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Deep Thought

Deep Thought.

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Deep Thought

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  1. Deep Thought 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.) BA 445 Lesson B.8 Beneficial Grim Punishment

  2. Readings • Readings • Baye “Repeated Games” (see the index) • Dixit Chapter 11 BA 445 Lesson B.8 Beneficial Grim Punishment

  3. Overview • Overview BA 445 Lesson B.8 Beneficial Grim Punishment

  4. Overview BA 445 Lesson B.8 Beneficial Grim Punishment

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

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

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

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

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

  10. Example 1: 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 playerearns 10 each period. BA 445 Lesson B.8 Beneficial Grim Punishment

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

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

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

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

  15. Example 1: 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 = -10starting the next month. That answer depends on the interest rate r between months. BA 445 Lesson B.8 Beneficial Grim Punishment

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

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

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

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

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

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

  22. Example 2: Uncertainty and the Effective Interest Rate 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 = 0.25625, which is 25.625%, 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. BA 445 Lesson B.8 Beneficial Grim Punishment

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

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

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

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

  27. Example 3: Non-Symmetric Games 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 Segaearn 180 and 48 each period. BA 445 Lesson B.8 Beneficial Grim Punishment

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

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

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

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

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

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

  34. Example 3: Non-Symmetric Games Recomputing, Sega can trust Nintendo to follow an agreement to use the Grim Strategy if the one period gain from cheating Cheat - Cooperate = 210-200 = 10 does not compensate for losses Punish - Cooperate = 200-180 = 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. BA 445 Lesson B.8 Beneficial Grim Punishment

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

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

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

  38. Example 4: Multiple Actions 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 – (QI + QA), where output quantities QI and QA are the thousands of processors produced monthly by Intel and AMD. Suppose unit costs of production are cI= 2 and cA= 2 for both Intel and AMD. Suppose Intel and AMD consider any quantities QI = 4 or 5 or 6 or 7, and QA = 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? BA 445 Lesson B.8 Beneficial Grim Punishment

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

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

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

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

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

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

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

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

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

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

  49. Example 5: Multiple Players • 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? BA 445 Lesson B.8 Beneficial Grim Punishment

  50. Example 5: Multiple Players 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 0each period. BA 445 Lesson B.8 Beneficial Grim Punishment

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