing Advertise, and so the best Reynolds can achieve is  Punish = -10, rather than the  Cooperate = 0 he would have had from Cooperation.

15 Example 1: Grim Punishment BA 445 Lesson B.8 Beneficial Grim Punishment Summing up, R.J. Reynolds can trust Philip Morris to cooperate and follow the Grim Strategy if the one period gain  Cheat -  Cooperate = 90 does not compensate for later loses  Punish -  Cooperate = -10 starting the next month. That answer depends on the interest rate r between months.

16 Example 1: Grim Punishment BA 445 Lesson B.8 Beneficial Grim Punishment Use the formula that $1 today is worth $r starting next month and continuing for each subsequent month, and that $1 starting next month and continuing for each subsequent month is worth $(1/r) today. R.J. Reynolds can trust Philip Morris to cooperate and follow the Grim Strategy if $90 < -$10/r That is, if r < $10/$90 =.1111, which is when the monthly interest rate is less than 11.11% Since the game is symmetric, when the monthly interest rate is less than 11.11%, Philip Morris can also trust R.J. Reynolds to cooperate and follow the Grim Strategy, and the Grim Strategy for each player a Nash Equilibrium. Finally, when each player chooses the Grim Strategy, each player does not advertise each period, which gives each player 0 payoff each month, which is mutually-better than if each player chose to Advertise.

17 Example 1: Grim Punishment BA 445 Lesson B.8 Beneficial Grim Punishment Comment: To prove the formula that $1 today is worth $i starting next month and continuing for each subsequent month, consider investing the $1 today and drawing out just the interest in each subsequent month. You keep the $1 invested forever, and you earn $i interest each month. Multiplying by X both sides of that formula above, $X today is worth $iX starting next month and continuing for each subsequent month. In particular, for X = 1/i, we get a second formula that $(1/i) today is worth $iX = $1 starting next month and continuing for each subsequent month.

18 BA 445 Lesson B.8 Beneficial Grim Punishment Example 2: Uncertainty and the Effective Interest Rate

19 BA 445 Lesson B.8 Beneficial Grim Punishment Overview Uncertainty and the Effective Interest Rate affect which prisoners’ dilemmas are solved by grim punishment. The interest rate discounts the effect of future punishment on current non-cooperation. Example 2: Uncertainty and the Effective Interest Rate

20 BA 445 Lesson B.8 Beneficial Grim Punishment Question: R.J. Reynolds Tobacco Corp. and Philip Morris Corp. must decide how much money to spend on advertising each month. They consider spending either $10,000 or zero each month. 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 monthly interest rate is 0.5%. And suppose advertising is initially legal, but there is uncertainty about the future; specifically, with probability 0.2, advertising will become illegal next month. Are there mutual gains from both players following the Grim Strategy for the repeated game (the game repeated month after month) rather than repeating the solution to the one-shot game? And is it a Nash Equilibrium for both players to follow the Grim Strategy? Example 2: Uncertainty and the Effective Interest Rate

21 BA 445 Lesson B.8 Beneficial Grim Punishment Comment: In the hypothetical case where the game with certainty continues from one period to the next, a loss next month is currently worth only 1/(1+r) times the amount lost, where r is the interest rate between periods. In the more realistic case where the game with probability p continues from one period to the next, a loss next month is currently expected to be worth only p/(1+r) times the amount lost. Put another way, a loss next month is currently worth only 1/(1+R) times the amount lost, where R is the effective interest rate determined by the equation 1/(1+R) = p/(1+r), or R = (1+r)/p-1. Example 2: Uncertainty and the Effective Interest Rate

22 BA 445 Lesson B.8 Beneficial Grim Punishment Answer: The only difference between this problem and Example 1 is that, now, we suppose each month there is uncertainty about whether the advertising game continues from one month to the next. Since the payoffs in the two problems are the same, R.J. Reynolds can trust Philip Morris to cooperate and follow the Grim Strategy if the monthly interest rate is less than 11.11%. Now, that condition is that the effective interest rate is less than 11.11%. But the effective interest rate is R = (1+r)/p-1 = 1.005/0.8-1 = , which is %, which is greater than 11.11%. So, R.J. Reynolds can not trust Philip Morris to cooperate and follow the Grim Strategy. Since the game is symmetric, Philip Morris can not trust R.J. Reynolds to cooperate, and the Grim Strategy for each player is not a Nash Equilibrium. Example 2: Uncertainty and the Effective Interest Rate

23 BA 445 Lesson B.8 Beneficial Grim Punishment Example 3: Non-Symmetric Games

24 BA 445 Lesson B.8 Beneficial Grim Punishment Overview Non-Symmetric Games complicate solving dilemmas. Player A must threaten enough to cause Player B to cooperate, and Player B must threaten enough to cause Player A to cooperate. Example 3: Non-Symmetric Games

25 BA 445 Lesson B.8 Beneficial Grim Punishment Question: Wii video game consoles are made by Nintendo, and some games are produced by 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). Suppose the monthly interest rate is 0.2%. And all demands are considered to last indefinitely. Are there mutual gains from both players following the Grim Strategy for the repeated game rather than repeating the solution to the one- shot game? And is it a Nash Equilibrium for both players to follow the Grim Strategy? Example 3: Non-Symmetric Games

26 BA 445 Lesson B.8 Beneficial Grim Punishment Answer: The essential data of the game includes the effective monthly interest rate R = (1+r)/p-1 = 1.002/1-1 = 0.002, where r is the inter- period interest rate and p = 1 is the probability that the game continues from one period to the next. So R is 0.2%. The one-shot version of the game each month is defined by the normal form. 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. Example 3: Non-Symmetric Games

27 BA 445 Lesson B.8 Beneficial Grim Punishment On the one hand, in the hypothetical one-shot game, Nintendo and Sega should choose $350 (Nintendo) or $50 (Sega) since they are dominate strategies for each player. Thus Nintendo and Sega earn 180 and 48. If players repeat the solution to the one-shot game, then Nintendo and Sega earn 180 and 48 each period. Example 3: Non-Symmetric Games

28 BA 445 Lesson B.8 Beneficial Grim Punishment On the other hand, since the game actually continues indefinitely, each player should consider the Grim Strategy. The Grim Strategy has two components. 1) The Cooperative choices of $250 and $40, which is mutually-better than the one-shot choice of $350 and $50. 2) The Punishment choice of $350 and $50 price, which gives the other player the worst payoff after that player chooses his best response to his punishment. Example 3: Non-Symmetric Games

29 BA 445 Lesson B.8 Beneficial Grim Punishment The Grim Strategy is thus, in each month, Cooperate and choose $250 (Nintendo) or $40 (Sega), as long as the other player has Cooperated and chosen $250 (Nintendo) or $40 (Sega) in every previous month. But otherwise then you punish by choosing $350 (Nintendo) or $50 (Sega) in the next month and in every month thereafter --- forever. Example 3: Non-Symmetric Games

30 BA 445 Lesson B.8 Beneficial Grim Punishment In particular, if both players follow the Grim Strategy for the repeated game, each month both choose $250 (Nintendo) or $40 (Sega), and so earn 200 (Nintendo) or 60 (Sega). And that is mutually-better than the 180 (Nintendo) or 48 (Sega) earned in the solution to the one-shot game. Example 3: Non-Symmetric Games

31 BA 445 Lesson B.8 Beneficial Grim Punishment Can Nintendo trust Sega to follow an agreement to use the Grim Strategy? To answer, suppose both players initially followed the Grim Strategy. Then, in Month X, consider the benefits of Sega deviating from the Grim Strategy and choosing $50 price while Nintendo continues to choose $250 price. In Month X, Sega gains  Cheat = 64 from $50 price rather than the  Cooperate = 60 it would have had from following the Grim Strategy Example 3: Non-Symmetric Games

32 BA 445 Lesson B.8 Beneficial Grim Punishment But starting in Month X+1 and continuing forever, Nintendo punishes Sega by choosing $350 price, and so the best Sega can achieve is  Punish = 48, rather than the  Cooperate = 60 it would have had if he had continued to follow the Grim Strategy. Example 3: Non-Symmetric Games

33 BA 445 Lesson B.8 Beneficial Grim Punishment Summing up, Nintendo can trust Sega to follow an agreement to use the Grim Strategy if the one period gain from cheating  Cheat -  Cooperate = 4 does not compensate for losses  Punish -  Cooperate = 12 starting the next period. Use the formula that $1 starting next month and continuing for each subsequent month is worth $(1/R) today. Nintendo can trust Sega to cooperate and follow the Grim Strategy if 4 < 12/R, which is when the effective monthly interest rate R < 12/4 = 3 is less than 300%. Since the effective interest rate is R = (1+r)/p-1 = 1.002/1-1 = 0.002, which is 0.2%, which is less than 300%, Nintendo can trust Sega to follow an agreement to use the Grim Strategy. Since the normal form is not symmetric, we must still check whether Sega can trust Nintendo. Example 3: Non-Symmetric Games

34 BA 445 Lesson B.8 Beneficial Grim Punishment Recomputing, Sega can trust Nintendo to follow an agreement to use the Grim Strategy if the one period gain from cheating  Cheat -  Cooperate = = 10 does not compensate for losses  Punish -  Cooperate = = 20 starting the next period. So, Sega can trust Nintendo if 10 < 20/R, which is when the effective monthly interest rate R < 20/10 = 2 is less than 200%, which is true. Example 3: Non-Symmetric Games

35 BA 445 Lesson B.8 Beneficial Grim Punishment Since each player can trust the other to follow the Grim Strategy, it is a Nash Equilibrium for both players to follow the Grim Strategy. Example 3: Non-Symmetric Games

36 BA 445 Lesson B.8 Beneficial Grim Punishment Example 4: Multiple Actions

37 BA 445 Lesson B.8 Beneficial Grim Punishment Overview Multiple Actions complicate solving dilemmas. The most effective punishment to non-cooperation is selected, from multiple alternative actions, to minimize the offending player’s payoff. Example 4: Multiple Actions

38 BA 445 Lesson B.8 Beneficial Grim Punishment Question: Intel and AMD simultaneously decide on the size of manufacturing plants for the next generation of microprocessors for consumer desktop computers. Suppose the firms’ goods are perfect substitutes, and market demand defines a linear inverse demand curve P = 20 – (Q I + Q A ), where output quantities Q I and Q A are the thousands of processors produced monthly by Intel and AMD. Suppose unit costs of production are c I = 2 and c A = 2 for both Intel and AMD. Suppose Intel and AMD consider any quantities Q I = 4 or 5 or 6 or 7, and Q A = 4 or 5 or 6 or 7. Suppose the monthly interest rate is 0.2%. And all demands are considered to last indefinitely. Are there mutual gains from both players following the Grim Strategy for the repeated game rather than repeating the solution to the one- shot game? And is it a Nash Equilibrium for both players to follow the Grim Strategy? Example 4: Multiple Actions

39 BA 445 Lesson B.8 Beneficial Grim Punishment Answer: The essential data of the game includes the effective monthly interest rate R = (1+r)/p-1 = 1.002/1-1 = 0.002, where r is the inter- period interest rate and p = 1 is the probability that the game continues from one period to the next. So R is 0.2%. The one-shot version of the game each month is defined by the normal form. For example, at Intel quantity 3 and AMD quantity 5, price = 20-8 = 12, so Intel profits = (12- 2)3 = 30 and AMD profits = (12-2)5 = 50. Example 4: Multiple Actions

40 BA 445 Lesson B.8 Beneficial Grim Punishment On the one hand, in the hypothetical one-shot game, Intel and AMD should each choose quantity 6 since they are the dominance solutions (first, eliminate 4 and 5 as being dominated by 6, then 6 dominates 7). Thus Intel and AMD each earn 36. If players repeat the solution to the one-shot game, then Intel and AMD each earn 36 each period. Example 4: Multiple Actions

41 BA 445 Lesson B.8 Beneficial Grim Punishment On the other hand, since the game actually continues indefinitely, each player should consider a Grim Strategy. A Grim Strategy has two components. 1) The Cooperative choices of 4 and 4 (or 5 and 5), which is mutually-better than the one- shot choice of 6 and 6. 2) The Punishment choice of 7, which gives the other player the worst payoff (30) after that player chooses his best response to his punishment. Example 4: Multiple Actions

42 BA 445 Lesson B.8 Beneficial Grim Punishment A Grim Strategy is thus, in each month, Cooperate and choose 4, as long as the other player has Cooperated and chosen 4 in every previous month. But otherwise then you punish by choosing 7 in the next month and in every month thereafter --- forever. Example 4: Multiple Actions

43 BA 445 Lesson B.8 Beneficial Grim Punishment In particular, if both players follow the Grim Strategy for the repeated game, each month both choose 4, and so earn 40. And that is mutually-better than the 36 earned in the dominance solution to the one-shot game. Example 4: Multiple Actions

44 BA 445 Lesson B.8 Beneficial Grim Punishment Can Intel trust AMD to follow an agreement to use the Grim Strategy? To answer, suppose both players initially followed the Grim Strategy. Then, in Month X, consider the benefits of AMD deviating from the Grim Strategy and choosing 7 while Intel continues to choose 4. In Month X, AMD gains  Cheat = 49 from 7 rather than the  Cooperate = 40 it would have had from following the Grim Strategy. Example 4: Multiple Actions

45 BA 445 Lesson B.8 Beneficial Grim Punishment But starting in Month X+1 and continuing forever, Intel punishes AMD by choosing 7, and so the best AMD can achieve is  Punish = 30, rather than the  Cooperate = 40 it would have had if he had continued to follow the Grim Strategy. Example 4: Multiple Actions

46 BA 445 Lesson B.8 Beneficial Grim Punishment Summing up, Intel can trust AMD to follow an agreement to use the Grim Strategy if the one period gain from cheating  Cheat -  Cooperate = 9 does not compensate for losses  Punish -  Cooperate = 10 starting the next period. Use the formula that $1 starting next month and continuing for each subsequent month is worth $(1/R) today. Intel can trust AMD to follow an agreement to use the Grim Strategy if 9 < 10/R, which is when the effective monthly interest rate R < 10/9 = 1.11 is less than 111%. Since the effective interest rate is R = (1+r)/p-1 = 1.002/1-1 = 0.002, which is 0.2%, which is less than 111%, Intel can trust AMD to follow an agreement to use the Grim Strategy. Since the game is symmetric, AMD can trust Intel to follow an agreement to use the Grim Strategy, and it is a Nash Equilibrium for both players to follow the Grim Strategy. Example 4: Multiple Actions

47 BA 445 Lesson B.8 Beneficial Grim Punishment Example 5: Multiple Players

48 BA 445 Lesson B.8 Beneficial Grim Punishment Overview Multiple Players complicate solving dilemmas. The most effective punishment to non-cooperation has every other player punishing the offending player. Example 5: Multiple Players

49 BA 445 Lesson B.8 Beneficial Grim Punishment Question: Consider a New York City street on which 25 small businesses are run, and which suffers from a serious litter problem that detracts customers. It costs $100 annually for each business to keep the front of their store clean. If a store owner decides to keep the front of their store clean, all businesses on the street will have improved sales and profits. Suppose every business on the street will have a $10 increase in annual profit for each business that decides to keep the front of their store clean. Suppose the yearly interest rate is 5%. And suppose all businesses last indefinitely. Are there mutual gains from all players following the Grim Strategy for the repeated game rather than repeating the solution to the one-shot game? And is it a Nash Equilibrium for both players to follow the Grim Strategy? Example 5: Multiple Players

50 BA 445 Lesson B.8 Beneficial Grim Punishment Answer: The essential data of the game includes the effective annual interest rate R = (1+r)/p-1 = 1.05/1-1 = 0.05, where r is the inter-period interest rate and p = 1 is the probability that the game continues from one period to the next. So R is 5%. The one-shot version of the game each year is defined by payoffs from simultaneous moves. On the one hand, in the hypothetical one-shot game, no one should clean since Not Cleaning is a dominate strategy. For any strategies by each of the other 24 stores, the extra payoff to Store Y from cleaning is a $10 increase minus a $100 cost, which makes the payoff $90 less than for Not Cleaning. If players repeat the solution to the one-shot game, then each store earns 0 each period. Example 5: Multiple Players

51 BA 445 Lesson B.8 Beneficial Grim Punishment On the other hand, since the game actually continues indefinitely, each player should consider a Grim Strategy. A Grim Strategy has two components. 1)The Cooperative choices of Cleaning, which is mutually- better than the one-shot choice of Not Cleaning. (If each of the 25 stores cleans, each receives a $10x25 increase minus a $100 cost, which makes the payoff $150 more than in the dominance solution.) 2)The Punishment choice of Not Cleaning, which gives the other player the worst payoff (0) after that player chooses his best response to his punishment. Example 5: Multiple Players

52 BA 445 Lesson B.8 Beneficial Grim Punishment The Grim Strategy is thus, in each year, Cooperate and choose Clean, as long as every other player has Cooperated and chosen Clean in every previous year. But otherwise then you punish by choosing Not Clean in the next year and in every year thereafter --- forever. In particular, if all 25 players follow the Grim Strategy for the repeated game, each year all choose Clean, and so each earns 150. And that is mutually-better than the 0 earned in the dominance solution to the one-shot game. Example 5: Multiple Players

53 BA 445 Lesson B.8 Beneficial Grim Punishment Can each store trust Store Y to follow an agreement to use the Grim Strategy? To answer, suppose all players initially followed the Grim Strategy. Then, in Year X, consider the benefits of Store Y deviating from the Grim Strategy and choosing Not Clean while all other stores continue to choose Clean. In Month X, Store Y gains  Cheat = 240 from Not Clean rather than the  Cooperate = 150 it would have had from following the Grim Strategy. But starting in Year X+1 and continuing forever, every other store punishes Store Y by choosing Not Clean, and so the best Store Y can achieve is  Punish = 0, rather than the  Cooperate = 150 it would have had if he had continued to follow the Grim Strategy. Example 5: Multiple Players

54 BA 445 Lesson B.8 Beneficial Grim Punishment Summing up, each store can trust Store Y to follow an agreement to use the Grim Strategy if the one period gain from cheating  Cheat -  Cooperate = 90 does not compensate for losses  Punish -  Cooperate = 150 starting the next period. Use the formula that $1 starting next period and continuing for each subsequent period is worth $(1/R) today. Each store can trust Store Y to follow an agreement to use the Grim Strategy if 90 < 150/R, which is when the effective interest rate R < 150/90 = 1.66 is less than 166%. Since the effective interest rate is R = (1+r)/p-1 = 1.05/1-1 = 0.05, which is 5%, which is less than 166%, each store can trust Store Y to follow an agreement to use the Grim Strategy. Since the game is symmetric, each store can trust each other store to follow an agreement to use the Grim Strategy, and it is a Nash Equilibrium for all players to follow the Grim Strategy. Example 5: Multiple Players

55 Review Questions BA 445 Lesson B.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.

56 BA 445 Lesson B.8 Beneficial Grim Punishment Review 1: Uncertainty and the Effective Interest Rate

57 BA 445 Lesson B.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%. And suppose each month, with probability 0.1, the product will no longer be sold next month. Are there mutual gains from both players following the Grim Strategy for the repeated game rather than repeating the solution to the one-shot game? And is it a Nash Equilibrium for both players to follow the Grim Strategy? Review 1: Uncertainty and the Effective Interest Rate

58 Review 1: Uncertainty and the Effective Interest Rate BA 445 Lesson B.8 Beneficial Grim Punishment Answer: The essential data of the game includes the effective monthly interest rate R = (1+r)/p-1 = 1.003/0.9-1 = , where r is the inter-period interest rate and p is the probability that the game continues from one period to the next. So R is 11.44%. The one-shot version of the game each month is 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)x100 = $1,000.

59 Review 1: Uncertainty and the Effective Interest Rate BA 445 Lesson B.8 Beneficial Grim Punishment On the one hand, consider the hypothetical case that the game were known with certainty to last only one period. In that one-shot game, each player should choose $85 price since it is the dominate strategy for each player. Thus each player earns 500. If players repeat the solution to the one-shot game, then each player earns 500 each period.

60 Review 1: Uncertainty and the Effective Interest Rate BA 445 Lesson B.8 Beneficial Grim Punishment On the other hand, since the game actually continues indefinitely, each player should consider the Grim Strategy. The Grim Strategy has two components. 1) The Cooperative choice of $95 price, which is mutually-better than the one-shot choice of $85 price. 2) The Punishment choice of $85 price, which gives the other player the worst payoff after that player chooses his best response to (makes the best of) his punishment.

61 Review 1: Uncertainty and the Effective Interest Rate BA 445 Lesson B.8 Beneficial Grim Punishment The Grim Strategy is thus, in each month, Cooperate and choose $95 price, as long as the other player has Cooperated and chosen $95 price in every previous month. But otherwise (if the other player has ever made a choice other than the $95 price), then you punish by choosing $85 price in the next month and in every month thereafter --- forever.

62 Review 1: Uncertainty and the Effective Interest Rate BA 445 Lesson B.8 Beneficial Grim Punishment In particular, if both players follow the Grim Strategy for the repeated game, each month both choose the $95 price, and so both earn 800. And that is mutually-better than the 500 each earns in the solution to the one-shot game.

63 Review 1: Uncertainty and the Effective Interest Rate BA 445 Lesson B.8 Beneficial Grim Punishment Can Sam’s trust Costco to follow an agreement to use the Grim Strategy? To answer, suppose both players initially followed the Grim Strategy. Then, in Month X, consider the benefits of Costco deviating from the Grim Strategy and choosing $85 price while Sam’s continues to choose $95 price. In Month X, Costco gains  Cheat = 1000 from $85 price rather than the  Cooperate = 800 he would have had from following the Grim Strategy.

64 Review 1: Uncertainty and the Effective Interest Rate BA 445 Lesson B.8 Beneficial Grim Punishment But starting in Month X+1 and continuing forever, Sam’s punishes Costco by choosing $85 price, and so the best Costco can achieve is  Punish = 500, rather than the  Cooperate = 800 he would have had if he had continued to follow the Grim Strategy.

65 Review 1: Uncertainty and the Effective Interest Rate BA 445 Lesson B.8 Beneficial Grim Punishment Summing up, Sam’s can trust Costco to follow an agreement to use the Grim Strategy if the one period gain from cheating  Cheat -  Cooperate = 200 does not compensate for losses  Punish -  Cooperate = 300 starting the next period. Use the formula that $1 starting next month and continuing for each subsequent month is worth $(1/R) today. Sam’s can trust Costco to cooperate and follow the Grim Strategy if 200 < 300/R That is, if R < 300/200 = 1.5, which is when the effective monthly interest rate is less than 150%. Since the effective interest rate is R = (1+r)/p-1 = 1.003/0.9-1 = , which is 11.44%, which is less than 150%, Sam’s can trust Costco to follow an agreement to use the Grim Strategy. By symmetry, Costco can trust Sam’s to follow the Grim Strategy, so it is a Nash Equilibrium for both players to follow the Grim Strategy.

66 End of Lesson B.8 BA 445 Managerial Economics BA 445 Lesson B.8 Beneficial Grim Punishment