Finish Q5 and Q6, and write the solutions in google colab in LaTeX form.
ECON 306 Page 3 of 4 2021 (b) If P3 observes signal s3 = L, what is the probability that the true state is A? i.e. what is Prob(s = A∣d1 = A, d2 = A, s3 = L)? Will P3 makes decisions following Bayes rule, will P3 choose d3 = A after observing a low signal? (c) If player 4 (P4) observes both P1 and P2 choose to adopt, will P4 be able to infer what signal P3 observed? (d) How will P4 choose? Briefly explain how this reasoning leads to a possible information cascade. 5. Beer-Quiche Game bNature0.1 0.9rwimpy senderQuiche Beer rsurly senderQuiche Beer p p p p p p p p p p p p p p p p p p p pReceiver p p p p p p p p p p p p p p p p p p p pReceiver r@@@ ��� [1− p] not duel r3, 0 r1, 1 r@@@ ��� [p] not duel r2, 0 r0,−1 r � � � @ @ @ [1− q] not duelr 2, 0 r 0, 1 r � � � @ @ @ [q] not duelr 3, 0 r 1,−1 (a) Verify that the following pooling outcome can be a part of PBE [(Beer if SS, Beer if WS)]. Describe the complete strategy profile and beliefs at the PBE. (hint: follow 3 steps as in slides) (b) Verify that the following separating outcome cannot be a part of PBE [(Quiche if SS, Beer if WS)]. (hint: follow 3 steps as in slides) 6. Corporate Investment Game • players—(E)ntrepreneur and (I)nvestor • types—one type of I; two types of E, tE ∈ {H, L}. Prob(tE = H) = θ. • The firm’s existing profit is v = ⎧⎪⎪⎨⎪⎪⎩ h if tE = H l if tE = L, which is private information for E, but cannot be credibly proved to I. • E requires an investment from I with the amount of k to fund a new project, which will yield a return of y for sure. • Timing 1. Nature draws tE, Prob(tE = L) = p 2. E learns tE, offers investor an equity stake 0 ≤ s ≤ 1 3. I observes s (but not tE) and decides to invest in the new project or not, d ∈ {A, R} ECON 306 Page 4 of 4 2021 4. If I accepts, the payoffs are πI(s, A; tE) = s(v + y) πE(s, A; tE) = (1− s)(v + y) If I rejects, the payoffs are πI(s, R; tE) = (1+ r)k πE(s, R; tE) = v • assume that the project is worth funding—(1+ r)k < y (a) if there is complete information, i.e. the investor knows the profitability of the firm perfectly, what is nash equilibrium of this game? (b) prove that, under the asymmetric information setup, there is no separating equilibrium (pbe) where both high-type entrepreneur (he) and low-type en- trepreneur (le) are funded (as the case in part (a)). follow these steps and use the 4 requirements of pbe to complete your proof. i. if i expects he and le to play as in the nash equil. in part (a), derive i’s belief about e’s types after observing different strategies. (for simplicity, assume that after observing any deviation from the expected strategy, i.e. off the equil. path, i believes that e is low-type with prob. 1.) ii. given i’s belief about e’s types, find i’s best response. (hint: i decides which types of offers to accept).1 iii. if he and le expect i to play the best response, is it best response for both he and le to play as the nash equil. in (a)? (c) prove that, under the asymmetric information setup, there is a separating equilibrium (pbe) where high-type (he) offers less than the nash equil. in part (a); and low-type (le) offer exactly as the nash equil. in part (a). as a result, only le is funded. follow these steps and use the 4 requirements of pbe to complete your proof. i. if i expects he and le to play the above separating strategies, derive i’s belief about e’s types after observing different strategies. (for simplicity, assume that after observing any deviation from the expected strategy, i.e. off the equil. path, i believes that e is low-type with prob. 1.) ii. given i’s belief about e’s types, find i’s best response. (hint: i decides which types of offers to accept) iii. if he and le expect i to play the best response, is it best response for both he and le to play these strategies? 1thanks to yanying (sophie) sheng for correction. y="" (a)="" if="" there="" is="" complete="" information,="" i.e.="" the="" investor="" knows="" the="" profitability="" of="" the="" firm="" perfectly,="" what="" is="" nash="" equilibrium="" of="" this="" game?="" (b)="" prove="" that,="" under="" the="" asymmetric="" information="" setup,="" there="" is="" no="" separating="" equilibrium="" (pbe)="" where="" both="" high-type="" entrepreneur="" (he)="" and="" low-type="" en-="" trepreneur="" (le)="" are="" funded="" (as="" the="" case="" in="" part="" (a)).="" follow="" these="" steps="" and="" use="" the="" 4="" requirements="" of="" pbe="" to="" complete="" your="" proof.="" i.="" if="" i="" expects="" he="" and="" le="" to="" play="" as="" in="" the="" nash="" equil.="" in="" part="" (a),="" derive="" i’s="" belief="" about="" e’s="" types="" after="" observing="" different="" strategies.="" (for="" simplicity,="" assume="" that="" after="" observing="" any="" deviation="" from="" the="" expected="" strategy,="" i.e.="" off="" the="" equil.="" path,="" i="" believes="" that="" e="" is="" low-type="" with="" prob.="" 1.)="" ii.="" given="" i’s="" belief="" about="" e’s="" types,="" find="" i’s="" best="" response.="" (hint:="" i="" decides="" which="" types="" of="" offers="" to="" accept).1="" iii.="" if="" he="" and="" le="" expect="" i="" to="" play="" the="" best="" response,="" is="" it="" best="" response="" for="" both="" he="" and="" le="" to="" play="" as="" the="" nash="" equil.="" in="" (a)?="" (c)="" prove="" that,="" under="" the="" asymmetric="" information="" setup,="" there="" is="" a="" separating="" equilibrium="" (pbe)="" where="" high-type="" (he)="" offers="" less="" than="" the="" nash="" equil.="" in="" part="" (a);="" and="" low-type="" (le)="" offer="" exactly="" as="" the="" nash="" equil.="" in="" part="" (a).="" as="" a="" result,="" only="" le="" is="" funded.="" follow="" these="" steps="" and="" use="" the="" 4="" requirements="" of="" pbe="" to="" complete="" your="" proof.="" i.="" if="" i="" expects="" he="" and="" le="" to="" play="" the="" above="" separating="" strategies,="" derive="" i’s="" belief="" about="" e’s="" types="" after="" observing="" different="" strategies.="" (for="" simplicity,="" assume="" that="" after="" observing="" any="" deviation="" from="" the="" expected="" strategy,="" i.e.="" off="" the="" equil.="" path,="" i="" believes="" that="" e="" is="" low-type="" with="" prob.="" 1.)="" ii.="" given="" i’s="" belief="" about="" e’s="" types,="" find="" i’s="" best="" response.="" (hint:="" i="" decides="" which="" types="" of="" offers="" to="" accept)="" iii.="" if="" he="" and="" le="" expect="" i="" to="" play="" the="" best="" response,="" is="" it="" best="" response="" for="" both="" he="" and="" le="" to="" play="" these="" strategies?="" 1thanks="" to="" yanying="" (sophie)="" sheng="" for="">