Economics in Action A Capstone Course
Economics in Action A Capstone Course ECON 4999
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Economics 4999 Logic and Mathematics in Economics Review questions on game theory 1 In a paragraph or so de ne game theory to a friend who has never taken an economics course N Prove that the Nash equilibrium is not always ef cient As part of your answer de ne ef cient answer An allocation payout is ef cient if there is no other allocation payout where at least one player is better OH and neither player is worse OH How to prove NE are not always efficient The easiest way is with a counter example That is an example of a game where the NE is not left right up 5 6 8 2 down 4 9 6 9 equilibrium but UL is not ef cient because at DR both players are better OH ef cient For example Why UL is a nash 03 Esther and Edgar met at a rave but were at the time incapable of exchanging phone numbers They desperately want to meet again There is another rave tonight and if they both go they will meet However they both have exams early tomorrow morning 6 am SO there would be some bene t to staying home and studying The payout matrix is Esther rave Edgar rave 10001000 study 0 0 each other They also want to pass their exams but passing up true love is hard to do Are there Nash equilibruim in pure strategies If so identify them Convince me that they will both end of at the rave my They think they might really like answer There are two Nash equilibrium in pure strategies Both at rave or both home studying Both at the rave is preferred to both home studying Both understand the game so gure out that the rational thing for the other player is to choose the rave r Roger and Michelle although they greatly enjoy each other s company have very diHerent preferences Roger s tastes run to ladies7 mud wrestling while Michelle prefers Italian opera They are planning what to do on Sat urday night Each has two options go to the Opera or go to mud wrestling Michelle wrestling Opera Roger wrestling 21 Opera 0 0 1 2 each other dearly and will be greatly upset if they choose diHerent venues The payout matrix is They love However Roger prefers they end up at mud wrestling and Michelle prefers La Scala ls this a zeroesum game As part of your answer de ne zeroesum game Does this game have a dominant strategy equilibrium Find this game s two pureestrategy Nash equilibrium If there is a mixedestrategy equilibrium my guess is will be each player playing each option 50 of the time Try to demonstrate that my intuition is correct This game is called the battle of the sexes ls one of these equilibrium more likely than the others If so which one and why answer This is a not a zerqsum game the two payouts in each box do not sum to zero The two Nash equilibrium in pure strategies are everyone at mud wrestling or everyone at the opera Therefore in this game EBR P7Tq2171 quml1 P7Tq01 171 77q1 2P7Tq1 171 77q and EBM q7rplql777p01iq7rp0liqli7rp2 q p21 q1 77p 219 1 7pm 7Tql 3m 7 ll and dig qu 2 1 7 1H1 m 377p 2l where p is the probability that Roger will choose mud wrestling q is the probability that Michelle will choose mud wrestling 7139q is Roger s subjective probability that Michelle will choose mud wrestling and 77p is Michelle s sube jective probability that Roger will choose mud wrestling Start with Roger s maximzation problem gt 3E3 R 3m 7 1 lt 3p 0 depending on the valueof 7139q Note that the derivative of the Roger s objective function with respect to its choice variable 17 is not a function of 17 So Roger can t choose 17 to make this equal zero There in no interior solution for Roger unless 7139q 333 However if 7139q 333 the Roger won t care what 17 he chooses By symmetry we know that Michelle won t care what q she chooses as long as 77p 666 Demonstrating that the mixedestrategy equilibrium is p 666 and q 333 They each choose their favorite 23 of the time What happened to my intuition Interestingly the answer book says the mixedestrategy equilibrium is p q 5 If Michelle or Roger played 5 what would the other one want to do The other one would want to play their favorite all of the time One can see this by plugging 5 for 739rq into E33 E33 p52 5 l 7p 5 517 This is continuously increasing in 177 so Roger will want to set 17 1 What is interesting is that the mixedestrategy equilibrium kind of sucks The probability that they will both end up at mud wrestling is 29 The probability that Roger will end up at mud wrestling and Michelle at the opera is 49 pl 7 q The probability that Roger will end up at the opera and Michelle at mud wrestling is 19 l 7 pq 13l37 and the probability that they will both end up at the opera is 29 1 i P1 i q 59 of the time they will end up apart What is the expected bene ts to both parties if they play the mixedestrategy Nash equilibrium EBR 223gtlt1sgt 13 23 23 EBM 1323 2 23 13 23 Compare this to the expected bene ts they each would get ifp q 5 E33 20555 5 5 75 EBM 55 2 5 5 75 Expectationally7 they both do better with p q 57 but it is not a Nash equilibrium 1 Consider the game Dueling Saabs It is played by adolescent females who attend Boulder High School They all own Saab convertibles with which they like to play chicken1Every time two females encounter one another they are compelled to play chicken They drive their Saabs at each other at breakneck speed and at the last second either swerve or not swerve If one swerves she is a wimp Those who do not swerve are considered aggressive Ceteris paribus7 aggressive is considered a desirable trait it attracts adolescents boys7 but aggressive can be costly if one encounters another aggressive driver Assume the payout matrix for this game in numer of boys attracted or repelled is swerve don t swerve swerve don t swerve 57 0 73 73 1 Care insurance is too costly for adolescent males so they remain careless 2The blood and guts resulting from accidents grosses out the boys While wandering through adolescence each female will encounter many other Saabs Suppose that she cannot tell in advance whether the other Saab driving adolescent will act aggressively or wimp out The payout to adopting either strategy depends on the proportion of the other females that act aggressively Convince me that there cannot be an equilibrium in which all female Saab drivers are wimps Convince me that there cannot be an equilibrium in which all female Saab drivers are aggressive Since there is not an equilibrium where every female chooses the same strategy determine whether there is an equilibrium where some proportion of the adolescent females choose to act aggressively and the rest wimp out a If the proportion of aggressive drivers is greater than the equilibrium of aggressive drivers which stragegy does better If the proportion of ag7 gressive drivers is less than the equilibrium of aggressive drivers which stragegy does better a If the proportion of aggressive drivers is not the equilibrium proportion will the proportion move back towards the equilibrium proportion as either wimps get more aggressive or aggressives become wimps H answer a If everyone is a wimp any individual driver can increase her expected bene ts by becoming aggressive EB if all wimps 1 your EB if your aggressive and everyone else is a wimp 5 0 all wimps is not a Nash equilibrium Given what everyone else is doing you want to change your behavior a If everyone is aggressive any individual driver can increase her expected bene ts by becoming a wimp EB if all aggressive 73 your EB if you are a wimp and everyone else is aggressive 0 So everyone aggressive is not a Nash equilibrium EBW 11 7 77a 071 1 7 77a and EBA 5177Ta 7377a 5757ra73739ra 5787Ta where 77a is the percent that are aggressive In equilibrium EBW EBA otherwise incentive to switch Solving 1 7 77a 5 7 877a Solution is 77a Graphically 1 7 77a H 0 2 Padent aggresme 0 8 1 Equilibrium to the wimp game If 77a gt wimps do better If 77a lt the aggressive drivers do better Yes As an aside is DL or UR Nash equilibrium in pure strategies Looks like it but what would it mean In a world of just two girls one always wimping and the other not is a Nash equilibrium in pure strategies What ifthere were hundred girls and fty were always wimps and 50 were always aggressive and everyone knew who was a wimp This would not be a Nash equil Because aggressive drivers would not want to be aggressive when they meet another aggressive The same for wimps Discuss game theory as a tool for modeling the behavior of ologopoly rms As part of your answer discuss the important factors one must consider in the modeling of the behavior of rms in a ologopolistic industry and explain how game theory might be used to model those factors As part of your answer provide a gameetheoretic model of ologopoly behavior As part of your answer discuss the notion of Nash equilibrium as it would apply in the context of equilibrium in an ologoplistic industry Game theory is designed to model the interactions among a small number of players where the actions of each affects all of the others Ologopoly industries have a small number of rms where the actions of each rm in the industry affects all of the other rms in the industry The payouts of the game are pro ts Game theory assumes rational behavior on the part of the players a reasonable starting assumption for ologopoly rms Nash equilibrium in this context means answer each rm in the industry is doing what all of the other rms in the industry expected it to do each rm is maximizing its pro ts given the behavior of the other rms in the industry example Joe Perrier is the sole owner of a spring that costlessly burbles forth as much mineral water as Joe cares to bottle It costs Joe SEQgallon to bottle the water green bottles are expensive The inverse demand function for Joe s water is p 20 7 2q where p E price per gallon and q E number of gallons sold Joe s pro ts as a function of q are 7711 7 20 7 2qq 7 211 7 18q 7 2q2 Start by nding the pro t maximizing level of output for this monopolist Look for critical points 377q 311 1874q Set this equal to zero and solve for q 187 4q 07 Solution is q 450 Check the second7order conditions 327701 74 lt 0 Bq2 q So qquot 45 and Joe s monopoly pro ts are 7745 7 1845 7 2452 7 405 Now suppose Joe s neighbor Wilbur Evian discovers a mineral spring on his property and no one can taste the difference between the two waters It costs 6gallon to pump and bottle Evian Total demand form mineral water remains as before 17 20 7 2q Both guys are dense7 assuming that the other guy s production decision is in7 dependent of what they do silly but true Find the Nash equilibrium Start by nding Perrier s supply function ap 7 20 7 201p 19 qp 7 211p 18qp 7 211 7 2qpqe Perrier s pro t maximizing level of output is found 8 7 18 7 4qp 7 2qe qu Set this equal to zero and solve for qp 1874qp72qe7 Solution is qp 450 7 05qe This is Joe Perrier s suppy function for Perrier as a function of qe Graphically7 450 7 05qe 0 1 zqes 4 5 Joe s reaction function This supply function is Joe s eaction function in that it tells us how much Joe Will choose to produce as a function of What Wilbur produces Now nd Wilbur Evian s supply function 77qe 20 7 2qp 19 qe 7 6 7 14119 7 2113 7 2qpqe Perrier s pro t maximizing level of output is found 8779 3 Set this equal to zero and solve for qe 14 7 e 7 2qp 07 Solution is qe 350 7 05 This is Wilburs suppy function for Perrier as a function of qp Graphically7 350 7 05gp 14 7 4qe 7 2qp 0 1 qus 4 5 Wilbur s reaction function The Nash equilibrium is the solution to the two reaction functions and qef39 393075qp 7 7 qp 7 450 7 05 Solutlon 1s qe 7 16 6677 qp 7 36 667 That 1s7 1f Wilbur bottles 16667 Evians and Joe bottles 36667 Perriers7 neither Will want to change their output level H Assume a twoeperson twoestrategy oneeshot game What characteristic s does the payout matrix have to have for this game to be a Prisoner s Dilemma Speci cally a List a condition that is necessary but not sufficient List the conditions that are necessary and sufficient List conditions that are sufficient but not necessary H answer A condition for PD that is neccesary is one whose absence 3 the game is not PD A condition is sufficient is one whose existence 3 PDThe following are each examples of conditions that are necessary but not sufficient for P a Player 1 has a dominant strategy a Player 2 has a dominant strategy Both players have dominant strategies None of these conditions are sufficient because it is possible to write down a game where the condition holds but the game is not PD Each of these conditions is necessary because it is impossible to write down a PD game where these three conditions do not hold Second part Conditions that are both necessary and sufficient for PD are the de ntion of a PD game that is de nition of PD ltgt conditions that are necc and suff for PD The necc and suff conditions are 1 there is a dominant stragegy equil and no other Nash equilibrium and 2 If both players play their nondominant strategy which is obviously not an equilibrium both players get a payout that is 2 his or her equilibrium payout Now express things in terms of the left right payout matrix Remember the game is the same is left lup is switched for right and up is switched for down Necessary and sufficient conditions for PD are 1 a gt e b gt d c 2 g f 2 h but either 0 gt g f gt h or both And 2 S bgh but either a lt 9 b lt h or both Why Note that a gt e and c 2 g 3 up is player s 1 s dominant stragegy and b gt d and f 2 h 3 left is player s 2 s dominant stragegy So UL is a dominant strategy equilibrium and UR and DL are not Nash equilib rium If one adds the additional condition that either 0 gt g f gt h or both then DR is not a Nash equilibrium3 So Di UL is a dominant strategy equilibrium and there are no other equilibrium Now look at 2 S b g h but either a lt 9 b lt h or both a S g and b S h 3 both players are at least as well off at DR as they are at UL and either a lt 9 b lt h or both 3 at least one of the players is strictly better off at DR So in summary 1 and 2 together are necessary and sufficient for the game to be a PD Now consider part 3 of the question An example will suf ce left right This game is PD so it is sufficient for PD but there are many PD games that do not have this payout matrix so it is not necessary This concludes my answer Note that l have de ned the PD game such that the weakly preferred outcome occurs when both players play their nondominant strategies One could generalize the de nition of the PD game such that their has to be an outcome that is preferred to the dominant strategy equilibrium but that is not necessarily the outcome with both player playing their nondome inant strategy Richard in class game up with an example where DL or UR was preferred to the dominant strategy equilibrium In this case the necc and sufficient conditions outlined above are sufficient but not necessary 3Note that if c g and f h then DR is Nash equilibrium