Boğaziçi University Deartment of Economics Sring 2017 EC 206 MICOECOOMICS II Problem Set 9 - Solutions 1. For the game below find the set of ooling and searating PBE s. (he first ayoff value is for the Sender and the second one is for the ) 0,0 m 1 S m 2 5,0-1,1 t 1 1/4-1,1 ature 0,1 t 2 3/4 1-1- 5,1-1,0 m 1 S m 2-1,0 ASWE: Consider ooling euilibria first. Suose that both tyes of the sender send message m 1. hen = 1/4, which makes otimal. ow if the action on the right hand info set is, then both tyes would deviate to m 2. So it must be on the right hand info set. he receiver can ick any as long as it makes otimal. One such is 1. hus, (m 1 m 1, ; = 1/4, = 1) is a ooling PBE. ow suose that both tyes send m 2. hen = 1/4, which makes otimal. ow if the action on the left hand info set is, then there is no deviation. In this case, would be seuentially rational, for instance, when = 0. hus (m 2 m 2, ; = 0, = 1/4) is a ooling PBE. If the action on the left hand info set is, then again there is no deviation. In this case, would be seuentially rational, for instance, when = 1. hus (m 2 m 2, ; = 1, = 1/4) is another ooling PBE. ow consider searating euilibria. Suose m(t 1 ) = m 1 and m(t 2 ) = m 2. hen = 1 and = 0. he otimal actions are on the left hand info set, and on the right hand info set. But then sender with tye t 1 deviates to m 2 since 5 > 1. hus this cannot be a PBE. ow suose m(t 1 ) = m 2 and m(t 2 ) = m 1. hen = 0 and = 1. he otimal actions are on the left hand info set, and on the right hand info set. But then again, the sender of tye t 1 deviates to message m 1. herefore there are no searating euilibria. 2. Consider the signaling game below. Suose each tye is eually likely. he first ayoff value is for the Sender and the second one is for the. 1
2,4 Sender m 1 t 1 m 2 1,3 3,2 r 3,2 4,1 3,1 2,2 0,2 t 2 m 1 m Sender 2 t 3,2 2,0 1,0 1-- m 1 m 2 t 3 Sender 1-r-t 2,2 (a) Find the set of ure-strategy ooling Perfect Bayesian euilibria. ASWE: Let s look at all three tyes sending m 1. In this case the information set on the left is on the euilibrium ath. Bayes ule says the s beliefs (robabilities) on this information set should be (1/3, 1/3, 1/3). Given these beliefs, the exected ayoff for the from is (4 + 2 + 1)/3 = 7/3 and the exected ayoff from is (2 + 2 + 0)/3 = 4/3. hus otimal action by the is on the left hand information set. he information set on the right is off the euilibrium ath. Let s say the action on this information set is. hen, t 1 deviates to m 2 (2 > 3). If it is, then t 3 deviates to m 2 (2 > 0). hus, there is no ure strategy ooling euilibrium with each tye sending m 1. ow let s look at all three tyes sending m 2. In this case the information set on the right is on the euilibrium ath. Bayes ule says the s beliefs (robabilities) on this information set should be (1/3, 1/3, 1/3). Given these beliefs, the exected ayoff for the from is (3 + 1 + 0)/3 = 4/3 and the exected ayoff from is (2 + 2 + 2)/3 = 2. hus otimal action by the is on the right hand information set. he information set on the left is off the euilibrium ath. If the action on this information set is, then tye t 2 deviates to m 1 (4 > 3). When it is on the left then no tye deviates. hus it must be. o make it seuentially rational, we can ick the belief on the left to be (0,1,0). hen, (m 2 m 2, ; (0, 1, 0), (1/3, 1/3, 1/3)) is the ooling euilibrium. hus, the set of ooling PBE is {(m 2 m 2, ; (0, 1, 0), (1/3, 1/3, 1/3))} (b) Find the set of ure-strategy Perfect Bayesian euilibria where tye t 1 and tye t 2 send the same message and tye t 3 sends another message. ASWE: Let s look at the case where t 1 and t 2 sends m 1, and t 3 sends message m 2. hen, both information sets are on the euilibrium ath. By Bayes rule, the beliefs on the left hand information set must be (1/2,1/2,0), and the beliefs on the right hand information set must be (0,0,1). hen the otimal action on the left must be and the otimal action on the right must be. But then t 2 deviates to m 2 (3 > 2). So there is no PBE of this sort. ow let s look at the case where t 1 and t 2 sends m 2, and t 3 sends message m 1. hen, both infor- 2
mation sets are on the euilibrium ath. By Bayes rule, the beliefs on the left hand information set must be (0,0,1), and the beliefs on the right hand information set must be (1/2,1/2,0). hen the otimal action on the left must be and the otimal action on the right is either or. Let s say on the right, then t 3 deviates to m 2 (2 > 0). So no such euilibrium. Let s say on the right, then t 2 deviates to m 1 (4 > 3). So no such euilibrium. So there is no PBE of this sort as well. hus, the set of PBE with the first two tyes sending the same message and t 3 sending another message is emty! 3. Suose that after you graduate you have two otions when alying for a osition at a firm: ou can either use your undergraduate diloma or ursue graduate studies for two years and receive an MA diloma and send it as art of your alication. our skill level is either High or Low. ou learn your skill level but the emloyer does not know your skill level. he emloyer only knows that your it is High with robability 1/4. After you learn your tye you either aly for the job with your undergraduate diloma or get an MA degree and aly with master s diloma. hus you have two ossible messages. ou already have a BA degree, so its cost is zero. he cost of MA degree is 2 if you are High tye and 5 if you are Low tye. he emloyer, observing the message you send but not your tye, decides what kind of job to offer you. here are two ossible jobs: ays 10 and ays 6. he net ayoff of the emloyer is 10 if you are High tye and emloyed at, 5 if you are High tye and emloyed at, 0 if you are Low tye and emloyed at, and 3 if you are Low tye and emloyed at. Is this game a chea talk game? Find all ooling and searating PBE s. ASWE: he extensive form is 10,10 8,10 BA ou MA 6,5 High 1/4 4,5 Emloyer ature Emloyer 10,0 Low 3/4 1-1- 5,0 6,3 BA ou MA 1,3 Consider ooling euilibria first. Suose your strategy is BA,BA; that is, send message BA for both tyes. hen = 1/4. Given = 1/4, emloyer offers (because Eu()=10/4 but 3
Eu()=14/4). ow look at the choice by the emloyer at the right hand info set. If the emloyer offers, then you would deviate to MA if your tye is High. hus it must be offered by the emloyer at the right hand info set. Here the emloyer can ick any to make her choice otimal, and =0 makes otimal. hus (BABA,; = 1/4, = 0) is a ooling PBE. ow look at the other ossible ooling euilibrium where your strategy is MA,MA; that is sending MA for both tyes. hen = 1/4, and given this belief the emloyer offers. ow if the emloyer offers at the left hand info set, that is when she observes BA as a signal, then you would deviate to MA for both tyes. If the emloyer offers then again you would deviate to BA for any tye of yourself. hus there is no ooling PBE with MA,MA. ow consider searating euilibria. Suose your strategy is BA,MA; that is send message BA when you are High tye, and MA when you are Low tye. hen = 1 and = 0. he emloyer with these beliefs offers you if she sees BA, and if she sees MA. But then Low tye would deviate to BA. hus this cannot be a PBE. Finally consider MA,BA; that is send message MA when you are High tye, and BA when you are Low tye. hen = 0 and = 1. he emloyer with these beliefs offers you if she sees BA, and if she sees MA. here is no deviation by any tye. hus (MABA,; = 0, = 1) is a searating PBE. 4. A loyal Boğaziçi University alumnus, A, has an extra ticket for the 2012 Chamions League final. he ticket is worth 1000L to whoever he sells it to. A otential buyer, B, aroaches him. A cannot tell whether B is a loyal Boğaziçi University alumnus or not, but believes the robability that B is a loyal alumnus to be 1/2. A asks B to contribute 500L to Boğaziçi University, and sees if B contributes 500L or not, and then sets the rice for the ticket. If A concludes that B is loyal then he sets the rice at 200L. If he concludes that B is not loyal then he sets the rice at 600L. If he cannot tell, he sets the rice at 400L. If B is loyal then a contribution costs her only half of the contribution amount because she receives some utility from being hay that Boğaziçi University has the money. If she is not loyal, then a contribution costs her the exact amount of the contribution. hus, the ayoff to B is 1000 - rice - (1/2)contribution if loyal, and 1000 - rice - contribution if not loyal. All of the above is common knowledge. (a) Find all ooling and searating PBE s. ASWE: Pooling euilibria: (1) (Cont.,Cont.): A sets the rice at 400L when he sees a contribution. When A observes no contribution, even he sets the rice at 600L, the not-loyal(l) tye would deviate to not contributing: Payoff of L tye when she contributes is 1000-500-400=100 which less than the ayoff when she does not contribute (even if rice is 600L) 1000-600=400. So there is no such ooling euilibrium. (2) (Don t Cont.,Don t Cont.): A sets the rice at 400L when he sees no contribution. And when he observes contribution, let s assume he believes the tye is loyal and hence sets the rice at 200L. hen, no tye deviates: he loyal tye gets 1000-400 from not contributing and 1000-200-500/2 from contributing, thus she does not deviate from not contributing. he not-loyal tye 4
gets 1000-400 from not contributing and 1000-200-500 from contributing, thus she also doesn t deviate. So this is a ooling PBE. Alternatively, you can also use other beliefs: Suose if he observes contribution he believes she is not-loyal for sure. hen he sets the rice at 600L. here will be no deviation by any tye (we already saw there is no deviation even when the rice is 200L). So this would also be a PBE. And finally if he believes that she is loyal with robability 1/2 then he sets the rice at 400L which also becomes a PBE. Searating euilibria: (3) (Cont., Don t Cont.): A sets the rice at 200L if he sees contribution (in which case she believes that the tye is loyal for sure), and 600L if he sees no contribution (in which case she believes that the tye is not-loyal for sure). hen, the loyal tye gets 1000-200-500/2=550L and the not-loyal tye gets 1000-600=400L. here is no deviation by any tye: Loyal tye would get 1000-600=400L (smaller than 550L) by not contributing, and the not-loyal tye would get 1000-200-500=300L (smaller than 400L) by contributing. hus this is a searating PBE. (4) (Don t Cont.,Cont.) A sets the rice at 200L if he sees no contribution (in which case she believes that the tye is loyal for sure), and 600L if he sees contribution (in which case she believes that the tye is not-loyal for sure). But then, the not-loyal tye gets 1000-600-500=- 100L, thus she would deviate to not contributing and would get 1000-200=800L (obviously larger than -100L), which is a rofitable deviation. hus this is not a PBE. hus we get, PBE={(Don t,don t; 200 if contribution, 400 if no contribution),(don t,don t; 400 if contribution, 400 if no contribution),(don t,don t; 600 if contribution, 400 if no contribution),(contribute, Don t Cont.; 200 if contribution, 600 if no contribution)} (b) ow suose that instead of asking B to contribute 500L, A asks B to contribute any amount she likes, so now the message sace is continuous. Find the set of ooling euilibria in ure strategies where A has the following beliefs: if B contributes at least c, then B is loyal with robability 1/2. If B contributes less than c, then B is for sure not loyal. ASWE: Suose both tyes contribute the same amount, say c. Given the above beliefs, A sets the rice at 400 if he sees at least c otherwise he sets the rice at 600. hen, we must have the following: 1000 400 (c /2) 1000 600 for the loyal tye not to deviate. Also, we must have 1000 400 c 1000 600 for the not-loyal tye not to deviate. hese ineualities boil down to c 400 and c 200. hus, any c 200 with the above beliefs is a ooling PBE. 5. Suose that you are working in a research-center. Suose the head of the center assigns you on a roject for which there is a deadline in three days. ou will be working with a co-worker who either knows how to handle the roject or has no idea. ou don t observe his knowledge about the roject, but you are a bit essimistic and believe that he knows how to handle the roject with robability 1/3. our co-worker tells you one of the two things: (1) rust me, the roject is easy, so we can do it the night before the deadline. Let s just go see a movie, or (2) It s a tough roject, so we should start working on it now in order to comlete it on time before the deadline. After hearing what he tells you, you decide whether to start working on it the night before the deadline () or today (). If you decide to do it the night before the deadline, then you both get 5 if your co-worker knows how to 5
handle the roject, and if he has no idea you get 0, he gets -1. If you decide to start working on it today, then you both get 3 if your co-worker knows how to handle the roject, and if he has no idea you get 2 and he gets 0. Also, if your co-worker tells you (1) and you choose to do it the night before the deadline, then you go to the movie with him, he buys the tickets which decreases his ayoff by 2. (a) Draw the extensive form game. ASWE: 3,5 (1) Co-worker (2) 5,5 3,3 knows 1/3 3,3 ou ature ou -3,0 has no idea 2/3 1-1- -1,0 0,2 (1) Co-worker (2) 0,2 (b) Is there a ure ooling PBE where both tyes send message (1)? ASWE: ES. (11, ; = 1/3, = 0) is the only such ure ooling euilibrium. (c) Is there a ure searating PBE where the tye who has no idea sends message (1)? ASWE: ES. (21, ; = 0, = 1) is the only such ure searating euilibrium. (d) Is there a ure ooling PBE where both tyes send message (2)? ASWE: ES. (22, ; = 0, = 1/3) and (22, ; = 1, = 1/3) are both ure ooling euilibria. (e) Is there a ure searating PBE where the tye who has no idea sends message (2)? ASWE: ES. (12, ; = 1, = 0) is the only such ure searating euilibrium. 6. Suose there are two tyes of car buyers in the world, eager ones and icky ones. he eager buyers get a ayoff of 100 from buying a car, while the icky ones get only 60. Half the buyers are eager and half are icky. Each buyer knows his tye, but car dealers don t know a given buyer s tye. A buyer comes to the dealershi and looks at a car. He can then either immediately go to the dealer and ask for a rice or he can wait for some fraction of the day before doing so. If he waits, he risks having some other buyer take the car before he can buy it. More secifically, the buyer chooses a number x between 0 and 1 which is the fraction of the day he waits before asking for a rice. If the fraction he waits is x, the robability the car is still available after this is 1 x. If it s no longer available, the buyer gets a ayoff of 0. he dealers all see how long a buyer waits before asking for a uote. Suose they are all cometitive and so will always set a rice eual to half of what their exectation is regarding the buyer s ayoff to buying the car. If the buyer urchases at this rice, he gets the value of the car minus the urchase rice as his ayoff. Find all searating and ooling PBE s. 6
ASWE: First note that, a message is the x the buyer chooses, that is, M = [0, 1]. Also note that the seller sets a rice of 30 if she believes the buyer is a icky tye, and 50 if she believes the buyer is an eager tye. If the seller does not learn anything, she simly charges 40=(1/2)30+(1/2)50. ow let s first consider the ooling euilibria. hat is, consider both tyes icking the same x. If a buyer asks for the rice before waiting x fraction, suose the seller believes the buyer is an eager tye, in which case she offers a rice of 50. We need waiting x fraction to be otimal for both tyes: For eager tye we need (1 x)(100 40) 100 50 = 50 and for the icky tye we need (1 x)(60 40) 60 50 = 10 he first ineuality gives x 1/6 and the second one gives x 1/2. hus, we have a set of ooling PBE s: Each buyer, eager or icky, waits till x 1/6, and if the buyer waits at least x, the seller infers the buyer is eager with robability 1/2 and icky with robability 1/2. If the buyer does not wait at least x, then the seller infers that the buyer is eager. he seller charges a rice 40 if the buyer waits at least x, and 50 if the buyer does not wait at least x. ow let s consider searating euilibria. In a searating euilibrium, the tyes will be revealed, and thus the eager tye will get a rice of 50 and the icky tye will get a rice of 30. Because the icky tyes will get a lower rice, they must have to wait to get it. Everyone would go for a lower rice without having to wait. So the eager tyes will not wait as much as the icky tyes. hus, the seller s inference must take the following form: Infer that the buyer is icky if the buyer waits at least x, eager otherwise. Given this, it is otimal for the eager tye to ask for a rice right away (and will be offered a rice of 50), otherwise they increase the risk of not getting the car without decreasing the rice. For this to be an euilibrium, we need two things: First, we need the eager tye to refer asking for a rice right away (x = 0) to waiting x > 0 and getting a rice 30, that is 100 50 = 50 (1 x)(100 30) = (1 x)70 Second, we need the icky tye to refer waiting x, that is (1 x)(60 30) = (1 x)30 10 = 60 50 So we need 2/7 x 2/3. hus we have a set of searating PBE s: Eager tyes ask for a rice immediately, and icky tyes wait till x [2/7, 2/3], and the seller infers the buyer is icky if the buyer waits till x [2/7, 2/3], offers rice 30, otherwise infers the buyer is eager and offers rice 50. 7