Advanced Microeconomic Theory I
Advanced Microeconomic Theory I ECON 805
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Date Created: 09/17/15
Econ 805 7 Advanced Micro Theory I Dan Quint Fall 2007 Lecture 16 7 Oct 30 2007 Everything we ve done so far has assumed that the number of bidders in an auction 71 is xed We can think of this as two separate assumptions rst the universe of potential bidders is xed and second there is no cost to participating in an auction so every potential bidder chooses to submit a bid Of course in many real world settings there is a cost to participating in an auction Could be c the time invested 0 cost of researching the item for sale and determining its worth 0 the costeffort of preparing a detailed bid eg in a procurement setting 0 the opportunity cost of participating in this eBay auction and not another one for an identical object In the case of the Glaxo Wellcome merger the cost of submitting a bid to acquire Wellcome was in the tens of millions of pounds Decision to enter before versus after learning your type 7 we ll focus on entry decision before learning type In most auction settings the ex ante expected pro t to each bidder is decreasing in the number of competitors he faces For example with symmetric independent private values either a rst or a second price auction with n bidders and no reserve price gives each bidder Et F 1sds which is decreasing in n This gives us two possible ways to model entry an expected pro t of 0 Potential bidders make sequential entry decisions 7 bidders enter one by one until there are just enough that the next one to enter would get expected pro ts lower than his entry costs 0 Potential bidders make simultaneous entry decisions perhaps playing independent mixed strategies between entering and not entering The earliest papers on entry in auctions used the rst approach Some others assumed that even if the entry game was played simultaneously it would result in an asymmetric equilibrium where the correct number of bidders entered However in large anonymous settings it s hard to see how bidders would coordinate and enter in the right number and since we re focusing on bidders who are ex ante symmetric it seems strange to focus on asymmetric equilibria in the entry game Today we ll focus on the second We begin with a paper by Dan Levin and James Smith Equilibrium in Auctions with Entry77 Levin and Smith model the following environment 0 N potential bidders Bidders are risk neutral and symmetric Bidders must incur some cost 0 gt 0 to participate in the auction and must decide whether to participate prior to learning their type So 0 can be thought of as the cost of researching the object and calculating their valuation for it The assumption that c is paid before bidders learn their types is crucial to the results In the rst stage of the game the potential bidders play a simultaneous move game where they decide whether or not to enter 0 In the second stage game some sort of auction is held 7 could be rst price second price or ascending could have a reserve price or an additional entry fee charged by the auctioneer on top of the cost 0 Since bidders who enter are symmetric LampS assume a symmetric increasing equi librium exists for the second stage auction The details of the second stage auction are common knowledge at the outset and the number of bidders who enter is revealed before the auction Value of the object to the seller is 0 Levin and Smith focus on the nontrivial case where entry by one bidder is pro table and entry by everyone is not pro table Milgrom de nes this as entry costs being moderate They focus on the symmetric equilibrium of the entry game where the potential bidders all play the same mixed strategy speci cally choosing to enter with probability q Obviously this is only an equilibrium if each bidder is indifferent between entering and not entering ie the expected gain to each bidder following entry in expectation over his own type everyone elses7 type and the number of other potential bidders who enter is 0 Let E7rln m denote a bidder s expected pro t from participating in the auction given a total of n entrants and a second stage auction denoted by in including paying the entry cost Then the condition for q to be a symmetric equilibrium in the entry game is N ZN71gtq 11qN E7rlmm 0 1 n71 since when everyone else mixes using q each bidder is indifferent between entering and not entering and therefore willing to mix Levin and Smith allow for any of the usual auctions we ve been looking at in the second stage provided they admit a symmetric increasing equilibrium They allow for the seller to charge an entry fee e on top of the exogenous entry cost 0 and a reserve price r Now if the terms of the second stage auction depend on the number of entrants it s possible to manipulate things so that E7r is not decreasing in n For example the auctioneer could set a higher reserve price when there are more entrants So it may be theoretically possible to have multiple symmetric equilibria in the entry game Levin and Smith pretty much assume this away but we ll nd it would generally not be in the seller s interest to do this anyway Since our equilibrium condition is that bidders earn zero expected pro ts total social surplus equals seller revenue Levin and Smith write this as N nq o SM n anTnmnm 7 m n1 where o q is the equilibrium entry level 0 Q is the environment including the second stage auction chosen 0 H the seller s expected revenue S social surplus N s N i i i i i o pn q 1 7 q n is the probability of n bidders entering 71 o TMRW is the probability that a sale occurs conditional on n bidders entering 0 V the expected value of the object to the highest bidder conditional on n entrants and sale occuring o n the average number of entrants Levin and Smith offer a couple of general results before looking at speci c cases Proposition 1 Any mechanism that maximizes expected reuenue induces socially optimal entry Such a mechanism may inuolue entry fees but not reserue prices The rst part is what we already said 7 since bidders earn 0 expected pro ts social surplus revenue To show that reserve prices are not optimal recall that we assumed the seller valued the object at 0 So at a given number of entrants n an auction with a positive reserve price is ex post inef cient By lowering the reserve price to 0 and raising the entry fee by some amount the seller could leave the bidders7 expected surplus in this auction constant and therefore not impact entry decisions but make more money Proposition 2 Any two mechanism that are reuenue equiualent with xed n and xed reserue price remain reuenue equiualent with induced entry For each n the two auctions give the same expected payoffs to each bidder So the two auctions would induce the same amount of entry the same q Since they are revenue equivalent for each realization of n they are therefore revenueequivalent in expectation Common Values Next we look at the special case of common values Since we ve established the optimal reserve price is O we ignore reserve prices V does not depend on the number of entrants so at a given level of entry social surplus and therefore revenue can be written as S 17p V7ch1717 NV7ch Maximizing this with respect to q gives the rst order condition N 7 N4VNt or 1 7 qN 1V c as the ef cient level of entry Proposition 3 In common ualue auctions the seller should discourage entry by charging a positiue entry fee but no reseruation price Without the entry fee entry would be excessiue from the social and priuate points of uiew To show this we recall that the bidders7 equilibrium condition can be rewritten as N N 7 1 7 7 V 7 W Z gtltqgt lt17qgtN 767 n 7 1 n n1 where Wn is the expected payment by the winner so that the left hand side is the bidder s expected pro t conditional on entering For optimality then N N71 V7W Z qn7117 qsN7n7n 7 6 17 qN71V n1 7739 7 1 n Since an auction with a single entrant and no reserve price sets W1 0 the n 1 case of the sum on the left is equal to the right hand side so N 6 7 Z N gt q 117qN 7V7nW gt0 n71 Proposition 4 If entry fees are not allowed the seller gains by setting a positive reserve price at least in the case of one entrant With one entrant the ex post optimal reserve price is greater than 0 since no entry fee and no reserve price leads to excessive entry lowering entry is also a good thing so this does better than nothing Levin and Smith give a proof that it s speci cally the n 1 case where the optimal reserve price is strictly positive Proposition 5 The revenue ranking of any two CV auctions that do not entail reservation prices or entry fees is preserved with equilibrium entry Suppose the seller prefers auction a to auction I when n is xed Then bidders must prefer I to a so 1 leads to higher bidder expected pro ts at each n and therefore leads to higher entry Go back to our earlier statement that HS171iqNVinc in any common value auction without reserve prices Differentiating twice shows that this is concave in q Since we just showed that any auction without entry fees or reserve prices induces excessive entry fltflt Since H is concave in q and q and qb are both above the optimum l lq 1 gt Hqb Levin and Smith also point out that by the same logic if the auctions being considered had reserve prices high enough to induce lower than e icient entry the revenue ranking would be reversed So for example with affiliated signals and common values a full information ascend ing auction is still preferable to a minimal information ascending auction which is still preferable to a rst price sealed bid auction Independent Private Values Proposition 6 Optimal entry for social surplus and for the seller occurs in IPV auctions when there is no entry fee and no reservation price The proof earlier that reserve prices should be 0 still holds so we only need to show that no entry fee is optimal We can write social surplus or revenue as N N 5 Z q 1qN Vn7ch n1 7739 where V the expected value of the object to the highest bidder is now increasing in n Differentiating with respect to q gives N 875 Z lt N gt We 7 q 7 N7 mm 7 com vi 7 NC 7iltzgtltm Grouping the N nq 1 7 qN terms as pm this gives N 7 2010 i q i N nqann 7 610 GONG 6117 q W1 1 N 01 7 qNPnV 7 q1 qgtNCgt n1 Now if e 0 bidders7 equilibrium entry condition can be rewritten as N ch anUn 7 W n1 since the right hand side is the total expected pro ts of all entrants and the left hand side is their total expected costs Recall that in lPV auctions by revenue equivalence payment equals the second highest bidder s value The expected payment given n bidders then is the expected value of the second highest of their types which is Wn Unn 7 lfvl 7 FUFU 2dv Separating this into two integrals and pulling out a couple of constants Wn nUn 7 lfvFU 2dv 7 n 7 l UnfvFv 1dv But n71fvFv 2 is the density of the highest of n7 1 so the rst term is just nVn71 similarly the second term is n71Vn So the expected payment is Wn nVn717 n71Vn and so Vn 7 WW nVn 7 Vn1 So N qNC annvn 7 Vnil n1 Plugging this into the expression for 858q gives at e O 8571ltilt Ngtvlt1gtiltv v gtgt 8Q i q1 q Fl n q pn n q Flpnn n WI and it ends up being just algebra to show that terms cancel and this is 0 So at e 0 entry is such that 0 Since we already showed that S is concave in q this means entry is efficient at e 0 Recall that with common values a positive entry fee was optimal and a positive reserve price was second best when entry fees were impossible In this case zero entry fee is optimal so zero reserve price is still optimal when entry fees are impossible This contradicts the usual xed n result that optimal reserve prices are positive in lPV auctions Af liated Private Values Proposition 7 If private valves are a liated then free entry with no reservation price is optimal for society and for the seller in a second price or ascending auction but excessive in a rst price auction In a second price auction with a iliated private values Wn 1 since bidding your type is a dominant strategy and therefore doesn t depend on independence so the arguments above still hold and entry is optimal However we saw that with xed n and affiliation second price auctions revenuedominate and so rst price auctions give higher bidder surplus So a rst price auction with no entry fee and no reserve price induces more entry than the second price auction and therefore induces excessive entry Coordination Costs Let n be the number of potential bidders at which bidders begin to mix everyone entering becomes unpro table given the entry cost 0 Levin and Smith show that in common value auctions as N increases beyond n equilibrium entry q decreases such that the probability of nobody entering is increasing in N and social welfare is decreasing in N when the seller uses an optimal mechanism that is uses the ef cient entry fee given N They pitch this as coordination costs77 7 that is as N increases the inef ciency due to uncertainty in the number of actual entrants increases as well Thus the intuition that thick markets 7 markets with lots of potential bidders 7 are always better is violated and potential bidders impose a cost on other bidders and the seller whether or not they actually participate in the auction Milgrom section 62 of PATW points out that based on this a seller can gain by limiting the number of bidders allowed to participate He considers a symmetric lPV environment First he shows that if 73 is a random number of entrants that takes at least three different values then at least one of the nonstochastic auctions with En bidders rounded either up or down must give higher expected social surplus than the stochastic one This is because the incremental value of an additional bidder is decreasing in the number of bidders already in the auction so total surplus is strictly concave in the number of bidders So in an lPV model with 450 potential bidders and entry costs that would induce an entry level of q at least one of the 4 and the 5 bidder auctions would give higher expected social surplus than the endogenous entry one Of course once entry is limited the bidders no longer automatically earn 0 pro ts so social surplus is not revenue But if after limiting the number of entrants the seller can still charge an entry fee then he can extract all the expected surplus Milgrom cites a result from a McAfee and McMillan 1987 paper on auctions with entry Proposition 8 The seller maximizes expected reuenue by limiting the number of bidders to the number that maximizes total surplus setting a reserue price of 0 and charging the entry fee that leaues bidders with 0 expected pro t That s basically it for endogenous entry To sum up we found that With endogenous entry bidders earn 0 expected pro ts so ef ciency and revenue maximization are the same problem Many of the usual revenue ranking results are robust to endogenous entry as long as entry fees and reserve prices are 0 0 With common values an entry fee is optimal to discourage excessive entry a positive reserve price is second best if an entry fee is impossible 0 With independent private values entry is ef cient if there is no entry fee or reserve price 0 With affiliated private values with no entry fee or reserve price entry is efficient in the second price auction but excessive in the rst price auction 0 With independent private values however the seller can gain further by limiting the number of entrants and charging an entry fee since the volatility in the actual number of entrants is inefficient Also recall that everything we ve done is based on the assumption that bidders must decide whether to enter before learning their types The problem gets much harder if bidders have meaningful private information prior to the entry decision The Milgrom chapter continues with a discussion of prequalifying bidders getting pre liminary77 bids and then holding a second price auction among those bidders with the highest preliminary bids as well as sequential negotiations when bidding is very costly Next up two papers on collusion among bidders Econ 805 7 Advanced Micro Theory I Dan Quint Fall 2008 Lecture 7 7 Sept 23 2008 Thus far we ve made four key assumptions that have greatly simpli ed our analysis 1 Risk neutral bidders 2 Ex ante symmetric bidders 3 Independent types 4 Private values These have bought us a lot 7 we proved revenue equivalence and solved for the optimal auction out of every possible feasible auction mechanism But they re also restrictive assumptions and revenue equivalence fails unless we have all ofthem There s a signi cant literature devoted to what happens when you relax each of these assumptions generally one or two at a time Since most common auctions are either rst price second price or ascending auctions much of this literature compares these formats to each other 7 in terms of revenue ef ciency etc 7 when each of these is relaxed When bidders are risk averse things get complicated Revenue equivalence fails There s a paper by Maskin and Riley on the syllabus characterizing the optimal auction under risk aversion which is a pretty complicated object Basically the seller can use the bidders7 risk aversion in two ways First he can effectively sell the bidders insurance 7 increase their payoffs when they lose the auction and decrease it when they win 7 and charge them for this And second he can use risk aversion to extract more surplus from the high types Since these are at odds which each other he can t do either perfectly but the optimal auction has aspects of both There s also a general result that when bidders are risk averse rst price auctions outperform second price auctions Intuition I may base a homework problem on this We already showed that when bidders are risk neutral risk averse sellers prefer rst price auc tions We ve already seen the optimal auction with asymmetric bidders There s also a nice paper by Maskin and Riley also on the syllabus comparing rst and second price auctions when bidders are asymmetric They nd that when one bidder s types are drawn from a stochastically higher distribution than the other s the strong bidder prefers second price auctions and the weak bidder prefers rst price auctions but which one raises more revenue depends The last two assumptions 7 independent signals and private values 7 are both relaxed in a fantastic paper by Milgrom and Weber 1982 Econometrica They introduce the affiliated inter dependent values framework lt s very general but in particular it nests two special cases that have received a lot of attention 0 Private values which we ve been looking at already but with values allowed to be positively correlated across bidders 0 Common values 7 where ex post the bidders all value the object the same but this true value is uncertain and each bidder has different information about it This is commonly used as a model of auctions for natural resource rights Right to drill for oil on a tract of government owned land is likely the same for every oil company depends on how much oil is underground each company might drill a couple of test holes to sample so each has a different estimate of the value of the object up for bid This week we ll develop the Milgrom and Weber model also covered in PATW section 54 After we study this model we ll then look at the special case of common values some of which predated the Milgrom and Weber results Before we introduce the Milgrom and Weber model however we need one result that I had hoped to cover earlier but didn t get to We say a probability distribution F rstorder stochastically dominates another one C if Ft Ct for every t Lemma 1 Let X and Y be random variables with distributions F and G IfF rst order stochas tically dominates G then EuX Z for any increasing function a When a is differentiable there s an elegant proof similar to the one we used for second order stochastic dominance De ne the step functions 1 i t gt k 1W If 7 0 if tlt k Note that if F rst order stochastically dominates G then 00 k 00 E1kX 1ktftdt Oftdt 1ftdt17 Fk17 Ck E1kY OO 700 k so we have the result when a is one of these step functions To prove it for general differentiable a write at a7oo 100 usds K as15tds ff utftdt ff K ff us15tds mm K ff us g 150 ftdtgt d5 K ff us ffooo19tgtdtgt d5 Euy So if X rst order stochastically dominates Y that is Ft Ct for all t then EuX 2 EaY for any increasing function a l Af liated Interdependent Values Model The general setup for the Milgrom Weber model is that there are N risk neutral bidders Each bidder i gets a signal 25 the value of the object to bidder i given these signals is 1147517 7527 7tN7 t0 where to indicates information that is not available to any of the bidders This could be information the seller has or information that nobody has to is allowed to be multi dimensional 7 that is it could consist of several different attributes ofthe good 7 but this doesn t end up making a difference so we ll treat it as a single variable for simplicity As before individual bidders signals ti must be onedimensional Milgrom and Weber make the following assumptions about 12 0 vi is nonnegative and continuous 0 vi is nondecreasing in all its arguments so good news for one bidder is good news for all the bidders 0 vi is symmetric in the following way WW 907M y to Z WW 957 Li y7to Z and Ul39ti 257139 yt0 Z Ul39ti 257139 0y to 2 where a is any permutation That is each bidder s valuation responds in the same way to to responds in the same way to his own signal responds in the same way to the other bidders signals and does not respond to which of his opponents had which signal just what they all are Given this symmetry we can rewrite bidder i s valuation as WW 907754 y7t0 Z 11907 ylt1gt7 ylt2gt7 7yltNTD7 to where 34 is the ith highest of the N7 1 elements of y That is we can rewrite bidder i s valuation as a function of his own signal to and the order statistics of his opponents signals and by the symmetry assumption this function is the same for every bidder Note that private values is simply the special case where vmy1y2 yN 1t0 gz for some function g and pure common values is the special case where 12tl m 257 342517 Z to Ujti tj 3425 Zt0 The nal two assumptions in the setup of the Milgrom Weber model are c The distribution of t0t1 tN is symmetric in the last N arguments o t0t1 tN are af liated Af liation is a technical condition which implies the variables are all positively correlated We ll spend the rest of today on af liation and its implications and then come back to the auction model on Thursday Sadly that means the rest of today is going to be a lot of gruntwork to lay the foundations for the very clean elegant stuff we ll do on Thursday Sorry Af liation Af liation is easiest to de ne for random variables with a density function The appendix of Milgrom and Weber gives a de nition for the other cases as well but we ll focus on this one Suppose you have n random variables jointly distributed in 9 according to a distribution function 1 Let x 1 zn and y y1yn be two generic points in 9 De ne o z A y m join y as the componentwise min z A y minm1y1 minmn o x V y m meet 3 as the componentwise max x V y maxz1y1 maxzn Draw it in 9 with the corners of a rectangle The n random variables with joint probability density 1 are af liated if and only if for every x and y 1 A yfw V y 2 ffy We ll show that this implies that a higher value of say 1 leads to stochastically higher values of the other variables 7 but this is a result not part of the de nition If we adopt the convention that logO foo and take logs of both sides this condition is the same as log x A9 logW Vy210gf10gfy which is exactly the requirement that logf is supermodular So random variables are af liated if and only if their density function is log supermodular For those of you who know supermodularity you already know that supermodularity of a function of n variables is equivalent to pairwise increasing differences in any two variables holding the others constant We restate this result in terms of log supermodularity Lemma 2 The function f is log supermodular if and only if for any m gt mg mi gt 9 and LM 6 W72 9 9073 907137 9027 9073 Lu 2 9027 9073 9Lijfi7 90 904439 The reason this has to do with increasing differences is that if we rewrite this with logs and rearrange this is equivalent to 10 901397 9 LN 10 K9027 9 7M Z 10 901397 90 957M 10g K9027 90 7M which is the same as saying that the difference log fm mi zij 7 log fz mi zij is increasing in 72 The pairwise condition we just gave can be rewritten as aw 95 Lm39 9027 9097 904439 mwwmem gt wwhm 7 That is for z gt mg and holding everything else xed the ratio must be increasing in zquot 7 Zw To see why the pairwise condition implies the af liation inequality consider two arbitrary points z and z For convenience reorder the arguments of 1 so that z 2 for i g k z lt for i gt k Then letting and be the max and min of mi and 2 respectively7 the affiliation inequality can be written as 95396377xZf f7 77ZZ miqmiwz y7mf f77907Z177Zgt or 0 0 flt9517795k795k17795ngtflt951vwk795k17795ngt fi77Z7177m f f777177x Now7 let X zinwzp and forj 2 k Xj zZ1mz 1z We can rewrite the left hand ratio as fltXangt fltXiXk1gtfltXiXk2 H fXXn fX7Xk fX7Xk fX7Xk1 fX7Xn71 and similarly7 if we let X0 z l zz we can rewrite the right hand ratio as fltX07Xngt fltX0Xk1gtfltX07Xk2gt H fX 7Xn fX 7Xk fX 7Xk fX 7Xk1 fXquot7Xn71 Now7 the pairwise condition we had above is exactly the condition that 90179027 7k7Xj1 f9617 9627 WWW X1 is increasing in M for i g k Applying this k times gives fX7Xj1 fX7 Xi fX 7Xj1 3 Maxi and the affiliation inequality follows The only if is trivial set x 27zj7zij and y mi7m97mij Note that this pairwise property is 10 901397 9 LN 10 K9027 9 7M Z 10 901397 90 957M 10g K9027 90 7M for zl gt and mi gt Rewriting M as e and dividing both sides by 6 gives log e zhz j 7 log zgm zii 39 gt 10 l E7mj7m ij 710gf7jviigtj e 7 e which7 taking 6 to O7 is the same as saying 3505f is increasing in mi So when 1 is twice differentiable7 the variables are affiliated if and only if 2 8 logf gt 0 8i8j 7 everywhere Next as with supermodularity we point out that affiliation is preserved by any order preserving transformation of the variables Lemma 3 If 91 gn are strictly increasing functions from 9 to 9 then 1 xn are a lz39ated if and only if 91z1 92z2 gnmn are a lz39ated This will be useful for us since equilibrium bid functions will be increasing it will imply that if bidder signals are affiliated then bids are affiliated as well To prove it let Fg be the joint cumulative distribution function of all the Since each gi is strictly increasing The probability that g 2 for all i is the probability that x 95121 for all i or F92172277Zn F9I1217791Zn If we assume the gi are all differentiable and differentiate n times once with respect to each 21 then 124217227 wzn f9f1217 79E1Zn9I1 219 1 22 99590 For simplicity we ll assume 1 is twice differentiable The result still holds without it but is harder to prove Then 91z1 gnzn are affiliated if and only if log f9 has positive mixed partials But log fg2122 2 Iogfltg121gtg1ltzngtgt 10g9f1 2110g9 1 22 logg1gt ltzngt The last n terms are functions of one variable and therefore have no mixed partial so 82 3 i 7 mlogmzhzz 2 9 1 2 But gig2i is positive and does not depend on 2739 so this is positive if and only if 12 is increasing in 27 So the transformed variables are affiliated if and only if the original variables are affiliated Lemma 4 If f 9 8 gt 9 2395 log supermodular then 1 99017 902 f872d5 0 2395 log supermodular Again this is the same as showing that 99017902 99037902 is increasing in 2 for any 1 gt We can write fwi f9gt2 d8 ml prwz 1 fsm 01 mpg gm1x2 f0m1fsm2ds 1 f 1f572d5 9i72 fox1 fsm2ds 011 fsm2ds Now here s the fun part Since f is log supermodular the fraction is increasing in 2 whenever s gt 1 and decreasing in 2 whenever s lt So the numerator in this last expression is increasing in 2 and the denominator is decreasing so the whole expression is increasing in 2 which is what we wanted to show Now here s why we wanted that last lemma Lemma 5 If fm1x2 is a log supermodular density function on 3amp2 then 1 the conditional density fm1lm2 is log supermodular 2 the conditional cumulatiue distribution function Fx1lx2 is log supermodular 3 the conditional cumulatiue distribution function Fx1lx2 is nonincreasing in 2 1 The conditional density function fm1lz2 can be written as fm1z2f2m2 where f2 is the unconditional marginal density of 2 Then log fm1lz2 logfz1m2 7 log f2m2 Super modularity is not affected by additive terms that do not include both variables so log fm1lz2 is supermodular if and only if logfm1 2 is supermodular 2 Fz1lm2 fowl fslz2ds which we just showed was log supermodular in our last result 3 Let 2 gt Since Fz1lm2 is log supermodular F1l2 lt Fil2 F1l 2 Fil 2 for z gt 1 Take z a 00 and the right hand side goes to 1 since lag eventually goes to 1 regardless of the value of 2 So F1l2 lt 1 F gala2 or Fz1lm2 for 2 gt This means that if 1 and 2 are affiliated then a higher value of 2 implies a higher distribution of 1 in the sense of rst order stochastic dominance That is the distribution of 1 conditional on a high value of 2 rst order stochastically dominates the distribution of 1 conditional on a lower value of 2 This is the sense in which affiliation implies positive correlation and it s pretty powerful since we already know that if one variable rst order stochastically dominates another it gives a higher expected value of any increasing function We ll come back to this in a bit Next we show that if a set of random variables are af liated then any subset of them are af liated as well Lemma 6 If fx1 mn is a log supermodular probability density then gz1 zn1 fz1 xn1 sds is a log supermodular probability density as well Again we ll show it for the case where f is twice differentiable so that log supermodularity is the same as positive mixed partials The result holds more generally 9 is log supermodular if 32 logg is increasing in mi Taking the derivative aimiffml7quot397nil7sds 7 ffim1mn1sds ffx1mn1sds ffx1mn1sds 8 8x1 10g917 7n71 7 7 f f rm sgtds ffx1mn1sds filt177n7175 E m m 10m1739nvmn7175 1 n 1 where the expectation is taken over 5 conditional on the rst n 7 1 variables We need to show this is increasing in mi An increase in mi has two effects First since 1 zn71 s are af liated an increase in mi increases the ratio directly When we x the values of all variables but mi and an it s clear that these remaining two are af liated since log 1 has positive mixed partials when other variables are held xed That means that an increase in mi leads to an increase in the conditional distribution of s in the rst order stochastic dominance sense by the last result But by a iliation is increasing in an and therefore its expectation over zn is increasing in mi So an increase in mi increases the whole expression which implies that g is log supermodular The next result is that conditional expectations of increasing functions of af liated variables behave predictably That is Lemma 7 Suppose 1 xn are a liated For any function g 9 gt 9 which is bounded and isotone increasing in all its arguments E9177mnl1 90 is increasing in m The proof is by induction on n and iterated expectations First suppose n 2 Then for m gt m 719 E99E72l1 95 Z E97902l901 7 90 Z E9907902l901 95 hx The rst inequality is because z gt m and g is increasing in both its arguments The second is because the distribution of 2 conditional on 1 z rst order stochastically dominates the distribution of 2 conditional on 1 z and we just showed the expected value of any increasing function is higher over a stochastically dominant distribution Now suppose the lemma is known to hold for functions of n 7 1 af liated variables De ne jWy Eg1727 7l901 907902 y and rewrite hz as E2 11 j7 2 By the inductive assumption for a xed m jz y is increasing in y and the distribution of 2 conditional on 1 m is increasing in x by FOSD So by the same logic for z gt z Mm E m2 m1 z The Milgrom and Weber paper actually give a different formulation of this result if m1 an are af liated and g is any increasing function then the expectation E 9m1 mnlm1 E a1b1z2 E a2b2 an 6 am bnl is increasing in all 2n of its arguments ai and bi Finally we will show that a bunch of random variables are a iliated if and only if their order statistics are a iliated Lemma 8 Suppose m1 xny have a joint density 1 which is symmetric in the rst n argu ments Then m1 mmy are a liated if and only if m1z2 m y are a liated where m is the ith highest of 1 xn The density of the n 1 random variables m1m2 am y is flt17 27 7n7 y nlflt 1727 7nvy1m1gtm2gtmgtmn This is because the density is 0 unless 1 gt 2 gt gt zn and Zfazy when this does hold where the sum is taken over the permutations of the elements of x but there are nl of these permutations and by assumption 1 takes the same value at all of them We need to show that f satis es the a iliation inequality everywhere if and only if 1 does We ll show the reverse First suppose f violates the a iliation inequality somewhere that is suppose there exist z y and m 3 such that mm 24 A m 7y f7y V 967 2 lt fm7yfx 7y There are two possible cases either one of the indicator functions 1 is zero on the left hand side or not Since the right hand side is strictly positive the indicator functions must both be 1 on the right hand side If not then nlf at all four points where it is being evaluated and so the a iliation inequality is also violated by f If so this means that although 1 gt 2 gt gt ml and 3 gt z z gt gt mg either the meet or the join are ordered incorrectly It is easy to show that this cannot occur so if f violates the a iliation inequality so does 1 On the other hand suppose f violates the a iliation inequality somewhere Then there is some point where fijij flti7 jvil jgt 1W 903 his NE 9597 957M By continuity we can nd such points like this such that mi and are very close together and similarly we can nd such points where M and are very close together By symmetry of 1 then reorder its arguments so that 1 gt 2 gt gt zn and we ll get a violation of the a iliation inequality for f at the same point So to sum up what we ve shown about affiliation 0 Random variables 12 mn are affiliated if and only if their joint density function is log supermodular o This is equivalent to NM 9 La f 95 9041 2 27177ijfi7277ij for z gt mg zj gt 9 any mik which is equivalent to the ratio 10901397907397904439 J WWMLM i i i increasing in 357 for any x gt z o Af liation is preserved by any order preserving transformation 12 mn are affiliated if and only if 91z1ggm2gnmn are affiliated where 9 are any strictly increasing functions If ml and 2 are affiliated then the conditional distribution of 2 when 1 z rst order stochastically dominates the conditional distribution of 2 when 1 m lt z o If m1 m2 mn are affiliated then any subset of them are affiliated as well o If m1 m2 mn are affiliated and g is any isotone function then is increasing in 961 0 Suppose the density Of12 zn y is symmetric in its rst 71 arguments Then m1 m2 zn y are affiliated if and only if m1m2 am y are affiliated That gives us the results we ll need to analyze the common auction types and prove some very general results in a world with affiliated signals and interdependent valuations That s what s coming on Thursday
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