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# INTRO GAME THEORY ECON 171

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This 14 page Class Notes was uploaded by Arno Leuschke on Thursday October 22, 2015. The Class Notes belongs to ECON 171 at University of California Santa Barbara taught by Z. Grossman in Fall. Since its upload, it has received 98 views. For similar materials see /class/227151/econ-171-university-of-california-santa-barbara in Economcs at University of California Santa Barbara.

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Date Created: 10/22/15

Econ 171 Spring 2010 Problem Set 2 Due May 6 Important band in only the two star problems The notation ab denotes problem number bfrom Chapter a in Watson Problem 1 Figure 1 An extensive game In the game in Figure 17 solve for the backwards induction strategies and outcomes when the four missing payoffs are lled in with each possible permutation of the four numbers 727717172 Problem 2 t7j k7l Figure 2 Another extensive game a List the strategies available to each player in the game shown in Figure 2 b Construct payoffs a7 b7e7 d7 e7 f7 97 h7t7j7 k7 l such that the terminal node r37 with payoffs k7 l7 will be the unique backwards induction outcome c Now construct a different set of payoffs such that the unique backwards induction outcome is r37 but the backwards induction strategies are different than in the game you constructed in the previous part gt1lt 96 96 96 96 96 96 Problem 3 Exercises from Watson that 1 consider to be one star problems 21 287 31 3473673771517153 Problem 4 Consider the game shown in Figure 27 with the payoffs a7b7c7d7e7f7g7h7 7j7k7l 27271737371737174767277 a What is the backwards induction strategy for each player b What is the backwards induction outcome c Two strategies can arguably be considered equivalen 77 ifthey designate the same behavior at all decision nodes that the player will reach with positive probability given her own previous choices as speci ed by both of the two strategies For each player7 state which if any strategies are equivalent to others Problem 5 Watson 152 Problem 6 Armies 1 and 2 are ghting over an island initially held by a battalion of army 2 Both armies have K battalions Whenever the island is occupied by one army the opposing army can launch an attack The outcome of the attack is that the occupying battalion and one of the attacking battalions are destroyed the attacking army wins and7 so long as it has battalions left7 occupies the island with one battalion Note7 that after K battles neither army occupies the island because no army has any battalions left The commander of each army is interested in maximizing the number of surviving bat talions but also regards the occupation of the island as worth more than one battalion but less than two Predict the winner using SPNE for all K 2 2 no formal proof required Explain the intuition for the result Hint try this for K 27 37 and 4 rst Problem 7 Watson 196 Problem 8 Watson 222 Problem 9 The trust game is a two player game with three periods Player 1 starts off with 10 Dollars She can send an amount 0 S x S 10 to player 2 The experimenter triples the sent amount such that player 2 receives 3 Player 2 can then send an amount 0 S y S 3m to player 1 a Draw a diagram of the extensive form of this game b Find all the SPNE in the trust game c Why is this game used to measure trust and trustworthiness Do you think it provides reasonable measures How would you interpret reciprocity in this game How would you detect it What is trust Come up with a operational de nition Problem 10 Consider a basic Cournot model of duopoly Two rms compete in quantities7 facing a linear demand curve p a 7 Q a 7 ql 7 qz and marginal cost 0 lt 13 Firm 17s board is looking for a new manager to run the company Three managers apply for the job TG Timid7 Bobby Prudent7 and QA Bombastic TG Timid is convinced that the cost of producing an airplane is double as high as it actually is and makes decisions accordingly Bobby Prudent has exactly correct beliefs QA Bombastic believes that the cost of production is only half of the actual cost Firm 2s managers know exactly the beliefs of a manager whom Firm 1 decides to hire a Which manage should Firm 1 choose to maximize pro ts b Suppose Firm 1 is a monopolist no Firm 2 What is the pro t maximizing quantity c Explain in words why Firm 1 might or might not want to hire a different type of manager depending on whether it faces a simple decision problem monopoly or a game duopoly with Firm 2 Econ 171 Spring 2010 Problem Set 2 Solutions to the twostar problems In some cases I ve provided more explanation than was asked of you Problem 4 96 96 Consider the game shown in the gure7 with the payoffs a7b707d7 e7f7g7h7i7j7k7 l 27271737371737174767277 a What is the backwards induction strategy for each player Solution The backwards induction strategy pro le is L1L27r1l2r3 b What is the backwards induction outcome Solution The backwards induction outcome is Lhrl with payoffs 27 2 c Two strategies can arguably be considered equivalent77 ifthey designate the same behavior at all decision nodes that the player will reach with positive probability given her own previous choices as speci ed by both of the two strategies For each player7 state which if any strategies are equivalent to others Solution For Player 17 R1112 and Rle are equivalent For Player 27 Tnglg and rlrgrg are equivalent and rllgrg and rllglg are equivalent Problem 5 Watson 152 Solution a The Nash equilibria are AC7 ZX7 BC7 VIY7 AD7 ZY7 BC7 VVX7 BD Of these7 only lVY7 AC and ZX7 B0 are subgame perfect b The Nash equilibria are UE7BD7 DE7BC397 UF7BD7 DE7AC397 but only UE7BD and DE7 B0 are subgame perfect Problem 6 Armies 1 and 2 are ghting over an island initially held by a battalion of army 2 Both armies have K battalions Whenever the island is occupied by one army the opposing army can launch an attack The outcome of the attack is that the occupying battalion and one of the attacking battalions are destroyed the attacking army wins and7 so long as it has battalions left7 occupies the island with one battalion Note7 that after K battles neither army occupies the island because no army has any battalions left The commander of each army is interested in maximizing the number of surviving bat talions but also regards the occupation of the island as worth more than one battalion but less than two Predict the winner using SPNE for all K 2 2 no formal proof required Explain the intuition for the result Hint try this for K 27 37 and 4 rst 96 96 Solution When K is even army 1 will attack and win the island When K is odd army 1 will not attack and army 2 will retain possession of the island To see why this result makes sense consider what happens for speci c small values of K 0 K 1 Army 1 does not attack because that would leave it with zero battalions Army 2 retains the island 0 K 2 Army 1 will attack and occupy the island Will army 2 try to retake the island Now army 2 nds itself in the same position as army 1 was when K 1 ie the previous subgame so it will not attack Thus army 1 wins the island K 3 If army 1 attacks it will occupy the island and army 2 will be in a subgame identical to the one described above This means army 2 would attack retake the island and win Knowing this and not wanting to needlessly lose battalions army 1 does not attack and army 2 wins 0 K 41f army 1 attacks it will occupy the island and army 2 will be in a subgame identical to the previous one and therefore not attack So army 1 attacks 2 does not attack back and army 1 wins Both armies know that whomever has possession when K 1 will win the island because the other army will not be able to re occupy the island after attacking Back wards inducting a non occupying army will only attack if it has an even number of armies Problem 7 Watson 196 Solution For simplicity let the offer always be stated in terms of the amount player 2 is to receive Let x be the offer in period 1 y be the offer in period 2 and 2 be the offer in period 3 1f period 3 is reached player 2 will offer 2 1 and player 1 will accept yielding a payoff of 1 for player 2 Thus in period 2 player 2 will accept any offer that gives her at least 6 Knowing this in period 2 if it is reached player 1 will offer just enough to make player 2 indifferent between accepting and rejecting to receive 1 in the next period Thus y 6 Finally in period 1 player 2 will accept any offer that gives her at least z 62 because she knows that she will only be offered y delta in period 2 So player 1 will offer z in the rst period that is just high enough to make player 2 indifferent between accepting and rejecting to receive 6 in the second period Thus player 1 offers z 62 and it is accepted Problem 7 Watson 222 Note You did not have to hand in this problem Econ 171 Spring 2010 Problem Set 3 Due Tuesday June 1 Important hand in only the two star problems There are no one star problems on this problem set The notation ab denotes problem number bfrom Chapter 1 in Watson Problem 1 Consider a two player Bayesian game where both players are not sure whether they are playing the game X or game Y7 and they both think that the two games are equally likely a This game has a unique Bayesian Nash equilibrium7 which involves only pure strategies What is it Hint start by looking for Player 27s best response to each of Player 17s actions b Now consider a variant of this game in which Player 2 knows which game is being played but Player 1 still does not This game also has a unique Bayesian Nash equilibrium What is it Hint Player 27s strategy must specify what she chooses in the case that the game is X and in the case that it is Y c Compare Player 27s payoff in from the rst two parts ofthis problem What seems strange about this Problem 2 Watson 252 Problem 3 Watson 266 Problem 4 Firm 1 is considering taking over Firm 2 It does not know Firm 27s current value7 but believes that is equally likely to be any dollar amount from 0 to 100 If Firm 1 takes over rm 27 it will be worth 50 more than its current value7 which Firm 2 knows to be x Firm 1 can bid any amount y to take over Firm 2 and Firm 2 can accept or reject this offer lf 2 accepts 17s offer7 17s payoff is x 7 y and 27s payoff is y lf 2 rejects 17s offer7 17s payoff is 0 and 27s payoff is x Find a Nash equilibrium of this game What does this situation have to do with dating and shopping for used cars Problem 5 Watson 276 Problem 4 Find all perfect Bayesian equilibria of the game in Figure 1 Figure 1 A variant of the entry game Problem 6 Consider two Cournot competitors facing demand PQ 1 7 Q with PQ 0 if Q gt 17 and each having unit cost 0 They both know that Firm 17s cost is 07 but while Firm 2 knows its own cost is 07 Firm 1 thinks that Firm 27s cost is equally likely to be CL and CH7 with CL lt CH Find a Bayesian Nash equilibrium of this game when CL and CH are close together close enough so that all outputs are positive in equilibrium Compare this equilibrium with the Bayesian Nash equilibrium of the game in which Firm 1 knows that 27s cost is CL and of the game in which 1 knows that 27s cost is CH Important problem set Econ 171 Spring 2010 Problem Set 3 Solutions Due Tuesday June 1 hand in only the two star problems There are no one star problems on this The notation ab denotes problem number bfrom Chapter a in Watson Problem 1 Consider a two player Bayesian game where both players are not sure whether they are playing the game X or game Y7 and they both think that the two games are equally likely a This game has a unique Bayesian Nash equilibrium7 which involves only pure A O V strategies What is it Hint start by looking for Player 27s best response to each of Player 1 s actions Solution The unique BNE is B7 L7 yielding each player a payoff of 2 Player 17s payoffs do not depend upon which version of the game is actually being played Her best response to L is to play B and T is a best response to M or R If 1 plays T7 then both M and R give Player 2 an expected utility of 157 so her best response is L Similarly7 Player 27s best response to B is L So in eppeeted utility7 L is a dominant strategy for 27 and 1 best responds with B Now consider a variant of this game in which Player 2 knows which game is being played but Player 1 still does not This game also has a unique Bayesian Nash equilibrium What is it Hint Player 27s strategy must specify what she chooses in the case that the game is X and in the case that it is Y Solution The unique BNE is T7 R7 Player 2 now knows the game that is being played7 and each type of Player 2 has a dominant strategy R for the type that knows the game is X and M for the type that knows that the game is Y Since there is no chance that 2 will play L7 Player 17s unique best response is to play T Compare Player 27s payoff in from the rst two parts ofthis problem What seems strange about this Problem 2 96 96 Solution In the rst part7 each player earned a payoff of 2 In the second part7 Player 2 actually has more information about what game is actually being played and ends up only earning 3 in either case At rst it may seem a bit strange that 2 is worse off knowing the game than she is not knowing it This happens because the uninformed Player 2 uses L as a compromise When she knows the game7 she will choose either M or R7 tailoring her action for t the game What hurts her is the fact that 1 knows that she knows this information Watson 252 Solution The probability of a successful project is p7 so we revise the incentive com patibility constraint to be pw bi 1 1ipw 1 2 w while the participation constraint becomes pw b 71 17pw 71 1 In the optimal contract7 these constraints bind7 so we need pwb1 1 pr 1 1 w which implies that b p lD This is decreasing in p7 so the more likely the project is to succeed7 the lower the bonus in equilibrium This suggests that higher powered incentives are required for riskier projects Problem 3 Watson 266 Solution LL 7 U Problem 4 Firm 1 is considering taking over Firm 2 It does not know Firm 27s current value7 but believes that is equally likely to be any dollar amount from 0 to 100 If Firm 1 takes over rm 27 it will be worth 50 more than its current value7 which Firm 2 knows to be x Firm 1 can bid any amount y to take over Firm 2 and Firm 2 can accept or reject this offer If 2 accepts 17s offer7 17s payoff is x 7 y and 27s payoff is y 1f 2 rejects 17s offer7 17s payoff is 0 and 27s payoff is x Find a Nash equilibrium of this game What does this situation have to do with dating and shopping for used cars Solution Firm 1 will bid zero and Firm 2 will accept any offer greater than or equal to x Firm 27s simply accepts offers that are higher than the rms own value Firm 1 knows that the value of a rm that accepts an offer of y is anywhere from 0 to y Thus7 the expected value of a rm that accepts is 1127 which means that Firm 17s expected payoff as a function of its bid is 7 y fig In other words7 it expects to lose money on any positive bid it makes lt7s best response7 then is to bid zero Just like in dating and the used car market7 this market is plagued by adverse selection7 which in this case leads the market to unravel completely Econ 171 Spring 2010 Problem Set 1 Due April 13 Important hand in only the two star problems Read the syllabus for the meaning of quotone star quottwo star etc The notation ab denotes problem number bfrom Chapter a in Watson Problem 1 a Create a large number of 2 gtlt 2 2 gtlt 3 and 3 gtlt 2 normal form games or use any of the ones provided in the rst 133 pages of Watson and determine what actions are rationalizable and list all Nash Equilibria Once we7ve learned about them to focus on mixed strategy Nash equilibria try the games that appear in Chapter 11 This may not start out as a one star problem but you should keep doing it until it becomes a one star problem Really do part a Now try to do the same with a few larger games eg 3 gtlt 3 3 gtlt 4 etc These often take a while to solve and will never be true one star problems but try to get comfortable with the process AA 0 C7 VV Problem 2 To follow up on Problem 1 here is a list of the exercises from Watson that I consider to be one star problems 32 35 36 41 42 43 44 45 61 62 63 64 65 67 68 71 91ab 92 93 Problem 3 Find the rationalizable strategies for this game For each iteration list the eliminated strategy and which strategy dominates it Problem 4 Find the pure strategy Nash equilibria for these two games lf before the PS is due we have covered mixed strategy Nash equilibrium nd all Nash equilibria ie including mixed strategy equilibria 96 96 Problem 5 Watson 97 96 96 Problem 6 Watson 99 96 96 Problem 7 Watson 113 You are only required to do this problem is we have covered mixed strategy Nash equilibrium before the problem set is due If we have not yet done so7 then you are encouraged to do try this problem after you turn in the problem set Problem 8 Watson 911ab and 119 Problem 9 Consider the Money Dilemma that we played on the second day of class C D O D a Why did 1 just call it Money Dilemma and not Prisoners7 Dilemma b Imagine that the people playing this game care not just about their own outcome7 but the others persons In particular7 Uselfself7 other 7 a 39 self 0 minself7 otherl7 where a E 01 is a xed parameter that is common knowledge to the players Redraw the situation as a standard 2 X 2 bi matrix game7 with utility entries that might depend on a do and dont depend on a and try to relate this to your interpretation of these preferences Problem 10 Consider the guess 23 of the average game from class a Assume that rationality is not common knowledge Suppose all players are ratio nal7 but believe that their opponents choose a number randomly7 with all numbers from 0 to 100 equally likely What would people choose b Assume that the players are rational and know that their opponents are ratio nal7 but for some reason believe that their opponents think that everyone else is randomizing What is the solution of the game now More generally7 what if all students iterate k times c Now assume an alternative model in which one third of the people are sophis A8 D oated and two thirds are naive Naive agents assume that their opponents are randomizing Sophisticated people realize that one third of their opponents are also sophisticated and apply iterated elimination of dominated strategies Which strategies will survive In general7 which strategies will survive if the share of sophisticated people is a Suppose the following bids were submitted by a group of people playing this game 177 127 257 227 247 177 187 307 07 607 337 237 30 In your opinion7 which of the preceding models best describes the data Econ 171 Spring 2010 Problem Set 1 Solutions to the twostar Problems In some cases I ve provided more explanation than was asked of you Problem 3 Find the rationalizable strategies for this game For each iteration7 list the eliminated strategy and which strategy dominates it The unique rationalizable strategy is 07 Z The logic is as follows 0 All actions are rational for Player 1 A is optimal if she believes 2 will play X with certainty7 B is optimal if she believes Y7 and C is optimal if she believes W will be played 0 Only lV7 X and Z are rational for Player 2 because Y is dominated by Z Y is eliminated o If Player 1 knows that 2 is rational7 she will never play B She knows Y wont be played A is optimal if she thinks X will be played and C is optimal given W ls B ever optimal Well7 if 1 believes that PrZ gt 14 then she prefers C to B and if she believes that PrZ lt 12 then she prefers A to B This means that B is never optimal7 which in this type of game tells us that it is dominated Any strategy that places probability q on A and 17 q on C with q E dominates B o If Player 2 only knows that 1 is rational7 then she cant be certain that 1 will eliminate any action7 but if she knows that 1 knows that she is rational7 then she knows 1 wont play B This means that X is dominated by W and can be eliminated o If Player 1 knows this7 then she knows that 2 will only play W or Z7 in which case A is dominated by O and is eliminated 0 Knowing this7 W is dominated by Z and is eliminated Problem 4 Find all Nash equilibria for these two games a There is no pure strategy Nash Equilibrium Any mixed strategy NE has to involve Player 2 mixing7 because otherwise Player 1 would not mix either Let pA p3 and p0 be the probabilities that 1 plays A7 B7 and C respectively Let q and 1 7 g be the probabilities with which 2 plays L and R7 respectively For Player 2 to be willing to mix7 it is necessary that p3 p0 because 2 strictly prefers L if C is more likely and prefers R if B is more likely 1 Case 1 Suppose p3 p0 gt 0 Player 1 will only be willing to mix B and C if L and R are equally likely q 12 but this would mean that A is preferred to both There is no NE in this case Case 2 Suppose p3 p0 0 A is preferred to both E and C for 13 3 q S 23 In this case pA 1 to which all g 6 01 is a best response for 2 Thus Aq is a NE for all q E g These are all of the NE of the game b Again there is no pure strategy NE One way to begin is to note that A is not rationalizable It is dominated by an even mix between B and G Let p be the probability with which Player 1 plays B and let q be the probability that 2 plays L For mixing to be sustained in equilibrium each player7s strategy must make the other indifferent between the two pure strategies in that persons mix Player 2 is indifferent when 21 7p 3p or p 25 Player 1 is indifferent between B and 0 when q 12 So the unique NE involves 1 playing 0 2535 and 2 playing 1212 Problem 5 Watson 97 That BX is a Nash equilibrium does not put any restrictions on x A Z is ef cient as long as x 2 4 For Y to be a best response to 91 i6 we need u201Y 3 2 22 1 u201 Z which is satis ed for x S 4 Thus for all three statements to be true it must be the case that z 4 Problem 6 Watson 99 a The Nash equilibria occur where the curves intersect ie 21 522 and 33 b The set of rationalizable strategy pro les is 23 gtlt 13 Explanation not re quired Looking at BB we see that no matter Player 17s beliefs 0 1 and 4 5 are never best responses and thus not rationalizable For Player 2 anything in the interval 4 5 is never a best response Player 1 knowing Player 2 is rational knows that 52 g 4 which eliminates 51 gt 35 Player 2 knowing 1 is rational knows that 1 S 51 S 4 which eliminates 52 gt 35 and 52 less than some value that looks to be around 08 We see from here that each successive iteration will further restrict the remaining set of pro les until it converges to the one stated above Problem 7 Watson 113 a N L and L N

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