A Mechanism for Providing Innovation Incentives for Digital Goods

Transcription

1 A Mechaism for Providig Iovatio Icetives for Digital Goods Erik Bryjolfsso ad Xiaoqua (Michael) Zhag {erikb, Sloa School of Maagemet, Massachusetts Istitute of Techology Cambridge, MA 0242 Abstract Digital goods ca be reproduced costlessly. Thus a price of zero would be ecoomically efficiet i terms of elimiatig deadweight loss. However, zero reveues would elimiate the ecoomic icetives for creatig such goods i the first place. We develop a ovel mechaism which solves this dilemma by decouplig the price of digital goods from the paymets to iovators while maitaiig budget balace ad icetive compatibility. Specifically, by sellig digital goods via large budles the margial price for cosumig a additioal good ca be made zero for most cosumers. Thus efficiecy is ehaced. Meawhile, we show how statistical samplig ca be combied with tiered coupos to reveal the idividual demads for each of the compoet goods i such a budle. This makes it possible to provide accurate paymets to creators which spurs further iovatio. I our aalysis of the proposed mechaism, we fid that it ca operate with a efficiecy loss of less tha 0.% of the efficiecy loss of the traditioal price-based system. Iovatio icetives i our mechaism are, of course, improved relative to the zero-price approach ofte favored by cotet cosumers. Surprisigly, we fid that iovatio icetives are also substatially better tha those provided by the traditioal system ofte favored by cotet owers which is based o excludability ad moopoly pricig of idividual goods. The techology ad legal framework for our proposed mechaism already exist ad portios of it have bee implemeted, although ot i ay coordiated fashio.

2 Itroductio Digital goods are differet. Ulike other goods, perfect copies ca be created at almost zero cost. With the advet of the Iteret, mobile telephoy, satellite commuicatios, broadbad ad related techologies, these goods ca be distributed to almost ayoe i the world at early zero cost as well. May idustries have bee profoudly affected. Two schools of thought have domiated the debate o the ecoomics of digital goods. Oe school stresses the beefits of the traditioal market system. Users who wish to beefit from a creatio must pay the creator. This paymet i tur assures that a) the goods go to those idividuals with the highest value for the good ad b) that the creator has icetives to cotiue to create valuable goods. Aother school of thought thiks that Iformatio wats to be free. The poit ca be made that sice digital goods ca be produced at zero margial cost, the textbook ecoomic priciple of efficiecy: price equals margial cost demads that price should ever be greater tha zero. After all, society as a whole is oly made worse off if a user is excluded from access to a digital good which could have bee provided without reducig the cosumptio of ayoe else. Either approach is demostrably suboptimal (e.g. Lessig, 2004). It would seem impossible to have both efficiecy ad iovatio whe it comes to digital goods. Improvig oe goal appears to be iextricably itertwied with hurtig the other goal. I this paper, we argue there is a third way. I particular, we develop ad aalyze a method for providig optimal icetives for iovatio to the creators of digital goods. We show that it is possible to decouple the paymets to the iovators from the charges to cosumers while still maitaiig budget balace. I this way, we ca slice the Gordia kot ad deliver strog icetives yet uhidered access to the goods for almost all iterested cosumers. I fact, we fid that our system actually provides better icetives for iovatio tha the traditioal price system. The essece of our mechaism is to a) aggregate a large umber of relevat digital goods together ad sell them as a budle ad the b) lear the social value of these goods, usig statistical samplig ad targeted coupos, ad c) allocate the reveues from this aggregatio to each of the cotributors to the budle i proportio to the value they cotribute. We do this i a way which is fully budget-balacig ad which provides accurate icetives for iovatio with ear zero efficiecy loss. The academic literature related to our aalysis is quite sparse. Some of the closest research is the work o a moopolist facig a ukow demad curve (e.g. Aghio et al, 99) where it is show that the seller ca experimet by pricig to differet buyers sequetially ad updatig the price accordigly. Our paper both itroduces a ovel mechaism ad rigorously aalyzes it, fidig that it is techically feasible ad that it ca domiate ay of the approaches debated thus far. Notably, if this iovatio succeeds, it should actually icrease the pace of future iovatios by improvig icetives for the creatio of useful digital goods. Model Setup We cosider a market with may providers of digital goods ad may potetial buyers. Suppose a moopolistic budler coects the producers ad the buyers by desigig a optimal pricig ad reveue distributio policy to maximize the budler s profit. Each buyer has (at most) uit demad for ay of the iformatio goods. Suppose a buyer s valuatios of the goods i the budle are draws from a radom variable V i the rage ormalized to [0,], ad that the radom variable has a cumulative distributio fuctio v), whose correspodig probability desity fuctio is f(v). At a price of p, the demad will be Q(p)=Prob(v>p)=-p), yieldig reveue of π(p)=p[-p)]. This implies that the iverse demad curve is P( q) = F ( q), ad the budler s problem is to solve: π = max{ p ( p))}. Takig first order coditio, p π p) = p) p = 0, we get: p f ( p ) p p =. For the moopolistic budler, it turs out that her profit p ) maximizig decisio is ot hard. As show i Bakos ad Bryjolfsso (999), the budler s job is to fid the optimal price for the sum of may radom variables ( S = i = v ). By the law of large umbers, it is easier to i fid a optimal price for the sum S tha for idividual goods v i, because the coefficiet of variatio of S is

3 decreasig i. I particular, it ca be show that for o-egative valuatios, the expected value ca be writte as E[ X ] = [ FX ( x)] dx. Iterestigly, this expressio ca be liked directly to the area uder the demad curve. 0 Whe price is v, demad is give by Q( v) = v), so the area uder the demad curve is just Q ( v) dv = E[ V ]. From this formulatio, we ca show that whe the cost of (re)producig the goods is close 0 to zero, budlig provides close-to-optimal allocatio of goods to cosumers (see also Bakos ad Bryjolfsso, 999). However the beefits come at a major cost. Budlig iheretly destroys iformatio about how each of the compoet goods are valued by cosumers. Without this iformatio, it is impossible to allocate reveues to the providers of cotet i a way that accurately ecourages value creatio. The Reveue Distributio Problem The ideal reveue distributio mechaism would be oe which somehow determied each good s demad curve, ad distributed the reveue amog the cotet providers i proportio to the social value of each good to all cosumers. This value ca be calculated by itegratig the area below each good s demad curve. Various mechaisms used to derive demad curve proposed i the literature all fail here because budle pricig does ot automatically provide way to observe the market s respose to a price chage of idividual goods. If the beefits created by each good caot be observed or calculated, the a host of iefficiecies may result. First, the cotet providers may ot have eough icetives to produce creative products, ad cosumers will evetually be harmed. Secod, without a good sigal of cosumers preferece, cotet providers may ot produce the cotet that best fits the cosumers taste. Third, i ay effort to overcome these problems, the cotet producers may force the potetial budler to adopt other strategies such as pay-per-view. However, such strategies re-itroduce the deadweight loss problem. The Coupoig Mechaism It has bee show that the ideal way to provide correct icetives is to lear cosumers valuatios for each good ad make correspodig paymets to the cotet providers. Sice budlig itself obscures cosumers valuatios for idividual goods, here we propose a mechaism to derive the demad curve for each good by issuig targeted coupos to a small, but statistically represetative, sample of cosumers. Our mechaism is substatially differet from the traditioal use of coupos as a marketig method to price discrimiate cosumers. Istead, coupos i this mechaism are similar to the price experimets suggested i the optimal pricig literature. Suppose the moopolistic budler offers a budle of iformatio goods to a set of cosumers. I order to derive the demad curve for oe of the compoets, we choose m represetative cosumers ad issue each of them a sigle coupo, where is the umber of price levels coverig the rage of the valuatios, which we call coupo levels (oe simple way to get these levels is to offer coupo values from V to V where V is the upper boud of cosumer valuatios for this good), ad m is the umber of coupos to be offered for each of the price levels, which we call sample poits (there will be m cosumers who ca receive a coupo with face value i V, i =,..., ). While m is large eough to make statistically valid ifereces, it is oetheless a very small fractio (e.g. /000 or less) of the total set of cosumers buyig the budle for ay large-scale implemetatio of the mechaism. If a cosumer receives a coupo with face value v ~, the he ca either choose to igore the coupo ad ejoy the complete budle or choose to redeem the coupo ad forfeit the right to use the idicated compoet ad access to it. So upo observig the cosumer s actio, the budler ca lear whether his valuatio is higher or lower tha the face value of the coupo. Aggregatig the m cosumers valuatios will give the budler a good estimate of the valuatios at that price, summarizig the results for the coupo levels, the budler ca plot a fairly accurate demad curve. The area uder the demad curve is the social valuatio for the particular good. 2

4 Usig the same method for all the compoets, the budler ca lear the social valuatio of each of the goods i the budle. She ca the distribute the reveue amog the cotet providers accordig to their share of the total valuatio. Let R be the total reveue from sellig budles, ad v i be the social value of the compoet i i the budle, cotet provider of i should be paid vi reveuei = R, where N is the total umber of cotet N j = v j providers. This mechaism compares favorable to the traditioal price mechaism. The traditioal price mechaism subjects 00% of cosumers to the iefficiecy of positive prices. However, oly data from a small fractio of cosumers are eeded to get extremely accurate estimates of the value created ad cotributed by each good. The greater precisio obtaied by icreasig the sample declies asymptotically to zero while the cost for subjectig each additio cosumer to a positive price remais just as high for the last cosumer sampled as the first oe. Whe balacig the costs ad beefits, the optimal sample size is almost surely less tha 00%. Secodly, Because multiple differet prices for coupos are offered, this mechaism provides a much more accurate overall picture of demad tha ca be obtaied the traditioal sigle price system. This has large ad importat implicatios for dyamic efficiecy ad iovatio icetives. Oe ca also compare our coupoig mechaism with that of the well-kow Vickrey-Clarke-Groves (VCG) mechaism. We fid that our coupoig mechaism does ot give us exact valuatios for each cosumer ulike the VCG. However, i geeral all approximate demad fuctios of the compoets will suffice, ad by icreasig the sample size, the accuracy ca be made almost arbitrarily precise. Our coupoig mechaism is superior to the VCG mechaism i several ways. () Truth-tellig is a robust ad strog equilibrium i the coupoig mechaism, i the sese that each cosumer simply compares his valuatio with the coupo s face value, he is ot required to assig correct beliefs o all other people s votes. (2) I the VCG, if oe respodet misreports his value (due to irratioality or due to error), the cosequece may be very severe for the rest of the people. Furthermore, coalitios of cosumers ca game the VCG to their advatage. However, i the coupoig mechaism, the effects o others from a cosumer s misreport are miimal. (3) The coupoig mechaism is fully budget balacig, ulike the VCG. (4) The coupoig mechaism is more ituitive tha the VCG for real world problems. The followig propositio asserts that the coupoig mechaism ideed gives us correct demad curve estimatios i expectatio. PROPOSITION : For ay oe of the compoets i the budle, give a large umber of radomly chose respodets ad level of coupos, the above mechaism gives a empirical demad fuctio Qˆ ( p) = Fˆ V ( p) that arbitrarily approximates the true demad fuctio: Q( p) = FV ( p). Proof: Attached i the appedix. Propositio gives a asymptotic result, we ru simulatios to verify the effectiveess of this mechaism (See Figure i the ext page). Based o the simulatio, a back-of-the-evelop calculatio suggests that samplig just 00 cosumers ca provide almost as accurate a estimate of demad as samplig the full populatio of the cosumers of the good, which could be i the millios. It turs out that there are some additioal beefits related to iovatio icetives of the cotet providers. 3

5 Iovatio Icetives Figure : Simulatio Results for CM We ca show that cotrary to commo belief, the traditioal price system based o excludability does ot provide correct iovatio icetives to producers. Thus, i additio to the efficiecy gais, the proposed coupoig mechaism ca be a socially desirable way to promote iovatio for digital goods. Specifically, we have the followig propositios: Propositio 2: If a iovatio ca icrease cosumers valuatios uiformly higher, the proposed mechaism (CM) gives the producer strictly greater icetives of iovatio tha does the traditioal market mechaism. Propositio 3: If a iovatio ca icrease oly some cosumers valuatios, the traditioal price system does ot provide correct icetives for the producer to iovate for people with relatively high or relatively low valuatios. I cotrast, the proposed mechaism always gives the producer socially desirable level of icetives to iovate. Propositio 4: The traditioal market gives the producer too high a icetive to iovate where it is most harmful to the social welfare, o icetive elsewhere; the proposed mechaism iduces the producer to make socially desirable iovatio efforts. Coclusio Revolutioary techologies ofte egeder iovatios i busiess orgaizatio. The digitizatio of iformatio is o exceptio. We seek to advace the debate o how best to allocate digital goods ad reward their creators by itroducig a ovel mechaism ad aalyzig its implicatios. Our approach elimiates the margial cost of cosumig digital iformatio goods for the vast majority of cosumers via massive budlig. For very large aggregatios, this preserves most of the static efficiecy which could be achieved with a zero price policy. However, i the log ru, the more importat issue is how to create icetives for ogoig iovatio. I this area, the proposed mechaism shows particular promise. We fid that our approach ca provide substatially better icetives for iovatio tha eve the a traditioal moopoly price system bolstered by artificial excludability (e.g. via DRM, laws, etc.). I particular, the traditioal price system, i which each good is sold for a specific price with the proceeds goig to the moopolist creator, focuses virtually o icetives o a very arrow bad of cosumers - those just o the margi of buyig. I fact, the price system provides too strog icetives for iovatios that help this arrow group of cosumers. Rets trasferred to the creator from such iovatios exceed the social beefits. I cotrast, our approach, usig statistical samplig ad coupoig, ca provide icetives which are early optimal for every type of iovatio. I summary the mechaism we itroduce, has potetially orders of magitude less iefficiecy tha the traditioal price system, is budget balacig, requirig o exteral iflows of moey, works with existig techology ad existig legal framework, 4

6 requires o coercio ad ca be completely volutary for all parties, sice it is fully icetive compatible, does t assume that iovators will cotiue iovate eve without fiacial rewards, ca be implemeted ad ru i real-time, ad is scalable to very large umbers of goods ad cosumers (i fact, works better for larger umbers), Our approach also has weakesses ad challeges. Compared to givig away all digital goods for free, our approach will exclude a small umber of cosumers ad create some iefficiecy as a result. More importatly, our approach does require the creatio of ew busiess istitutios or models, which is ever easy. Specifically, a etity is eeded to maage the statistical samplig ad coupoig, aalyze the resultig data, ad allocate paymets to the cotet owers accordigly. Near misses for this type of etity already exist. For istace, ASCAP does much the same thig already for broadcast music, but without accurate price iformatio. Nielse ad similar orgaizatios provide usage iformatio, but agai without accurate price iformatio. There are also orgaizatios which regularly collect ad distributed large sums of moey to member compaies based o various algorithms. The Federal Deposit Isurace Corporatio which does this for baks is oe example. Some cooperatives are also ru this way. Last but perhaps ot least, the govermet regularly makes these types of trasactios. However, it should be stressed, that our mechaism does ot require ay govermet role sice all of the participats (cosumers, cotet creators, budlers) have icetives to participate completely volutarily. This stads i cotrasts to the proposal by Fisher (2004) or the varied proposals to chage copyright or other laws. By offerig this ew framework ad aalysis, with a ew set of opportuities ad challeges, we hope to lay the foudatio for future research o the critical questio of providig icetives for iovatio i the creatio of digital cotet ad implemetig mechaisms to deliver that cotet to cosumers efficietly. We expect that the ext 0 years will witess a scale of orgaizatioal iovatio for creatig ad distributig digital goods surpassig eve the remarkable pace of the last 0 years. New coordiatio mechaisms, such as the iovatio icetive approach described ad aalyzed i this paper will flourish. With a proactive attitude toward techology-eabled orgaizatioal iovatio, we believe that academia ca speed this process by framig the issues, ad by providig tools, cookbooks ad aalyses. Refereces: Aghio, P; Bolto, P., Harris, C, ad Jullie, B., Optimal learig by experimetatio, Review of Ecoomic Studies, 58(4), (99), Bakos, J. Y., ad Bryjolfsso, E., Budlig Iformatio Goods: Pricig, Profits ad Efficiecy. Maagemet Sciece, 45(2), (999), Fisher, W., Promises to Keep: Techology, Law, ad the Future of Etertaimet, (forthcomig 2004), Staford Uiversity Press. Lessig, L., Free Culture: How Big Media Uses Techology ad the Law to Lock Dow Creativity, (2004), Pegui Press. 5

7 Appedix: Proof of Propositio : We prove propositio i two steps. First, we show that for each price level, the mechaism offers a cosistet estimate of the true demad at that level. Secod, we show give eough price levels, the demad curve ca be arbitrarily closely approximated. For oe particular compoet, the seller first chooses the umber of coupo levels, the, for each coupo level, seds m coupos to m radomly chose cosumers. For a coupo with face value v ~ for the compoet, the respodet will take it oly if he has a valuatio lower tha v ~. The probability of the coupo gettig accepted is prob( V v~ ) = F ( v ~ V ). We ow defie idicator variables Y,...,Y where m Y i is if the coupo with face value v ~ is accepted by the i th cosumer, 0 otherwise. We have X k v~, where k =,..., m. Note that Yk = 0 X k > v~ prob( Y = ) = prob( X v~ ) F ( v ~ ), ad prob( Y = 0) = prob( X > v~ ) = F ( v ~ ). For all the m people to whom we k = V k set coupo v ~, we kow the umber of the experimets tellig us what percetage of people accepts the coupo v ~. We ca show the expected value of the empirical cdf is the true ukow cdf. ˆ [ ] ( ~ am m E Y E[ F v )] = E[ ] = = E[ Y ] = 0 prob( Y = 0) + prob( Y = ) = F ( v ~ V ). That completes the step. m m Next cosider the iterval betwee ay eighborig coupo s value levels. For explaatory purpose, we ow assume that the seller sets equi-distace itervals o the value rage [0,], that is, the coupo values are 0,,...,. Our result does ot rely o this assumptio, it holds as log as the distaces are all weakly shrikig whe addig more coupo levels. V Figure A. The upper boud of error i estimatig demad For eighborig coupo levels i ad i +, the seller may estimate poits A ad C from step. She ca simply coect the estimated poits to approximate the demad curve betwee the two poits. Sice the demad curve is mootoically decreasig from to 0, whe estimatig the area below the demad curve, the triagle ABC is the upper boud for the error. The area of ABC is i + i ˆ i ˆ i + ABC = ( )[ ) )]. We kow ˆ i ˆ i + F ( ) ), 2 ad give the assumptio that F V (x) is cotiuously differetiable. We have ˆ i ˆ i + lim [ ) )] = 0, so we have lim ABC = ˆ i ˆ i + (lim ) [lim ( ) ))] = 0, which says that whe is large eough, the error i 2 estimatio will coverge to 0. Q.E.D.

8 Proof of Propositio 2: Figure 2: A uiform upward shift of demad curve Suppose the iovatio ca icrease each cosumer s valuatio by δ, this is equivalet to movig the demad curve upward by δ. Case (Traditioal Market): whe the demad is shifted upward, the moopolistic seller ca do some combiatio of two thigs: charge a higher price of p +d or still charge the price p ad sell to more people (the demad will be q ' = p δ ) ow). We ext show, i lemma, that both strategies lead to the same expected profit for the seller. Lemma : Margially, the iovative moopolist seller ca charge a higher price or ejoy a icreased demad, ad the two strategies are equivalet i terms of expected profit. Lemma says that i Figure 2, the area A ad the area B should be equal. Essetially, by takig some effort, the seller ca get expected margial gai A = B = δ [ p )]. Case 2 (Budlig with coupo mechaism): whe the demad is shifted upward, the seller ca get paid virtually the π = E[ V ] = full amout of the extra valuatio it created for the cosumers. Let the origial profit be [ v)] dv 0, she ca ow ear π ' = δ + π. Sice p is the optimal price, ad ca ever be zero, we have p ) <. So the margial profit of iovatio from budlig π T = δ is strictly greater tha the margial profit of iovatio from traditioal market π T = δ [ p )]. Accordigly, we coclude that sellers iovatio icetive i the budlig scheme is higher tha that i the traditioal market. Proof of Propositio 3: The total potetial social value of the good equals the area uder the demad curve. If the seller makes a targeted iovatio for some cosumers with valuatio v %, the social gai of the iovatio is thus deoted by the area ABC i Figure 3. Whe δ is small, ABC δ[ v ~ + δ ) v ~ ) δ 2 f ( v ~ ). 2 2

9 Figure 3: Social beefit/loss vis-a-vis seller iovatio. We shall eed the followig techical assumptio to get a well-behaved demad curve. Assumptio: F (v) is twice cotiuously differetiable with F ( 0) =, ) =, f ( v) > 0, v > 0 ad is strictly covex for v (0,). v) This assumptio is implied by log-cocavity of v), which itself is implied by log-cocavity of f (v). The ituitive meaig of the assumptio that the radom variable V has a log-cocave desity fuctio is that it has a uique global maximum. Note that this assumptio implies that the profit fuctio p [ p)] is cocave. Give ay iovatio that icreases some cosumers valuatio by δ, there exists some v, such that the seller is idifferet betwee carryig out the iovatio ad ot carryig out the iovatio. For the seller, ( v + δ )[ v)] = p [ p )]. Solvig for v, we have two values, v L ad v H, such that p ( v L, v H ). Also, v ( v L, v H ), it must be that ( )[ ( )] [ ( v + δ F v < p F p )], so the seller has o icetive at all to iovate for cosumers with valuatios outside the rage ( v L, v H ). This is very ituitive, i the traditioal market, if the seller sells goods to cosumers with valuatio higher tha v H, it makes o sese to icrease their valuatios further because that will oly cotribute to cosumer surplus, ad the seller will ot be able to extract the added value. Similarly, for the potetial cosumers with lower valuatios (lower tha v L, to be precise), the seller will ot take the effort to iovate because they will ot be coverted to cosumers. For small δ, the rage ( v L, v H ) is very small, ad eve i this rage, iovatio may ot be socially desirable. For cosumers with valuatio i the rage ( v L, v ), oe ca look at three distict cases: H () v ( vl, p δ ) I this case, the seller would wat to charge a price p = v + δ ad ear profit π = ( v + δ )[ v)]. By lemma, π > ( vl + δ )[ vl)] > p [ p )]. So the seller prefer to lower the price from p to p = v + δ, ad ear a higher profit. The reductio i price has two socially desirable effects. First the cosumer surplus is icreased. For people with valuatio i the rage ( v + δ, p ), they are o loger excluded from accessig the good; ad for people with valuatio i the rage ( p, + ), they ca each ejoy a icrease i cosumer surplus of CS = p ( v + δ ). Secod, deadweight loss is reduced, the chage i dead-weight-loss is DWL = [ p ) v + δ )]( v + δ ). The reductio i deadweight loss is composed of two parts: first, for people with valuatio i the rage ( v + δ, p ), apart from the icrease i cosumer surplus, there is also reductio i DWL due to the fact that their demad is satisfied, secod, for people with valuatio i the rage ( v, v + δ ), the iovatio icreases their valuatio, ad they are o loger excluded from purchasig the good. (2) v p, v ) ( H 2

10 I this case, the seller iovates for people with valuatio just higher tha the optimal price. By lemma, we kow it is worthwhile for her to icrease the price to v + δ, there are two socially udesirable effects associated with this. First, for cosumers origially havig a valuatio above v + δ, they each lose cosumer surplus by CS = v + δ p. Also, for people with valuatio i the rage ( v, v + δ ), although their valuatio is icreased due to the iovatio, they o loger ejoy a surplus ow. Secod, for people with valuatio i the rage ( p, v ), they ca o loger afford to buy the good ow, so there is a icrease i deadweight loss. (3) v ( p δ, p ) This case has mixed effects. O oe had, there are socially desirable effects as i, ad o the other had, there are also socially udesirable effects as i 2. For people with valuatio higher tha v + δ, they suffer a reductio i cosumer surplus by CS = v + δ p. I Figure 3, the loss is idicated by the area ADEI. For people with valuatio i the rage ( p,v + δ ), due to iovatio, they have a higher valuatio ow, but due to the icreased price, they o loger ejoy a surplus (the area AIF). For people with valuatio i the rage ( v, p ), their valuatio is icreased to v + δ, but agai, the seller gleas all the surplus due to iovatio. A socially desirable side effect is that the deadweight-loss is reduced because these group of people are able to use the product ow. The area FGHC idicates the social gai from reduced deadweight loss. I total, cosumer surplus is hurt by the area ADEF, deadweight loss is reduced by the area FGHC, ad the seller ejoys the extra value created by iovatio idicated by area ABC. Proof of Propositio 4: To see the socially wasteful icetive of iovatio i the traditioal price system, cosider the case of the cosumers valuatios ear the optimal price. For example, if the seller takes a effort to iovate ad icreases the valuatio for some cosumers from p to p + δ, the her gai is δ [ p )]. The ratio of her gai over her cotributio is icetive _ ratio Traditioal = 2 p ) 2 p δ[ F ( p )]/[ δ f ( p )] = 2 =, ad 2 δf ( p ) δ lim δ 0 icetive _ ratio Traditioa l =. For the case of the proposed mechaism, the 2 ( ~ 2 icetive_ ratio ) /( ( ~ Budlig = δ f v δ f v )) =, which is fair. 2 2 Proof of Lemma : We eed to compare π p = ( p + δ )[ p )] ad π q = p [ p δ )], ad show that as δ 0, they are equal. Equivaletly we eed to show: limδ 0( p + δ )[ p )] = limδ 0 p [ p δ )], which is p ) p δ ) p ) p ) lim 0 = f ( p ) =, which is true due to the optimality coditio. δ δ p p Q.E.D 3