PS5+Solutions - Strategic Thinking and Game Theory 890 24.3...

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Unformatted text preview: Strategic Thinking and Game Theory 890 24.3 Consider a simultaneous game in which both players choose between the actions "Cooperate’i de- noted by C, and “Defect”, denoted by D. A: Suppose that the payofiis in the game are as follows: If both players play C, each gets a payofi’ of 1 ; if both play D, both players get 0; and if one player plays C and the other plays D, the cooperating player gets a while the defecting player gets [3. (a) Illustrate the payofi’ matrix for this game. Answer: This is illustrated in Table 24.5. Player 2 C 1,1 0.3/3 Player 1 D .6,“ 00 Table 24.5: Potentially a Prisoners’ Dilemma Game (b) What restrictions on a and fl would you have to impose in order for this game to be a Prison- ers’ dilemma? Assume from now on that these restrictions are in fact met. Answer: In order for this to be a Prisoners’ Dilemma game, it must be that D is a dominant strategy for both players. Consider player 2 first: If player 1 plays C, it must be that it is optimal for player 2 to play D — which implies it must be that [3 > 1. Similarly, if player 1 plays D, it must be that D is optimal for player 2 — which implies it must be that a < 0. Player 1’s situation is symmetric — so the same restrictions will work to make D a dominant strategy for player 1. This is therefore a prisoners’ dilemma game if a < 0 and [3 > 1. B: Now consider a repeated version of this game in which players 1 and 2 meet2 times. Suppose you were player 1 in this game, and suppose that you knew that player 2 was a "Tit-for- Tat” player — i.e. a player that does not behave strategically but rather is simply programed to play the Tit-for- Tat strategy. (a) Assuming you do notdiscount the future, would you ever cooperate with this player? Answer: You know that player 2 will play C the first time and will then mimic what you do the first time when you meet the second time. You have four possible pure strategies to play: (C, C), (C, D), (D, C) and (D, D). Assuming you do not discount the future, your payoffs from these will be (1+1), (1 + [3), ([3 + a) and ([3 + 0) respectively. Since fl > 1 and a < 0, we know that (1+fi)>(1+1), (fi+0)>(fi+a) and (1+fi)>(fi+0). (24.1) The first inequality implies that (C, D) strictly dominates (C, C) and you will therefore not play (C, C). The second inequality implies that (D, D) strictly dominates (D, C) and you will therefore not play (D, C). And the third equality implies that (C, D) strictly dominates (D, D) and you will therefore not play (D, D). Thus, you will play (C, D) — and you will therefore cooperate in the first stage. (b) Suppose you discount a dollar in period 2 by 6 where 0 < 6 < 1. Under what condition will you cooperate in this game? Answer: Your payoffs from strategies (C,C), (C, D), (D, C) and (D, D) will now be (1 + 6), (1 + 613), ([3 + 66:) and ([3 + 0) respectively. We can then say unambiguously that (l+§fi)>(1+§) and (fi+0)>(fi+§a), (24.2) which implies that (C, D) strictly dominates (C, C) and (D, D) strictly dominates (D, C). You will therefore definitely never play (C, C) or (D, C). This leaves only (C, D) and (D, D), with payoffs of (1 + 613) and ([3 + 0) respectively. The former is larger than the latter so long as 1 fi<(1—6)’ (24.3) 891 Strategic Thinking and Game Theory which implies you will play (C, D) when this condition holds and (D, D) otherwise. (Both are possible if the equation holds with equality.) Thus, if f} is sufficiently large relative to the discount factor 6, you will still cooperate in the first stage. (c) Suppose instead that the game was repeated 3 rather than 2 times. Would you ever cooperate with this player (assuming again that you don’t discount the future)? (Hint: Use the fact that you should know the best action in period 3 to cut down on the number of possibilities you have to investigate.) Answer: In the third encounter, it will always be best to play D since D is a dominant strategy in the single shot game. Thus, the only question is what you would do in the first and second encounter. We therefore have 4 possible strategies to consider: (C, C, D), (C, D, D), (D, C, D) and (D, D, D). The payoffs (given that the other player plays Tit-for-Tat) for these strategies are: (1 + 1 + [3), (1 + [i + 0), ([3 + a + [3) and ([3 + 0 + 0) respectively Since (1+1+fi)>(1+fi+0)>(fi+0+0), (24.4) we know that (C, C, D) dominates (C, D, D) which dominates (D, D, D) and thus neither (C, D, D) nor (D, D, D) get played. That leaves only (C, C, D) and (D, C, D) to consider, and (C, C, D) dominates (D, C, D) so long as (1 + 1 + [3) > ([3 + a + [3). This simplifies to [3 < 2 — a, (24.5) and as long as this condition holds, you will play (C, C, D). If, however, [i > 2 — a, you will play (D, C, D) (and if f} = 2 — or, either of the two strategies will maximize your payoff.) In either case, you will cooperate at some point, though only under (C, C, D) do both players ever cooperate at the same time. (d) In the repeated game with 3 encounters, what is the intuitive reason why you might play D in the first stage? Answer: If the reward from defecting when your opponent cooperates is sufficiently large relative to the loss one takes when cooperating in the face of the opponent defecting, it makes sense to take advantage of the Tit-for-Tat opponent right away, and then trick him into cooperating again in the last stage. (e) If p layer2 is strategic, would he ever play the "Tit-fo r- Tat” strategy in either of the two repeated games? Answer: No, it would not make sense because player 2 should realize that player 1 will play D in the last encounter. This implies that the last encounter does not matter for what action will be played in the second to last encounter — which again implies player 1 will play D. Thus, the logic of subgame perfection should imply that player 2 will play D always. (f) Suppose that each time the two players meet, they know they will meet again with probability 7' > 0. Explain intuitively why "Tit-for-Tat” can be an equilibrium strategy for both players if y is relatively large (i.e. close to 1) but not if it is relatively small (i.e. close to 0). Answer: If y is close to 1, the probability of meeting again is large. Thus, the short term payoff for player 1 from deviating from "Tit-for-Tat” and playing D without provocation is outweighed by the fact that the opponent will now play D until player 1 unilaterally starts playing C again. Put differently, rather than getting a payoff of 1 by playing C against the "Tit-for-Tat” strategy this period, player 1 can get [i > 1, but it also implies that player 1 will face payoffs of 0 (rather than 1) from now on as both players switch to D, or player 1 will have to incur a payoff of a < 1 (rather than 1) in a future period in order to get his opponent to cooperate again. If the chance of meeting again is sufficiently large, that’s not worth it. If it is sufficiently small, however, it makes sense to graph [3 while you can. Thus, “Tit-for- Tat” can be a best response to “Tit-for-Tat” only if the chance of another encounter is large enough. 903 Strategic Thinking and Game Theory 24.8 Everyday Application: Burning Money, Iterated Dominance and the Battle of the Sexes: Consider again the "Battle of the Sexes” game described in exercise 24.4. Recall that you and your partner have to decide whether to show up at the opera or a football game for your date — with both of you getting a payofi’ of 0 if you show up at difi’erent events and therefore aren’t together: If both of you show up at the opera, you geta payofi’ of 1 0 and your partner gets a payojfof5, with these reversed if you both show up at the football game. A: In this part of the exercise, you will have a chance to test your understanding of some basic build- ing blocks of complete information games whereas in part B we introduce a new concept related to dominant strategies. Neither part requires any material from Section B of the chapter: (a) Suppose your partner works the night shift and you work during the day — and, as a result, you miss each other in the morning as you leave for work just before your partner gets home. Neither of you is reachable at work — and you come straight from work to your date. Unable to consultone another before your date, each of you simply has to individually decide whether to show up at the opera or at the football game. Depict the resulting game in the form of a payofi” matrix. Answer: This is depicted in Table 24.7. Partner Opera Football Opera 10,5 0,0 You Football 0,0 5,10 Table 24.7: Battle of the Sexes (b) In what sense is this an example of a "coordination game”? Answer: It is a coordination game in the sense that both of you would prefer to coordinate to be in the same place even though you disagree which is the better place. As illustrated in the next part, there are therefore two pure strategy Nash equilibria corresponding to the two ways we can get to the same place. (c) What are the pure strategy Nash equilibria of the game. Answer: The pure strategy equilibria are {Opera, Opera} and {Football, Football} — where the first item in each set stands for your strategy and the second stands for your partner’s strategy. (Given you go to the Opera, it is a best response for your partner to go to the Opera and vice verse; similarly with Football.) (d) After missing each other on too many dates, you come up with a clever idea: Before leaving for work in the morning, you can choose to burn $5 on your partner’s nightstand — or you can decide not to. Your partner will observe whether or not you burned $5. So we now have a sequential game where you first decide whether or not to burn $5, and you and your partner then simultaneously have to decide where to show up for your date (after knowing whether or not you burned the $5). What are your four strategies in this new game? Answer: Your four strategies in this game are then (Bur n, Opera), (Bur n, Foo tbal 1), (Don’ t Burn, Opera) and (Don't Burn,Football). (e) What are your partner’s four strategies in this new game (given that your partner may or may not observe the evidence of the burnt money depending on whether or not you chose to burn the money.) Answer: Your partner observes either Burn or Don’ tBurn from you. A strategy is then a complete plan of action — a plan for what to do after observing Burn and a plan for what to do after observing Don’ tBurn. Let 0 stand for Opera and F for Football. With the first action in each parenthesis indicating the plan in the event that Burn has been observed and the second indicating the plan in the event that Don’ tBurn has been observed, your partner’s strategies are then (0,0), (O,F), (F, O) and (F,F). The last strategy, for instance, says “always go to the football game regardless of what happened” and the first strategy says “always go to the opera”. But the second strategy says “go to the opera if there is burnt Strategic Thinking and Game Theory 904 money on the night stand but go to the football game if there isn’t”, and the third strategy says “go to the football game if there is burnt money and to the opera if there isn’t.” (f) Illustrate the payofi’ matrix of the new game assuming that the original payofis were denom- inated in dollars. What are the pure strategy Nash Equilibria? Answer: The payoff matrix is given in Table 24.8. Partner (0, 0) (OF) (F, O) (F, F) (Burn, Opera) 5,5 5,5 -5,0 -5,0 (Burn,F00tball) -5,0 -5,0 0,10 0,10 Y 0“ (D0n’tBurn,Opera) 10,5 0,0 10,5 0,0 (D0n’tBurn,F00tball) 0,0 5,10 0,0 5,10 Table 24.8: Battle of the Sexes with Money Burning B: In the text, we defined a dominant strategy as a strategy under which a player does better no matter what his opponent does than he does under any other strategy he could play. Consider now a weaker version of th is: We will say thata strategy B is weakly dominated by a strategy A for a player if the player does at least as well playingA as he would playingB regardless of what the opponent does. (a) Are there any weakly dominated strategies for you in the payofi” matrix you derived in AO‘)? Are there any such weakly dominated strategies for your partner? Answer: For you, the strategy (Bur n, Football) is weakly dominated by the strategy (Don' t Burn, Opera) — because the first number in each cell of the row (Don't Burn,0pera) is greater than or equal to the first number in the cell immediately above it. There are no weakly dominated strategies foryour partner. (b) It seems reasonable that neither of you expects the other to play a weakly dominated strategy. So take your payofi’ matrix and strike out all weakly dominated strategies. The game you are left with is called a reduced game. Are there any strategies for either you or your partner that are weakly dominated in this reduced game? If so, strike them out and derive an even more reduced game. Keep doing this until you can do it no more — what are you left with in the end? Answer: After eliminating (Bur n, Football), we are left with the game in Table 24.9. In this game, there are no weakly dominated strategies for you. But (F, O) is weakly dominated for your partner — because he would do at least as well and sometimes better by playing (0, O). Partner (0,0) (O,F) (E0) (RF) (Burn, Opera) 5,5 5,5 -5,0 -5,0 You (Don’t Burn, Opera) 10,5 0,0 10,5 0,0 (Don’t Burn,F00tball) 0,0 5,10 0,0 5,10 Table 24.9: Reduced Game 1 Eliminating (F, 0) from the matrix, we get the game in Table 24.10. In this new game, (F, F) is weakly dominated by (O, F) for your partner. Eliminating (F, F), we get a further reduced game depicted in Table 24.1 1. Nowyour strategy (Don't Burn, Football) is weakly dominated by (Burn, Opera). Eliminating (Don' t Bur n, Foo tbal l), we get a further reduced game depicted in Table 24.12. But now (0, F) is weakly dominated by (O, O) for your partner. 905 Strategic Thinking and Game Theory Partner (0,0) (OF) (RF) (Burn, Opera) 5,5 5,5 -5,0 You (Don’ t Burn, Opera) 10,5 0,0 0,0 (D0n’tBurn,F00tball) 0,0 5,10 5,10 Table 24.10: Reduced Game 2 Partner (0, 0) (0, F) (Burn, Opera) 5,5 5,5 You (Don’t Burn, Opera) 10,5 0,0 (D0n’tBurn,F00tball) 0,0 5,10 Table 24.1 1: Reduced Game 3 Partner (0, 0) (0, F) You (Burn, Opera) 5,5 5,5 (Don’t Burn, Opera) Table 24.12: Reduced Game 4 10,5 0,0 Eliminating (O, F), we get the further reduced game in Table 24.13. And now (Bur n, Opera) is strictly dominated by (Don' t Bur n, Opera). Partner (0, O) (Burn, Opera) 5,5 You (Don’t Burn, Opera) 10,5 Table 24.13: Reduced Game 5 Eliminating (Don' t Burn, Opera), we are left with the trivial game in Table 24.14. (c) After repeatedly eliminating weakly dominated strategies, you should have ended up with a single strategy left for each player: Are these strategies an equilibrium in the game fi'omAO‘) that you started with? Answer: Yes, they are. But there are lots of other Nash equilibria in the original game as well. A list of all the pure strategy Nash equilibria is as follows: (1) {(Don' t Burn, Opera), (0, 0)}; (2) {(Don't Burn, Opera), (F, 0)}; (3) {(Don'tBurn,Football), (O,F)}; (4) {(Don't Burn,Football), (F,F)}; and (1) {(Burn, Opera), (O,F)}. (d) Selectingamong multiple Nash equilibria to a game by repeatedly getting rid of weakly dom- inated strategies is known as applying the idea of iterative dominance. Consider the initial game fi'omA(a) (before we introduced the possibility ofyou burning money). Would applying the same idea of iterative dominance narrow the set ofNash equilibria in that game? Answer: No, there are no weakly dominated strategies in the initial game (without money burning). Strategic Thinking and Game Theory 906 Partner (0,0) You (D0n’tBurn,Opera) | 10,5 Table 24.14: Reduced Game 6 (e) True or False: By introducing an action that ends up not being used, you have made it more likely that you and your partner will end up at the opera. Answer: If we believe that eliminating Nash equilibria through iterative dominance gives us the right prediction of what will be played, then the statement is true. The action — money burning that was introduced does not get used in the Nash equilibrium that survives iterated dominance. But it’s introduction allowed the application of iterated dominance. In the context of this game, by introducing the idea of burning money, you are able to get to your most preferred Nash Equilibrium (that results in both you and your partner getting to the opera) — without actually having to burn the money. (Strange, isn’t it?) Externalities in Competitive Markets 798 21.1 1 Policy Application: Pollution that increases firm casts — Policy Solutions: This exercise continues to build on exercises 21.9 and 21.1 0. Assume the same basic setup affirms located ara und a lake producing pollution that causes the fixed casts of all firms to increase. A: Continue to assume that each output unit that is produced results in an increase of fixed costs of 6 far all firms in the industry. (a) Begin by illustrating the market demand and long run industry supply curves, labeling the market equilibrium as A. Answer: This is illustrated in panel (a) of Graph 21.11 where SLR is the long run industry supply curve and X * is the market outcome. Graph 21.11: Policy on the Lake (b) Next, without drawing any additional curves, indicate the pointB in your graph where the market would be producing if firms were taking the fill cost of the pollution they emit into account. Answer: This is also done in panel (a) of Graph 21.11. (The point would arise from the intersection of demand with SB — the long run industry supply curve that would emerge if all firms accounted for the full cost of the pollution they produce (as Barney does in the previous exercise).) (c) Illustrate the Pigouvian tax that would be necessary to get the market to move to equilibrium B. Answer: This, too, is illustrated in panel (a) of Graph 21.11 where the tax is t — resulting in output level X “P t, price pd for consumers and price p s for producers. (d) SupposeN* is the number affirms in the industry in the market outcome, N “P t is the optimal number affirms and 6 continues to be as defined throughout. What does the government have to know in order to implement this Pigouvian tax? Ls what the government needs to know easily observable prior to the tax? Answer: Each unit of output results in an increase in fixed costs by 6 for each of the firms in the industry. For the tax to be equal to the marginal social cost of pollution, it must therefore be equal to 6N for the N that will exist in the optimal outcome. The optimal Pigouvian tax is then t = 6 N on t — implying the government must know how many firms should exist. Simply observing how many firms N * do exist is not enough — implying that it is not sufficient for the government to know what is easily observable in the absence of the tax. (e) Where in your graph does consumer surplus before and after the tax lie? Answer: The consumer surplus before the tax is illustrated as area (a + b + c) in panel (a) of Graph 21.11, while the consumer surplus after the tax is illustrated by area (a). 799 Externalities in Competitive Markets (f) Keeping in mind what you concluded in exercise 21.9, has (long run) producer surplus — or long run industry profit— changed as a result of the tax? Answer: No — long run industry profit is zero before and after the tax because of entry and exit. Our usual practice of measuring producer surplus on the supply curve below the price does not apply here because the reason for the upward sloping industry supply curve is not that some firms are better at producing outcome than others — all firms here are equal, and the upward slope comes from the externality. Thus, all firms make zero profit before and after because of entry and exit. (g) True or False: The pollution cost under the Pigouvian tax is, in this example, equal to the tax revenue that is raised under the tax. Answer: This is true. The total pollution cost when the industry settles at B is equal to 6 N 0” tX 0” t because the pollution damage is 6 N “P t for each unit of output produced in the industry. The tax revenue is thpt — and t = 6 N “P t. (h) Is there additional pollution damage under the marketoutcome ( in the absence of the tax)? Answer: Yes, there is additional pollution damage. In the market outcome, the total pol- lution damage is 6 N * X * — and both the number of firms and the overall output level are higher at A than they are at B. One way to approximate this additional damage is through area (c+ e+ f) in panel (a) of Graph 21.11 — but this is actually an underestimate of the ad- ditional damage because the marginal social cost with N * firms is 6 N * while the marginal cost with N on t firms is 6N 0” t; i.e. the marginal social damage for the first X on t units of output is larger when there are N * firms — and thus larger than (17 + d), and the additional damage for the units between X “P t is also larger than indicated by the areas for the same reason. The externality cost under the market outcome (in the absence of taxes) is therefore larger than the area (17+ c+ d + e+f). (i) Is there a deadweight loss from not using the tax? Answer: Yes, there is. We concluded above that consumer surplus shrinks from (a + b + c) to (a) under the tax, the pollution cost decreases from something larger than (17 + c+ d + e+ f) to (b + d) and the tax revenue under the tax is (b + d). Given that industry profits are zero before and after, we get total surplus increasing from something less than (a — d — e — f) to (a) — implying a deadweight loss larger than (d + e + f) in the absence of the tax. (j) Suppose the government instead wanted to impose a cap-and-trade system on this lake — with pollution permits that allow a producer to produce the amount of pollution necessary to produceone unit of output. What is the “cap” on pollution permits the government would want to impose to achieve the efficient outcome? What would be the rental rate of such a permit when it is traded? Answer: The government would then want to set the overall pollution permits (or vouchers) to X “P t — creating a market in pollution voucher that would settle at an equilibrium rental rate of r* = t (where t is the optimal Pigouvian tax). This voucher market is illustrated in panel (b) of Graph 21.1 1. (k) What would the governmenthave to know to set the optimal cap on the number of pollution permits? Answer: The government would have to know X “P t — the optimal output level in the in- dustry. This appears different than the information required for the creation of an optimal Pigouvian tax— which was 6 and N “P t. In the end, however, the government would have to know 6 and N * to really estimate the optimal level of industry output — so there is not that much difference in how much the government needs to know to set optimal tax or voucher policies in this example — and in the real world it probably never has enough information to really conclude what is “optimal”. The advantage of tradable pollution permits does not ac- tually show up in this example: Conditional on deciding how much pollution is acceptable generally, tradable vouchers have the advantage that firms that might differ in their ability to reduce pollution will self select — with those for whom it is difficult buying permits and those for whom it is easy simply reducing pollution instead. B: Continue with the functional forms for costs and demand as given in exercises 21.9 and 21.10. Suppose, as you did in parts of the previous exercises, that f} = 1, 6 = 0.1 and A = 10, 580 throughout this exercise. Externalities in Competitive Markets 800 (a) If you have not already done so in part (f) of exercise 21.9, determine the Pigouvian tax that would cause producers to behave the way the social planner would wish for them to behave. What price will consumers end up paying and what price will firms end up keeping under this tax? Answer: As we calculated in the previous exercise, the Pigouvian tax is t = 18.78 — with consumer prices rising from $46 to $56.34 and producer price falling from $46 to $37.56. (b) Calculate (for our numerical example) consumer surplus with and without the Pigouvian tax. (Skip this ifyou are not comfortable with integral calculus.) Why is (long run) producer surplus — or long run profit in the industry— unchanged by the tax? Answer: Answer: Since we know from the text note in the previous exercise that the demand curve is one that can arise from a single representative consumer for whom x is quasilinear, we know we can treat the demand curve as a marginal willingness to pay curve — and the demand function xd = (10580/p)2 as a compensated demand function. This implies that consumer surplus is 0° 10580 2 105802 f — dp= . (21.85) p P P Evaluating this at the original price of 46 and the after tax price of 56.34, we get csbefore = 2,433,400 and CSafte, = 1,986,802. (21.86) (V\T1thout rounding error, CS a f re, = 1, 986, 863.) Producer surplus — or long run profit — remains zero in both cases as discussed in partA(f) of this exercise. (c) Determine the total cost of pollution before and after the tax is imposed. Answer: In the previous exercise, we calculated the number of firms to be N * = 2,300 and the industry output to be X * = 52,900 in the absence of taxes. The total cost of pollution before the tax is imposed is 6N* X* = 0.01 (2300) (52900) = 662, 288. (21.87) We also calculated the optimal number of firms to be N “P t = 1, 878 and the optimal industry output to be X “P t = 52,267. The total cost after the optimal tax is imposed is therefore §N°Ptx°Pt = 0010878) (35267) = 662,314. (21.88) (VVlthout rounding error, the latter would be 662,288 instead.) (d) Determine tax revenue from the Pigouvian tax. Answer: The tax rate we calculated is t = 18.78 per unit of output, and the after-tax industry output level was calculated as X “P t = 52,267, giving us a tax revenue of TR = 18.78(53, 267) = 662,314. (21.89) (In the absence of rounding error, this would be 662,288.) (e) What is the total surplus before and after the tax — and how much deadweight loss does this imply in the absence of the tax? Answer: Subtracting the pollution cost from the consumer surplus prior to the tax, we get total surplus before the tax equal to $1,216,700. After the tax, the tax revenue is exactly offset by the pollution cost — leaving us with total surplus equal to consumer surplus. This is $1,986,802 (or, without rounding error, $1,986,863). Subtracting the surplus before from the surplus after, we get the deadweight loss (from not imposing the Pigouvian tax) equal to $770,102 (or $770,163 in the absence of rounding error). (f) Suppose next that the government instead creates a tradable pollution permits — or voucher — system in which one voucher allows a firm to produce the amount of pollution that gets emitted from the production of 1 unit of output. Derive the demand curve for such vouchers. 801 Externalities in Competitive Markets Answer: The pollution level allowed by each voucher is set such that 1 voucher v allows production of 1 unit of x. We can thus replace x with v — and the demand will be the difference between the market demand curve for x (p = A/x0'5) and the long run industry supply curve for x (p = 2(fi6x)°-5). The rental rate r for a voucher is therefore _ A 1/2 _ “L580 1/2 7'0!) — W —2(fi§l/) — vllz —0.2x . (21.90) (g) What is the optimal level of vouchers for the government to sell — and what will he the rental rate of the vouchers if the government does this? Answer: The optimal level of vouchers is the level that results in the optimal level of industry output (which we calculated to be 35,267). Thus, v0”t = 35,267. Plugging this into the demand curve for vouchers, we get a rental rate of 10, 580 (35, 267)“2 Note that this is exactly equal to the optimal per-unit Pigouvian tax because we defined the amount of pollution permitted by one voucher to be sufficient to produce one unit of output. r(35,267) = — 0.2(35,267)1l2 = 18.78. (21.91) ...
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