A self-organized criticality participative pricing mechanism for selling zero-marginal cost products
SSelf-organized criticality auction model for selling products in real time
Daniel Fraiman
Departamento de Matem´atica y Ciencias, Universidad de San Andr´es, Buenos Aires, Argentina, andCONICET, Argentina. ∗ Consumer markets are quickly growing, creating the need to design new sales mechanisms. Here weintroduce a new auction model for selling products in real time and without production limitations.Interested buyers continuously offer bids and if the price is “right”, the bid is accepted. The modelexhibits self-organized criticality; it presents a critical price from which a bid is accepted withprobability one, and avalanches of sales above this value are observed. We also discuss how toimplement the model and consider the impact of information sharing on total income, as well as theimpact of setting a base price.
PACS numbers: 05.65.+b, 87.10.-e, 87.23-n, 89.75-k, 02.50.Cw
I. INTRODUCTION
In certain situations, the act of buying and selling maybe considered an art. As we will see, special consider-ations arise in cases where the buyer and seller shareno or minimal interaction. In an auction [1], the “true”value of a product is unknown to the seller, and the salesprice is discovered during the course of open competitivebidding. The most well-known auction mechanism is theEnglish auction. In this form of open ascending pricingauction, the auctioneer calls out a low price for a singleor multi-item product and raises it until there is onlyone interested buyer remaining. Traditionally used tosell rare collectibles and antiques, nowadays auctions arewidely used with many types of products and services,both in the traditional way (live) and on the Web (e.g.Amazon and eBay). For example, Google, Microsoft, andYahoo! use auctions for search advertising and Amazonuses an auction mechanism for selling computing time inthe Cloud.Products or services may have a zero or non-zeromarginal cost (i.e., the cost of producing a single addi-tional unit is zero). An important part of today’s economyis based on doing business with products or goods thathave zero or almost zero marginal cost. Examples of theseproducts include adding a student to an online course,selling one more mobile application, adding a Facebookaccount, offering a new home on Airbnb, responding to anew customer via a virtual agent, having a car in UBER,and, in the future, analyzing a medical image using analgorithm. Establishing the price of these products is nottrivial, and perhaps one of the most important tasks ismaking the product known on the market. Just think ofa new useful mobile app: the selling price may be verylow or zero but it may be difficult for customers to find itin a “sea of applications”.Here we present an auction model for selling zero or non-zero marginal cost products in an infinite stock system ∗ Electronic address: [email protected] that involves a certain degree of hurry in making thedecision if the bid is accepted.
II. THE MODEL
Let us suppose that the seller sells a product that has noproduction limitation (i.e., infinity stock). For example, adownloadable software or mobile application. In additionto earning a profit, the company that sells this productis interested in increasing its visibility in the marketplaceand earning shares when it faces competitors. Next, letus suppose that each interested buyer can only make oneprice offer, and that the decision to purchase must bemade with some degree of hurry. As an example, imaginea customer who buys a computer without an operatingsystem (OS). This customer is interested in buying the“A” OS so he/she offers a bid price for it. The companythat sells the “A” OS must quickly decide if they will sellit at that price or whether the customer will never haveaccess to the “A” OS on that computer. This type ofcustomer appears all the time and the decision must bemade almost in real time. Our auction model takes theseconditions into account.Buyers appear at different times, which are described bysome general stochastic process (e.g., an inhomogeneousPoisson Process with rate λ ( t ), or any other). Once abuyer appears, he/she offers a bid price for the product.This value cannot be subsequently modified, and thebuyer cannot participate again in the buy-sell process.The buyer does not know the bids made by the previousinterested buyers, as occurs in a blind auction. The sellerwill sell the product to the buyer with the best bid, andthe transaction decision must occur almost in real time.Let us suppose that the selling process starts at timezero and buyers start to appear. The first potential buyer(that appears at some arbitrary time t ) offers a value X for the product; the second potential buyer offers X ,and so on. The selling rule is the following: at each bidappearance time, the highest remaining bid is executed,except when the new bid exceeds this value. Wheneverthis occurs, no transaction occurs and the new bid remainsin the bid queue until the time it becomes the highest a r X i v : . [ q -f i n . T R ] M a y value. For example, let us suppose that the following bidvalues were offered in this order:14$ , $ , $ , , $ , , . In this example, the first transaction occurs when thefourth person (bid) appears. When he/she offers a priceof 13 $ that is lower than the maximum at that time (18 $ ),the third transaction offering 18 $ is executed. The fifthbid value is the greatest value of the pending offers, whichis why no transaction is done at that moment. However,this last bid is executed during the next bid appearancetime, when a smaller bid value of 12 $ is presented. Finally,the bid value of 15 $ is executed because it turns out tobe the highest offer at the time the last bidder appears.As we saw in this example, only the three largest offerswere executed. This sale process continues (up to infinity)because new buyers appear continuously, and the maingoal is to understand how much money the companywill earn and how many products will be sold during agiven period of time. In particular, we are interested inunderstanding what the stationary regime of this processis.As in any auction, the interested buyer makes an offerbased on his/her valuation of the product (based on thedemand for that product). Each bidder makes his/her ownvaluation and ignores that of the other potential buyers.Each participant offers a price valuation, X , that is welldescribed by a certain probability density function f ( x )with cumulative probability F ( x ). In this sales model, theoffered bids represent a sequence of independent randomvariables X , X , X , . . . , X n , . . . with probability law F . The key is to understand the consequences of the newselling rule described above over this bid sequence.A simple algorithm for this process is shown below. x =randomF(1); y =vector(); x =vector(); x [1] = x ;k=1 for i in 2:N do z = max ( x ); out = f ind (˜ x == z ) ; x = randomF (1) ; i th-bid price˜ x = c (˜ x, x ) ; if x < z then ˜ x = ˜ x [ − out ] ; y [ k ] = z ;k=k+1; endend Vector ˜ x contains the remaining bids and vector y contains the purchase prices of the accepted bids. The Total Income earned by the company, T I , is described by,
T I = ˜ N (cid:88) k =1 Y k , (1)where ˜ N is the random variable number of sales when N interested parties have made offers.Next, we show that at some time long after the startof the auction, let us say when N >> − p c ) N bidderswill have made a purchase, (cid:104) ˜ N (cid:105) = (1 − p c ) N, (2)and each one has paid a value greater than a value x c which verifies F ( x c ) = p c , (3)with [10] p c ≈ e − . (4)Moreover, the expected total income earned by the com-pany will be (cid:104) T I (cid:105) = N (cid:90) ∞ x c xf ( x ) dx. (5)For example, if we suppose that the bid price distribution f ( x ) is exponential, with a rate of λ , f ( x ) = λe − λx , thenthe mean total income is ( x c + 1 /λ ) e − λx c with x c = − λ ln (1 − p c ). It is straightforward to compute the mean T I for other price distributions such as Log-Normal, orheavy tail Power Law with α >
2. The case α > x c (represented by a dashed line) thatverifies eq. 3. The right price histogram enhances thedistribution of accepted versus not accepted bids.Importantly, after a short period of time, the meannumber of active bids does not continue to grow. Thatis why the probability density function of the remainingor frozen offered prices g (˜ x ), in the limit of N going toinfinity, converges to g (˜ x ) = (cid:26) F ( x c ) f (˜ x ) if x ≤ x c x > x c . (6)Thus, the complement of the remaining offers, the salesprices, Y , have a probability density function, h ( y ), de-scribed by h ( y ) = (cid:26) y ≤ x c − F ( x c ) f ( y ) if y > x c . (7)The remaining price distribution ( g (˜ x )) for the previousexample is depicted in Fig. 1 in grey and the distributionof the accepted prices ( h ( y )) is depicted in red. Note thatthere is a clear cutoff even for as few as 1,000 bidders.Now that we have calculated the accepted bid distri-bution, it is straightforward to obtain the mean TotalIncome per bid in the limit of N going to infinity,lim N →∞ N (cid:104) T I (cid:105) = (cid:90) ∞ x c yf ( y ) dy. (8)So far we have shown that if N is very large, the mean T I can reach N (cid:82) ∞ x c yf ( y ) dy . But, what about the varianceof T I ? How does it grow with N ? The variance verifieslim N →∞ N V ar ( T I ) = a f , (9)where a f is a number that depends on the price distribu-tion ( f ). This behavior is good because for large N values,it scales linear with N which is not much variability. a f can be computed in a empirical way. Nevertheless, aswe will see a first order approximation can be done forobtaining a quick estimation of a f . The total income isa random sum of independent random variables (eq. 1).Therefore, if we calculate the variance as if ˜ N and the Y k sequence were independent (which is not the case) wefind that V ar ( T I ) = (cid:104) ˜ N (cid:105) V ar ( Y ) + (cid:104) Y (cid:105) V ar (cid:16) ˜ N (cid:17) . (10)Finally, using eq. 2 and the empirical fact that for large N the V ar (cid:16) ˜ N (cid:17) goes as bN with b ≈ . N →∞ N V ar ( T I ) = (1 − p c ) V ar ( Y ) + b (cid:104) Y (cid:105) ≈ a f . (11)Therefore, as a first order approximation the asymptotic T I variance can be calculated (from eq. 7 and 11) oncethe price distribution is known. The empirical a f for theprevious example sale process with a Log-Normal pricedistribution is 0.093, and the given by eq.11 is equal to0.102 (see Supp. Fig. 3).Figure 1 (B) shows the T I behavior as a function ofthe number of bidders. The empirical interval
T I ± (cid:112) V ar ( T I ) from 200 simulations is shown in red. Morethan 99% of the simulations fall within this interval [12]The black line corresponds to the (theoretical) mean value
T I given by eq. 5. Equation 8 describes the Total Incomeper bid for large times (
N >>
1) after the start of the saleprocess. Equation 5, on the other hand, is written for fixed N ; although N must be large, as mentioned earlier, it isinteresting to note the extent to which the descriptionis good for small N . A close-up of Fig. 1 for smallervalues of N is shown in the inset of Fig. 1. This graphichighlights the non-linear behavior at the beginning of thesales process, particularly when the number of bidders isless than 50. For N larger than 50, eq. 5 yields a goodapproximation. The same happens when we compare theempirical T I variance with the theoretical variance, givenby eq. 11 ( V ar ( T I ) = a f N ). There is some differencebetween both variances for N <
50, and this difference isnegligible for larger N values (see Fig. Supp. 3).The model presented here and its asymptotic behaviorresembles the Bak-Sneppen model [5–8]. In this well-known and elegant model, the number of species N (inour case the number of bidders) is fixed, and the modelis studied at the thermodynamic limit ( N → ∞ ); thecommon limit in statistical physics. In our case, we havea growing model where the number of bidders goes toinfinity. But, is this sales model a self-organized criticalitymodel as is the Bak-Sneppen model? Self-organized criti-cality processes are characterized by stationary regimesthat present avalanches. Are there avalanches in the salesmodel? Avalanches do in fact exist in the model presentedhere. If one studies the number of consecutive sales, τ ,above x c (or the number of sales between two successivepurchases below price x c ), a power law behavior is ob-served (see Fig. 2). Above we said that all purchases areabove x c , but this is not exactly true. A few purchasesare around or slightly below x c . However, the numberof these purchases is negligible (when N → ∞ ), whichis why we previously considered them as nonexistent forlarge N . They are negligible because the mean durationof a sales avalanche, (cid:104) τ (cid:105) , is infinity due to the heavy tailbehavior of τ ( P ( τ > k ) ∼ k − . ). III. MATHEMATICAL DESCRIPTION
Let X , X , X , . . . be the sequence of bid prices in theorder they appear. As is true of all auctions, some offersare effective, some will be effective in the future, andothers will never be effective. Let Y , Y , Y , . . . be theprices of the effective transactions. This sequence is asubsequence of the X sequence and can be described asfollows: Let us define1. The set of the first k bid prices accepted,Θ k = { Y ; Y ; . . . ; Y k } .
2. The sequence of remaining bids,˜ X j,k = X j { X j / ∈ Θ k } .
3. The maximun remaining bid,
FIG. 1: (A) The auction process for a Log-Normal pricedistribution, f ( x ) = xσ √ π e − ( ln ( x ) − µ )22 σ , with µ = 0 and σ =0 . x c ( F ( x c ) = p c = e − and therefore x c ≈ . average ± is shown.The theoretical value (cid:104) T I (cid:105) is shown with a black line ( (cid:104)
T I (cid:105) = (cid:82) ∞ x c yf ( y ) dyN = 0 . N ). In the inset we show a close-upof the first values together with the theoretical value. ˜ X maxk ( t ) = max { ˜ X ,k ; ˜ X ,k ; . . . ; ˜ X t,k } The accepted n − th bid verifies Y n = X k n , (12)with k n = argmax ≤ j ≤ h n ˜ X j,n − h n = min { h > k n − + 1 : ˜ X maxn − ( h ) = ˜ X maxn − ( h + 1) } . l l 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k P ( t > k ) FIG. 2: Avalanche distribution. The probability that theduration of a sales avalanche (of prices larger than x c ) isgreater than an arbitrary value k as a function of k , P ( τ > k ).The red line corresponds to a robust fit of the tail distributionin the range of 100 < k < − . ± .
05. The tail of the distribution decays abruptlynear to 100000 because there is a finite size scaling problem.Simulations were done for a number of bidders (N) equal to2000000.
Let ˜ N ( k ) be the number of accepted bids when k bidshave been offered. This value evolves in the followingway:˜ N ( k + 1) = (cid:40) ˜ N ( k ) if X k +1 ≥ ˜ X max ˜ N ( k ) ( k )˜ N ( k ) + 1 if X k +1 < ˜ X max ˜ N ( k ) ( k ) , (13)with ˜ N (1) = 0. The critical value can be obtained bytaking the following limitlim N →∞ (cid:104) ˜ N ( N ) (cid:105) N = 1 − p c , (14)and the asymptotic variance can be calculated in a similarway, lim N →∞ V ar (cid:16) ˜ N ( N ) (cid:17) N = b ≈ . . (15)An avalanche is a sequence of events where the acceptedbid values are greater than x c . It starts at k + 1 if Y k < x c and Y k +1 > x c , and has a duration τ if Y k +2 > x c , Y k +3 >x c ,..., Y k + τ > x c , and Y k + τ +1 ≤ x c . The τ distributiondisplays an asymptotic behavior (for large s ), P ( τ = s ) ∼ s − α or P ( τ > s ) ∼ s − α +1 , (16)with α ≈ .
5, and therefore (cid:104) τ (cid:105) = ∞ . IV. MODIFICATIONS OF THE MODEL
The model may be modified to have a different criticalvalue and similar dynamics. For example, a bid wouldbe accepted if it is the highest bid compared with thosepresented before and after it two consecutive . This lastmodel yields a critical value p c ≈ . V. DISCUSSION
Herein we have introduced a novel model for sellingzero or non-zero marginal cost products in real time. Thismodel exhibits self-organized criticality [9]. The modelcan be applied for selling any product that has infinitystock or products that can be produced at the same (orsimilar) rate of the demand, for example, for selling onlinepublicity, electronic posters, software, etc. In the modelfor almost all bids, the decision is made quickly and theacceptance price is above a (critical) value x c that onlydepends on the bid price distribution. Approximately63.2%, or exactly (1 − e − )100%, of the bidders will buythe product (bids accepted). This value may be modifiedby slightly altering the model, as we have shown. Oneof the advantages of the model is that the average totalincome can be estimated with high accuracy.We believe this model may motivate quantitative re-searchers to further explore this topic. As a result, neweconomic models may emerge for situations where thereis little or no interaction between the economic agents.On the following subsection we discuss some consid-erations for the implementation of the model, and somerelevant questions to the field of behavioral economics:what is the impact of information sharing in the model?What is the empirical price distribution? Is it convenientto set a base price? A. Considerations for the implementation of themodel
One important consideration for implementing themodel is that each interested buyer must apply to the auc-tion process through a unique bid, as in closed envelopetenders. Therefore, in practice, it is necessary to verifybidder identities, which is not a difficult problem. Onemight ask, why not sell to everyone who makes an offer(instead of defining an auction mechanism), as the bandRadioHead did 10 years ago? The band sold their album
In Rainbows using a “pay what you want” method, whichproved to be a success. In terms of bids, this means that all the offered prices were accepted. The key point is thatunder this hypothesis, if we define the sequence of “surebids”, X sure , X sure , X sure , . . . , these bids probably willbe very different ( f sure ( x ) (cid:54) = f ( x )). It is reasonable tosuppose that X sure will be smaller than the value X ofthe auction model presented here because there is no riskof not obtaining the product (i.e. probably X sure ≤ st X ),which is equivalent to saying that its corresponding cu-mulative distributions will verify F sure ( x ) ≥ F ( x ) for all x . Therefore, the total income will most likely be smallerwith the “pay what you want” (or donation) option, yetthis method has the advantage of socializing the product.
1. The price distribution
Clearly, the bidder behavior is influenced by his income,the opportunities presented by the economic environment,the valuation of the product, and the auction rule, amongother things. That is why bidders presents different bids.If we choose randomly one of the bidders, she/he willmake an offer that is describe a random variable X withdistribution F ( x ). The profit depends on this last distri-bution. We cannot advocate for any one price distributionover another, but suggest that novel products or serviceswould have an exponential or power law distribution,while products that are well-known on the market wouldhave a Normal or Log-Normal distribution. It is interest-ing to note that if the price distribution has a a powerlaw tail with α <
2. Targeting prices
With additional information about the bidders, onecan categorize them according to country, sex, age, andany other relevant sociodemographic variable. Bidderscould compete with other bidders from the same economicsegment, which would yield more equity opportunitiesfor acquiring the product. The (1 − p c )100% most inter-ested targeted buyers (based on the bids) will obtain theproduct.
3. Base price or not?
The same auction procedure may be applied with abase price. In this case, the bids received ( X ) will belarger than the base price. Companies that sell non-zeroproducts may be tempted to use a base price. Is thisa good strategy? Will the profit be larger? This is notan easy question to answer. Once again, this most likelydepends on the novelty of the product. However, settinga base price may have a priming effect, a phenomenonwell-known to the cognitive neuroscience and behavioraleconomics communities. In priming [3, 4], exposure toone stimulus influences a response to a subsequent stimu-lus without conscious guidance or intention. How muchawareness the bidder has regarding the priming effect onhis/her bid value is a matter of debate. ll l ll NO BASE LOW BASE HIGH BASE10100100010000100000 B I D P R I C E [ $ ] FIG. 3: Bid prices for each of the experimental conditions.Each base price is shown with a red dashed line.
To further explore this matter, we conducted an experi-ment using a rudimentary protocol. Eighty-one universitystudents were asked to make a bid for a new product thatsignificantly improves memory. The specific question was:“How much would you pay to use this product for 1 hour a day for 2 years?” All students, except two, showedinterest to purchase the product. Details about the prod-uct and questionnaire are described in Supp. Mat. Eachsubject was randomly assigned to one of three groups.In group 1, students make a bid to obtain the service;in group 2, students make a bid with a pre-establishedcheap base price; and in group 3, students make a bidwith a pre-established larger base price. Figure 3 showsa boxplot of the bid prices for each group in log-scaletogether with the base price (dashed lines). As shownin Fig. 3, the larger the base price, the higher the bids.Moreover, “no base price” yields similar bids than lowbase prices. Note that subjects who consider the baseprice high offer lower bids.
4. What about information sharing?
Here we discuss and speculate what would happen if thepurchase price distribution is known by the new interestedbuyers. Let us suppose that after some time, when theprocess has stabilized, the company decides to share thepurchase price distribution (i.e., a histogram with all thesales conducted until that moment). Alternatively (andperhaps more realistically), buyers who purchased theproduct share their purchase price on some webpage orsocial network that is accessible by new potential buyers.How would this influence the bids of future bidders? Webelieve this could lead to a surprising effect: a new criticalprice x newc will appear, which will be larger than theprevious one ( x c ). This will happen because most ofthe accepted bids will be greater than x c , and thus anew customer interested in the product will offer a valuegreater than x c , thus pushing the critical price to a largervalue. How sharing information changes the rest of theoriginal price distribution is a mystery. Note that evenwhen x newc is greater than x c and the rate of biddersis equal, the new total income T I new may be smalleror larger than the previous one (
T I ). The value
T I new depends on the details of the new price distribution, F new . [1] V. Krishna Auction theory. Academic press (2009).[2] S. Lahaie, D. M. Pennock, A. Saberi, and R. V. Vohra.Sponsored search auctions.
Algorithmic Game Theory ,699-716 (2007).[3] E. Tulving, D. Schacter. Priming and human memorysystems.