Equilibrium Pricing with Positive Externalities

Nima AhmadiPourAnari∗ Shayan Ehsani† Mohammad Ghodsi†‡

Nima Haghpanah§ Nicole Immorlica§ Hamid Mahini†

Vahab Mirrokni¶

Abstract

We study the problem of selling an item to strategic buyers inthe presenceof positivehistorical externalities, where the value of a product increases asmore people buy and use it. This increase in the value of the product isthe result of resolving bugs or security holes after more usage. We consider acontinuum of buyers that are partitioned intotypeswhere each type has a val-uation function based on the actions of other buyers. Given afixed sequenceof prices, orprice trajectory, buyers choose a day on which to purchase theproduct, i.e. they have to decide whether to purchase the product early in thegame or later after more people already own it. We model this strategic set-ting as a game, study existence and uniqueness of the equilibria, and designan FPTAS to compute an approximately revenue-maximizing pricing trajec-tory for the seller in two special cases: thesymmetricsettings in which thereis just a single buyer type, and thelinear settings that are characterized byan initial type-independent bias and a linear type-dependent influenceabilitycoefficient.

Keywords: Social Networks, Game, Revenue Maximization, Equilibrium,Historical Externalities, Pricing

∗University of California at Berkeley, EECS Department,nima.ahmadi.pour.anari@gmail.com

†Sharif University of Technology, Computer Engineering Department,{ehsani,mahini}@ce.sharif.edu

‡Institute for Research in Fundamental Sciences (IPM), P.O.Box: 19395-5746, Tehran, Iran,ghodsi@sharif.edu, This author’s work has been partially supported by IPM School of CS(contract: CS1388-2-01)

§Northwestern University, EECS Department,{nima.haghpanah,nicimm}@gmail.com¶Google Research NYC, 76 9th Ave, New York, NY 10011,mirrokni@google.com

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1 Introduction

Many products like software, electronics, or automobiles evolve over time. Whena consumer considers buying such a product, he faces a tradeoff between buyinga possibly sub-par early version versus waiting for a fully functional later version.Consider, for example, the dilemma faced by a consumer who wishes to purchasethe latest Windows operating system. By buying early, the consumer takes fulladvantage of all the new features. However, operating systems may have morebugs and security holes at the beginning, and hence a consumer may prefer to waitwith the rationale that, if more people already own the operating system, then morebugs will have already been uncovered and corrected. The keyobservation is, themore people that use the operating system, or any product forthat matter, the moreinherent value it accrues. In other words, the product exhibits a particular type ofexternality, a so-calledhistorical externality1.

How should a company price a product in the presence of historical exter-nalities? A low introductory price may attract early adopters and hence help thecompany extract greater revenue from future customers. On the other hand, toolow a price will result in significant revenue loss from the initial sales. Often, whenfaced with such a dilemma, a company will offer an initial promotional price at theproduct’s release in a limited-time offer, and then raise the price after some time.For example, when releasing Windows 7, Microsoft announceda two-week pre-order option for the Home Premium Upgrade version at a discounted price of $50;thereafter the price rose to $120, where it has remained since the pre-sale ended onJuly 11th, 2009. Additionally, beta testers, who can be interpreted as consumerswho “bought” the product even prior to release, received therelease version ofWindows 7 for free (as is often the case with software beta-testers).2

We study this phenomenon in the following stylized model: a monopolisticseller wishes to derive a pricing and marketing plan for a product with historicalexternalities. To this end, she commits to a price trajectory. Potential consumersobserve the price trajectory of the seller and make simultaneous decisions regardingthe day on which they will buy the product (and whether to buy at all). The payoffof a consumer is a function of the day on which he bought the product, the price onthat day, and the set of consumers who bought before him. We compute the equi-libria of the resulting sequential game and observe that therevenue-maximizing

1Note that this is different from the more well-studied notion of externalities in the computerscience literature where a product (e.g., a cell phone) accrues value as more consumers buy it simplybecause the product is used in conjunction with other consumers.

2Historical prices, announced upon the press release, can befound in archived versions of varioustechnology news websites such as Ars Technia [22] and the Microsoft blog [16]. The current priceswere accessed on Microsoft’s website at the time of submission.

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price trajectories for the seller are increasing, as in the Windows 7 example above.A few words are in order about our model. First, we focus on settings in which

the seller has the ability to commit to a price trajectory. Such commitments areobserved in many settings especially at the outset of a new product (see the Win-dows 7 example described above) and have been assumed in prior models in theeconomics literature on pricing as well as in other games in the form of Stackel-berg strategies (see Section 1.2). Further, commitment increases revenue: clearlya seller who commits to price trajectories can extract at least as much revenue as aseller who does not (or can not). We further observe via example in Section 2 thatin fact commitment enables a seller to extract unboundedly higher revenue thanin settings without commitment. Second, we assume a consumer’s payoff is onlya function of past purchases; i.e. consumers have no utilityfor future purchases.We motivate this in the Windows 7 example by arguing that bugsare resolved inproportion to usage rates. Of course, strictly speaking, consumers of Windows7 benefit from future purchases as well via software updates and the like. How-ever, this forward-looking benefit is substantially dampened in comparison to pastbenefits by safety and security risks, and time commitments involved in updates.Another justification for our payoff model comes from consumers’ uncertaintiesregarding products. In many settings, consumers have signals regarding the valueof a product (say an electronic gadget like the iPad for example), but do not ob-serve its precise value until the time of purchase. Past purchases and the ensuingonline reviews may help consumers improve their estimates of their values prior topurchase, an especially important factor for risk-adversebuyers.

1.1 Our Results

We focus on the non-atomic setting in which we have a continuum of consumersso that each consumer is infinitesimally small and thereforehis own action has anegligible effect on the actions of others. Consumers are drawn from a (possi-bly infinite) set of types. These types capture varying behavior among consumergroups. We study a sequential game in which the seller first commits to a price tra-jectory and then the consumers simultaneously choose when and whether to buy inthe induced normal-form game among them. We study subgame perfect equilibria.We first observe in Section 3 that equilibria exist due to a slight generalization of apaper of Mas-Colell [18].

We then turn to the question of uniqueness in Section 4. We focus onwell-behaved equilibriain which consumers with non-negative utility always purchasethe product (thus indifferent consumers purchase the product). In general multi-ple such equilibria may exist. However, in an aggregate model in which the valuefunction of each consumer type depends only on theaggregatebehavior of the pop-

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ulation (i.e. the total fraction of potential consumers that have bought the productand not the total fraction of various types), then we are ableto show that whenthey exist well-behaved equilibria of this game are unique in the sense that thefraction of purchases per-type-per-day is fixed among all equilibria. This enablesus to search for the revenue-maximizing price trajectory. In Section 5, we addressthis question in settings in which we either have just one type or there are multi-ple types whose valuation functions are linear in the aggregate, both of which arespecial cases of the aggregate model discussed above. For each price trajectory,we define its revenue to be the amount of money consumers spendon the product.We then design an FPTAS to find the revenue-maximizing price trajectory for amonopolistic seller in these settings. We do this via a reduction to a novel rectan-gular covering problem in which we must find the discounted area-maximizing setof rectangles that fit underneath a given curve. We study the rectanglular coveringproblem in Section 6.

In summary, we get the following main result for the settingsdescribed above:first, every price trajectory has well-behaved equilibria that are revenue-uniqueand revenue-maximal among all possible consumer equilibria. Second, it is possi-ble to (approximately) compute the revenue-maximizing price trajectory. Hence,the strategy tuple in which the seller announces this (approximately) revenue-maximizing price trajectory and the consumers respond by playing a well-behavedequilibrium is an (approximately) subgame perfect equilibrium of our game.3

A preliminary version of this paper appeared in WINE2010 [1].

1.2 Related Work

Our work falls in the long line of literature investigating pricing and marketing ofproducts that exhibit externalities [2, 3, 4, 5, 7, 10, 11, 12, 14, 15, 19, 20, 23].Among these, the paper of Bensaid and Lesne [5] is most closely related to ourown work. Bensaid and Lesne [5] analyzed the two and infinite period pricingproblems in the presence of linear historical externalities and the study equilib-ria of the induced games both with and without commitment. They observe, aswe do, that optimal price trajectories are increasing. The historical externalitiesthat we study generalize the externalities of Bensaid and Lesne [5], and in thismore general model, we solve for the optimal price sequence for any fixed numberof price periods. Most of the remaining externalities literature studies externali-ties in which consumers care about the total population of users of a product andhence their utility is affected by future sales as well as past sales. Although the

3We must also define what happens off the equilibrium path in subgame perfect equilibria; herewe assume that for any price trajectory announced by the seller, the consumer respond by playing awell-behaved equilibrium.

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phenomenon studied is different from ours, some of the modeling assumptions inthese papers are similar to ours. For example, in the economics literature, Cabral,Salant, and Woroch [7] also consider a seller that commits toa price trajectory andthen observe that the revenue-maximizing price sequence with fully rational con-sumers (playing a Bayesian equilibrium) is increasing. Similar to our model, theystudy the pricing problem in the presence of a continuum of consumers.

In the computer science literature Akhlaghpour et al. [2], Hartline et al. [12],and Bimpikis et al. [6] study algorithmic questions regarding revenue maximizationover social networks for products with externalities. However, their models assumenaive behavior for consumers. Namely, they assume consumers act myopically,buying the product on the first day in which it offers them positive utility withoutreasoning about future prices and sales that could affect optimal buying behaviorand long-term utility. Furthermore, Hartline et al. [12] allow the seller to use adap-tive price discrimination. In contrast, we model consumersas fully rational agentsthat strategically choose the day on which to buy based on full information regard-ing all future states of the world and a sequence of public posted prices. While thecorrect model of pricing and consumer behavior probably lies somewhere betweenthese two extremes, we believe studying fully rational consumers is an importantfirst step in relaxing myopic assumptions. Bimpikis et al. [6] study such a model inwhich players act strategically. They study a two stage pricing problem in whichthe seller first determines the prices for adivisible good, and then the buyers si-multaneously decide about the amount of consumption. In contrast to our modeland similar to Hartline et al. [12], the seller is allowed to use price discrimination.Also, the strategy of buyers in their setting is the amount ofconsumption, not thetime in which consumption happens. They assume that each player’s utility is anon-decreasing function of the consumption of other buyers. Similar to our model,they study the subgame perfect equilibria of the two-stage game. They study thestructural properties of the optimal prices, as well as the algorithmic problem offinding the optimal price sequence.

Our work can be viewed as a analysis of the optimal commitmentstrategyof a seller in the presence of historical externalities. Stackelberg [24] first stud-ied optimal commitment strategies, called Stackelberg strategies, in the Cournotsduopoly [9] model, proving that a firm who can commit has strategic advantage.The computational aspects of Stackelberg strategies have been studied in genericnormal form [8, 21] and extensive form games [17]. Letchfordand Conitzer [17]suggested polynomial-time algorithms for various models of perfect informationextensive form games and proved NP-hardness of the problem with imperfect in-formation even with two players. A perfect information extensive form game isone in which at every stage of the game, every player is exactly aware of what hashappened earlier in the game. This does not happen in games with simultaneous

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move, as in our game.

2 Model

We wish to study the sale of a good by a monopolistic seller over k days to a set ofpotential consumers or buyers. We model our setting as a sequential game whoseplayers consist of the monopolistic seller and a continuum of potential consumersor buyersb ∈ [0, 1]. In our game, the seller moves first, selecting aprice trajectoryp = (p1, . . . , pk) wherepi ∈ ℜ assigning a (possibly negative) pricepi to each dayi. The buyers move next, selecting a day on which to buy the product given thecomplete price trajectory, as described below.

The buyers are partitioned inton typesT1, . . . , Tn where eachTt is a subinter-val of [0, 1]4. The strategy setA = {1, . . . , k}∪{∅} indicates the day on which theproduct is bought (∅ is used to indicate that the product was not purchased). Hencethe strategy profile of the buyer population can be represented by a(k+1)×n ma-trix X = {Xi,t}i=1,...,k+1;t=1,...,n where entryXi,t indicates the fraction of buyersthat are of typet and buy the productbeforeday i, and we defineX1,t = 0 for allt. Note that by normalization

∑

t Xk+1,t ≤ 1 and1 −∑

t Xk+1,t is the fraction ofbuyers that don’t buy the product at any time. Correspondingto this matrixX wealso define themarginal strategy profilematrix x = {xi,t}i=1,...,k;t=1,...,n wherexi,t = Xi+1,t − Xi,t is the fraction of buyers who are of typet and buy on dayi.In the special case when there is only1 type, we useXi as a scalar to denote thefraction of buyers who bought before dayi andxi as a scalar to denote the fractionof buyers who buy on dayi.

Given a strategy profileX, we define the value of buyers of typet buying onday i by a value functionF t

i (Xi) whereXi is the i’th row of X (hence buyersare indifferent to future buying decisions). Note the explicit dependence ofF ontime, which allowsF t

i (Xi) to be different thanF tj (Xj), for i 6= j. The revenue-

maximization results in Section 5 further assume that the dependence ofF ti (Xi)

on i is of the formF ti (Xi) = βiF t(Xi) for β ∈ [0, 1]. This special case is of

particular interest as theβ factor models settings in which the value degrades overtime due to, for example, a reduction in the novelty of the product.

Given a strategy profileX, the payoff of buyers of typet who buy on dayi isdefined to beF t

i (Xi) − pi. We additionally allow buyers to have a discount factorα such that their payoff is(1 − α)i(F t

i (Xi) − pi). Thusα represents the way inwhich agents discount future payoffs with respect to present payoffs. We say thata strategy profileX is a Nash equilibrium of the induced subgame given by pricetrajectoryp, or equivalentlyX ∈ NE(p), if for any buyer of typet who buys on

4Later, we will generalize this to infinitely many types.

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dayi we havei ∈ arg maxj(Ftj (Xj)− pj)(1−α)j , and the strategy is∅ whenever

the maximum is not positive (in which case the buyer’s payoffis zero). We call anequilibrium well-behavedif all indifferent buyers buy, i.e. a buyer does not buy ifand only if his payoff(1−α)i(F t

i (Xi)− pi) is negative on all days1 ≤ i ≤ k. Wesay that(p,X) is a (well-behaved) equilibrium if the profileX is a (well-behaved)Nash equilibrium for the subgame of price trajectoryp. Equivalently, a marginalstrategy profilex of a stragety profileX is a (well-behaved) Nash equilibriumfor the subgame of price trajectoryp if the profile X is a (well-behaved) Nashequilibrium for the subgame of price trajectoryp.

Given a price trajectoryp and a marginal strategy profilex that arises in thesubgame induced byp, we define the payoff of the seller to be therevenueof x forp, which isR(p, x) =

∑ki=1

∑nt=1 xi,tpi(1 − α)i. A subgame perfect equilibrium

of the sequential game is then a price trajectoryp∗ and a set of marginal strategyprofilesxp for each possible price trajectoryp such that: (1)xp is a Nash equilib-rium of the subgame induced byp, and (2)p∗ maximizesR(p, xp). Theoutcomeof this subgame perfect equilibrium is(p∗, xp∗) and its revenue isR(p∗, xp∗).

We are interested in computing the outcome in a revenue-maximizing subgameperfect equilibrium. To do so, we must compute a price trajectory which maxi-mizes the revenue of the seller in equilibrium. Note that this is equal to finding thebest response of the seller given the strategies{xp} of the buyers. We solve thisproblem for special settings in which there exist revenue-maximizing well-behavedequilibria inNE(p) for any price trajectoryp, allowing us to maximize over them.These settings are as follows. For the purpose of these definitions, we allow eachbuyer to have a unique type and hence there are infinitely manytypes. We will useb ∈ [0, 1] to denote the type of buyerb.

Definition 1 The Aggregate Model: The value function of each type in this modelis a function of the aggregate behavior of the population andis invariant withrespect to the behavior of each separate type. That is, the value function of buyerb is a function ofXi only, whereXi is a scalar indicating the total fraction of allbuyers who buy before dayi. In this instance, we overload the notation for thevalue function and letF b

i (Xi) indicate the value of buyerb (henceF bi (·) now maps

the unit interval to the non-negative reals).

Definition 2 The Linear Model: This is a special case of the aggregate modelwhich is defined by a functionFi, an initial bias I, and a functionC so that thevalue of buyerb is F b

i (Xi) = I + C(b) · Fi(Xi). We further define the commonly-known distributionC : R → [0, 1] such thatC(c∗) indicates the fraction of buyersb with C(b) ≤ c∗.

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Definition 3 The Symmetric Model: In this version we only have one type, that is,F b

i = Fi for all b.

We note that alternatively, one could model this pricing game as a sequentialgame with multiple stages where in each dayi the seller selects a pricepi and thenbuyers simultaneously choose whether to buy or not. Such a model is appropriatewhen it is not possible for a seller to commit to a price trajectory in advance.Again, in this setting, one could study the subgame perfect equilibria and analyzethe resulting revenue. Clearly the revenue with commitmentis at least as highas that without commitment. The below example shows that therevenue withoutcommitment can actually be unboundedly less.

Example. We consider buyersb ∈ [0, 1] with utility function F b(X) = (1+a|X|)bfor a non-negative constanta, and focus on a setting withk = 2 days. In a strategyprofile, the seller picks a pricep1 for day1 and a pricep2(S) for any subsetS ⊆2[0,1] of buyers that remain in day2. Each buyer decides whether to buy on day1given any possible announcementp1 of the seller, and also whether to buy on day2given any possible announcementp2 of the seller and decisionsS of other buyers.A strategy profile is a subgame perfect equilibrium if each player is playing a bestresponse given the strategies of others. In particular, this implies that the sellermust setp2(S) equal to the revenue-maximizing price for buyersS.

Assume a subgame perfect equilibrium in which the prices on path of play arep1 andp2, and subsetsS1, S2 ⊆ 2[0,1] buy at days1 and2, respectively. We claimp1 ≥ p2. Suppose for contradiction thatp1 < p2. First note that if|S1| = 0, thenbuying on day1 gives higher utility that buying on day2, and thereforeS2 = ∅.But this gives the seller zero revenue which is not his best strategy as he can lowerthe price of day2 and gain positive revenue. So assume from now on that|S1| > 0.The set of buyers who buy at the first day are the ones who have non-negativeutility of buying on day1, and also more utility of buying on day1 than day2.The first constraint is equal to havingb ≥ p1, and the second to havingb such thatb(1 + 0) − p1 > b(1 + a|S1|) − p2, or equivalantlyb < p2−p1

a|S1|(remember that we

assumed|S1| > 0). Let b∗ = p2−p1

a|S1|. The set of buyers buying on day1 is equal to

the buyers withb satisfyingp1 ≤ b and alsob < b∗. Our assumption that|S1| > 0now implies thatp1 < b∗. We observe thatp2 ∈ (p1(1 + a|S1|), b

∗(1 + a|S1|)), ascan be seen from the equalityp2 = a|S1|b

∗ + p1 and the inequalityb∗ > p1. Bythe properties of subgame-perfect equilibria, the seller has to play his best responsein the subgame induced at day2. In day 2 the utilities of players are outside theinterval (p1(1 + a|S1|), b

∗(1 + a|S1|)). In fact playerb utility at day 2 is in theinterval (p1(1 + a|S1|), b

∗(1 + a|S1|)) if and only if p1 < b andb < b∗ whichimplies the playerb bought in day1. So p2 ∈ (p1(1 + a|S1|), b

∗(1 + a|S1|))

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can not be a revenue-maximizing price. We conclude that in any subgame perfectequilibrium, the seller has to set the first price at least equal to second price, andno player will buy in the first day. The highest revenue for theseller is to set theprice of the second day equal to the monopoly price (and any weakly higher pricefor the first day), which gives revenue1/4.

Now consider the game with commitment in which the seller selects a pricetrajectory(p1, p2) and then buyers choose their actions. It is easy to check thatp1 = 1/2, p2 = 8+3a

16 , S1 = [1/2, 3/4], S2 = [3/4, 1] is an equilibrium, and itsrevenue is1/4 + 3a/64. We conclude by observing that the ratio of the revenuewith commitment to that without goes to infinity asa grows.

An important observation is that without externalities (whena = 0), both equilibriawould have been the same, with revenue equal to the optimal revenue of the 1-stagegame. This is in fact true for any form of utility functions.

3 Existence

For the aggregate model, the existence of equilibria follows from a result of Mas-Colell [18]. Here we use Kakutani’s fixed point theorem [13] to prove a slight gen-eralization of this result applicable to models with finitely many types and contin-uous valuation functions. We also give examples of games with a non-continuousvaluation function in which no equilibria exist.

Note that while the above theorems prove existence of equilibria, they do notguarantee that the equilibria are necessarily well-behaved (i.e. all indifferent buy-ers buy at some point). Well-behaved equilibria may not exist. Furthermore, evenwhen they exist they do not necessarily maximize revenue. However, for the linearand symmetric models that we focus on, we can show that well-behaved equilib-ria do in fact exist and maximize revenue. The existence results are presented inSection 3.1; the results in Section 5 imply the revenue-maximizing property.

We prove our pricing game has an equilibrium whenever buyershave finitelymany types and continuous valuation functionsF t

i , 1 ≤ t ≤ n, 1 ≤ i ≤ k. Todo so, we define a set-valued function on the space of marginalstrategy profilematrices whose fixed point is an equilibrium of our game and show the existenceof a fixed point using Kakutani’s fixed point theorem (KFPT). Letφ : X → 2Y bea set-valued function, i.e. a function fromX to the power set ofY . We say thatφhas a closed graph if the set{(x, y)|y ∈ φ(x)} is a closed subset ofX × Y in theproduct topology.

Theorem 1 (Kakutani Fixed Point Theorem (KFPT) [13]) LetS be a non-empty,compact and convex subset of Euclidean spaceRn for somen. Letφ : S → 2S be

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a set-valued function onS with a closed graph and the property thatφ(x) is non-empty and convex for allx ∈ S. Thenφ has a fixed pointx such thatx ∈ φ(x).

We are now ready to prove the equilibrium existence. We defineφ to be the cor-respondence that maps strategy profile matrices to the set ofbest-response matricesand then use KFPT to show that this mapping has a fixed point.

Theorem 2 If valuation functions are increasing and continuous, thenour gamehas an equilibrium.Proof : Let S be a subset of the Euclidean spaceR

k×n consisting of all validmarginal profile matricesx. Also letµ(t) be the length ofTt for each typet (recallthatTt is a subinterval of[0, 1]). Eachx ∈ S is a marginal strategy profile matrixx = (x1,1, · · · , xk,n), wherexi,t is the fraction of buyers who are of typet andchoose to buy on dayi, with the constraint that for each1 ≤ t ≤ n, the inequality∑

i xi,t ≤ µ(t) holds. Defineφ : S → 2S to be the function assigning eachx ∈ Sthe set of all marginal strategy profile matricesy ∈ φ(x) which are simultaneousbest-responses to the profilex. Formally, φ(x) consists of ally satisfying thefollowing conditions:

1. A buyer buys iny only if they get non-negative utility inx:∑

i yi,t > 0 onlyif there existsj such thatF t

j (Xj) − pj ≥ 0,

2. If some type has a positive utility inx, then they all buy iny: if F tj (Xj) −

pj > 0 then∑

i yi,t ≥ µ(t),

3. If a buyer buys iny, then he does so on a day which gives him maxi-mum utility in x i.e. for every dayi and typet, if yi,t > 0 then i ∈arg maxj F t

j (Xj) − pj.

If the conditions of the KFPT hold, we get a fixed pointx ∈ S; i.e. a pointx for which x ∈ φ(x). It is easy to check that any such fixed point is an equilib-rium of our game. Now let us prove that the setS and the functionφ satisfy theconditions of KFPT. SetS can be defined as the set of pointsx ∈ R

k×n satisfyingthe following linear inequalities, i.e.∀i, t : xi,t ≥ 0, and∀t :

∑

i xi,t ≤ µ(t).As a result,S, being the intersection of half-spaces, is a polyhedron, and clearly isclosed and convex. The setS is also bounded, because eachxi,t lies in the interval[0, µ(t)]. SoS is a compact and convex subset ofR

k×n.Let x be an arbitrary point inS. The setφ(x) can be defined as the intersection

of S, and a set of (possibly open) half-spaces defined by linear inequalities listedin the conditions above. Thus,φ(x) is a convex set. It is also nonempty as eachbuyer of each typet has a well-defined set of best-responses toX, the cumulative

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corresponding to marginal profilex, which is either some dayj if F tj (Xj)−pj ≥ 0

or the empty strategy (not buying) otherwise.It only remains to show that the graph ofφ is a closed subset ofR2(k×n). We

will show that each(x, y) lying outside the graph is contained in an open neigh-borhood which also lies outside the graph. This neighborhood will be of the formA × B, whereA is an open neighborhood ofx andB is an open neighborhood ofy. Since(x, y) is not in the graph, either(x, y) is not inS ×S or y does not satisfyone of the conditions definingφ(x). In the former case, sinceS × S is closed, wecan find a suitable neighborhood of(x, y) having no intersection withS × S andby extension the graph ofφ.

So assume thaty does not satisfy at least one of the constraints definingφ(x).Let U : R

k×n → Rk×n be the function assigning eachx ∈ S the matrix of utilities

{ui,t}, whereui,t = F ti (Xi)− pi denotes the utility of buying on dayi for a buyer

of type t. Assuming the valuation functions are continuous and increasing,U andhenceU−1 is continuous and invertible. Asy 6∈ φ(x), there is some typet suchthat either:

1.∑

j yj,t > 0 and for all daysj, uj,t < 0: Let A = {U−1({ui,t}) | ∀j : uj,t <0}, B = {{yi,t} |

∑

j yj,t > 0},

2. Ormaxj uj,t > 0 and∑

i yi,t < µ(t): Let A = {U−1({ui,t}) | ∃i : ui,t >0}, B = {{yi,t} |

∑

i yi,t < µ(t)},

3. Or there existsj∗ 6∈ arg maxj uj,t andyj∗,t > 0: Let A = {U−1({ui,t}) |∃i : ui,t > uj∗,t}, B = {{yi,t} | yj∗,t > 0}.

ThenA×B is an open neighborhood containing(x, y) sinceU−1 is continuousand invertible and we are applying it to an open subset of the domain in each case.Also,A×B has no intersection with the graph ofφ, which proves that the graph ofφ is closed. Hence all assumptions of the KFPT hold and we have an equilibrium.2

We next show via example that the game may have no equilibriumpoint whenvaluation functions are not continuous.

Example 3 Suppose there are two daysk = 2 and only one type in the market.Let Fi(X) = 1 if X ≤ 1

2 andFi(X) = 32 + X otherwise, fori = 1, 2. We claim

price trajectoryp = (0, 1) has no equilibrium. Consider any strategy profileX.If x1 ≤ 1/2, then the payoff of day1 is 1 and day2 is zero, so buyers who don’tbuy in day one are not playing a best-response soX is not an equilibrium. On theother hand, ifx1 > 1/2 then the payoff on day1 is still 1 but the payoff on day2is now greater than3/2 + 1/2 − 1 ≥ 1. Hence the buyers who buy in day one arenot playing a best-response and soX is not an equilibrium.

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2

0.25

F 1

Xi

1

0.51 0.6

3

F 2

Xi

Figure 1: The valuation function of type1 and2.

While the previous example is enough to show continuity is necessary for ex-istence of equilibria, one might notice that we can resolve the issue by changingthe function toFi(X) = 1 if X < 1

2 andF ti (X) = 3

2 + X otherwise, fori = 1, 2.In this casex = (1/2, 1/2) is an equilibrium for price trajectoryp = (0, 1). Thefollowing example is more robust to the changes in the valuation function:

Example 4 Assume there are three types and the fraction of each type is13 . The

valuation functionF ti (X) = 2 if Xt′ < 1

3 and F ti (X) = 4, wheret′ = 2, 3, 1

respectively fort = 1, 2, 3. Then similar reasoning shows price trajectoryp =(1, 2) has no equilibrium.

3.1 Existence of Well-Behaved Equilibria

Our revenue results assume the existence of revenue-maximizing well-behavedequilibria for all price sequences. Unfortunately, this does not hold even for theaggregate model, as the following example shows:

Example 5 Suppose there are three daysk = 3 and two types in the market, eachof which contain half the total population. LetF 1

i (Xi) andF 2i (Xi) be as shown

in Figure 1 for i = 1, 2, 3. Note that these are aggregate valuation functions.Consider price trajectoryp = (1, 2, 3).

If we assume that a buyer may decide not to buy the product whenthe utilityof buying is0, then we have an equilibrium point in this example. The vectorx = ((0, 0.25), (0.35, 0), (0, 0.25)) is an equilibrium point and a0.15 fraction oftype1 will not buy the product. However a simple case analysis shows there areno well-behaved equilibria (i.e. equilibria in which a buyer buys when the utilityof buying is0).

Fortunately, we can show that for the linear and symmetric models, well-behaved equilibria exists. We actually show something a bitmore general; namely

12

we derive a condition on the utilities that is sufficient to guarantee existence ofwell-behaved equilibria in which eitherall or no buyers buy. We will present theproof for finitely many types; it is clear that it extends to infinitely many types.

Theorem 6 If either mint,i F ti (0) ≥ mini pi or maxt,i F

ti (0) < mini pi holds,

then there exists a well-behaved equilibrium.

Proof : If maxt,i Fti (0) < mini pi then it is an equilibrium when no person buys.

This is an equilibrium because when no one is buying, utilities are all of the formF t

i (0)−pi which is negative. Hence everyone has a negative utility andthe strategyprofile is a well-behaved equilibrium in which no buyer buys.

So assume thatmint,i F ti (0) ≥ mini pi. Consider the altered price trajectory

(p1 − ǫ, . . . , pk − ǫ) for any ǫ > 0 (recall our model permits negative prices).Using this price sequence, all buyers buy on some day since the utility of buyingon the day with the minimum price is strictly positive. Hencethere is a well-behaved equilibrium for this new price trajectory. We will show that this samewell-behaved equilibrium is also an equilibrium for the original price trajectory.When prices are all decreased by the same amount all utilities are also decreasedby that same amount. Hence the relative ordering of utilities is the same for bothprice trajectories. So after changing the prices back to theoriginal ones, everyoneis still buying an optimal day. The only equilibrium condition that might not holdanymore is the one asserting that buyers only buy when the optimum utility isnon-negative. However, sincemint,i F

ti (0) ≥ mini pi, everyone has non-negative

utility on the day with the minimum price and so has non-negative optimal utility.Therefore this is a well-behaved equilibrium for the original price trajectory inwhich all buyers buy. 2

If F ti (0) is the same for alli, t, thenmaxt,i F t

i (0) = mint,i F ti (0). So at least

one of the conditions of Theorem 6 holds and we can conclude Corollary 7. Notefor both the symmetric and linear models,F t

i (0) is the same for alli, t.

Corollary 7 If F ti (0) is the same for alli, t then there is always a well-behaved

equilibrium.

4 Uniqueness of Equilibria

The following example shows that our game might have more than one equilibria,with different revenues for the seller, if we allow the valuation functions to be sen-sitive to the behavior of each type separately, even when allthe valuation functionsare continuous.

13

Example 8 Assume there are two types andF ti (Y ) = Yt′ + 2 wheret = 1, 2 and

t′ 6= t. In other words a social valuation function is only depends on fraction ofbuyers of another type. The population of type1 buyers are0.3 and the populationof type2 buyers are0.7. Suppose the seller wants to sell the product in two daysandp1 = 1 andp2 = 1.2.

It is clear that two vectorsX = ((0, 0), (0.3, 0)) and X ′ = ((0, 0), (0, 0.7))are equilibrium. In the first equilibrium all the type1(2) buyers have bought theproduct on day1(2). The revenue is0.3×1+0.7×1.2 = 1.14 in this equilibrium.In the second one all the type1(2) buyers have bought the product on day2(1). Therevenue is0.7 × 1 + 0.3 × 1.2 = 1.06 in this equilibrium.

Here, we prove that if there exists awell-behaved equilibrium, that is an equi-librium in which everyone with non-negative utility buys onsome day, then it isunique. We show this for an infinite number of types in the aggregate model whichgeneralizes both the linear and symmetric models.

Recall that we allow for each buyerb ∈ [0, 1] to have a unique type in theaggregate model such that the valuation function of buyerb is F b

i . We will showthat in all of the well-behaved equilibrium points the fraction of people buyingon each day is the same. In turn, it implies that the revenue ofall well-behavedequilibrium points is the same and hence the well-behaved equilibria are revenue-unique. In what follows, we consider the equilibria of a fixedprice sequencep. Westart with a definition: Consider two well-behaved equilibria x andy. Partition theset ofk days to two sets as follows: We call a dayi a level 1day, and denote it byi ∈ D1(x, y), if Xi < Yi. Otherwise, ifXi ≥ Yi, we calli a level 2day and denoteit by i ∈ D2(x, y).

Lemma 9 Assume that there exist two distinct well-behaved equilibria x and y.Then there exists a buyer whose strategy inx is a dayi such thati ∈ D1(x, y) andwhose strategy iny is j ∈ D2(x, y).

Proof : Sincex 6= y, assume WLOG that there exists a dayi so thatXi < Yi. LetS1 be the set of buyers who have bought on a level 1 day inx andS2 be the set ofbuyers who have bought on a level 2 day iny. Further defineS = S1 ∪ S2. Weshow that|S1| + |S2| > |S|. ThereforeS1 andS2 have a nonempty intersectionwhose elements are the buyers we are looking for. We do this byshowing that|S1| + |S2| > min(Xk+1, Yk+1) ≥ |S|.

First observe that|S| ≤ min(Xk+1, Yk+1) since

• If b ∈ S1 thenb bought on some level 1 dayi ∈ D1(x, y) in x. Thereforehis utility on dayi in y is non-negative. Asy is a well-behaved equilibrium,this means thatb must buy on some day iny. Thusb buys in bothx andy.

14

• If b ∈ S2 thenb bought on some level 2 day iny and so similar reasoningshows that asx is a well-behaved equilibrium,b must also buy on some dayin x. Thus againb buys in bothx andy.

Hence|S| ≤ min(Xk+1, Yk+1) as claimed.Next letzi be equal toxi for i ∈ D1(x, y), and equal toyi for i ∈ D2(x, y).5

Note that|S1|+ |S2| =∑

i∈D1(x,y) xi +∑

i∈D2(x,y) yi =∑k

i=1 zi. Thus it suffices

to show that∑k

i=1 zi > |S|. Let Zi = z1 + . . . + zi−1. So we must argue thatZk+1 > min(Xk+1, Yk+1). To do so we use the following claim:

Claim 10 For each dayi if Zi ≥ (respectively>) min(Xi, Yi), then we haveZi+1 ≥ (respectively>) min(Xi+1, Yi+1).Proof : For dayi, there are four possibilities: Ifi andi − 1 are level 1 days, thenZi+1 = Zi + zi = Zi + xi ≥ (>)Xi + xi = Xi+1. If i is a level 1 day andi− 1 isa level 2 day, thenZi+1 = Zi + zi = Zi + xi ≥ (>)Yi + xi > Xi + xi = Xi+1.Note that in this case we get strict inequality even assumingweak inequality fori.If i is a level 2 day andi − 1 is a level 1 day, thenZi+1 = Zi + zi = Zi + yi ≥(>)Xi + yi ≥ Yi + yi = Yi+1. Finally, if i andi − 1 are both level 2 days, thenZi+1 = Zi + zi = Zi + yi ≥ (>)Yi + yi = Yi+1. 2

Now we complete the proof of lemma by making the following twoobserva-tions: First, fori = 1, we haveX1 = Y1 = Z1 = 0 so by induction for alli we haveZi ≥ min(Xi, Yi). Second, there exists a day of level 1, and sinceday 1 is level 2, there must be a dayi that falls in the second case of the claimfor which we must haveZi > min(Xi, Yi). Then by induction we will haveZk+1 > min(Xk+1, Yk+1). 2

Theorem 11 LetF bi (X) be a strictly increasing function for each buyerb and day

i. For a price sequencep and two well-behaved equilibrium pointsx and y, wehaveXi = Yi, i.e. the fraction of buyers who have bought the product before dayiis unique.Proof : Assume for contradiction that we have two well-behaved equilibriumpointsx andy and a dayi for whichXi 6= Yi. Again assume without loss of gener-ality thatXi < Yi. By Lemma 9 we know that there exists a buyerb who buys on alevel 1 day inx and buys on a level 2 day iny. Assume thatb buys on dayi in x andon dayj in y. ThenF b

i (Xi)− pi ≥ F bj (Xj)− pj andF b

j (Yj)− pj ≥ F bi (Yi)− pi.

Adding the two inequalities we get:F bi (Xi) + F b

j (Yj) ≥ F bj (Xj) + F b

i (Yi).On the other hand sincei is a level 1 day,Xi < Yi; hence by monotonicityF b

i (Xi) < F bi (Yi). Sincej is a level 2 day,Xj ≥ Yj; henceF b

j (Yj) ≤ F bj (Xj).

The addition of these two inequalities contradicts the previous one. 2

5Strictly speakingz is not a valid marginal strategy profile as some buyers will buy in two differ-ent days inz.

15

5 Revenue Maximization

In this section, we solve the revenue-maximizing problem intwo special cases: thediscounted version of the symmetric model, and the general linear model withoutdiscount factors. In both cases, we provide an FPTAS to compute the revenue-maximizing price sequence.

5.1 Symmetric Setting

Since all players in this model have the same valuation function F , the marginalstrategy profile matrix will reduce to the vectorx = (x1, . . . , xk). Also, fixingp andx, the utility of buyerb for the item on dayi is F b

i (Xi) = F (Xi)βi(1 −

α)i − pi(1 − α)i, and the revenueR(p, x) =∑

i xipi(1 − α)i. By renamingqi = pi(1 − α)i andγ = β(1 − α), the utility of buyerb for the item on dayiwill be F (Xi)γ

i − qi, and the revenue becomes∑

i xiqi. Using this new notation,we may assume without loss of generality that the only discount factor isγ. Forconvenience, we usep for the discounted pricesq.

Since we only have one type in this model, we know that the utility of buyingin day i is equal among all players. We use the termutility of a dayi, denoted byui, for ui = F (Xi)γ

i − pi. Defineu(p, x) = maxi ui. Consider a price sequenceand its equilibrium strategy profilex. We get the following properties immediatelyfrom the facts that players are utility maximizing: (i) players are allowed to chooseinaction and have utility zero, (ii) they choose to buy if there is a day with a strictlypositive utility. First, if there is ani with xi > 0, thenu(p, x) ≥ 0 andui =u(p, x). Second, if there is a dayi with xi > 0, then

∑ki=1 xi = 1.

First, we observe the following lemma:

Lemma 12 Let p be the revenue-maximizing price vector that results in equilib-rium x. Thenu(p, x) = 0.Proof : Assume for contradiction thatu(p, x) = w > 0. Let w = (w, . . . , w)be a vector of lengthk with all of its elements equal tow, and consider the pricesequencep+w and vectorx. The utility of each day decreases by the same amountw, and the set of maximizers have positive utility, that is,u(p + w, x) ≥ 0. There-fore, each dayi with xi > 0 is still a maximizer withui ≥ 0. We conclude thatx is an equilibrium forp + w. Since

∑

i xi = 1, this price sequence has strictlygreater revenue, contradicting the optimality ofp. 2

Assume that there is a price sequencep with equilibrium x andu(p, x) = 0such that for some dayi, we havexi = 0 andxi+1 > 0. Then we can define anew price sequencep which is equal top except thatpj = pj+1/γ for eachj ≥ i.Also define the vectorx to be equal tox except thatxj = xj+1 for eachj ≥ i,and xk = 0. One can observe that the pair(p, x) is an equilibrium with no less

16

x

price

F (x)

F−1(p′t) F−1(p′t+1)

p′tpt

1

Figure 2: The discounted RCP problem. The blue area is the total area covered byrectangles, discounted by the index of each rectangle).

revenue. So we can assume WLOG that for a revenue maximizing price sequencep associated withx, there exists ak′ ≤ k such thatxi 6= 0 if and only if i ≤ k′.For such a price sequence, Lemma 12 shows thatF (Xi)γ

i − pi = 0 for each1 ≤ i ≤ k′. As a result, we haveXi = F−1(pi/γ

i), which is well-defined asF is increasing. Now setp′t = pt/γ

t. The fraction of people buying on dayiand paying pricepi is equal toxi = F−1(p′i+1) − F−1(p′i). So the revenue is∑

i xipi =∑

i(F−1(p′i+1) − F−1(p′i))p

′iγ

i. This sum is equal to the sum of theareas of a number of rectangles, discounted byγ, that are fit under the graph ofF (see Figure 2). So the revenue maximization problem reducesto the followingrectangular coveringproblem.

Definition 4 Rectangular Covering Problem (RCP)Given an increasing func-tion F and an integerk, find a sequencep of size at mostk that maximizes thediscounted total area of the rectangles fit under the graph ofF , that is, p ∈arg maxp′

∑

t(F−1(p′t+1) − F−1(p′t))p

′tγ

t.

In Section 6.1, we present an FPTAS to solve the rectangular covering problemfor concave valuation function and then show how to generalize the proof to non-concave functions in Section 6.2. See Section 6 for a generaltreatment of differentversions of the RCP.

5.2 Linear Version

To solve the problem in the linear version, we first study the properties of anyequilibrium in Section 5.2.1. We also show how to derive prices and the totalrevenue from vectorx. Then we study the revenue maximization problem in linearversion in Section 5.2.2. We prove that we can approximate the maximum revenue

17

by solving the Rectangular Covering Problem for a specific curve. This curve isobtained from the functionF and the distribution functionC.

5.2.1 Equilibrium Properties:

Fix a price sequencep with an equilibriumx. Since we do not have discounts,we can assume without loss that for somek′ ≤ k, a purchase happens in dayiif and only if i ≤ k′ (just remove the days with no purchase to the end). So wecan assume that for alli < j ≤ k′, we haveXi < Xj . The utility of a buyerb for buying on dayi is Ib + CbF (Xi) − pi. In order to concentrate on networkexternalities, for now we restrict the model and assume thatIb is always equal to afixed constantI for all buyers. Then the utility can be written asI +CbF (Xi)−pi.

Consider the set of pointsqi = (F (Xi), pi), 1 ≤ i ≤ k′. Define s(i, j) tobe equal to(pi − pj)/(F (Xi) − F (Xj)), which is the slope of the line betweenpointsqi andqj. Also lets(i) = s(i, i + 1). In Lemma 13, we prove thats is non-decreasing ini. Lemma 14 then shows that the utility of buyerb will be maximizedon dayi if and only ifCb ∈ [s(i−1), s(i)]. Finally, we use these lemmas in Lemma15 to show how to find a price vector givenx. We use these properties in the nextsection in order to find the desirable equilibrium.

For a fixedb, let i > j be two distinct days. The player prefers dayi to j if I +CbF (Xi)−pi ≥ I +CbF (Xj)−pj . The above inequality can be written as (recallthat we knowXi > Xj , and thereforeF (Xi) > F (Xj)):Cb ≥

pi−pj

F (Xi)−F (Xj). The

converse is also true. IfCb is less than(pi − pj)/(F (Xi) − F (Xj)), then dayjwill be preferred to dayi.

Lemma 13 For the functions as defined above,s(i) is non-decreasing ini.

Proof : Let i, i + 1, i + 2 be three consecutive days, and letb be a buyer whochooses to buy on dayi + 1. For b, day i + 1 is at least as good as daysi, i + 2.HenceCb must be greater than or equal tos(i) and less than or equal tos(i + 1).We conclude thats(i + 1) ≥ s(i). 2

Lemma 14 If Cb ∈ [s(i− 1), s(i)] thenb will have the maximum utility buying ondayi.

Proof : Assume thatCb ∈ [s(i − 1), s(i)]. For eachj > i, the utility of dayiis at least as good as that of dayj, becauseCb ≤ s(i) ≤ s(i, j). Similarly foreachj < i, the utility of dayi is at least as good as that of dayj, becauseCb ≥s(i − 1) ≥ s(j, i). The two special casesCb ∈ [0, s(1)] andCb ∈ [s(k − 1),∞)are dealt with the same arguments. 2

This lemma enables us to find the key relations between pricesand vectorx.

18

Lemma 15 In an equilibrium, the following holds for each2 ≤ i ≤ k: pi−pi−1 =(F (Xi) − F (Xi−1))C

−1(Xi).

Proof : The fraction of people who buy on dayi is exactly the fraction whoseCb’slie inside interval[s(i−1), s(i)]. So we havexi = Xi+1−Xi = C(s(i)−s(i−1)).The two sequences{X1, . . . ,Xk} and{C(s(0)), C(s(1)), . . . , C(s(k − 1))} haveidentical differences of consecutive terms. They also haveidentical initial elementsX1 = C(s(0)) = 0. Hence they are identical and we haveXi = C(s(i − 1)).

On the other hands(i − 1) is equal to pi−pi−1

F (Xi)−F (Xi−1) by definition. Thereforewe can conclude the desirable result. 2

5.2.2 Revenue Maximization:

We have analyzed the properties of any equilibrium. In this part, we study proper-ties of equilibria which are candidates for the revenue-maximizing one. We show inthe revenue-maximizing equilibrium, the first pricep1 will be equal toI in Lemma16. Using this result, we express total revenue in the revenue-maximizing equi-librium as a function of vectorx. Details are in Lemma 17. Finally, in Lemma18, we prove that the revenue maximization problem can be reduced to Rectangu-lar Covering Problem, and therefore there exists an FPTAS tosolve the revenuemaximization problem (Theorem 22).

Lemma 16 In a revenue-maximizing equilibrium,p1 = I.

Proof : Let x, p be the defining vectors of an equilibrium. Obviouslyp1 ≤ I,because people who buy on the first day, have nonnegative utility. Now raise allelements ofp by I − p1 to get p′. It’s easy to check thatx, p′ still define anequilibrium. The new equilibrium’s revenue is greater thanthe original one. Hencep1 = I in the revenue-maximizing equilibrium. 2

Note that by calculating allXi with respect to vectorx, we know the valuespi − pi−1 using Lemma 15. On the other hand,p1 = I in the revenue-maximizingequilibrium. So we would know allpi’s. Sox uniquely determines prices. LetPrice(x) be price vector which has been determined by vectorx. It is easy toverify that vectorsx and Price(x) make an equilibrium. Hence it suffices to vieweverything as functions of the free variablex. Now we express the revenue in acandidate equilibrium in terms of vectorx.

Lemma 17 If x andp correspond to the revenue-maximizing equilibrium, the totalrevenue can be expressed by the following formula:

R(p, x) = I +k

∑

i=2

(1 − Xi)C−1(Xi)(F (Xi) − F (Xi−1))

19

Proof : Since the utility of buying on the first day is nonnegative foreverybody, allbuyers would choose to buy. i.e.Xk+1 = 1. The total revenue can be written as:

R(p, x) =k

∑

i=1

pixi = p1

k∑

j=1

xj

+k

∑

i=2

(pi − pi−1)

k∑

j=i

xj

To interpret the above formula, note that the price change from dayi − 1 to dayiis exerted on all buyers who have bought on dayi or later. Since

∑kj=1 xj = 1,

we can rewrite the above formula asR(p, x) = p1 +∑k

i=2(pi − pi−1)(1 − Xi).Now we can substitute forp1 andpi − pi−1 from Lemmas 16 and 15 and writerevenue as a function of vectorx. Therefore we haveR(p, x) = I +

∑ki=2(1 −

Xi)C−1(Xi)(F (Xi) − F (Xi−1)), which is the desired result. 2

The next step is to reduce the problem of maximizing revenue to RectangularCovering Problem. Note thatI is a constant in Lemma 17 which does not affectrevenue maximization.

Lemma 18 The problem of maximizing∑k

i=2(1−Xi)C−1(Xi)(F (Xi)−F (Xi−1))

can be reduced to the Rectangular Covering Problem.Proof : SinceF is a monotone and hence bijective function, we can write therevenue as

R =

k∑

i=2

(1 − F−1(F (Xi)))C−1(F−1(F (Xi))(F (Xi) − F (Xi−1))

The above formula which only depends onF (Xi)’s can be interpreted as a Rect-angular Covering Problem instance. LetFmin = F (0), Fmax = F (1). We defineH : [Fmin, Fmax] → R

≥0 as follows :H(x) = (1 − F−1(x))C−1(F−1(x)). Notethat the revenue is exactlyR =

∑ki=2 H(F (Xi))(F (Xi) − F (Xi−1)). Therefore

we want to find valuesti = F (Xi)’s in order to maximizeR. RevenueR can beshown as total area of some rectangles with one corner on curveH (See Figure 3).

The problem of finding these rectangles is very similar to Rectangular CoveringProblem. If we defineH ′(x) = H(−x) and t′i = −tk−i then we can write therevenue as:

R =∑

H(ti)(ti − ti−1) =∑

H ′(−ti)(ti − ti−1)

=∑

H ′(t′k−i)(t′k−i+1 − t′k−i) =

∑

H ′(t′j)(t′j+1 − t′j))

So we should solve Rectangular Covering Problem for function H ′. 2

We have proved that the revenue maximization problem can be solved usingRectangular Covering Problem. Therefore, there exists an FPTAS to solve thisproblem (Theorem 22 in Section 6.2).

20

H

ti−1 ti

H(ti)

Figure 3: Revenue with respect to non-monotone functionH.

6 Rectangular Covering Problem

Definition 5 Rectangular Covering Problem: A function F : [0, 1] → R isgiven. We want to findk indicesx1, x2, · · · , xk in order to maximize the goalfunction

∑ki=0(xi+1 − xi)F (xi)γ

i, wherex0 = 0 andxk+1 = 1.

In the rectangular covering problem we want to placek rectangles under thecurveF , and maximizing the discounted covering area (see Figure 2). We proposean FPTAS algorithm for this problem for concave functionF in Section 6.1 andfor bounded slope functionF in Section 6.2. Algorithm described in Lemma 19can be used to solve the rectangular covering problem for general functionF withF (0) > 0. The running-time of the algorithm ispoly(k, log1+ǫ F (1)/F (0)) inthis case. In the remaining part of paper we assume thatF is non-decreasing andF (1) = 1. We prove that these assumptions do not hurt the generality in Section6.3.

Now we define some restricted versions of the rectangular covering problemwhich are useful in the paper.

Definition 6 δ-Rectangular Covering Problem: it is an instance of rectangularcovering problem in which everyF (xi) should be greater than or equal toδ.

Definition 7 (δ2, δ1)-Rectangular Covering Problem: it is an instance of rect-angular covering problem in which everyF (xi) should be out of a given interval(δ2, δ1).

6.1 Concave FunctionF

In this section we propose an FPTAS algorithm for the rectangular covering prob-lem when the functionF is concave.

21

F (x)

oi

F (oi)γi

F (xi)γi

xix

Figure 4: The Region with AreaA

Lemma 19 For everyǫ > 0 and δ ≥ 0, theδ-rectangular covering problem canbe solved inpoly(k, log1+ǫ(F (1)/δ)) time, with approximation factor1 + ǫ.Proof : At first we define a sequenceS = (s1, s2, · · · , sm) and then we prove thatif we choose indicesx1, x2, · · · , xk from sequenceS, we could approximate theoptimum. Letsi = F−1((1 + ǫ)i−1δ). In factF (si+1) = (1 + ǫ)F (si) for everyi < m, s0 = 0 andsm ≤ 1 < sm+1.

Assume optimum indices areo1, o2, · · · , ok in the rectangular covering prob-lem. The goal function is equal toO =

∑ki=0(oi+1 − oi)F (oi)γ

i in the optimumsolution. Letxi be the maximum index of sequenceS with value no more thanoi.SoF (oi) ≤ F (xi)(1 + ǫ). We can boundO as follows: (In all equations assumex0 = o0 = 0 andok+1 = xk+1 = 1)

O =k

∑

i=0

(oi+1 − oi)F (oi)γi ≤

k∑

i=0

(oi+1 − oi)(1 + ǫ)F (xi)γi = (1 + ǫ)A

(1)

The region with areaA has been shown in Figure 4. It is clear thatA is lessthan or equal to

∑ki=0(xi+1 − xi)F (xi)γ

i. So if we choose indicesx1, x2, · · · , xk

from sequenceS we can approximate the optimum with factor1 + ǫ.Now we design a dynamic programming algorithm to find indicesx1, x2, · · · , xk

from S. Let A[n, r] is equal to the best solution when we want to selectr indicesfrom sequence(s1, s2, · · · , sn) and alsosn has been selected. We have:

A[n, r] = maxn′<n{A[n′, r − 1] + (sn − sn′)F (sn′)γr}

The running time of the algorithm isΘ(poly(m,k)), wherem = Θ(log1+ǫ(F (1)/δ))2

22

Now we are ready to propose an FPTAS algorithm for the rectangular coveringproblem with concave functionF .

Theorem 20 For everyǫ > 0, the rectangular covering problem with concavefunctionF can be solved inpoly(k, 1/ log(1 + ǫ), log(1/ǫ)) time with approxima-tion factor1 + ǫ.Proof : Let ǫ′ = ǫ/2. Assume we want to solve theδ-rectangular covering problem

for function F with δ = F (1)( 11+ǫ′ )

n, wheren = log1+ǫ′4(1+2ǫ′)

ǫ′ . BecauseF isconcave the optimum solution for the rectangular covering problem (OPT ) is atleastF (1)/4. Let the optimum for theδ-rectangular covering problem beOPTδ

and the solution found by Lemma 19 beAδ. We have proved thatAδ ≥ OPTδ

1+ǫ′ .On the other hand, the optimum solution for the rectangular covering problem is atmostOPTδ + δ. SoAδ ≥ OPT−δ

1+ǫ′ = OPT1+ǫ′ −

F (1)(1+ǫ′)n+1 ≥ OPT

1+ǫ′ −4OPT

(1+ǫ′)n+1 . If we

replace(1 + ǫ′)n by 4(1+2ǫ′)ǫ′ , we can concludeAδ ≥ OPT

1+ǫ .We have used Lemma 19 withǫ′ andδ = F (1)( 1

1+ǫ′ )n. So the algorithm runs

in Θ(k, log1+ǫ′ F (1)/δ) = Θ(k, n) time, wheren = Θ( log(1/ǫ)log(1+ǫ)). 2

6.2 FunctionF With Bounded Slope

In this section we propose an FPTAS algorithm to solve the rectangular coveringproblem for functionF whenγ = 1 andF ′(x) ≤ L.

Lemma 21 For everyδ1, δ2, ǫ > 0, the (δ2, δ1)-Rectangular Covering Problemcan be solved inpoly(k, log1+ǫ ( 1

1−F−1(δ2)), log1+ǫ ( 1−δ2δ1−δ2

)) time, with approxi-mation factor1 + ǫ

Proof : At first we define a sequenceS = (s1, s2, · · · , sm) and then we prove thatif we choose indicesx1, x2, · · · , xk from sequenceS, we could approximate theoptimum. The sequenceS consists of two sequencesS′ andS

′′, which have been

constructed as follows:

• SequenceS′ = (s′1, s′2, · · · , s′m′) consist of indices withF (s′i) ≥ δ1. Let

s′1 = F−1(δ1) ands′i+1 = F−1((1 + ǫ)(F (s′i) − δ2) + δ2) ands′m′ = 1. Infact the we haveF (s′i+1)− δ2 ≤ (1 + ǫ)(F (s′i)− δ2) for everyi < m′. Thelength of sequenceS′ is m′ = log1+ǫ(

1−δ2δ1−δ2

).

• SequenceS′′

= (s′′

1 , s′′

2 , · · · , s′′

m′′ ) consist of indices withF (s′′

i ) ≤ δ2. Let

s′′

1 = 0 and1−s′′

i = ( 11+ǫ)

i−1 ands′′

m′′ = F−1(δ2). In fact we have1−s

′′

i ≤

(1 + ǫ)(1 − s′′

i+1). The length of sequenceS′′ is m′′ = log1+ǫ ( 11−F−1(δ2)

).

23

F (x)

oj oixj xi

δ2

δ1

x

Figure 5: The Region with AreaA

Assume optimum indices areo1, o2, · · · , ok in the (δ2, δ1)-Rectangular Cov-ering Problem. And let the goal function beO =

∑ki=1(oi+1 − oi)F (oi) in the

optimum solution. We know that everyF (oi) is out of interval(δ2, δ1). For everyoi ≤ F−1(δ2), Let xi be the minimum index in sequenceS (S

′′) with value not

less thanoi, and for everyoi ≥ F−1(δ1), Let xi be the maximum index in se-quenceS (S′) with value no more thanoi. Assumexi ∈ S

′′for every indexi < k′

andxi ∈ S′ for every indexi ≥ k′. We can boundO as follows: (In all equationsassumex0 = o0 = 0 andok+1 = xk+1 = 1)

O =

k∑

i=0

(oi+1 − oi)F (oi) =

k′−1∑

i=0

(oi+1 − oi)F (oi) +

k∑

i=k′

(oi+1 − oi)F (oi) (2)

The right hand side of above equation can be written as:

k′−1∑

i=0

(oi+1 − oi)F (oi) =k′−1∑

i=0

(1 − oi)(F (oi) − F (oi−1)) − F (ok′−1)(1 − ok′) (3)

And the left hand side can be written as:

k∑

i=k′

(oi+1 − oi)F (oi) =

k∑

i=k′

(oi+1 − oi)(F (oi) − F (ok′−1)) + F (ok′−1)(1 − ok′) (4)

Rewrite equality 2 with respect to equality 3 and 4.

O =k′−1∑

i=0

(1 − oi)(F (oi) − F (oi−1)) +k

∑

i=k′

(oi+1 − oi)(F (oi) − F (ok′−1)) (5)

24

For everyi < k′, we have(1− oi) ≤ (1 + ǫ)(1− xi). On the other hand for everyi ≥ k′, we haveF (oi) − δ2 ≤ (1 + ǫ)(F (xi) − δ2). BecauseF (ok′−1) ≤ δ2 wehaveF (oi) − F (ok′−1) ≤ (1 + ǫ)(F (xi) − F (ok′−1)), for everyi ≥ k′. Withrespect to these facts we can bound the optimum as follows:

O ≤ (1 + ǫ)

k′−1∑

i=0

(1 − xi)(F (oi) − F (oi−1))

+ (1 + ǫ)

k∑

i=k′

(oi+1 − oi)(F (xi) − F (ok′−1))

= (1 + ǫ)A (6)

The region with areaA has been shown in Figure 5. It is clear thatA is lessthan or equal to

∑ki=1(xi+1 − xi)F (xi). So if we choose indicesx1, x2, · · · , xk

from sequenceS we can approximate the optimum with factor1 + ǫ. Now we de-sign a dynamic programming algorithm to find indicesx1, x2, · · · , xk from S. LetA[n, t] is equal to the best solution when we want to select indices from sequence(s1, s2, · · · , sn) and alsosn has been selected. We have:

A[n, t] = maxn′<n{A[n′, t − 1] + (sn − sn′)F (sn′)}

2

Consider an instance of Rectangular Covering Problem with optimum goalOPT . Constructk2 + 1 instances of(δ2, δ1)-Rectangular Covering Problem fromRectangular Covering Problem instance. In thei-th instance we setδ2 = i−1

k2+1and

δ1 = ik2+1

and assume the value of goal function in the optimum solutionof thisinstance isOPTi. Assumeo1, o2, ..., ok are the optimum indices in the Rectangu-lar Covering Problem instance. It is clear that there exist1 ≤ j ≤ k2 + 1 suchthat everyoi is out of interval( j−1

k2+1 , jk2+1). Therefore we haveOPT = OPTj.

Assume we solve all of thek2 + 1 instances of(δ2, δ1)-Rectangular CoveringProblem using Lemma 21. LetAi be the output of the algorithm fori-th in-stance. We proved in Lemma 21 thatOPTi ≤ (1 + ǫ)Ai. So if returnmaxiAi

as the output for the Rectangular Covering Problem instance, we can approx-imate the optimum with factor1 + ǫ. Now we prove that this algorithm runin polynomial time. In order to prove this fact we should prove that if we setδ2 = i−1

k2+1 and δ1 = ik2+1 then log1+ǫ ( 1

1−F−1(δ2)), log1+ǫ ( 1−δ2δ1−δ2

) are polyno-

mial. First we have1−δ2δ1−δ2

= k2 + 2 − i which is polynomial with respect to

k. On the other hand we haveδ2 +∫ 1F−1(δ2) F ′(x)dx = F (1) = 1. Therefore

∫ 1F−1(δ2) F ′(x)dx = 1 − δ2 ≥ 1

k2+1. If we assume thatF ′(x) ≤ L we can con-

25

clude that1 − F−1(δ2) ≥ 1L(k+1) . So 1

1−F−1(δ2)is at mostL(k2 + 1) which is

polynomial with respect tok andL. Now we can conclude Theorem 22.

Theorem 22 For everyǫ > 0, the Rectangular Covering Problem withF ′(x) ≤ Lcan be solved inpoly(k, log1+ǫ k, log1+ǫ L) time, with approximation factor1+ ǫ.

6.3 Assumptions about functionF

In this section, we prove that some restrictions can be assumed about functionFin the rectangular covering problemWLOG. These restrictions are:

• F is non-decreasing: Let G(x) = max0≤y≤x F (y). We prove that the bestrectangular covering ofG andF are exactly the same. We prove this state-ment by showing every optimum solution of rectangular covering problemfor F is a solution forG with same objective value and vice versa. First as-sumex1, x2, · · · , xk are optimum indices corresponding to some rectangularcovering for curveF . We prove thatF (xi) = G(xi), for every1 ≤ i ≤ k. Ifit is not the case, letj be the smallest index whichF (xj) < G(xj). Let xj′

be the smallest index whichG(xj′) = G(xj). It is clear thatF (xj′) is thebiggest value in interval[0, xj ]. Now if we replacexj by xj′ in the optimumindices, the change in the objective function will be:

(xj′ − xj−1)F (xj−1)γj−1 + (xj+1 − xj′)F (xj′)γ

j

− (xj − xj−1)F (xj−1)γj−1 + (xj+1 − xj)F (xj)γ

j

Note thatF (xj′)γj is greater thatF (xj)γ

j andF (xj−1)γj−1. Therefore we

can conclude that the amount of change in the objective function is positive,which is a contradiction with optimality of indicesx1, x2, ·, xk. So indicesx1, x2, ·, xk is a solution for curveG with the same objective value.

On the other hand assumex1, x2, ·, xk are optimum indices correspondingto some rectangular coveringfor curveG. Note thatG is non-decreasing. Ifthere are two indicesxj′ andxj such thatxj′ < xj andF (xj′) = F (xj),thenxj will not appear in any optimum solution ofG. If remove indiceswith this property from the search space only indicesxi with G(xi) = F (xi)remain. So indicesx1, x2, ·, xk is a solution for curveF with the same ob-jective value.

• F (1) = 1: Let F (1) = c 6= 1. DefineG(x) = F (x)c and solve the problem

for G. Assumex1, x2, · · · , xk are indices corresponding to some rectangular

26

coveringfor curveG. It is clear that:

k∑

i=0

(xi+1 − xi)G(xi)γi =

1

c

k∑

i=0

(xi+1 − xi)F (xi)γi

So we can convert every solution for covering area under curve G to a so-lution for covering area under curveF and vice versa. So by solving therectangular covering problemfor curveG we can solve the problem for curveF .

7 Conclusion

In this paper, we studied the problem of item pricing in the presence of historicalnetwork externalities and strategic buyers. We provided necessary and sufficientconditions on the valuation functions for the existence andthe uniqueness of theequilibria. We defined the revenue maximization problem in settings in which theequilibrium exists and is unique. We provided nearly optimal algorithms for thetwo special cases of linear and symmetric valuation functions. The problem ofproviding algorithms or proving inapproximability for themore general aggregatemodel remains open.

We studied the historical externality setting in which the valuation for a pur-chase on a given day can depend only on the purchases on previous days. However,in many settings, users can benefit from purchases that happen in future. Character-izing equilibrium and studying the algorithmic pricing problem seems to be anotherinteresting direction for future research.

Finally, one can define and study the externalities model with competing sell-ers, and compare the behavior of equilibria with and withoutthe presence of com-peting sellers. It will be interesting to study the extensions of classical models ofduopoly from algorithmic perspective.

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