Notes on Expected Revenue from Auctions

Transcription

1 Notes o Expected Reveue from Auctios Ecoomics 1C, Professor Bergstrom These otes spell out some of the mathematical details aout first ad secod price sealed id auctios that were discussed i Thursday s lecture You ca fid a eve more detailed discussio i Steve Matthews Techical Primer, which is liked to the class wesite Although there is quite a it of detail, you will see that the oly mathematics used is simple proaility theory, a little algera ad some easy calculus of a sigle variale differetiatig ad itegratig polyomial fuctios The aalysis for 2 idders is fairly simple ad I urge you to try to follow the argumet here for this case The idder case requires some more calculatio, ut uses the same road map to its results Eve if you do t follow the etire argumet for the idder case, it is valuale to compare these results with the two idder case ad to uderstad what happes as you add more idders Private Value Auctios Suppose that a supplier has a sigle oject for sale ad that there are possile uyers The value of the oject to perso i is some umer v i that is kow to i ut ot kow to ayoe else If Perso i gets the oject for price p, his profit will e v i p Let us suppose that everyody except perso i thiks that i s value for the oject is a radom variale draw from the same proaility distriutio If radom variale is cotiuous, we let F (x) deote the proaility that this radom variale is less tha or equal to x ad f(x) = F (x) e the correspodig desity fuctio We will ofte work with the example of a uiform distriutio o the iterval [, 1] Sometimes it is coveiet to thik of a discrete versio of this radom variale i which the radom variale takes o oly iteger values ad where the proaility the that x = for ay iteger etwee 1 ad 1 is f() = 1/1 The the proaility that x must e F () = /1 Here we will work with the cotiuous uiform distriutio o [, 1] I this case, for ay real umer x etwee ad 1, we have F (x) = x/1 ad f(x) = F (x) = 1/1 1

2 How to id i a secod-price sealed id auctio I a secod-price sealed id auctio, the oject is sold to the high idder The high idder does ot pay his ow id, ut istead pays the amout id y the secod-highest idder If your id is ot the high id, you do t get the oject ad you do t have to pay aythig I this auctio, your ow id does ot affect the amout that you would pay if you got the oject It oly determies whether you get the oject A remarkale feature of the secod-price auctio is that o matter what you elieve aout other people s strategies, the est thig for you to do is to id your true value Why is this so? Whatever happes i the auctio, there will e some id z that is the highest id made y the other guys If your ow value is v, the whatever the value of z is, your profit will e v z if your id is greater tha z (ecause you get the oject ad pay z for it) ad it will e if you id less tha z (ecause i this case you do t get the oject ad do t pay for it) What differece would it make if you tried to overid y iddig > v? If z < v, it would t matter at all, ecause you would wi the oject ad pay z just as you would if you id your true value v But what if > z > v? I this case, you would ot wi the oject if you id your true value v, ut you will wi the oject with your id of But if > z > v, the your profit is v z < You will e payig more for the oject tha it is worth to you ad will e worse off tha if you had id the truth ad ot wo the oject This meas that you ca ever make yourself etter off ad you may make yourself worse off y overiddig tha y iddig your true values What differece would it make if you tried uderiddig, y iddig less tha your true valuatio? If your id is < v ad if z <, it will make o differece that you uderid, sice you will still wi the oject ad you will still pay z for it But if it happes that < z < v, the with a id of, you will ot wi the oject ad will get profit If you had id your true value v, you would have wo the oject ad had a profit of v z > So you caot help yourself ad you may hurt yourself y iddig less tha your true value It follows that o matter what the other idders are doig, the est thig for you to do is to id your true value Biddig aythig else caot help you ad may hurt you A strategy for which this is the case is called a weakly domiat strategy Thus iddig your true value i a secod-price sealed id auctio is a weakly domiat strategy 2

3 How to id i a first-price sealed id auctio I a first-price sealed id auctio, the oject is sold to the high idder at the high idder s id I this case, it is easy to see that there will ot e a weakly domiat strategy (A strategy is weakly domiat if oe could ot do etter eve if oe kew what the other players are doig I the case of a first-price auctio, the est thig to do depeds o what the other idders are doig For example if the value to you is $1 ad the highest id y ayoe else is $4, you are est off iddig just a tiy it more tha $4, ut if the highest id y ayoe else is $5, you are est off iddig just a it more tha $5 So if there is o weakly domiat strategy, what do you do whe you do t kow what the other idders are doig? A reasoale approach is to maximize the expected value of your profit If you id ad your value is v, your expected profit will (v )P () where P () is the proaility that you get the oject if you id If higher ids are more likely to wi the oject, the P () will e a icreasig fuctio of while v will e a decreasig fuctio of You are goig to have to make a tradeoff Let s see how this would work if there are oly two idders ad each of them elieves that the other s value is a radom variale draw from the uiform distriutio o the iterval [, 1] You realize that it would e foolish to id your true value, sice i this case your profit is whether you wi the oject or ot To get started, assume that the other guy will id some fractio c of his true value The if you id, the proaility that you will wi the oject will e the proaility that cx < where x is the other idder s value This is the same as the proaility that x < /c where x is the other guy s value Sice have assumed that the other guy s value is a radom variale from the uiform distriutio o [, 1], it follows that the other guy s value is less tha /c is F (/c) = 1c So we have P () = 1c ad expected profits of (v ) 1c = 1 ( ) v 2 1c You wat to choose to maximize this expected profit calculus coditio for this maximizatio is d 1 ( ) v 2 = 1 (v 2) = d 1c 1c The first order (You should check the secod order coditio to see that this is really a maximum) The solutio of this equatio is v = 2, or equivaletly, = v 2 3

4 So if you elieve that the other guy is goig to id a fractio c of his value, the est thig for you to do is to id 1/2 of your value If he elieves the same thig aout you, the est thig for him is to id 1/2 of his value So i the case of two idders with values uiformly distriuted o a iterval, there is a equilirium i which each ids half of his value We ote that the oject would the always e sold to the idder with the higher uyer value ad it would e sold at a price equal to half of that perso s value Let s work out the case for idders Suppose that you id What is the proaility that your id is the highest oe? It is the proaility that all 1 of the other guys id less tha Suppose that each of them ids some fractio c of his true value The the proaility that ay oe of them ids less tha is F (/c) = tha is 1c F (/c) 1 = The if you id, your expected profit is ad the proaility that all of them id less ( ) 1 = 1 1c (1c) 1 1 (v ) (1c) = 1 ( v 1 ) 1 (1c) 1 To maximize your expected profit, you would choose so that d 1 ( v 1 ) = d (1c) 1 This is equivalet to 1 (1c) 1 ( ( 1) 2 v 1) = ( 1) 2 v 1 =, which is equivalet to = 1 v Thus we see, that as the umer of idders gets larger, each idder ids closer to his actual value If there are 3 idders, the i equilirium, each ids 2/3 of his value If there are 4 idders, each ids 3/4 of his value If there are 1 idders, each ids 9/1 of his value ad so o Expected Reveue If he does t kow the values of the idders, a seller will ot kow i advace whether oe type of auctio will give him more reveue tha aother His 4

5 retur from either type of auctio is a radom variale But give his eliefs aout the proaility distriutio of idders values, he ca calculate his expected reveue from ay type of auctio Expected Reveue from First-Price Sealed Bid Auctio We will work this out first for the case of 2 idders where the values of idders are idepedet draws from a uiform distriutio o the iterval [, 1] From our earlier discussio, we have leared that i equilirium, each idder will id half of his value Thus, the price at which the oject will e sold is 1/2 of the higher of the two idders values To fid the proaility distriutio of the seller s reveue, we first wat to fid the proaility distriutio of the higher of the two idder s values Let us fid the proaility g(x, 2) that the highest of 2 ids is x Now x ca e the highest id i two possile ways Oe way is that idder 1 has value x ad idder 2 has value less tha or equal to x The proaility of this evet is f(x)f (x) Sice f(x) = 1 x ad F (x) =, the proaility of this outcome is 1 x The other way i which x ca e the highest id is that idder 2 has value x ad idder 1 has value less tha or equal to x This also happes with proaility 1 x Therefore the proaility that x is the highest value from two idders 1 1 is ( ) ( ) 2 x g(x, 2) = 1 1 We ca calculate the expected reveue of the seller as follows Recall that with two idders, each idder ids half of his value So the expected reveue of the seller must e 1/2 of the expected value of the radom variale which is the highest value idder s value Sice g(x, 2) is the proaility that the highest value is x, the expected value of the highest value is 1 xg(x, 2)dx = 1 ( ) ( ) 2 x x dx = 2 1 x 2 dx Now 1 x 2 dx = 1 3 x3 1 = 13 3 Therefore the expected value of the highest idder value is 1 xg(x, 2) = ( ) 1 3 =

6 Recall that i equilirium, idders id oly half their values, so the expected reveue of the seller is 1/2 of the expected value of the higher idder value Hece the expected reveue of the seller is just What if there are idders? What is the proaility g(x, ) that the high id is x if there are idders? The x could e the highest uyer value i differet ways, oe way for each perso who could e high idder Each of these ways has proaility ad so it must e that ( 1 ) ( x ) ( ) ( ) 1 x 1 g(x, ) = 1 1 The expected value of the high id will e the itegral 1 Now, sice xg(x, )dx = 1 1 ( ) ( ) 1 x 1 ( ) 1 1 x dx = x dx x dx = x+1 1 = , we have 1 xg(x, )dx = Recall that that whe there are idders, each ids the fractio 1 of his value Thus where there are idders, the expected value of the high idder s id is 1 times the expected value of the highest value Thus the seller s expected reveue must e 1 1 ( 1 xg(x, )dx = ) ( ) 1 = + 1 ( ) This is cosistet with our previous result for 2 idders, where expected reveue was (1/3)1 Now we see that if there are 3 idders, expected reveue will e (1/2)1, if there are 4 idders, expected reveue will e (3/5)1 ad if there are 1 idders, expected reveue will e (9/11)1 6

7 Expected Reveue from Secod-price sealed id auctio Recall that i a sealed id secod-price auctio, the est strategy for ay player is to id her true value Thus the expected reveue of the seller will e the expected value of the secod highest value Agai, we will work this out first for the case of 2 idders where the values of idders are idepedet draws from a uiform distriutio o the iterval [, 1] Let h(x, 2) e the proaility that the secod highest id is x This ca happe i two differet ways Bidder 1 ca have a value of x, while the value of the other idder is greater tha or equal to x The proaility that a idder has a value greater tha or equal to x is 1 F (x) where F (x) is the proaility that a idder has value less tha or equal to x Therefore the proaility that idder 1 has value x ad idder 2 has value at least x is ( ) ( 1 f(x)(1 F (x)) = 1 x ) 1 1 The other way i which the secod highest id ca e x is if idder 2 has value x ad idder 1 has value at least x This also happes with proaility f(x)(1 F (x)) ad so h(x, 2) = 2f(x)(1 F (x)) = The expected value of h(x, 2) is 1 Now xh(x, 2)dx = ( 1 1 x x2 1 ( ) ( 2 1 x ) 1 1 ( ) ( 2 x 1 x ) dx = ) ( 1 ( dx = x2 2 x = 1 6) 2 Sustitutig ito our previous expressio, we see that 1 xh(x, 2)dx = x x2 1 Notice that this is exactly the same as the expected reveue of the seller from a first-price auctio i which oth idders use equilirium strategies Sayig this i aother way, whe there are two idders, the expected value of the secod highest uyer value is equal to half of the expected value of the highest uyer value ) dx 7

8 What if there are idders? Calculatig the expected reveue is a little more complicated ow If perso 1 is the secod highest idder, the oe of the 1 other idders must id at least as much as he does ad all of the 2 other idders have to id o more tha he does The proaility of this happeig is f(x)( 1)(1 F (x))f (x) 2 Sice there are differet people who could e the secod highest idder, the proaility that the secod highest id is x must e h(x, ) = ( 1)f(x)(1 F (x))f (x) 2 Now the expected value of h(x, ) will e 1 1 ( ) ( 1 xh(x, )dx = ( 1) x 1 x ) ( ) x 2 dx Simplifyig this expressio, we have 1 xh(x, )dx = ( ) ( 1) ) (x 1 x dx 1 Now 1 ) ( (x 1 x 1 dx = ) ( ) = ( + 1) Sustitutig ito the previous expressio, we fid that the expected value of the secod highest idder s value is 1 xh(x, )dx = Sice i equilirium for the secod-idder auctio, everyoe ids his value, this is also the expected reveue of the seller Note that this is exactly the same as the expected reveue of a seller i a first-price sealed id auctio The Reveue Equivalece Theorem Most people are very surprised to lear that i the examples we looked at, if idders are i equilirium, the reveue from a first-price auctio is exactly the same as that from the secod-price auctio Is this just a fluke? Would the result disappear if the distriutios of values were ot uiform o some 8

9 iterval? It turs out that this result is a special case of a much more geeral theorem aout private value auctios The equilirium expected reveue from first price ad secod price auctios would e the same so log as the distriutios of values are cotiuous ad idepedet etwee idividuals Eve more remarkaly, reveue from may other kids of auctios would e the same as that from the first ad secod price auctios, so log as distriutios of values are idepedet ad the auctios have the followig features 1) The good always goes to the perso with highest uyer value 2) The perso with the lowest uyer value is guarateed to make zero profits i the auctio This result is kow as the reveue equivalece theorem 9