aa r X i v : . [ c s . G T ] F e b Heavy Tails Make Happy Buyers
ERIC BAX
In a second-price auction with i.i.d. (independent identically distributed) bidder valuations, adding biddersincreases expected buyer surplus if the distribution of valuations has a sufficiently heavy right tail. While thisdoes not imply that a bidder in an auction should prefer for more bidders to join the auction, it does implythat a bidder should prefer it in exchange for the bidder being allowed to participate in more auctions. Also,for a heavy-tailed valuation distribution, marginal expected seller revenue per added bidder remains strongeven when there are already many bidders.
In a second-price sealed-bid auction, also known as a Vickrey auction [30], bidders submit theirbids, the item is awarded to the bidder with the highest bid, and the winner pays the runner-upbid. The seller’s revenue is the runner-up bid. Adding a bidder increases expected revenue for theseller by increasing competition: if the new bid is the highest bid, then the new bidder wins andpays the previous highest bid; if the new bid is the runner-up, then it increases the price for thewinner.In this paper, we consider whether adding a bidder can also increase expected buyer surplus.Assuming bidders bid their valuations, which is a dominant strategy [13, 16, 30], the buyer’s surplusis the difference between the highest bid (the buyer’s valuation) and the runner-up bid (the buyer’sprice). If the new bid is the highest bid, then the new bidder becomes the new buyer and buyersurplus increases if the difference between the new bid and the previous highest bid is greater thanthe difference between the previous highest and runner-up bids. So buyer surplus benefits fromvaluations that are right-tail outliers. We show that exponential distributions maintain expectedbuyer surplus as bidders are added, and Pareto distributions increase expected buyer surplus. Wealso show that as tails get heavier the ratio of expected buyer surplus to expected seller revenueincreases, even with optimal reserve prices.Ideally, we would show that adding a bidder is a Pareto improvement [20] (in expectation),meaning that it benefits the seller or one of the bidders and it harms no seller or bidder. But this isimpossible: adding a bidder decreases expected surplus for the previous bidders, because the newbid may reduce the previous winner’s surplus by being the new runner-up bid or eliminate theprevious winner’s surplus by being the new winning bid. However, we show that adding a bidder isa Pareto improvement in expectation if the additional bidder is symmetric to the previous bidders.For example, if adding a bidder increases expected buyer surplus, then it is a Pareto improvementto have all bidders participate in an auction rather than select a subset of bidders at random andexclude the others. Similarly, for a set of auctions with each bidder excluded from one auction andall others participating, it is a Pareto improvement for all bidders to participate in all auctions.Our analysis uses order statistics [7] of the distribution of bidder valuations. Order statistics areoften used to analyze auctions [13, 16, 18], since, for truthful second-price auctions, the winningbid is the greatest order statistic among valuations and the price is the next-greatest order statistic.Second-price auctions (and variations on them) are the basis for online marketplaces for adver-tising [8, 29] and for goods and services [16]. Distributions of valuations are a subject of muchstudy, because they determine optimal reserve prices [17, 24], and estimates of the distributionsdetermine reserve prices for actual marketplaces [3, 6, 19].Our results depend on the shape of the valuation distribution. With a uniform distribution oversome range, adding more bidders drives both the average winning bid and the average runner-upbid toward the right of the range, pushing these averages closer to each other, which squeezes the
Manuscript submitted for review to GAMES 2020. ric Bax
Assume there are n + X , . . . , X n + be the bidders’ valuations, drawn i.i.d. froma distribution D . Let X ( ) , . . . , X ( n + ) be the order statistics for X , . . . , X n + – the values rankedfrom least to greatest, with any ties broken randomly.Let p n + be the buyer’s surplus in a second-price auction if all n + p n be the buyer’s surplus if only the first n bidders participate. Assume each bidder bids theirprivate value, since the second-price auction is truthful [13, 16, 30]. Recall that buyer surplus is thedifference between the highest bid and the runner-up bid. The following theorem gives a conditionfor adding bidder n + n + D . Theorem 2.1. E ( p n + − p n ) = n + [ E ( X ( n + ) − X ( n ) ) − E ( X ( n ) − X ( n − ) )] , where all expectations are over ( X , . . . , X n + ) ∼ D n + . Proof.
To assess the impact of adding bidder n + n bidders, we will assessthe (opposite) impact of removing bidder n + n + p n + = X ( n + ) − X ( n ) . With probability n + , X n + is X ( n + ) (the highest bid), so removing bidder n + X ( n ) , and the new runner-up bid is X ( n − ) .So buyer surplus becomes p n = X ( n ) − X ( n − ) . Also with probability n + , X n + is X ( n ) . If so, then removing bidder n + X ( n ) to X ( n − ) and increasing buyer surplus to p n = X ( n + ) − X ( n − ) . Alternatively, with probability n − n + , X n + is neither the highest nor the runner-up bid. Then remov-ing bidder n + p n = p n + .The cases are: (a) X n + is X ( n + ) , (b) X n + is X ( n ) , and (c) X n + is one of X ( ) , . . . , X ( n − ) . For eachcase, multiply the probability of occurrence by the expectation of p n + − p n given the occurrence: E ( p n + − p n ) = n + E [( X ( n + ) − X ( n ) ) − ( X ( n ) − X ( n − ) )] + n + E [( X ( n + ) − X ( n ) ) − ( X ( n + ) − X ( n − ) )] + n − n + E [ ] . ric Bax = n + E [ X ( n + ) − X ( n ) + X ( n − ) ] . Use linearity of expectation again and rearrange terms: = n + [ E ( X ( n + ) − X ( n ) ) − E ( X ( n ) − X ( n − ) )] . (cid:3) The theorem says that adding a bidder increases expected buyer surplus if the expected differ-ence between the top two bids with n + D must tend toproduce winners that are outliers. Exponential distributions have cdf: F ( x ) = (cid:26) − e − λx if x ≥
00 if x < , where λ > Corollary 2.2. If D is an exponential distribution, then E ( p n + − p n ) = . Proof.
We will show that E ( X ( n + ) − X ( n ) ) − E ( X ( n ) − X ( n − ) ) = . The order statistics for an exponential distribution have known distributions [23] [10] (pg. 20) [7](Ch. 2): X ( i ) ∼ λ i Õ j = Z j n − j + ! , where Z j are exponential random variables with λ =
1. Since the mean of an exponential distribu-tion is λ , ∀ j : EZ j =
1. So E ( X ( n + ) − X ( n ) ) − E ( X ( n ) − X ( n − ) ) = λ n + Õ j = n − j + − λ n Õ j = n − j + ! − λ n Õ j = n − j + − λ n − Õ j = n − j + ! = λ n + Õ j = n + n − j + − n Õ j = n n − j + ! = λ (cid:18) − ( ) (cid:19) = . (cid:3) ric Bax Theorem 2.3.
For second-price auctions with n + bidders having i.i.d. exponential valuationdistributions, the ratio of expected seller revenue to expected buyer surplus is approximately ln ( n + ) + γ − , where γ is the Euler - Mascheroni constant ( γ ≈ . ). Proof.
Expected buyer surplus is the expected difference between the top two bids: E ( X ( n + ) − X ( n ) ) = λ n + Õ j = n − j + − λ n Õ j = n − j + ! = λ . Expected seller revenue is the expected second bid: EX ( n ) = λ n Õ j = n − j + = λ (cid:18) + + . . . + n + (cid:19) ≈ λ ( ln ( n + ) + γ − ) because 1 + + . . . + n ≈ ln n + γ . (cid:3) The seller takes the lion’s share of the total auction value as the number of bidders increases: as n → ∞ , ln n → ∞ , but expected buyer surplus remains constant. Can the buyer instead take thelion’s share? Yes, if the tail is heavy enough, as we show later.But first, consider how using an optimal auction [17, 24] with a reserve price, instead of a straightsecond-price auction, would affect our results. In Appendix A, we show that with the Myerson-optimal reserve price of λ , the expected buyer surplus with n + λ [ − ( − e − ) n + ] . This converges from below to the result for the second-price auction, λ , as n increases, becausethe reserve price is less likely to have an impact as more bidders are added. So expected buyersurplus increases with each new bidder, but the amount of increase decreases with each new bidder,making the increase in expected buyer surplus nearly zero for large n .Both auction types have the same sum of seller revenue and buyer surplus; it is the top bid.Since the optimal auction charges the buyer a bit more in expectation, it increases expected sellerrevenue by the same amount: λ ( − e − ) n + , and expected seller revenue still dominates expectedbuyer surplus as n increases. Type I Pareto distributions have cdf: F ( x ) = (cid:26) − (cid:0) ax (cid:1) v if x ≥ a x < a , where a > v > ric Bax n + EX ( i ) = a ( n + ) ! ( n + − i ) ! Γ ( n + − i − v ) Γ ( n + − v ) , where Γ () is the gamma function, which is a continuous version of the factorial: Γ ( n + ) = n ! forall integers n and Γ ( z + ) = z Γ ( z ) for all real z >
0. We use both of these equalities to simplifyexpectations for order statistics in the next proof.
Corollary 2.4. If D is a Type I Pareto distribution, then adding a bidder increases expected buyersurplus: E ( p n + − p n ) > . Proof.
We will show that E ( X ( n + ) − X ( n ) ) − E ( X ( n ) − X ( n − ) ) > . Let д ( n , a , v ) ≡ a Γ ( n + ) Γ ( n + − v ) Γ ( − v ) . Then EX ( n + ) = д ( n , a , v ) , EX ( n ) = д ( n , a , v )( − v ) , and EX ( n − ) = д ( n , a , v )( − v )( − v ) . So E ( X ( n + ) − X ( n ) ) − E ( X ( n ) − X ( n − ) ) = д ( n , a , v ) (cid:18)(cid:20) − ( − v ) (cid:21) − (cid:20) ( − v ) − ( − v )( − v ) (cid:21) (cid:19) The terms in д () are all positive. The term in parentheses is:1 v − ( − v )( − ( − v )) = v − ( − v ) v = v > . (cid:3) Compare expected buyer surplus to expected seller revenue. Expected buyer surplus is EX ( n + ) − EX ( n ) = д ( n , a , v ) (cid:18) − ( − v ) (cid:19) = д ( n , a , v ) v . Expected seller revenue is EX ( n ) = д ( n , a , v )( − v ) . ric Bax Γ ( n + ) Γ ( n + − v ) ≈ ( n + ) v , expected buyer surplus is ≈ a v ( n + ) v Γ ( − v ) , and expected seller revenue is ≈ a ( − v )( n + ) v Γ ( − v ) . Both increase with the number of bidders.The ratio of expected seller revenue to expected buyer surplus is:1 − v v = v − . So expected seller revenue equals expected buyer surplus at v =
2. As v → ∞ , the seller takesmore of the value; as v →
1, the buyer takes more.
Consider what happens as v → EX ( n ) = д ( n , a , v )( − v ) = a Γ ( n + ) Γ ( n + − v ) Γ ( − v )( − v ) , and Γ ( − v )( − v ) = Γ ( − v ) , so = a Γ ( n + ) Γ ( n + − v ) Γ ( − v ) . Since Γ ( n + ) = n ! for all integers n and Γ ( ) =
1, expected seller revenue approaches a ( n + ) .For expected buyer surplus: lim v → v д ( n , a , v ) = lim v → v a Γ ( n + ) Γ ( n + − v ) Γ ( − v ) = a ( n + ) lim v → Γ ( − v ) . And lim z → + Γ ( z ) = ∞ . So expected buyer surplus approaches infinity.Can the seller impose a reserve price to raise expected seller revenue? The answer is no, becausewith v = r and v = n bidders: n ∫ ∞ r r F ( r ) n − f ( x ) dx + n ( n − ) ∫ ∞ r xF ( x ) n − f ( x )[ − F ( x )] dx ric Bax n possible top bidders, with revenue r if the other n − r (probability F ( r ) n − ) and the top bid is above r (probability f ( x ) dx integratedover x from r to ∞ ). The second term is revenue from the second price: n ( n − ) possible top andsecond bidder combinations, with revenue the second price x if the other n − x (probability F ( x ) n − ) and the top bid is above x (probability [ − F ( x )] ).The first term (revenue from reserve price) is = anr ( − ar ) n − ∫ ∞ r x − dx = anr ( − ar ) n − (cid:2) − x − (cid:3)(cid:12)(cid:12) ∞ r = an ( − ar ) n − . The second term (revenue from second price) is = n ( n − ) ∫ ∞ r a ( − ax ) n − dx = an h ( − ax ) n − i(cid:12)(cid:12)(cid:12) ∞ r = an − an ( − ar ) n − . The two terms sum to an ; the reserve price has no effect. For this reason, the distribution with v = If adding a bidder increases expected surplus for the winner, then should n bidders welcome onemore bidder to an auction? No, because the new bidder can decrease surplus for the previouswinner – by being the new winner or the new runner-up – and cannot increase surplus for theprevious bidders. Stated another way: adding a bidder decreases other bidders’ probabilities ofwinning and may potentially raise the winner’s price. In fact: Theorem 3.1.
For any bidder valuation distribution D that has support over at least two differentvalues, expected surplus per participating bidder decreases with each additional bidder: Ep n + n + < Ep n n . Proof.
Because the new bidder can take the win from the previous winner or increase theirprice, adding a bidder reduces expected surplus for the previous bidders. By symmetry amongbidders, the new bidder has the same expected surplus as the other bidders. So the added biddermust decrease expected surplus per participating bidder.This argument fails only if the top two bids among the previous bidders are equal with probabil-ity one. In that case, there is zero expected surplus, and adding a new bidder does not change that.But the theorem requires D to have support over at least two different values, so the probabilityof a tie among the top two bids is less than one. (cid:3) If we consider the new bidder and previous bidders as a class, then the expected surplus perclass member increases with wider participation if Ep n + > Ep n , because that clearly implies: Ep n + n + > Ep n n + . ric Bax n + Ep n + n + . But it changes each pre-vious bidder’s expected surplus from Ep n n to Ep n + n + , which is a decrease according to Theorem 3.1.So this is not a Pareto improvement, because it harms the previous bidders.However, there is a Pareto improvement if each of the n + Ep n + n + − Ep n n + . Similarly, suppose there is a set of n + n + ( n + ) Ep n + n + − n Ep n n = Ep n + − Ep n . For a more general framework to evaluate whether a change in auction participation over multi-ple auctions is a Pareto improvement in expectation, assume that all bidders draw their valuationsi.i.d. for each auction, though there may be different valuation distributions for different auctions.Let i index potential bidders. Let j index auctions. Let I ( i , j ) be one if bidder i is allowed to bid inauction j and zero otherwise. For each auction j , let b j = Õ i I ( i , j ) be the number of bidders. Let Ep j ( b ) be the expected buyer surplus for auction j if b bidders par-ticipate. Then the expected surplus for potential bidder i over all the auctions is Õ j I ( i , j ) Ep j ( b j ) b j . Let Er j ( b ) be the seller’s expected revenue for auction j if b bidders participate. If there is a singleseller for all auctions, then their expected revenue over all auctions is Õ j Er j ( b j ) . For a new bidding participation arrangement, let I ′ ( i , j ) indicate whether bidder i is allowed to bidin auction j . For each auction j , let b ′ j = Õ i I ′ ( i , j ) be the new number of bidders. Then the new arrangement is a Pareto improvement in expectationif ∀ i : Õ j I ′ ( i , j ) Ep j ( b ′ j ) b ′ j ≥ Õ j I ( i , j ) Ep j ( b j ) b j , and, if there is a single seller, Õ j Er j ( b ′ j ) ≥ Õ j Er j ( b j ) , and the inequality is strict for at least one bidder or the seller. If each auction has a different seller,then we require ∀ j : Er j ( b ′ j ) ≥ Er j ( b j ) ric Bax Consider how the number of bidders affects expected seller revenue. In accord with common sense,having more bidders increases competition, which increases expected seller revenue:
Theorem 4.1.
For any valuation distribution D that has support over multiple values, adding abidder to a second-price auction increases expected seller revenue. Proof.
Seller revenue is the second-highest bidder valuation. Regardless of the rank of the addi-tional bid among itself and the previous bids, the second-highest bid cannot decrease. Let s be thesecond-highest bid, excluding the additional one. Since there is support over multiple values, thereis positive probability that s is not a maximum among the distribution values that have support.If it is not a maximum, then there is positive probability that the additional bid is greater than s ,which increases the second-highest bid if we include the additional bid. (cid:3) In general, heavier-tailed valuation distributions produce more expected seller revenue peradded bidder:
Theorem 4.2.
Let s n + be expected seller revenue for n + bidders and let s n be expected sellerrevenue for n bidders. Then: • For a uniform valuation distribution: D ∼ U [ , ] , s n + − s n ∈ O (cid:16) n (cid:17) . • For an exponential valuation distribution, s n + − s n ∈ O (cid:0) n (cid:1) . • For a Type I Pareto valuation distribution, s n + − s n ∈ O (cid:16) n − v (cid:17) . Proof.
To examine the expected increase in expected seller revenue from adding a bidder toa competition with n bidders, analyze the difference in expected second-highest value over n + n values for the valuation distributions. For the (non-heavy-tailed) uniform distribution,the expectation of the i th order statistic for n samples is in + [7]. So the expected increase in sellerrevenue is s n + − s n = nn + − n − n + = n + n + ∈ O (cid:18) n (cid:19) . Next, consider the exponential distribution. Recall that s n + = λ (cid:18) + + . . . + n + (cid:19) . So s n + − s n = λ (cid:18) n + (cid:19) ∈ O (cid:18) n (cid:19) . For the Type I Pareto distribution, s n + ≈ a ( − v )( n + ) v Γ ( − v ) . So s n + − s n ≈ a ( − v ) Γ ( − v ) h ( n + ) v − n v i . Since ∂∂ nn v = v n − v , s n + − s n ∈ O (cid:18) n − v (cid:19) . ric Bax (cid:3) For the Pareto Type I distribution, as v → ∞ , the limit is O (cid:0) n (cid:1) , as for the exponential distri-bution. For v =
2, it is O (cid:16) √ n (cid:17) . As v → + , the limit is O ( ) , indicating that as the tail becomesheavier, the seller gains nearly as much from adding a bidder when there are already many biddersas from adding a bidder when there are only a few. We have shown that expected buyer surplus can increase with increased competition if the val-uation distribution is sufficiently heavy-tailed. While adding a new bidder can never increaseper-bidder expected surplus, it can increase expected surplus over the class of new and previousbidders. As a result, under sufficiently heavy-tailed valuation distributions, each potential biddershould prefer wider participation in auctions for themselves and their fellow bidders over narrowerparticipation for all, if the restrictions on their participation are symmetric to those on other bid-ders.In the future, it would be interesting to explore whether or how heavy-tailed valuation distri-butions occur in practice. They seem most likely to occur in scenarios in which great gains arepossible. For example, if we model approaches to curing serious and widespread health condi-tions (such as cancer or aging) as bids and the value to society for solutions as valuations, thena heavy-tailed valuation distribution corresponds to the possibility that some creative approachwill yield great progress. In such cases, adding efforts to develop creative solutions increases theexpected profit for the provider of the best solution (corresponding to buyer surplus if the providercan charge as much as the value of the next-best solution), in addition to increasing the expectedbenefit to society.In some cases, valuation distributions may mimic heavy-tailed distributions over some range,but lack support above some finite upper bound. In these cases, it would be interesting to analyzemarginal expected buyer surplus per added bidder as a function of the number of bidders. If thevaluation distribution is unknown, but there is bid information from previous auctions with thesame distribution, it would be interesting to explore whether it is possible to accurately estimatethe number of bidders needed to maximize expected buyer surplus. It would also be interesting toexamine how non-i.i.d. valuation distributions affect the results in this paper, especially if somepotential bidders have heavy-tailed valuation distributions and others do not.
A HEAVY TAILS AND RESERVE PRICES
In this appendix, we examine how using a reserve price to convert the second price auction tothe Myerson [17] optimal auction affects our results in this paper. For the exponential distribution,a reserve price decreases expected buyer surplus, but the difference between second-price andMyerson-optimal expected buyer surplus decreases exponentially in the number of bidders. Inter-estingly, for the Pareto Type I distribution, a reserve price does not make sense, because no reserveprice increases expected seller revenue to more than that from a straight second-price auction.
A.1 Reserves for Exponential
First, observe that the exponential distribution meets the required Myerson regularity condition:that x − − F ( x ) f ( x ) = x − λ ric Bax x . Set this equal to zero and solve for the optimal reserve price: r = λ . Theorem A.1.
For n bidders with valuations drawn i.i.d. from an exponential distribution, optimal-auction expected buyer surplus is λ [ − ( − e − ) n ] . Proof.
The expected buyer surplus is the expected difference between the top bid and the re-serve price if the second bid is below the reserve price, and the expected difference between thetop two bids otherwise. Using t for the top bid and s for the second-highest bid, this is n ∫ ∞ t = r ( t − r ) F ( r ) n − f ( t ) dt (1) + n ( n − ) ∫ ∞ s = r F ( s ) n − f ( s ) ds ∫ ∞ t = s ( t − s ) f ( t ) dt . (2)The first integral is = nF ( r ) n − ∫ ∞ t = r ( t − r ) f ( t ) dt = n ( − e − λr ) n − ∫ ∞ t = r ( t − r ) λe − λt dt . Change the variable of integration from t to t + r : ∫ ∞ t = r ( t − r ) λe − λt dt (3) = ∫ ∞ t = tλe − λ ( t + r ) dt = e − λr ∫ ∞ t = tλe − λt dt . The integral is the mean of the exponential distribution, so it is λ , and Expression 1 is = nλ ( − e − λr ) n − e − λr . (4)For Expression 2, first examine the inner integral. It is Expression 3, with s instead of r . So it is = e − λs λ . So Expression 2 is = n ( n − ) ∫ ∞ s = r F ( s ) n − λ f ( s ) ds e − λs = n ( n − ) ∫ ∞ s = r ( − e − λs ) n − e − λs ds e − λs . Use integration by parts ( ∫ u dv = uv − ∫ v du ), with u = e − λs and dv = ( − e − λs ) n − e − λs ds , so du = − λe − λs ds and v = ( n − ) λ ( − e − λs ) n − : = ne − λs λ ( − e − λs ) n − (cid:12)(cid:12)(cid:12)(cid:12) ∞ s = r + n ∫ ∞ s = r ( − e − λs ) n − e − λs ds ric Bax = ne − λs λ ( − e − λs ) n − (cid:12)(cid:12)(cid:12)(cid:12) ∞ s = r + λ ( − e − λs ) n (cid:12)(cid:12)(cid:12)(cid:12) ∞ s = r . Recall that r = λ : = [ − nλ ( − e − ) n − e − ] + [ λ − λ ( − e − ) n ] = λ [ − ( − e − ) n ] − nλ ( − e − ) n − e − . The right term is the same as Expression 4, but with a minus sign. So they cancel, giving expectedbuyer surplus: 1 λ [ − ( − e − ) n ] . (cid:3) For comparison, without a reserve price, the expected seller revenue is λ . A.2 Reserves for Pareto Type I
Recall that Type I Pareto distributions have cdf: F ( x ) = (cid:26) − (cid:0) ax (cid:1) v if x ≥ a x < a , where a > v > A.2.1 Single Bidder.
For a single bidder and reserve price r , expected seller revenue is r [ − F ( r )] :the seller receives r if the bid is at or above r and zero otherwise. (We assume the bidder bids theirvaluation, because truth-telling is a Nash equilibrium for the second price auction with reserveprices. [17, 24, 30]) For a bid drawn from a Pareto Type I distribution, if r ≤ a then F ( r ) =
0, soexpected seller revenue is r , indicating that we should set the reserve to at least a .For r ≥ a , r [ − F ( r )] = a v r v − . The derivative of expected seller revenue with respect to r is negative for r ≥ a : [ a v r v − ] ′ = −( v − ) a v r v . So a is the optimal reserve price for a single bidder.Note that lim v → + −( v − ) a v r v = . So as v → + , any reserve price becomes almost as effective as r = a . ric Bax A.2.2 Multiple Bidders.
But is it possible that a higher reserve price would generate higher ex-pected seller revenue with more bidders? We will show that the answer is no. The expected sellerrevenue with reserve price r for n > h ( r ) = n ∫ ∞ r r F ( r ) n − f ( x ) dx + n ( n − ) ∫ ∞ r xF ( x ) n − f ( x )[ − F ( x )] dx . The first term is revenue from the reserve price: n possible top bidders, with revenue r if the other n − r (probability F ( r ) n − ) and the top bid is above r (probability f ( x ) dx integratedover x from r to ∞ ). The second term is revenue from the second price: n ( n − ) possible top andsecond bidder combinations, with revenue the second price x if the other n − x (probability F ( x ) n − ) and the top bid is above x (probability [ − F ( x )] ).Following the derivation of optimal reserve prices [17, 24], we will find h ′ ( r ) , the derivative ofexpected seller revenue h with respect to reserve price r . If it is negative for all r > a , then settingany reserve price above the minimum of the distribution decreases expected seller revenue. Pullconstants out of the first integral and reverse the direction of integration of the second: h ( r ) = nr F ( r ) n − ∫ ∞ r f ( x ) dx − n ( n − ) ∫ r ∞ xF ( x ) n − f ( x )[ − F ( x )] dx . To take the derivative, apply the product rule to the first term to get the first three terms belowand the fundamental theorem of calculus to the second term to get the last term below: h ′ ( r ) = nF ( r ) n − ∫ ∞ r f ( x ) dx + nr ( n − ) F ( r ) n − f ( r ) ∫ ∞ r f ( x ) dx + nr F ( r ) n − (cid:20)∫ ∞ r f ( x ) dx (cid:21) ′ − n ( n − ) r F ( r ) n − f ( r )[ − F ( r )] . Note that ∫ ∞ r f ( x ) dx = − F ( r ) , and reverse the direction of integration then apply the fundamental theorem of calculus to thederivative of the integral in the third term: h ′ ( r ) = nF ( r ) n − [ − F ( r )] + nr ( n − ) F ( r ) n − f ( r )[ − F ( r )]− nr F ( r ) n − f ( r )− n ( n − ) r F ( r ) n − f ( r )[ − F ( r )] . The second and fourth term cancel. Combine the other two terms: h ′ ( r ) = nF ( r ) n − [ − F ( r ) − r f ( r )] . ric Bax r gives the well-known expression for the optimalreserve price: r ∗ = − F ( r ) f ( r ) . A.2.3 Reserve Price Harms Revenue for Pareto.
For the Pareto Type I distribution, the pdf is f ( x ) = v (cid:16) ax (cid:17) v x . So h ′ ( r ) = nF ( r ) n − [ − F ( r ) − rv (cid:16) ar (cid:17) v r ] = nF ( r ) n − [ − F ( r ) − v (cid:16) ar (cid:17) v ] . Since 1 − F ( r ) = (cid:16) ar (cid:17) v , we have h ′ ( r ) = nF ( r ) n − [ − F ( r )]( − v ) . For v >
1, this is negative for all r > a , so no reserve price benefits the seller in expectation. A.2.4 Reserves and Competition.
Does adding bidders make the expected losses from a reserveprice larger or smaller? Take the ratio of h ′ ( r ) for n + n : ( n + ) F ( r ) n [ − F ( r )]( − v ) nF ( r ) n − [ − F ( r )]( − v ) = n + n F ( r ) . If this ratio is greater than one, n + n F ( r ) ≥ , then adding a bidder makes the expected revenue loss worse for the seller. Using the definition of F and solving for r gives: r ≤ ( n + ) v a . Equivalently, solving for n, n ≥ (cid:16) ra (cid:17) v − . So, for v >
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