Search Prominence and Recall Costs

Transcription

1 Search Prominence and Recall Costs Arthr Fishman Bar Ilan University Dmitry Lbensky Indiana University October 15, 2016 Abstract Althogh it is widely held that sellers prefer to appear early in a consmer s search, some evidence from online markets sggests this need not be the case. We develop a model which incorporates costly search and costly recall and demonstrate that when both frictions are small, appearing later may be better. When recall is free, prominence is desirable by standard logic, however costly recall indces a tradeoff it benefits the prominent seller by redcing the initial search bt also benefits the later seller by preventing retrn conditional on search. We show that later can be better whenever vales are sfficiently correlated, or whenever vales are drawn independently from a distribtion in which two draws are likely to be near one another. arthr.fishman@gmail.com dmitry.lbensky@gmail.com

2 1 Introdction In the consmer search literatre, it is often assmed that the order in which consmers search is random. However, there are many sitations in which the order of search is not random (Armstrong (2016)). For example, online search reslts are arranged by relevance andconsmers typically beginat the topofthe pageand work their way down (Urs (2016)). Casal intition sggests that when consmer search is ordered in this way, sellers that are higher p onthe list have an advantageover those which are lower on thelist, andrecent theoretical work has confirmed this. Whether prodcts are homogeneos (Arbatskaya (2007)) or differentiated (Armstrong et al. (2009), Armstrong and Zho (2011a), Zho (2011)), the search cost allows a prominent seller to win consmers even in sitations in which he offers a worse tility than competitors. However, the data presented in Figre 1 (from De los Santos and Kolayev (2012), p. 31) seem to tell a different story. Depicted are the nmber of clicks for search reslts on Kayak.com for hotel stays in the city of Chicago dring May For each search there were 15 reslts shown per page, with mltiple pages available to the consmer by clicking Next, and the figre shows the total nmber of clicks at each position. Reslts were retrned in decreasing order of poplarity as is evidenced by the decreasing click rate within each page. However the monotonicity is broken across pages. For instance, the 16th reslt which appears at the top of page two received more clicks than any of the reslts 4-15 on the first page, and similarly the 31st reslt which appears at the top of page three received more clicks than any of the reslts on the second page. In other words, the data sggests that often later is better. Motivated by this example, we develop a more comprehensive model of search prominence which can accommodate both settings in which prominence confers a competitive advantage, as in the aforementioned literatre, as well as settings in which later is better. Specifically, it is typically assmed that searching for a new option is costly bt that it is costless to retrn to options previosly observed if frther search proves disappointing. While this assmption is made for convenience and is innocos when the nmber of sellers is infinite, Janssen and Parakhonyak (2014) shows it can be conseqential when the nmber of seller is finite, which is precisely when search prominence has bite (Armstrong et al. (2009)). In reality recall is rarely costless, since often the very costs associated with the travel and time necessary to obtain a new offer also apply to retrning and accepting an offer from a previos seller. Frthermore, even when the recall cost is minimal as with many online markets, it may still play an important role in the same sense that a minimal search costs plays a role in Diamond (1971). We ths introdce a cost of recall into a setting of search with prominence and investigate how this cost affects the advantage of prominence. Specifically, we consider a differentiated prodct dopoly in the style of Wolinsky (1986) in which 1

3 !*!#$ &) %( '&!$ $ * #!%!!* &* %( '!!# & $ )!#!%" )& %( ')!) * '!!'!#* &% %% '$!$!(!!!!!#& $( #! ''!) &!( '!(!*) $* #$ '& "!# % % +,-./01- / " $ & ) '!# $( %' '"!( $ % #!"#!!$ %& '% '( ) & # '%#!!# )* '"!&!' * & $! *% '#!#!( ) # '&* * % # '! '&# '"' %"! %)#!(* &$ ')!"!# *!!') )) #% '%!#!( #!*) &$ #) #(!(!% ) '!% $" %$ '%!'!( & #((!'# &" %)!" '! " ( "(" (! ' # % * ) & $ / :.;,-:-.<=4>?.@>>7::-A Figre 1: Clicks for hotel searches a consmer pays a search cost to visit each firm to learn the match vale, and mst visit the prominent firm first and the follower firm second. In addition, a consmer that visits both firms and then wishes to by from the initial prominent firm mst pay a retrn cost to do so. In this setting a cost of recall has two opposing effects. On the one hand, conditional on the consmer searching past the prominent seller, costly recall makes retrning to that seller less attractive which redces the profitability of prominence. On the other hand, anticipating that retrn will be costly, a consmer is less inclined to search, which works to the advantage of prominence. We show that when the retrn cost is small the first effect may dominate and later may be better. More precisely, we identify two characterizations of match distribtions nder which the prominent firm may be less profitable when the search cost and the retrn cost are small. The first, called chnkiness, measres the likelihood that two independent draws from the same distribtion are near one another. We show that when the 2

4 match distribtion is sfficiently chnky then for small search and retrn costs later is better. Similarly, when the distribtion is not chnky bt the two draws are sfficiently correlated, then again later is better for small search and retrn costs. The intition is as follows. Small search and retrn costs encorage consmers to search a second time in the hope of getting a better match, ths redcing the benefit of prominence. Frthermore, if the distribtion is sfficiently chnky or the tilities are correlated, then there is a high probability that the two sellers offers are tied, or nearly tied, in which case the retrn cost breaks the tie in favor of the second seller. In these cases, prominence is a competitive disadvantage and later is better. Or analysis takes thesearch orderasgiven, however themodel we solve mayalso bethoght of as the second stage of a game in markets in which firms compete in the first stage. Sellers may compete for prominence throgh advertising (Haan and Moraga-González (2011)), price setting (Haan et al. (2015), Armstrong and Zho (2011b)), obfscation or disclosre (Wilson (2010)), or other methods. Or reslts sggest that the intensity of the first stage competition strongly depends on the magnitdes of the search and retrn costs and the distribtional properties of match vales, and that in fact sometimes no competition shold occr. We take as given the distribtions of match qalities, which too cold be inflenced by sellers in the first stage throgh price setting, investments in qality, or other means. Becase we wish to characterize the effects of prominence regardless of the process throgh which it emerges, we remain agnostic abot these particlars of the first stage. We begin with a simple example in Section 2 and demonstrate that for all symmetric threepoint distribtions later is better when search and recall costs are small. In Section 3 we introdce the general setting and describe the optimal search strategy and ensing seller payoffs when recall costs are positive. In Section 4 we restrict attention to cases in which match vales are drawn independently, as is typically assmed in search models, and in Section 5 we instead allow for draws to be correlated. Section 6 then incldes a discssion of the application of or reslts to several topics of interest. 2 An Example A consmer with nit demand may by from either seller A or seller B. The consmer s tility for either seller i s prodct is ncertain and is given by L with probability x i = M with probability 1 2x, H with probability x with A and B drawn independently, H M = M L = δ, and x 1. The consmer does 2 not initially know her tilities bt may learn them by searching seqentially. The order of 3

5 search is exogenosly determined, sch that she first searches at the prominent seller A and then at the other seller B, with search cost s 0 for each visit. In addition there is a recall cost r > 0 if the consmer wishes to retrn and accept at A after having visited B. The consmer has no otside option and mst by either from A or from B. Utilities are ths distribted symmetrically arond M, with x = 1 corresponding to even 3 weights on all three otcomes, x < 1 corresponding to distribtions with a single peak, and 3 x > 1 corresponding to distribtions with a single trogh. We now show that regardless 3 of the shape of the distribtion, when search and recall costs are low the consmer is more likely to by from B than from A. Proposition 1 For any x and δ, if s and r > 0 are sfficiently small so that s +r < δ then x the consmer is more likely to by from B then from A. Proof Consider the consmer s search strategy. Since r < δ by assmption, it is optimal to retrn to A at the end of search whenever A s offer is strictly better than B s. Having observed A = L, the consmer searches since her payoff from this is xl + (1 2x)(L + δ) + x(l + 2δ) s = L + δ s > L, with the ineqality following from the assmption. Having observed A = M, the consmer also searches since her expected payoff from this is (x)(m+δ)+(1 2x)(M)+(x)(M r) s = M+x(δ r) s > M, with the ineqality again following from the assmption. Finally having observed A = H it is optimal for the consmer not to search, since the payoff to doing so is (x)(h r)+(1 2x)(H r)+(x)(h) s = H (1 x)r s < H. With this search strategy, the probability that A makes the sale is Q A = Pr( A = H) + Pr( A = M)Pr( B = L) = x+(1 2x)x = 2x(1 x), and ths Q A 1 2 since x 1 2. Then, since Q B = 1 Q A it is more likely that B makes the sale. The intition for the reslt is illstrated in Figre 2. Both the leader A and the follower B make the sale whenever their tility is highest, and sales differ only in the cases of ties. In the northeast HH region the search friction breaks the tie in favor of A, and indeed this is the effect captred by Armstrong et al. (2009). For the center MM and sothwest LL regions the tie is broken by the retrn friction in favor of seller B. Since for the given family of symmetric distribtions the LL and HH regions are eqally likely, B sells more often than A. The retrn cost indces a tradeoff for search prominence, helping the leader by decreasing the benefit of searching and helping the follower by decreasing the benefit of retrning conditional on searching. To see this consider a sitation with s = 0 and initially also r = 0, in which ties are broken evenly. The consmer always observes both offers and the sellers are on eqal footing regardless of the order of search; they make the sale whenever their tility is higher and they evenly split sales in case of ties. Now, when the retrn cost increases to 4

6 B H M Q B Q A L L M H A Figre 2: Sales by A and B for the discrete distribtion 0 < r < δ the consmer s search strategy changes so that she immediately accepts A = H bt does not retrn if A = B = M or A = B = L, as depicted in Figre 2. Ths, even in the absence of a search cost the retrn cost benefits A by breaking HH ties in its favor, and benefits B by breaking MM and LL ties in its favor. By contrast, if r > δ then it becomes optimal for the consmer not to contine searching both when A = H and A = M. Now the prominent seller A both captres MM ties and also wins when ( A, B ) = (M,H), and sells with a higher likelihood than B. Ths, in this example retrn costs introdce a tradeoff for whether it pays to be prominent, with the net effect positive for small r and negative for large r. We now consider a more general setting and derive the conditions nder which prominence is not desirable. 3 General Model As in the preceding example there is a prominent seller A and a follower seller B. Aconsmer with search cost s 0 and retrn cost r > 0 has nit demand and her vales for the prodcts of the two sellers are ncertain and drawn from a joint distribtion F( A, B ) with (possibly infinite) spport on [ A,ū A ] [ B,ū B ] R 2, conditional distribtions F A ( A B ) and F B ( B A ), marginal distribtions F A ( A ) and F B ( B ), and and well-defined densities for each. To simplify the analysis we assme as in Anderson and Renalt (1999) that the consmer has no otside option and mst choose from among A and B. 5

7 Search Strategy Having observed both A and B the consmer optimally chooses max( A r, B ). Having observed only A r, the consmer s contination vale is B V( A ) = s + B f B ( B A ) d B + F B ( A r A )( A r) A r } {{ } } {{ } accept B retrn to A = A (s+r) + B A r (1 F B ( B A )) d B. That is, if the consmer searches she pays search cost s, accepts all offers B A r, and retrns to A whenever B < A r. The second line then follows from rearranging terms and applying integration by parts. Assmption 1 w( A,r) B A r (1 F B ( B A )) d B is strictly decreasing in A. As in Rothschild (1974), this assmption is needed to ensre that a threshold strategy is optimal. Specifically, the term w( A,r) captres the expected benefit from searching and is affected by A in two ways. First, when the consmer observes a higher A there are fewer observations of B that make searching worthwhile. Second, when observing a higher A the consmer may pdate her beliefs abot the distribtion of B. If the two tilities are independent or negatively correlated then w is garanteed to decrease in A. However, if the tilities are positively correlated then for w to decrease it mst be that first effect dominates ūb F B ( B A ) the second, or more precisely that d B 1 F B ( A r A ). A A r Given Assmption 1 the consmer ses a threshold û for search, which at interior soltions (i.e. û +r) satisfies 0 = (s+r)+w(û,r). (1) To garantee that an interior soltion exists we assme that search frictions are sfficiently small as follows. B Assmption 2 s+r < min A (1 F B ( B A ))d B. If this assmption holds with a strict ineqality then at A = r the benefit of searching w( A, A ) otweighs the cost s + r and ths the threshold û > + r. Observe that the 6

8 B Q B Q A r û A Figre 3: Sales by A and B for continos distribtions assmption garantees that both search and retrn happen with positive probability. The probability that the consmer bys from A is ths Q A = û +r F B ( A r A )f A ( A ) d A + (1 F A (û)), (2) where in the first term the consmer searches at B and retrns only if B A r, and in the second term the consmer accepts immediately at A. Similarly, the probability the consmer bys from B is B ( ) Q B = min(û, B +r) f B ( B ) d B, (3) B F A where B makes the sale only if A s offer is not better by more than r and is not higher than the threshold û. Figre 3 illstrates the tradeoff associated with prominence. The ble area depicts the set of otcomes in which A < B bt A still makes the sale since A > û, while the red area depicts otcomes in which B < A bt B makes the sales de to the retrn cost. The net effect in general depends on the probability mass in each of these areas, with û also changing as the masses change. We now derive conditions nder which the red area dominates and ths that later is better. 4 Independent Utilities Here we consider a setting in which tilities are independently drawn from the same distribtion F A () = F B () = F(), as is common in the consmer search literatre. Assmption 7

9 1 is satisfied atomatically for any F, and ths the search threshold in (1) is well-defined. Becase the distribtions are identical, the comparison between being visited first and second simplifies to Q 1 2 (Q B Q A ) = [ +r F()f() d + û +r ( ) ] F() F( r) f()d û (1 F())f()d. (4) AsdepictedFigre3,Abenefitsfrombeingfirstwhensearchispre-emptedandconsmersdo not discover B s higher offer, as captred by the third term, while B benefits from being last when retrn to A is preempted, as captred by the two terms in brackets. The comparison of the two effects is not immediate since û is endogenosly determined by the distribtion F and the search and retrn costs. A decrease in retrn cost r has contervailing effects it increases the search threshold û (increasing the second term and redcing the third term) thereby making later better, bt also decreases the likelihood of retrn conditional on search (redcing the integrand of the second term). While in or leading example we cold ensre that later is better when s and r are small, as we now demonstrate there are some distribtions for which it is always better to be first for any vales of the search frictions. Proposition 2 If F is niform on [0,1] then Q 0 for all s and r. Proof To obtain reslt, observe that the search threshold in (1) simplifies to 0 = (s+r)+ 1 û r (1 ) d û = 1 2s+2r+r, and that plgging into (4) obtains Q = (rû 12 ) r2 1 2 (1 û)2 = s. Ths the vale of prominence is fixed at s. This reslt may seem srprising in light of the opening example, in which for every symmetric three-point distribtion, inclding the niform analog with x = 1, later is better for small 3 search and recall costs. However, the key difference across the two settings is the probability mass on realizations ( A, B ) in which the recall cost is pivotal in directing the consmer to by from B rather than retrning to A. In the discrete example these otcomes are the exact ties LL and MM, and the mass on these otcomes is fixed for any r > 0 no matter how small. By contrast, in the continos niform example the set of these otcomes is depicted in Figre 3 by the red band of height r arond the diagonal, and as r decreases so too does the probability mass associated with this band. To ensre that for small enogh r later is better, there mst be sfficient likelihood that otcomes A and B are close. 8

10 Given that tilities are drawn independently, one way to indce close otcomes is to redce the nmber of likely otcomes. For instance, in a steep nimodal distribtion it is likely that A and B are both in the neighborhood of the peak, and if the peak is narrow they are close to one another. This might sggest that the benefit of being last is captred by a standard measre of dispersion sch as variance, bt this is not qite so. In particlar, a bimodal distribtion with the two peaks separated by a large distance has a high variance bt also a high probability of close otcomes. That is, the fact that the two peaks are far apart is not relevant for the probability of close otcomes. With this in mind we propose the following measre. Definition 1 Distribtion G is chnkier than distribtion F if it has the same (possibly infinite) spport [, ū] and g 2 ()d f 2 ()d. Chnkiness measres the average height of the density fnction and in this sense the continos niform distribtion is extreme for a fixed spport it has the minimal chnkiness of all distribtions, and ths it is always better to be first. However, as we now show if the distribtion is sfficiently chnky then for small enogh search and recall costs later is better. Proposition 3 If f 2 ()d > f(ū) then for small enogh r and s later is better. The proof ses the fact that Q = 0 at s = r = 0, and demonstrates that at this point Q increases in r whenever the condition of the proposition is satisfied. Observe that in the preceding niform example f 2 ()d = f(ū) = 1 and ths Q remains fixed in r. It also follows that later is better for any distribtion with f(ū) = 0, sch as the normal distribtion or in fact any distribtion with an nbonded right tail. Frthermore, applying integration by parts to the above condition yields f ()F()d > 0, which implies that the proposition holds for any distribtion with a downward sloping density or a symmetric single-peaked density. Finally, for two distribtions F and G with f(ū) = g(ū), if G is chnkier than F then if later is better for F when r and s are small enogh, then later is also better for G when r and s are small enogh. To get a clearer sense of when later is better and to verify that this is not a knife-edge case holding only in the limit, we nmerically compte the following example. Sppose the density is { 2 if [ ] 0, 1 2 f() = 2 2 if [ 1,1], 2 9

11 r which satisfies the hypothesis of Proposition 3 since f(ū) = f(1) = 0. Figre 4 demonstrates the set of combinations of (r,s) for which Q > 0. Observe that when s is sfficiently small then for small vales of r later is better, bt, once r rises frther it becomes better to be prominent. This in fact is analogos to the leading discrete example, where for small r the search threshold is A = H and later is better, while for larger r the threshold drops to A = M and then early is better Triangle Distribtion s Figre 4: Vales of s and r for which later is better (dark gray) Chnkiness ths improves the payoff to the second seller by making close otcomes of A and B more likely when tilities are drawn independently. Alternatively this may be achieved if draws are correlated, which we explore next. 5 Correlated Utilities Sppose there are two independent variables ε 1 and ε 2, both drawn from distribtion F with spport [,ū]. Utilities are A = ε 1 and { ε 1 with probability ρ B = ε 2 with probability 1 ρ, 10

12 with ρ [0, 1) measring the correlation of the two payoffs. The consmer s search threshold is described by û = s+ρû+(1 ρ) û r f()d + (1 ρ)f(û r)(û r), which incorporates into (1) the fact that the conditional distribtion F( B A ) now has an atom of size ρ at A. Simplifying this condition then yields 0 = s 1 ρ r + û r (1 F())d, (5) reflecting the fact that when the consmer pays search cost s she only receives a new draw with probability 1 ρ. The difference between second and first is now ( Q = ρ F(û) 1 ) 2 ( [ +r û ( ) ] ) ū +(1 ρ) F()f() d + F() F( r) f()d (1 F())f()d, +r where in the first line with probability ρ tilities are identical and A wins the tie above û while B wins the tie below û, and in the second line tilities are independent and the expression is identical to that in (4). If F(û) > 1, then the first term is increasing in ρ, 2 and ths in these cases more correlation benefits seller B. An increase in ρ also redces û by (5), and the net effect is ambigos. However, if the search cost s falls sfficiently as ρ is increased then B improves, as demonstrated in the following. Proposition 4 If s and r are sfficiently small then there exists ρ(s,r) [0,1) so that later is better whenever ρ [ρ(s,r),1). In the proof, as ρ is increased the search cost s is decreased so that remains fixed, and ths so too does û from (5). As ρ approaches one, only otcomes along the A = B diagonal become relevant, and ths later isbetter aslong asthe diagonalis longer below ûthanabove. 6 Discssion While it is commonly held that appearing earlier in a consmer s search is desirable for a seller, we demonstrate here that this need not be the case. By allowing for a more realistic environment in which it is both costly to obtain new offers and to recall previos offers, we show that it is sometimes better to be later in the search order. Althogh it is well known that the search cost redces the byer s threshold and ths benefits the prominent seller, the 11 s 1 ρ û (6)

13 retrn costs has contervailing effects. A consmer anticipating costly retrn is less likely to search, bt also less likely to retrn conditional on search. We show that the latter effect can dominate whenever search and retrn costs are small and match vales at both sellers are likely to be similar, and ths in these cases later is better. Or analysis applies to several settings in which search order plays an important role. For example consider search for online content. Users enter a topic into a search engine and reslts are displayed in an order determined by an algorithm. The link to each article is accompanied by a short descriptive blrb, a signal of the qality of the match of the article with the ser s interests, and sers seqentially read the blrbs from the top down, eventally clicking on the desired article. How valable is it for the content providers to appear higher in the list of reslts? Or analysis sggests that the answer largely depends on the type of content being provided. For instance, if the search is for a movie review then articles may be highly correlated, and if the ser is patient and expects a possibility of a particlarly insightfl review, later may be better. On the other hand, if the ser searches for fnny cat videos, the hmor of one reslt is likely ncorrelated with the other reslts, and cat videos that appear earlier likely receive more clicks. Relatedly, if the platform sells top search positions then nderstanding the characteristics of the match vale distribtion for different types of content helps determine the demand for top spots. Pricing for prominence in search reslts shold depend not only on the volme of traffic for the content, bt also on the properties of the match vale distribtions that we describe. These considerations can gide the platform not only in the sale of prominent spots bt also in the platform s design and organization. For instance, allowing for longer, more detailed blrbs provides a more informative signal for sers, thereby leading to a less chnky distribtion of match vales and a higher retrn to prominence, while shortening or eliminating blrbs all together has the opposite effect. The job market is another application. An employer searching for a sitable employee interviews job candidates sccessively. Other things being eqal, how mch more likely is a candidate to be hired if she is interviewed first and ths how mch effort shold she expend to be among the first interviewees? Or shold she, to the contrary, strive to be interviewed last? If the distribtion of candidates in the pool is chnky, for instance if most candidates fit the position similarly bt with a small chance the match is otstanding, an employee is more likely to be hired if she is interviewed late rather than early. Here too the employer s design of the interview process affects the probability that earlier interviewees are hired. For example, reqiring more or less detailed pre- interview information and docmentation from applicants affects the distribtion of match vales between the employer and job applicants, which in trn determines the extent to which earlier applicants are more likely to be hired. 12

14 References Anderson, S. P. and R. Renalt (1999): Pricing, prodct diversity, and search costs: a Bertrand-Chamberlin-Diamond model, The RAND Jornal of Economics, 30, pp Arbatskaya, M. (2007): Ordered search, The RAND Jornal of Economics, 38, Armstrong, M. (2016): Ordered Consmer Search,. Armstrong, M., J. Vickers, and J. Zho (2009): Prominence and consmer search, The RAND Jornal of Economics, 40, Armstrong, M. and J. Zho (2011a): Paying for prominence, The Economic Jornal, 121, F368 F395. (2011b): Paying for prominence, The Economic Jornal, 121, F368 F395. De los Santos, B. and S. Kolayev (2012): Using Consmer Preferences to Improve pon Poplarity Rankings in Online Markets, Working Paper. Diamond, P. A. (1971): A model of price adjstment, Jornal of Economic Theory, 3, Haan, M., J. L. Moraga-González, and V. Petrikaite (2015): Price and matchvale advertising with directed consmer search, Tech. rep., Citeseer. Haan, M. A. and J. L. Moraga-González (2011): Advertising for attention in a consmer search model, The Economic Jornal, 121, Janssen, M. C. and A. Parakhonyak (2014): Consmer search markets with costly revisits, Economic Theory, 55, Rothschild, M. (1974): Searching for the lowest price when the distribtion of prices is nknown, The Jornal of Political Economy, 82, pp Urs, R. M. (2016): The Power of Rankings, Mimeo. Wilson, C. M. (2010): Ordered search and eqilibrim obfscation, International Jornal of Indstrial Organization, 28, Wolinsky, A. (1986): Tre monopolistic competition as a reslt of imperfect information, The Qarterly Jornal of Economics, 101, pp Zho, J. (2011): Ordered search in differentiated markets, International Jornal of Indstrial Organization, 29,

15 Appendix Proof of Proposition 3 The aim is to show that at d( Q) > 0 at s = 0 and r = 0, which is sfficient since at these dr vales Q A = Q B. Differentiating (4) and simplifying yields d( Q) dr = û +r f( r)f()d + f(û)(1 F(û r)) dû dr. Then totaldifferentiation of(1) yields dû = F(û r) dr frther to 1 F(û r), andplgging into theabove simplifies d( Q) dr = û +r f( r)f()d f(û)f(û r). Now at r = 0 and s = 0, the threshold û = ū, and ths d( Q) dr = f 2 ()d f(ū) > 0, with the ineqality following by the proposition s assmption. Finally, becase Q = 0 at s = 0 and r = 0 and is continos in s and r, there exists a neighborhood with s > 0 and r > 0 in which later is better. Proof of Proposition 4 Fix ρ [0,1), define v so that 2F(v) 1 = 0, and let { A s,r 0 < s } 1 ρ r + (1 F())d, v r be the set of (s,r) combinations for which û > v. To verify that A is nonempty, observe that given ũ and r = 0 there exists a s > 0 that satisfies (5), and that therefore there also exists an (s,r) hyperplane with s < s and r > 0 that also satisfies (5). A is ths the set of (s,r) combination above zero and below this hyperplane. Now, fix (s, r) A with corresponding threshold û > v from (5). Sppose that we simltaneosly increase ρ and decrease s so that remains fixed, and ths û remains fixed. s 1 ρ Then in expression (6) the terms mltiplying ρ in the first line and (1 ρ) in the second line remain fixed. Frthermore, by constrction the first term is positive and ths when ρ is sfficiently large Q becomes positive, and remains positive for any higher ρ. 14