Problem Set 2 Solutions

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1 FDPE Mcroeconomcs 4: Informaton Economcs Sprng 07 Juuso Välmäk TA: Chrstan Krestel Problem Set Solutons Problem Prove the followng clam: Let g,h : [0, ) R be contnuous and dfferentable such that ) g (0) h(0). ) x > 0,{g (x) < h(x)} {g (x) h (x)}. Then g (x) h(x), x 0. (Hnt: Draw pctures to see the content of the clam). Soluton. Assume by way of contradcton that g ( ˆx) < h( ˆx) for some ˆx 0. By ) we know that ˆx > 0. Let s defne f : g h. Clearly, f s contnuous and dfferentable on [0,+ ). Now let x 0 0 be the largest x < ˆx such that f (x) 0. The exstence of such pont x 0 follows drectly from the ntermedate value (or Bolzano s) theorem. Now, by the mean value theorem there exsts a pont a (x 0, ˆx) such that f (a) f ( ˆx) f (x 0) ˆx x 0 < 0, () where the nequalty follows from the defnton of f, our choce of x 0 and the assumpton g ( ˆx) < h( ˆx). But then combnng () wth the fact that f (a) g (a) h (a) we obtan g (a) < h (a), whch s a contradcton to ). Problem Consder a frst-prce aucton where a sngle ndvsble object s sold to one of n bdders. The bdders have ndependent prvate monetary valuatons v for the object. The valuaton s drawn from the common cdf F ( ) on [0,]. If bdder wns the aucton at bd b, her payoff s u (v b ). where u (0) 0, 0 < u < and u 0 Losng bdders do not get the object and make no payments. Show that the followng symmetrc bddng strateges descrbe an equlbrum of the frst-prce aucton n ths case. b (v ) (n ) f (v ) F (v ) u (v b (v )) u, b (0) 0. (v b (v )) (Hnt: If all others use b ( ), consder the payoff to wth valuaton v from b ( v ). In equlbrum, ths must be maxmzed at v v. Next argue for the boundary condton.) Soluton. Let b( ) be a monotoncally ncreasng, dfferentable, and symmetrc equlbrum bddng strategy. If players j bd accordng to b(v j ), then the expected payoff to player wth type v > 0 from bddng b(v ) s u (b(v ),b ; v ) u(v b(v )) F n (v ), () These solutons are based on those prepared by Jula Salm n 05 and Mchele Crescenz 06. Ther permsson to use the materal s gratefully acknowledged.

2 where F n (v ) s the probablty that wns the aucton. Now, the frst order condton to maxmze (6) wth respect to v s (n )F (v )n f (v )u(v b(v )) F (v )n u (v b(v ))b (v ) 0. Snce at equlbrum we must have v b (b(v )) v, pluggng ths n the frst order condton above and rearrangng yelds b (v ) (n )f (v ) u(v b(v )) F (v ) u (v b(v ). To show that b(0) 0, t suffces to observe that u (0,b ;0) 0 whle u (b,b ;0) < 0 for every b > 0. Problem 3 Use Problems. and. to argue that rsk averse bdders bd more aggressvely than rsk-neutral bdders n IPV symmetrc frst-prce auctons. To do ths, let β(v ) denote the symmetrc equlbrum bddng strategy under rsk-neutralty: β (v ) (n ) f (v ) ( v β(v ) ). F (v ) If β(v ) b (v ) where b (v ) s the equlbrum bddng strategy for a strctly concave u ( ), then b (v ) > β (v ).(Show ths.) Then use Problem 3 to reach the concluson. Soluton. Let s start out by provng the prelmnary statement n the text, namely f β(v ) b (v ), then b (v ) > β (v ). To do ths, assume by way of contradcton that β(v ) b (v ) and b (v ) β (v ). Thus we have b (v ) β (v ) u (v b (v )) u (v b (v )) v β(v ). (3) for every x > 0, or equva- Now, snce u( ) s a strctly concave functon, and u(0) 0, t follows that u (x) < u(x) x lently x < u(x) u (x). Usng ths n (3), we obtan v b(v ) < u (v b (v )) u (v b (v )) v β(v ), whch mples b(v ) > β(v ), so leadng to a contradcton. Now, the clam that we have just proved mples condton ) of the lemma n Problem 3. Snce β(0) b(0) 0, condton ) of the lemma s satsfed as well. Therefore, we can nvoke the lemma of Problem 3 to conclude that b(v ) β(v ) for every v 0. Problem 4 Consder the optmal Myerson aucton dscussed n the lectures. (a) Show that the optmal Myerson aucton can be desgned so that the bdders have domnant strateges. We can take v as our decson varable nstead of b snce b s one-to-one when restrctng the co-doman to [0,b()]. We can do ths snce t cannot be optmal to bd more than b().

3 Soluton. The allocaton rule of the optmal Myerson aucton s: x (θ f θ F (θ ) f,θ ) (θ ) θ j F j (θ j ) f j (θ j ) for every j, and θ F (θ ) f (θ ) 0 0 otherwse Assumng symmetrc type dstrbutons and that the vrtual valuaton s ncreasng for each type, ths s mplementable through a second prce aucton wth a reserve prce,.e. the auctoneer accepts all bds for whch θ ( F (θ ))/f (θ ) 0 and then sells the object to the hghest bdder at prce that equals the second hghest bd. By standard arguments, t s weakly domnant for the bdders to bd ther true type,.e. b (θ ) θ. (b) In contrast to second-prce auctons, optmal Myerson auctons depend on the dstrbuton of valuatons F ( ) on [0,]. To evaluate how much the seller gans n revenue from usng the optmal aucton, compute the expected revenue for 3 bdders n the optmal Myerson aucton when F (θ ) θ (.e. unform dstrbutons) and the valuatons are ndependent. Contrast ths to the expected revenue from 4 ndependent bdders n the second-prce aucton wthout reserve prces. Soluton. Frst of all, note that the densty for the hghest type s f (θ) nf n (θ)f (θ) nθ n and the second hghest s f (θ) n(n )F (θ) n f (θ)( F(θ)) n(n )θ n ( θ). We wll need ths later. Also note that for the unform dstrbuton the expected kth hghest value from a sample of sze n s: E(θ k ) θ + n + k (θ θ). n + Usng that, the expected revenue from the second prce aucton wth 4 bdders s the expected second hghest valuaton, whch turns out to be R SPA (5 )/5 3/5. Now, to calculate the expected revenue from the Myerson aucton, we need to fnd the reserve prce, whch s: b 0 ( F (b 0)) f (b 0 ) 0 b 0. Recallng that only bdders wth valuatons above the reserve prce wll partcpate n the Myerson auc- 3

4 ton, and by the revenue equvalence theorem, the expected revenue s R M θ θ n n 0.5 (θ F (θ) )f (θ)dθ) f (θ) (θ )nθ n dθ 0.5 (θ n θ n )dθ ( n + n n n + ( ( ) n+ + ( ) n ) n + ( ) n. ) n n n + + n When n 3, we thus have R M 7 3. Clearly, R SPA > RM. (c) Can you generalze part b to a comparson between optmal Myerson aucton wth n bdders and secondprce aucton (wthout reserve prces) for n + bdders? Soluton. To answer ths queston, we take frst a short detour to monopoly theory to provde an economc nterpretaton of the vrtual valuatons of the bdders. The relevant artcle for the whole analyss s Bulow and Klemperer (AER 996). Consder a monopolst who s sellng to a contnuum of consumers wth unt demand so that there are F (θ) consumers wth value θ. Then the demand s q(θ) F (θ), whch mples that the nverse demand s v(q) F ( q). Revenue s v(q)q, so margnal revenue dqf ( q) dq F ( q) + q df ( q) dq F ( q) q f (F ( q)). Now substtutng n F ( q) and q gves MR(θ) θ F (θ) f (θ) whch s essentally the same as the vrtual valuaton that we have encountered n the aucton settng. Now we can use ths to rewrte the expected revenue as an expectaton over the margnal revenues from dfferent bdders R Θ Θ (θ F (θ ) )x (θ ) f (θ )dθ f (θ ) MR (θ )x (θ ) f (θ )dθ 4

5 And now note that we are n symmetrc settng and that both of our auctons allocate the good to the bdder wth hghest vrtual valuaton (x (θ) for the hghest type) so that the above expresson becomes just the expected maxmum of the margnal revenues R E(max{MR(θ ),..., MR(θ n )}) Ths means that we can compare two auctons just based on the expected maxmal margnal revenue. In our case, we want to compare the reserve prce aucton to SPA wth one addtonal bdder. The reserve prce was found by settng MR(θ) 0, so n the aucton wth a reserve prce the expected revenue s thus R E(max{MR(θ ),..., MR(θ n ),0}) (4) Where as n SPA wth n + bdders we have R E(max{MR(θ ),..., MR(θ n ), MR(θ n+ )}) (5) One can prove (check the orgnal paper) that (5) cannot be less than (4), so mplyng that optmal auctons are domnated by second-prce auctons wth an addtonal bdder. Problem 5 Consder a frst-prce aucton wth two bdders where the valuaton of bdder, θ 0 wth probablty π, and θ wth probablty π. The valuaton of bdder s known to be b >. (a) Can you fnd π and b such that the game has an equlbrum n pure strateges? Soluton. The answer depends on the te-breakng rule adopted. If we assume that, n case of a te, the object s assgned through a ffty-ffty lottery, then no equlbra n pure strateges exst. If we assume nstead that the object s assgned to the bdder wth the hghest valuaton, then pure-strategy equlbra do exst n ths game. One of those equlbra s such that player bds 0 f her valuaton s 0 and bds f valuaton s,.e. b (0) 0 and b (), whle the equlbrum strategy for player s to bd provded that b πb, whch s equvalent to π b. (b) Suppose 0 < b <. Fnd an equlbrum for the game. Soluton. You can easly verfy that there are no pure strategy equlbra n ths case. Let us construct a mxed strategy equlbrum. To do that, let s assume that both player and player wll bd on a contnuous support [0, s], where 0 < s b. Let s also assume that player wll bd 0 when θ 0 (weakly domnant). 5

6 Wth arbtrary bds b,b [0, s], player s expected payoff s Eu (b,b ) Pr (θ 0)(b b ) + Pr (θ )Pr (b b )(b b ) π(b b ) + ( π)f (b )(b b ), (6) where F ( ) s player s mxed strategy when θ. Now we can use the fact that each player must be ndfferent between any pure strategy n the support of the mxed strategy that s beng played n equlbrum. More specfcally, bddng zero ensures player a payoff of πb and, usng ths n (6), we obtan F (0) 0. Furthermore, snce F (s), we have π(b s) + ( π)(b s) πb s ( π)b. Hence both bdders bd on [0,( π)b]. Would player want to devate above ( π)b? No, snce she could wn wth certanty even wth ( π)b. Usng agan the ndfference condton, we can pn down the equlbrum mxed strategy for player, whch s F (b ) πb ( π)(b b ). We can use the same procedure to derve the strategy F ( ) of player. If θ, player wll get ( π)b for certan f she bds ( π)b, and usng ths we obtan Eu (b,b ) F (b )( b ) ( ( π)b) F (b ) ( π)b b. F (( π)b) and F (0) ( ( π)b) so that player has an atom at 0. Takng everythng together, the bdders strateges on [0,( π)b] are πx F (x) ( π)(b x) σ (θ ) f θ 0 f θ 0 F (x) ( ( π)b) ( x) wth probablty ( π)b σ 0 wth probablty ( π)b. (c) Compare the expected payoff n part b to the expected payoff n second-prce aucton. Soluton. We cannot use the revenue equvalence theorem here because the type space s dscrete. However, we can easly calculate the revenue from the second prce aucton snce the bdders strateges wll 6

7 be just to bd ther own valuaton: R SPA π 0 + ( π)b ( π)b. The expected revenue from the frst-prce aucton s: R F PA π[( π)be(b )] + ( π)[e(max{b,b })] < π( π)b( π)b + ( π)( π)b < ( π)b R SPA. where the frst nequalty follows from the fact that mxed strateges for both players have support [0,( π)b], whle the second nequalty follows from the assumpton that b <. Problem 6 Consder a symmetrc aucton for a sngle object where n bdders have prvate valuatons θ drawn ndependently form a common dstrbuton F ( ) on [0,]. (a) Use the Revenue Equvalence Theorem to compute the symmetrc equlbrum strateges of an all pay aucton where the bdder wns the object and all bdders pay ther own bd (regardless of who wns). Soluton. From the Revenue Equvalence Theorem, we have: θ V (θ ) V (θ ) + θ X (s)ds. (7) Assumng an equlbrum wth strctly ncreasng strateges, we have X (θ ) F n (θ ) and V (θ ) 0. Furthermore, the expected payoff to player s V (θ ) θ X (θ ) b (θ ) θ F n (θ ) b (θ ). Hence we can rewrte (7) as follows: θ θ F n (θ ) b (θ ) F n (s)ds, θ from whch we mmedately get b (θ ) θ F n (θ ) θ F n (s)ds. θ (b) (Harder) What can you say about the dstrbuton of the revenue to the seller as n >? How does ths compare to the frst- and second-prce auctons? By revenue equvalence of the auctons, we mmedately know that expected revenue must be the same for all auctons. Suppose we have n ndependent draws from a dstrbuton. We next order the draws accordng to ther sze θ (n) θ (n) θ n (n). The random varables θ (n) are called order statstcs. 7

8 It s easy to show that the dstrbuton F (y) of the hgest order statstc gven n draws s gven by F (y) Pr(θ (n) y) Pr(θ y, ) F (y) n where the last equalty follows from the fact that the draws are..d. For the second hghest order statstc, we have the dstrbuton F (y) F (y) n + nf (y) n ( F (y)) (Hnt: F (y) n s the probablty that all draws are smaller than y, the second reflects the probablty that one draw s above y and all others have a value smaller or equal to y). Let θ (n) be once more the kth order k statstc for a random sample of sze n. Snce P(θ ) 0, we have that lm n P(θ(n) < ) lm n ( F () n + n( F ()) n F () ) Therefore θ (n) converges n dstrbuton to the dstrbuton of, whch s a constant. (To complete the argument: For any 0 < y <, F (y) F (y) n y nf (s) n f (s)ds nf (y) n y f (s)ds and snce lmn F (y) n 0, lm n nf (y) n 0. But then lm n F (y) lm n ( F (y) n + n( F (y))f (y) n ) 0 for any 0 < y <.). Therefore the expected revenue from a second prce aucton converges to one. Invoke revenue equvalence to show that the expected revenue n the FPA and APA also converges to one. What can you say about the varance of the revenue n a SPA and FPA? For the varance of the revenue n an APA thngs are less clear. We specalze to the case where F ( ) s the unform dstrbuton and compute the varance of the revenue for the seller. Ths s nterestng n ts own rght. Snce everythng s symmetrc, we can drop the subscrpt to smplfy notaton. The unform assumpton mples that F (θ) θ. The symmetrc equlbrum bddng strategy n the APA s: The expected revenue to the seller s b(θ) θ n θ 0 s n ds n n θn. R APA ne(b(θ)) ( n n 0 n n n +. θn ) dθ 8

9 As for the varance, we have V AR(R APA ) nv AR(b(θ)) n ( E(b(θ) ) (E(b(θ))) ) ( (n ) ) n (n + ) θ n (n ) dθ 0 n(n + ) ( ) n ( n ). n + n + Now let us consder the frst-prce aucton. Agan, we can use the Revenue Equvalence Theorem to fnd the equlbrum bddng strategy: (θ b (θ))θ n θn n b (θ) θ n n. To calculate the expected revenue, recall that the densty for the hghest value s f () (θ) nθ n. Thus we have The correspondng varance s: R F PA E(b () (θ)) ( θ n n 0 n n +. ) nθ n dθ V AR F PA V AR(b () (θ)) E(b () (θ) ) (E(b () (θ))) ( ) n ( n θ nθ n dθ 0 n n + ( ) (n ) (n + ). n(n + ) Therefore, even though both auctons yelds the same expected revenue to the seller, the frst-prce aucton has a sgnfcantly lower varance. ) Problem 6 A seller has two dentcal goods for sale. Three bdders wth unt demand are nterested n the objects. Model ths stuaton as an ndependent prvate values aucton and fnd the expected revenue maxmzng format. Suppose that the seller decdes to hold two consecutve second prce auctons wth reserve prces to sell the goods. Is t a domnat strategy to bd the value n the frst aucton? Recall that the losers stll have the opton 9

10 of partcpatng n the second aucton. How does the expected revenue compare to the optmal mechansm. Consder only the symmetrc case. Soluton. Each bder has unt demand. Therefore let us ntroduce q (θ) the probablty that the frst good s allocated to bdder and q (θ) the probablty that the second good s allocated to bdder gven the type profle s θ. We then defne the allocaton rule x (θ) as follows: x (θ) q (θ) + ( q (θ))q (θ) (8) Ths allocaton rule for bdder takes care of unt demands. If bdder gets the frst good, the probablty of gettng the second good s zero. If bdder does not get the frst good, the probablty of gettng the second good s q (θ). We then let X (θ ) E θ x (θ,θ ) and proceed exactly as n the lecture notes to fnd the allocaton rule for the Myerson optmal aucton. We obtan E n T (θ ) Θ Θ [ x (θ) θ F ] (θ ) f (θ )dθ f (θ ) [ (q (θ) + ( q (θ))q (θ)) θ F ] (θ ) f (θ )dθ f (θ ) where the second lne uses (8). But ths has an easy soluton: Gve good one to the bdder wth the hghest vrtual valuaton (f non-negatve), gve good two to the bdder wth the second hghest vrtual valuaton (f non-negatve). Ths clearly maxmzes ths expresson. Hence, denotng the vrtual valuaton by ψ (θ ), we have the allocaton rule (8) wth q (θ) f ψ (θ ) ψ j (θ j ), j, and ψ (θ ) 0 0 otherwse q (θ) f k : ψ k (θ k ) ψ (θ ),ψ (θ ) ψ j (θ j ), j,k and ψ (θ ) 0 0 otherwse We have seen that when bdders are symmetrc the Myerson aucton can be mplemented as a SPA wth F (r ) reserve r solvng r f (r ). Here each bdder submts one bd, the frst good goes to the bdder wth the hghest vrtual valuton (f postve), the second good to the bdder wth the second hghest vrtual valuaton (f postve). In the SPA the bdder wth the hghest bd pays the second hghest bd, the bdder wth the second hghest bd pays the thrd hghest (f above the reserve prce). We denote the order statstcs by θ (n),... as before. Expected revenue s,θ(n) ( ) [ ] ( ) [ ] R M Pr θ (n) > r E max{θ (n) θ,r } (n) > r + Pr θ (n) > r ) E max{θ (n) θ 3,r } (n) > r For the two-stage SPA: Clearly t s weakly domnant to bd truthful at the second stage. Suppose that all players except bd truthful at the frst stage, that s b (θ ) θ. Suppose that bdder has the hghest valuaton, and that, wthout loss of generalty, bdder j has the second hghest, bdder k the thrd hghest valuton. If (9) 0

11 bds truthfully at the frst stage, she wns and payoff s θ θ j. Suppose then that bds b < θ j at the frst, θ at the second stage. Then she loses at the frst stage and wns at the second stage. Payoff s θ θ k > θ θ j. Hence t s not a domnant strategy to bd the own valuaton at stage one. Let r be the reserve prce n the Myerson aucton. Let b (θ ) be the bd of bdder, who has valuaton θ, n the frst stage. We restrct attenton to symmetrc equlbra, hence b (θ ) b(θ ) for all. We assume that b( ) s strctly monotone n the type and that the reserve prce s r n both stages, one and two. To economse on notaton t s convenent to defne the followng varables: Y θ (n ) and Y θ (n ). Y s the hghest type among the n opponents of a bdder, Y s the second hghest type of the n opponents of ths bdder. Recall that the CDF of Y s F (y) F (y) n (0) and the PDF f (y) (n )F (y) n f (y) () Y and Y are not ndependent. Ther jont dstrbuton s f (n ), (n )(n )f (y )f (y )F (y ) n 3 () Convnce yourself that a type wth valuaton θ r optmally bds b(r ), that s r b(r ). Fx an arbtrary bdder, say bdder wth valuaton θ. Suppose she ntends to place a bd b(θ ) r. The expected payoff n the frst stage, π, from ths bd s gven by π (θ,θ ) Pr ( b(θ ) > max{b(y ),r )} ) E [ θ max{b(y ),r } b(θ ) > max{b(y ),r ) ] We can rewrte ths as follows: π (θ,θ ) Pr ( b(θ ) > b(y ) > r ) E [ θ b(y ) b(θ ) > b(y ) > r ] + Pr ( b(θ ) > r > b(y ) ) E [ θ r b(θ ) > r > b(y ) ] The frst lne s the expected payoff of bdder when she wns and she has to pay the second hghest bd, because the second hghest bd s larger than the reserve prce. The second lne s the expected payoff when she wns but only has to pay the reserve prce. We next use that, by assumpton, b( ) s strctly ncreasng. Hence the events {b(θ ) > b(y ) > r } and {θ > Y > b (r )} are equvalent. The expected payoff n the frst stage can

12 then be wrtten as θ π (θ,θ ) b (r ) θ b (r ) (θ b(y))f (y)d y + F (b (r ))(θ r ) (θ b(y))(n )F (y) n f (y)d y + F (b (r ))(θ r ) where we have made use of (0) and (). Ths formula s ntutve. If all n opponents of bdder make a bd below the reservaton prce, whch happens wth probablty F (b (r )) n, bdder wns and obtans a payoff of θ r. The frst term captures the expected payoff n the case where the hghest bd of the n opponents of bdder s above the reserve prce and bdder wns. Bdder partcpates n the second stage f she loses the frst stage, ths s the event {θ < Y }. The event that she wns the second stage s {θ > max{y,r }}, because at the second stage t s weakly domnant to bd the own valuaton. Hence {θ < Y,θ > max{y,r }} s the event that bdder partcpates n the second stage and wns the second stage payng ether Y or the reserve prce, whchever s hgher. We thus have the followng expected payoff n stage : π (θ,θ ) Pr ( θ < Y,θ > max{y,r } ) E [ θ max{y,r } θ < Y,θ > max{y,r } ] Usng () allows us to wrte ths as π (θ,θ ) θ mn{θ,y } Bdder s expected payoff s thus θ θ (θ max{y,r })(n )(n )f (y )f (y )F (y ) n 3 d y d y π(θ,θ ) π (θ,θ ) + π (θ,θ ) + θ (θ b(y))f (y)d y + F (b (r ))(θ r ) b (r ) θ mn{θ,y } θ θ (θ max{y,r })(n )(n )f (y )f (y )F (y ) n 3 d y d y Dfferentatng ths wth respect to θ gves the frst order condton mn{θ (θ b(θ ))f (θ,θ ) } (θ max{y,r })(n )(n )f (θ )f (y )F (y ) n 3 d y 0 In equlbrum b(θ ) b(θ ), hence θ (θ b(θ ))f (θ ) θ θ (θ max{y,r })(n )(n )f (θ )f (y )F (y ) n 3 d y 0 The thrd term s always strctly negatve (θ r ), hence t follows that b(θ ) < θ n equlbrum. (From ths condton you can explctely solve for the bddng strategy. Does t remnd you of some bddng strategy n another aucton wth reserve prce?) The expected revenue n the two-stage SPA wth reserve prce r n both

13 perods s thus ( ) [ ] R T SSPA Pr θ (n) > r E max{b(y ),r } θ (n) > r + Pr(Y > r )E [ max{y,r } Y > r ] Ths s, of course, equvalent to ( ) [ ] ( ) [ ] R T SSPA Pr θ (n) > r E max{b(θ (n) θ ),r } (n) > r + Pr θ (n) > r E max{θ (n) θ 3,r } (n) > r Comparng ths wth (9) and notng that b(θ) < θ for all θ > r, shows that R T SSPA < R M. 3