Pricing Problems for Data Sets

CSE544: Project Report

Dimitrios C. Gklezakos

University of Washingtongklezd@cs.washington.edu

Abstract. In this paper the problem of pricing in online auctions witha stochastic assumption is discussed, with the main focus on approachesthat are suitable for pricing access to data sets in the context of datamarkets. The performance of existing algorithms is evaluated empirically.We present two simple approaches to pricing, when maintaining multiplecandidate prices is not suitable. Last, the issue of revenue variance isdiscussed as a measure of risk.

1 Introduction

Current data hosting platforms such as Microsoft Azure enable data set ownersto reach a much broader client base and in a more efficient and elegant way. Theowners make their data available in the market and interested clients purchaseaccess. However, this setting creates some issues related to the way these datasets are priced. In this paper, we consider the pricing problem of the owner of arecently created data set. The owner wants to make it available via the market-place, but is not aware of the actual value of the product. Moreover there areno similar data sets to compare against, which means that the owner does nothave a point of reference to help price the data.

Suppose that there are various potential customers that value the data differ-ently. We will refer to these customers as “bidders”. The bidders arrive at themarketplace one at a time and demand a quote from the data owner. The ownersuggests a price and the bidder will buy access to the data if she values the datamore than the price. Otherwise she leaves, without providing any informationon that valuation. The owner’s goal is to maximize his revenue. How is he goingto approach this situation? The problem is challenging since not only the owneris unaware of how bidders value the product, but there is also limited feedbackfrom them.

2 The Model

Let V = (V1, ..., Vm), be a vector of valuations, where the i-th bidder to arrivehas valuation Vi. The meaning of Vi is that bidder i will purchase access at pricep only if p ≤ Vi. Let’s further assume that all valuations are bounded by some

known quantity h. Then an online pricing scheme is defined as a sequence offunctions:

gi : (x1, ..., xi−1)→ [0, h]

where ∀j < i, xj equals 1 if the offer zj = gj(x1, ..., xj−1) was accepted and 0otherwise. Notice that the owner does not have any information about the actualvaluations of the bidders apart from accept/reject responses to previously offeredprices.

2.1 Adversarial valuations

So far we haven’t made any assumption on the nature of V . The valuations in Vcan be assumed to be chosen by an “adversary” whose goal is to maximize ourloss relative to some benchmark. In this case, the benchmark that we are goingto use is the best possible fixed posted price in L ; a set of candidate prices.Specifically, let Gl = l|{vi : vi ≥ l}| and let Gmax = maxl∈L(Gl). For anyarbitrary sequence of valuations V , we want to minimize:

Gmax −m∑i=1

xigi(x1, ...xi−1) = Gmax −m∑i=1

xizi

2.2 Random valuations

Instead of assuming that the choice of V is made by an adversary, in the contextof a real data set pricing application, it is more natural to model vi’s as randomvariables drawn from some distribution. Therefore each vi ∼ F , where F is anunknown to us regular distribution that is common for all i. This is a settingthat is easier to deal with, since the accept/reject responses to previous offers arecorrelated with the valuations of the future incoming bidders, in direct contrastwith the adversarial valuations setting.The expected revenue raised from one bidder by offering price p is:

R(p) = p(1− F (p))

It is clear that the best posted price is p∗ = arg maxp∈[0,h](R(p)). For regular

distributions, p∗ can be found by solving for ∂R(p)∂p = x − 1−F (x)

f(x) = 0 where

f(x) is the density function of the distribution. The expected revenue from thatoptimal posted price is Rmax = mR(p∗). Our goal is to minimize:

Rmax −m∑i=1

xizi

2.3 Related work

For both assumptions on valuations there are powerful results.

Adversarial valuations The main line of work on adversarial valuations mod-els this problem as a non-stochastic multi-armed bandit problem. All the pricesand valuations lie in the [0, 1] interval. The main approach is to apply the EXP3

algorithm as introduced in [2]. There is a set of candidate prices {pi}li=1 thatcorrespond to the “levers” in the original definition of the problem. EXP3 main-tains a weight vector w(t) on candidate prices. At each round price pi is chosenwith probability qi(t) = (1 − γ)wi(t) + γ

l . Suppose that this price results inrevenue ri(t) when offered to the bidder at time step t. Then its new weight is:

wi(t+ 1) = wi(t)(1 + α)ri(t)

qi(t)

where α is a parameter of the algorithm. At each round EXP3 enriches thecurrent weight vector by chosing a price according to it with probability (1− γ)and from the uniform distribution on prices with probability γ. EXP3 guaranteesrevenue:

rEXP3 ≥Gmaxβ−O(K logK)

where K is the number of candidate prices and β is a constant. In subsequentwork that utilizes EXP3 for online posted price auctions [4] (which is exactlythe setting that we are considering), K is set to be O(log(h)). Moreover rewardsare scaled from [0, 1] to [0, h]. This transforms the additive loss from O(K logK)to O(h log h log log h). In more recent results presented in [3], a clever choiceof enriching the weight vector by using a geometric distribution instead of theuniform gets rid of a log h factor in the additive loss above.

Random valuations Similar results for random valuations are presented in [5]utilizing the UCB algorithm. UCB starts by choosing each price once. Then ateach round t+ 1 the algorithm computes the following coefficient for each price:

ci(t) = xi(t) +

√2 ln(t)

ni(t)

where xi(t) is the average revenue so far and ni(t), the number of times that theprice was picked in the past. The price with the largest ci(t) is chosen.

2.4 Limitations & Contributions

Limitations Even though the current state of the art offers powerful algorithms,these results can sometimes be unsuitable for use in the context of a data market-place. For example, EXP3 and UCB maintain distributions on prices and offer aprice sampled from these distributions. Therefore the price offered at each timestep is not deterministic, even if the algorithm has “seen” considerable part ofthe input. This property is unattractive for real applications since clients preferpredictable prices.

Another unattractive characteristic of the approaches discussed is that revenue,as a random variable, not only has a mean, but also variance. Offering differentprices results in different variance of revenue. In a data market setting, owners donot only consider how high the revenue is, but also how predictable and reliableit is. In other words, owners care about risk. A common measure of risk is thestandard deviation (volatility) of the income. Therefore owners might be willingto sacrifice a fraction of their expected revenue, in order to reduce its volatility.Most of the common approaches do not consider risk-averse entities and there-fore restrict their focus on maximization of expected revenue. Although there issome discussion in [1] about variance of rewards in UCB, they focus more onimproving the volatility of the “regret” of the algorithm as defined previously.

Contributions We simulated UCB and EXP3 for different distributions andnumbers of clients and measured the ratio between the revenue of the algorithmand the theoretically optimal one. UCB was modified to stop after a constantfraction of the input and pick the best price so far. Experiments show thatthis algorithm has little difference in revenue from UCB, but surely results ina single price for the rest of the input. Moreover we implemented an algorithmthat naively estimates the cdf of the distribution and chooses the empiricallybest price. Last but not least we consider the problem of minimizing varianceof revenue for a single agent with known distribution, while maintaining a pre-specified level of expected revenue. All implementations were done in R.

3 Revenue Approximation

3.1 UCB performance

We performed simulations on 10, 102, 103, 104, 105 agents for valuations distributednormally, uniformly, exponentially or according to the student-t1 distribution.We measured the performance of EXP3 and UCB. As expected EXP3 performedworse than UCB (figure 1(a)) and this is simply because we are in a stochas-tic setting with a prior, whereas EXP3 cannot assume a prior distribution onthe valuations. Henceforth, we are only going to consider UCB. UCB performedreally well against the test distributions. In various simulation it consistentlyapproximated optimal revenue to a large fraction. In figure 1(b) we can see theperformance of UCB against normally distributed valuations with standard de-viations 1, 2, 4, 8, 16, 32. Notice that the response to variance is good, since for asuitably large number of agents the ratios converge. Again for a large number ofagents, UCB recovered about 3

4 of the optimal revenue. This makes UCB a veryattractive choice.

3.2 Single Posted Price

To address the need to offer a single price, UCB was modified to stop afterhandling 1

10 of the input and pick the price with the best revenue so far (STOP

1 Which was chosen for its fatter tails.

(a) UCB (red) perfmormed better than EXP3(black) as expected. This figure identifies thistrend for normally distributed variables.

(b) UCB for normally distributed valuations of various standard deviations.

algorithm). Moreover the following algorithm, that naively estimates the cdf ofthe distribution, was implemented:

1. Split the support 2 of the distribution in K =(d nlogne

) 14

equidistant prices3 pi.

2. Offer each pi to m10K of the clients and record how many offers where ac-

cepted. Record the frequencies fi.3. Fix local non-monotonicity introduced by sampling errors, naively, by equat-

ing adjacent non-monotone frequencies. For fi, fj : j = i + 1 where fj < fiset fj = fi. This is the empirical cdf.

4. Pick the optimal price according to the empirical cdf and offer it to the restof the clients.

The experiments show that STOP is almost identical to UCB ; the differencesin revenue are negligible. This means that in practice UCB seems to convergeto a good price, quite quickly. Maybe the nicest positive result of these twoapproaches is that for a suitably large number of agents, naive estimation per-formed just as well or sometimes better than UCB and STOP. This is useful,since naive estimation is a very simple and natural algorithm. In figure 1(c), theratios of the three different algorithms can be seen for exponentially distributedvaluations.

4 Revenue Variance

In this section we examine a problem related to the variance of the revenue ofthe data owner. Suppose that the valuation distribution of the clients is availableto the owner, via marketing research or potentially via estimation by one of thepreviously discussed algorithms. Then for each potential price that the ownercan offer to a client, the associated expected revenue is a random variable withknown expectation. Moreover, for different prices, the variance of the revenue isdifferent. This observation reveals an interesting question related to the choiceof price:

What happens if the variance of the revenue of the optimal price is unaccept-ably high for the owner?

Intuitively offering a specific price carries an associated risk. The revenue fromthis choice of price might be optimal, by being high but infrequent. In practice,the data owner will be a risk averse entity that might prefer a lower expectedrevenue if it is more “stable”, in the sense that it is more predictable. We will usevariance (or standard deviation) as the standard measure of risk. In this section

2 Previously we discussed knowing a bound h on the valuation. Here we assume thatwe know the supremum of the support of the distribution. For distributions withinfinite support the 0.999-th quantile was chosen.

3 K was picked according to the choice of [5] for UCB.

(c) UCB (black), STOP (red), naive estimation (blue). For a small number of agents UCBand STOP are identical and perform better than the naive approach. However for a largenumber of agents, naive estimation clearly dominates UCB and STOP. The difference be-tween UCB and STOP is very small and in the favor of UCB (as expected). Note thatlambda is the parameter of the exponential distribution.

we will investigate the relation between expected revenue and risk when dealingwith a single client.

4.1 Preliminaries

We will start by defining our setting. We have to sell access to the data set to asingle interested client. Let v be the valuation of the client for such access. Weassume that v ∼ F from some known and regular F . Then as discussed in theprevious sections, the revenue curve depending on the price p is given by:

R(p) = p(1− F (p))

Let gp be the r.v. that denoted the realized revenue from offering p. Then:

gp =

{p w.p. 1− F (p)

0 o.w.

We known that E[gp] = R(p). Its variance is given by:

V ar(gp) = E[g2p]−R2(p) = p2(1− F (p))−R2(p)

= R(p)(p− p(1− F (p))) = pF (p)R(p)

Let p∗ be the optimal price discussed in the previous sections. Then any level ofexpected revenue µ ∈ [0, R(p∗)] is achievable. Suppose that we want to answer

the following question:

Given a desired level of revenue µ what is the auction that results in that expectedrevenue, but with minimum variance.

This question has a dual one; for any given level of risk, find the auction thatresults in the maximum revenue. Answering this question provides the ownerwith the ability to consider different configurations of risk/gain and choose themost suitable one for his needs. This question is not trivial for two reasons:

1. There can be multiple prices that result in the same revenue. To be precisefor each µ ∈ [0, R(p∗)) there are two prices with expected revenue µ.

2. Instead of choosing to offer a specific price deterministically, randomizationcan be used. Specifically we can define a distribution on various prices, sam-ple a price from that distribution and offer that to the client.

We will address both these problems.

4.2 Double prices

To see why there are two prices for every possible µ (apart from the optimal),we can re-write the revenue function in quantile space:

R(q) = qv(q)

where q ∈ [0, 1] is a quantile and v(q) = F−1(q). The revenue curve in quantilespace is known to be concave for regular distributions, therefore any equationof the form R(q) = µ, µ ∈ [0, R(p∗)) has exactly two solutions x1, x2 : x1 > x2.Let p1 = v(x1), p2 = v(x2) be the corresponding prices that are solutions to theequation R(p) = µ. Note that p1 < p2, since x1 > x2, which means that p1

is more likely to result in a successful sale, therefore it must correspond to thelower of the two prices. Notice that:

R(p1) = R(p2)

p1 < p2

F (p1) < F (p2)

Therefore:

V ar(gp1) = p1F (p1)R(p1) < p2F (p2)R(p2) = V ar(gp2)

Thus, among the two prices that achieve expected revenue µ the lowest onealways results in lower variance. Thus we can restrict our attention only on the[0, p∗) interval. Since R(p) is increasing on that interval (by regularity), thisallows us to invert it. Henceforth we refer to that inverse as R−1(µ), µ ∈ [0, p∗).

4.3 Distributions on prices

Identifying the minimum variance price suffices only when we discuss deter-ministic variance minimizing strategies. However, we can define the followingrandomized strategy:

1. Define a distribution D on a set of candidate prices {pi}li=1 3 R−1(µ).2. Sample p ∼ D and offer it to the client.

The distribution identifies non-negative weights xi, which correspond to theprobability of offering price pi. Let GD be the r.v. that represents the revenuefrom pricing according to D. Let E[GD] = µD. Then:

V ar(GD) =

l∑i=1

xipiR(pi)− µ2D

If we are given a desired expected revenue goal µ, then we can find the distribu-tion that results in that revenue, but minimizes variance by solving the followinglinear-program:

Input:{pi}li=1, µ

min

l∑i=1

xipiR(pi)

s.t.l∑i=1

xiR(pi) = µ

l∑i=1

xi = 1

xi ≥ 0,∀i ∈ [l]

We empirically solved this LP for the distributions used in the evaluation of UCB,for different parameters and the result was that a single price minimizesvariance. This result is useful because it says that if the data owner wantsto minimize variance, given some revenue goal, all that has to be done is toinvert R(p) on the [0, p∗) interval and pick the corresponding price. There isno distribution that can achieve lower variance. Although we were not able tovalidate this via a rigorous proof, we were able to solve it for restricted casesand we conjecture that it is true.

5 Future Directions

This paper presents the first steps towards an empirical and practical studyof pricing algorithms suited to the needs of data markets. The assumptions onwhich the results were made are not necessary realistic. A good direction forfuture research would be to adjust the above algorithms and results to complyto assumptions directly specified by the needs of the industry.

Moreover, we dealt with clients that want to purchase lifetime access to thedata set. This is rarely the case. It would be very interesting to see results forsubscription based models. Another large step would be to approximate revenuein settings where clients draw their valuations from different distributions. Byoffering free trials and monitoring their access patterns to the database, we couldclassify them in groups and price them differently.

Another interesting direction would be to characterize the relation between rev-enue and variance for more agents or again for agents that belong to differentcategories.

References

1. Jean-Yves Audibert, Remi Munos, and Csaba Szepesvari. Exploration-exploitationtradeoff using variance estimates in multi-armed bandits. Theor. Comput. Sci.,410(19):1876–1902, April 2009.

2. Peter Auer, Nicolo Cesa-Bianchi, Yoav Freund, and Robert E. Schapire. The non-stochastic multiarmed bandit problem. SIAM J. Comput., 32(1):48–77, January2003.

3. Avrim Blum and Jason D. Hartline. Near-optimal online auctions. In In Proceedingsof the 16th Annual ACM-SIAM Symposium on Discrete Algorithms, pages 1156–1163, 2005.

4. Avrim Blum, Vijay Kumar, Atri Rudra, and Felix Wu. Online learning in onlineauctions. In Proceedings of the fourteenth annual ACM-SIAM symposium on Dis-crete algorithms, SODA ’03, pages 202–204, Philadelphia, PA, USA, 2003. Societyfor Industrial and Applied Mathematics.

5. Robert Kleinberg and Tom Leighton. The value of knowing a demand curve: Boundson regret for online posted-price auctions. In Proceedings of the 44th Annual IEEESymposium on Foundations of Computer Science, FOCS ’03, pages 594–, Washing-ton, DC, USA, 2003. IEEE Computer Society.