Game Theory Solutions to Problem Set. A seller owns an object that a buyer wants to buy. The alue of the object to the seller is c: The alue of the object to the buyer is priate information. The buyer s aluation is a random ariable distributed oer the interal [; V ] according to the (continuous) c.d.f. F: Assume that [ F ()] f () is a decreasing function of : The on Neumann-Morgenstern utility of a type from getting a unit at price p is p and the utility of no purchase in : (i) Suppose the seller is constrained to charge just one price. Show that the pro t maximizing price satis es p c + [ F (p)] f (p) : (ii) Suppose that the seller can commit to a menu of o ers [q () ; p ()] ; where q () is the probability with which a consumer who chooses o er will get a unit, and p () is the consumer s (expected) payment. Proe that the menu that maximizes the seller s pro t consists of a single price, which is the one found in (i), and that any buyer can get the good at this price with probability : (i)the seller problem is to choose a price p [; V ] to maximize: (p) [ F (p)](p c) (P) Notice that (p) is di erentiable (thus continuous) and [; V ] is compact. c () < (V ): Thus, the any solution is interior and requires: Thus: (p ) f(p )(p c) + [ F (p )] F (p p ) c f(p ) Furthermore, The LHS of () is a strictly increasing function of p while the RHS of () is decreasing (by assumption), thus the solution is unique. (ii) The seller problem is to choose a menu fq(); p()g ; ( where p : [; V ]! < and q : [; V ]! [; ] ) to maximize: ()
8 < s:t: : : R V max fq();p()g [p() q()c]f()d U() q() p() q( ) p( ) for all (; ) [; V ] U() for all [; V ] q nondecreasing Using the results presented in the lectures we know that U () q(): Substituting into the IC and imposing U() we hae U() R q(~)d~: Thus: p() q() Z Substituting into P we hae the following problem: (P) q(~)d~ () max q()[;] q nondecreasing Integrating (3) by parts we hae: [q() Z q(~)d~ q()c]f()d (3) By assumption h F () f() [q() Z q(~)d~ q()( c)d + q()( q() c)d c Z q()c]f()d q() [ F () f() q(~)d~d( F ()] d f()d F ()) i is decreasing. Thus there is a unique (; V ) such that c F ()?,? f() Thus pointwise maximization implies: if q () if < Notice that q is obiously nondecreasing, what guarantees the optimality of the presented solution. Finally, () implies that the price paid by all types that obtain the object is :. Consider the following auction enironment. A seller has a single object for sale and can commit to any selling mechanism (the seller s aluation of the object is zero). There are two potential bidders, indexed by i ; : The aluation of the object
of bidder i ; is denoted by i and is distributed uniformly oer the unit interal. Valuations are independent between the two bidders. Bidder knows her own aluation : Howeer, bidder does not know. The bidders payo s are as follows. Suppose bidder i ; has type i and pays the amount t i to the seller. Her payo is equal to i t i if she gets the object, and equal to t i otherwise. (i) Construct the optimal direct mechanism for the seller (i.e., nd the incentie compatible, indiidually rational mechanism that maximizes the seller s expected reenues). Compute the seller s reenues. (ii) Can you nd a simple indirect mechanism that gies to the seller the same expected reenues as the optimal direct mechanism? i) From the fact that bidders are risk-neutral bidder behaes as if his aluation were : The expected reenues of the seller are: F ( ) Q ( ) + Q ( ) f( )d : (4) f( ) Using the fact that F () (4) can be rewritten as: Q ( ) ( ) + Q ( ) f( )d (5) Maximizing (5) pointwise subject to Q i ( ) ; Q ( )+ Q ( ) we obtain: Q ( ) if > 3 4 otherwise Q ( ) Q ( ) The solution is clearly nondecreasing. Finally notice that T ( ) 3 4 f > : In order to 3 4g calculate T just notice that bidder will obtain no rent. Thus: T Pr (Q ) 3 4 3 : 8 ii) Indirect Mechanism: ask a price of 3 to bidder and to bidder. Sell to only if 4 refuses to pay the price. 3. A seller has a unit for sale. Its quality is either high (H) or low (L) : The quality is known to the seller but not to the buyer, whose prior probability that the quality is high is : Their aluations of the unit are as follows. Quality H Quality L Buyer V Seller 7 3 (6)
where V > 7: Thus, the utility to the buyer of getting the unit at price p is p if it is of the low quality, and V p if it is of the high quality. Similarly, the utility to the seller is p and p 7; respectiely. (i) Find the ex-post e cient outcomes. (ii) Identify the range of V (aboe 7) for which there is, and the range of V for which there is no incentie compatible, indiidual rational mechanism that will achiee the ex-post e cient outcome. (iii) Describe the best outcome (in the maximizing of the sum of expected utilities) that can be achieed for each V (aboe 7) and the mechanism that achiees it. HINT: A mechanism for this Bayesian bargaining problem consists of a pair of functions q : fl; Hg! [; ] and t : fl; Hg! R; where q (i) is the probability that the object will be sold to the buyer and t (i) is the expected net payment from the buyer to the seller if i L; H is the type reported by the seller to a mediator. i) Ex-post e ciency means that in eery state the party that alues more the good, the buyer, always obtains the object. ii) Since the object should always go to the buyer, the transfer made from the buyer to the seller, T; should be independent of the state (from IC). From (IR) T has to be as least 7:We now check under what alues of V there exists a transfer greater than 7 satisfying the (IR) of the buyer. We need: V + 7, V Thus we need V : iii) We will nd the best menu: fp (L); T (L); P (H); T (H)g where P (L) (P (H)) is the probability that the seller keeps the object when he reports low (high) aluation and T (L) (T (H)) is the transfer that the seller receies from the buyer when he reports low (high) aluation. We need to check the following constraints for the seller: and only the rationality from the buyer: P (H)7 + T (H) P (L)7 + T (L) (ICHS) T (L) T (H) (ICLS) P (H)7 + T (H) 7 (IRHS) T (L) (IRLS) ( P (H))V + ( P (L)) (IRB) Thus, we can set up surplus maximization: problem 4
max P (H);P (L)[;] T (L);T (H) 8 < s:t: : : ( P (H)) (V 7) + ( P (L)) T (L) T (H). P (H)7 + T (H) 7 ( P (H))V + ( P (L)) T (L) T (H) Notice that if V we are in (ii), then we assume V (7; ) :This problem can be easily soled by KT techniques. Rather, we gie a somewhat more informal deriation using obserations (a),(b), (c) and (d) below: (a) P (L) : This follows because both the objectie function and (IRB) are strictly decreasing in P (L). Thus, from (ii) we know that in any solution we need P (H) > : (b) T (L) T (H): Otherwise one can increase T (H) by ", decrease T (L) by " and decrease P (H) by " : For " small enough this is feasible and increases the objectie function. 7 (c)p (H)7 + T (H) 7: Otherwise P (H) can be decreased by some small "; what increases the alue the objectie function. (d)( P (H))V + T (L) T (H) : Otherwise we can increase both T (L) and T (H) by some small " and decrease P (H) by ", what increases the alue the objectie function. 7 From (b) to (d) we 3 equations and 3 unknowns. Soling the system we hae: T H T H 4 V ; P (H) : From (a) P (L) : 4 V 4 V 4. A seller owns an object that a buyer wants to buy. The quality of the object is a random ariable ; with support [; ] and distribution function F () ; where > : The seller knows the quality of the object but the buyer does not. When the quality of the object is ; the alue of the object is to the seller and z to the buyer, where z >. Thus, if the object of quality is traded at price p; the seller gets p and the buyer gets z p. Both players hae utility equal to zero if there is no trade. Consider the function G : (; )(; )! [; ] de ned as follows. For each pair (; z) construct the incentie-compatible indiidually rational mechanism that maximizes the (ex-ante) probability of trade. Denote this probability by G (; z) : Derie the function G: (N.B. If the probability of trade is q () when the quality is ; then the (ex-ante) probability of trade is equal to R q () df () : Let q() be the probability of trade gien and t() the payment from the buyer to the seller gien : The seller (IC), the seller (IR) and the buyer (IR) are respectiely: t() q() t( ) q( ) (ICS) U() t() q() (IRS) 5
(zq() t()) f ()d (IRB) where f () : Notice that (ICS) and (IRS) immediately imply that if U() then U() for all : Furthermore usual analysis implies U () q() and q() nonincreasing. From (IRS) we hae; Thus: U() q(x)dx + U() t() q() + Substituting (7) into (IRB) we hae: zq() q() From (8) we can set U() : Integrating by parts we hae: q(x)dx + U() (7) q(x)dx U() f ()d (8) q(x)dxf ()d q(x)dxdf () (9) q()f ()d q() d Substituting (9) into (8) we hae: q() z q() z f ()d f ()d + Thus if z which holds i z we hae q() for eery ; otherwise q() for eery : Therefore the probability of trade is: if z + G (; z) otherwise 6