Wine 2010


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Wine 2010

  1. 1. 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 in the presence of positive historical externalities, where the value of a product increases as more people buy and use it. This increase in the value of the product is the result of resolving bugs or security holes after more usage. We consider a continuum of buyers that are partitioned into types where each type has a val- uation function based on the actions of other buyers. Given a fixed sequence of prices, or price trajectory, buyers choose a day on which to purchase the product, i.e. they have to decide whether to purchase the product early in the game 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 design an FPTAS to compute an approximately revenue-maximizing pricing trajec- tory for the seller in two special cases: the symmetric settings in which there is just a single buyer type, and the linear settings that are characterized by an initial type-independent bias and a linear type-dependent influenceability coefficient. Keywords: Social Networks, Game, Revenue Maximization, Equilibrium,Historical Externalities, Pricing ∗ University of California at Berkeley, EECS Department, † Sharif University of Technology, Computer Engineering Department,{ehsani,mahini} ‡ Institute for Research in Fundamental Sciences (IPM), P.O. Box: 19395-5746, Tehran, Iran,, This author’s work has been partially supported by IPM School of CS(contract: CS1388-2-01) § Northwestern University, EECS Department, {nima.haghpanah,nicimm} ¶ Google Research NYC, 76 9th Ave, New York, NY 10011, 1
  2. 2. 1 IntroductionMany 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 key observation is, themore people that use the operating system, or any product for that matter, the moreinherent value it accrues. In other words, the product exhibits a particular type ofexternality, a so-called historical 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 announced a 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 the release 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 the revenue-maximizing 1 Note 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. 2 Historical prices, announced upon the press release, can be found 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. 2
  3. 3. 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 whichthe 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 utility for future purchases.We motivate this in the Windows 7 example by arguing that bugs are 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-adverse buyers.1.1 Our ResultsWe focus on the non-atomic setting in which we have a continuum of consumersso that each consumer is infinitesimally small and therefore his 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 on well-behaved equilibria in 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 the aggregate behavior of the pop- 3
  4. 4. ulation (i.e. the total fraction of potential consumers that have bought the productand not the total fraction of various types), then we are able to 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 spend on 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 settings described 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 WorkOur 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 3 We 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. 4
  5. 5. 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 to a 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 consumers as 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 a divisible 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 of consumption, 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 commitment strategyof 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]. Letchford and 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 5
  6. 6. move, as in our game.2 ModelWe 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 buyers b ∈ [0, 1]. In our game, the seller moves first, selecting a price trajectoryp = (p1 , . . . , pk ) where pi ∈ ℜ assigning a (possibly negative) price pi 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 into n types T1 , . . . , Tn where each Tt is a subinter-val of [0, 1]4 . The strategy set A = {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 entry Xi,t indicates the fraction of buyersthat are of type t and buy the product before day i, and we define X1,t = 0 for allt. Note that by normalization t Xk+1,t ≤ 1 and 1 − t Xk+1,t is the fraction ofbuyers that don’t buy the product at any time. Corresponding to this matrix X wealso define the marginal strategy profile matrix x = {xi,t }i=1,...,k;t=1,...,n wherexi,t = Xi+1,t − Xi,t is the fraction of buyers who are of type t and buy on day i.In the special case when there is only 1 type, we use Xi as a scalar to denote thefraction of buyers who bought before day i and xi as a scalar to denote the fractionof buyers who buy on day i. Given a strategy profile X, we define the value of buyers of type t buying onday i by a value function Fit (Xi ) where Xi is the i’th row of X (hence buyersare indifferent to future buying decisions). Note the explicit dependence of F ontime, which allows Fit (Xi ) to be different than Fjt (Xj ), for i = j. The revenue-maximization results in Section 5 further assume that the dependence of Fit (Xi )on i is of the form Fit (Xi ) = β i F t (Xi ) for β ∈ [0, 1]. This special case is ofparticular interest as the β factor models settings in which the value degrades overtime due to, for example, a reduction in the novelty of the product. Given a strategy profile X, the payoff of buyers of type t who buy on day i isdefined to be Fit (Xi ) − pi . We additionally allow buyers to have a discount factorα such that their payoff is (1 − α)i (Fit (Xi ) − pi ). Thus α represents the way inwhich agents discount future payoffs with respect to present payoffs. We say thata strategy profile X is a Nash equilibrium of the induced subgame given by pricetrajectory p, or equivalently X ∈ N E(p), if for any buyer of type t who buys on 4 Later, we will generalize this to infinitely many types. 6
  7. 7. day i we have i ∈ arg maxj (Fjt (Xj ) − pj )(1 − α)j , and the strategy is ∅ wheneverthe maximum is not positive (in which case the buyer’s payoff is zero). We call anequilibrium well-behaved if all indifferent buyers buy, i.e. a buyer does not buy ifand only if his payoff (1 − α)i (Fit (Xi ) − pi ) is negative on all days 1 ≤ i ≤ k. Wesay that (p, X) is a (well-behaved) equilibrium if the profile X is a (well-behaved)Nash equilibrium for the subgame of price trajectory p. Equivalently, a marginalstrategy profile x of a stragety profile X is a (well-behaved) Nash equilibriumfor the subgame of price trajectory p if the profile X is a (well-behaved) Nashequilibrium for the subgame of price trajectory p. Given a price trajectory p and a marginal strategy profile x that arises in thesubgame induced by p, we define the payoff of the seller to be the revenue of x forp, which is R(p, x) = k i=1 n i t=1 xi,t pi (1 − α) . A subgame perfect equilibriumof the sequential game is then a price trajectory p∗ and a set of marginal strategyprofiles xp for each possible price trajectory p such that: (1) xp is a Nash equilib-rium of the subgame induced by p, and (2) p∗ maximizes R(p, xp ). The outcomeof this subgame perfect equilibrium is (p∗ , xp∗ ) and its revenue is R(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 in N E(p) for any price trajectory p, 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 many types. We will useb ∈ [0, 1] to denote the type of buyer b.Definition 1 The Aggregate Model: The value function of each type in this modelis a function of the aggregate behavior of the population and is invariant withrespect to the behavior of each separate type. That is, the value function of buyerb is a function of Xi only, where Xi is a scalar indicating the total fraction of allbuyers who buy before day i. In this instance, we overload the notation for thevalue function and let Fib (Xi ) indicate the value of buyer b (hence Fib (·) now mapsthe 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 function Fi , an initial bias I, and a function C so that thevalue of buyer b is Fib (Xi ) = I + C(b) · Fi (Xi ). We further define the commonly-known distribution C : R → [0, 1] such that C(c∗ ) indicates the fraction of buyersb with C(b) ≤ c∗ . 7
  8. 8. Definition 3 The Symmetric Model: In this version we only have one type, that is,Fib = Fi for all b. We note that alternatively, one could model this pricing game as a sequentialgame with multiple stages where in each day i the seller selects a price pi 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 commitment is at least as highas that without commitment. The below example shows that the revenue withoutcommitment can actually be unboundedly less.Example. We consider buyers b ∈ [0, 1] with utility function F b (X) = (1+a|X|)bfor a non-negative constant a, and focus on a setting with k = 2 days. In a strategyprofile, the seller picks a price p1 for day 1 and a price p2 (S) for any subset S ⊆2[0,1] of buyers that remain in day 2. Each buyer decides whether to buy on day 1given any possible announcement p1 of the seller, and also whether to buy on day 2given any possible announcement p2 of the seller and decisions S 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 set p2 (S) equal to the revenue-maximizing price for buyers S. Assume a subgame perfect equilibrium in which the prices on path of play arep1 and p2 , and subsets S1 , S2 ⊆ 2[0,1] buy at days 1 and 2, respectively. We claimp1 ≥ p2 . Suppose for contradiction that p1 < p2 . First note that if |S1 | = 0, thenbuying on day 1 gives higher utility that buying on day 2, and therefore S2 = ∅.But this gives the seller zero revenue which is not his best strategy as he can lowerthe price of day 2 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 day 1, and also more utility of buying on day 1 than day 2.The first constraint is equal to having b ≥ p1 , and the second to having b such thatb(1 + 0) − p1 > b(1 + a|S1 |) − p2 , or equivalantly b < pa|S1 |1 (remember that we 2 −passumed |S1 | > 0). Let b∗ = pa|S1 |1 . The set of buyers buying on day 1 is equal to 2 −pthe buyers with b satisfying p1 ≤ b and also b < b∗ . Our assumption that |S1 | > 0now implies that p1 < b∗ . We observe that p2 ∈ (p1 (1 + a|S1 |), b∗ (1 + a|S1 |)), ascan be seen from the equality p2 = a|S1 |b∗ + p1 and the inequality b∗ > p1 . Bythe properties of subgame-perfect equilibria, the seller has to play his best responsein the subgame induced at day 2. In day 2 the utilities of players are outside theinterval (p1 (1 + a|S1 |), b∗ (1 + a|S1 |)). In fact player b utility at day 2 is in theinterval (p1 (1 + a|S1 |), b∗ (1 + a|S1 |)) if and only if p1 < b and b < b∗ whichimplies the player b bought in day 1. So p2 ∈ (p1 (1 + a|S1 |), b∗ (1 + a|S1 |)) 8
  9. 9. 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 the seller is to set theprice of the second day equal to the monopoly price (and any weakly higher pricefor the first day), which gives revenue 1/4. 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 , S1 = [1/2, 3/4], S2 = [3/4, 1] is an equilibrium, and its 16revenue is 1/4 + 3a/64. We conclude by observing that the ratio of the revenuewith commitment to that without goes to infinity as a grows.An important observation is that without externalities (when a = 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 ExistenceFor 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 buyers have finitelymany types and continuous valuation functions Fit , 1 ≤ t ≤ n, 1 ≤ i ≤ k. Todo so, we define a set-valued function on the space of marginal strategy profilematrices whose fixed point is an equilibrium of our game and show the existenceof a fixed point using Kakutani’s fixed point theorem (KFPT). Let φ : X → 2Y bea set-valued function, i.e. a function from X to the power set of Y . We say that φhas a closed graph if the set {(x, y)|y ∈ φ(x)} is a closed subset of X × Y in theproduct topology.Theorem 1 (Kakutani Fixed Point Theorem (KFPT) [13]) Let S be a non-empty,compact and convex subset of Euclidean space Rn for some n. Let φ : S → 2S be 9
  10. 10. a set-valued function on S with a closed graph and the property that φ(x) is non-empty and convex for all x ∈ S. Then φ has a fixed point x such that x ∈ φ(x). We are now ready to prove the equilibrium existence. We define φ to be the cor-respondence that maps strategy profile matrices to the set of best-response matricesand then use KFPT to show that this mapping has a fixed point.Theorem 2 If valuation functions are increasing and continuous, then our gamehas an equilibrium.Proof : Let S be a subset of the Euclidean space Rk×n consisting of all validmarginal profile matrices x. Also let µ(t) be the length of Tt for each type t (recallthat Tt is a subinterval of [0, 1]). Each x ∈ S is a marginal strategy profile matrixx = (x1,1 , · · · , xk,n ), where xi,t is the fraction of buyers who are of type t andchoose to buy on day i, with the constraint that for each 1 ≤ t ≤ n, the inequality S i xi,t ≤ µ(t) holds. Define φ : S → 2 to be the function assigning each x ∈ Sthe set of all marginal strategy profile matrices y ∈ φ(x) which are simultaneousbest-responses to the profile x. Formally, φ(x) consists of all y satisfying thefollowing conditions: 1. A buyer buys in y only if they get non-negative utility in x: i yi,t > 0 only if there exists j such that Fjt (Xj ) − pj ≥ 0, 2. If some type has a positive utility in x, then they all buy in y: if Fjt (Xj ) − pj > 0 then i yi,t ≥ µ(t), 3. If a buyer buys in y, then he does so on a day which gives him maxi- mum utility in x i.e. for every day i and type t, if yi,t > 0 then i ∈ arg maxj Fjt (Xj ) − pj . If the conditions of the KFPT hold, we get a fixed point x ∈ 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 set S and the function φ satisfy theconditions of KFPT. Set S can be defined as the set of points x ∈ Rk×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 set S is also bounded, because each xi,t lies in the interval[0, µ(t)]. So S is a compact and convex subset of Rk×n . Let x be an arbitrary point in S. The set φ(x) can be defined as the intersectionof 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 type t has a well-defined set of best-responses to X, the cumulative 10
  11. 11. corresponding to marginal profile x, which is either some day j if Fjt (Xj )−pj ≥ 0or the empty strategy (not buying) otherwise. It only remains to show that the graph of φ is a closed subset of R2(k×n) . Wewill 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, where A is an open neighborhood of x and B is an open neighborhood ofy. Since (x, y) is not in the graph, either (x, y) is not in S × S or y does not satisfyone of the conditions defining φ(x). In the former case, since S × S is closed, wecan find a suitable neighborhood of (x, y) having no intersection with S × S andby extension the graph of φ. So assume that y does not satisfy at least one of the constraints defining φ(x).Let U : Rk×n → Rk×n be the function assigning each x ∈ S the matrix of utilities{ui,t }, where ui,t = Fit (Xi ) − pi denotes the utility of buying on day i for a buyerof type t. Assuming the valuation functions are continuous and increasing, U andhence U −1 is continuous and invertible. As y ∈ φ(x), there is some type t suchthat either: 1. j yj,t > 0 and for all days j, uj,t < 0: Let A = {U −1 ({ui,t }) | ∀j : uj,t < 0}, B = {{yi,t } | j yj,t > 0}, 2. Or maxj 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 exists j ∗ ∈ arg maxj uj,t and yj ∗,t > 0: Let A = {U −1 ({ui,t }) | ∃i : ui,t > uj ∗ ,t }, B = {{yi,t } | yj ∗ ,t > 0}. Then A×B is an open neighborhood containing (x, y) since U −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 of φ, 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 equilibrium point whenvaluation functions are not continuous.Example 3 Suppose there are two days k = 2 and only one type in the market.Let Fi (X) = 1 if X ≤ 1 and Fi (X) = 3 + X otherwise, for i = 1, 2. We claim 2 2price trajectory p = (0, 1) has no equilibrium. Consider any strategy profile X.If x1 ≤ 1/2, then the payoff of day 1 is 1 and day 2 is zero, so buyers who don’tbuy in day one are not playing a best-response so X is not an equilibrium. On theother hand, if x1 > 1/2 then the payoff on day 1 is still 1 but the payoff on day 2is now greater than 3/2 + 1/2 − 1 ≥ 1. Hence the buyers who buy in day one arenot playing a best-response and so X is not an equilibrium. 11
  12. 12. F1 F2 3 2 1 Xi Xi 0.25 0.51 0.6 Figure 1: The valuation function of type 1 and 2. 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 to Fi (X) = 1 if X < 1 and Fit (X) = 3 + X otherwise, for i = 1, 2. 2 2In this case x = (1/2, 1/2) is an equilibrium for price trajectory p = (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 is 1 . The 3valuation function Fit (X) = 2 if Xt′ < 1 and Fit (X) = 4, where t′ = 2, 3, 1 3respectively for t = 1, 2, 3. Then similar reasoning shows price trajectory p =(1, 2) has no equilibrium.3.1 Existence of Well-Behaved EquilibriaOur 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 days k = 3 and two types in the market, eachof which contain half the total population. Let Fi1 (Xi ) and Fi2 (Xi ) be as shownin Figure 1 for i = 1, 2, 3. Note that these are aggregate valuation functions.Consider price trajectory p = (1, 2, 3). If we assume that a buyer may decide not to buy the product when the utilityof buying is 0, 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 a 0.15 fraction oftype 1 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 is 0). Fortunately, we can show that for the linear and symmetric models, well-behaved equilibria exists. We actually show something a bit more general; namely 12
  13. 13. we derive a condition on the utilities that is sufficient to guarantee existence ofwell-behaved equilibria in which either all 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 Fit (0) ≥ mini pi or maxt,i Fit (0) < mini pi holds,then there exists a well-behaved equilibrium.Proof : If maxt,i Fit (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 formFit (0)−pi which is negative. Hence everyone has a negative utility and the strategyprofile is a well-behaved equilibrium in which no buyer buys. So assume that mint,i Fit (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. Hence there 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 the original 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, since mint,i Fit (0) ≥ mini pi , everyone has non-negativeutility 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 Fit (0) is the same for all i, t, then max F t (0) = min F t (0). So at least t,i i t,i ione of the conditions of Theorem 6 holds and we can conclude Corollary 7. Notefor both the symmetric and linear models, Fit (0) is the same for all i, t.Corollary 7 If Fit (0) is the same for all i, t then there is always a well-behavedequilibrium.4 Uniqueness of EquilibriaThe 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 all the valuation functionsare continuous. 13
  14. 14. Example 8 Assume there are two types and Fit (Y ) = Yt′ + 2 where t = 1, 2 andt′ = t. In other words a social valuation function is only depends on fraction ofbuyers of another type. The population of type 1 buyers are 0.3 and the populationof type 2 buyers are 0.7. Suppose the seller wants to sell the product in two daysand p1 = 1 and p2 = 1.2. It is clear that two vectors X = ((0, 0), (0.3, 0)) and X ′ = ((0, 0), (0, 0.7))are equilibrium. In the first equilibrium all the type 1(2) buyers have bought theproduct on day 1(2). The revenue is 0.3 × 1 + 0.7 × 1.2 = 1.14 in this equilibrium.In the second one all the type 1(2) buyers have bought the product on day 2(1). Therevenue is 0.7 × 1 + 0.3 × 1.2 = 1.06 in this equilibrium. Here, we prove that if there exists a well-behaved equilibrium, that is an equi-librium in which everyone with non-negative utility buys on some 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 buyer b ∈ [0, 1] to have a unique type in theaggregate model such that the valuation function of buyer b is Fib . 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 of all well-behavedequilibrium points is the same and hence the well-behaved equilibria are revenue-unique. In what follows, we consider the equilibria of a fixed price sequence p. Westart with a definition: Consider two well-behaved equilibria x and y. Partition theset of k days to two sets as follows: We call a day i a level 1 day, and denote it byi ∈ D1 (x, y), if Xi < Yi . Otherwise, if Xi ≥ Yi , we call i a level 2 day 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 in x is a day i such that i ∈ D1 (x, y) andwhose strategy in y is j ∈ D2 (x, y).Proof : Since x = y, assume WLOG that there exists a day i so that Xi < Yi . LetS1 be the set of buyers who have bought on a level 1 day in x and S2 be the set ofbuyers who have bought on a level 2 day in y. Further define S = S1 ∪ S2 . Weshow that |S1 | + |S2 | > |S|. Therefore S1 and S2 have a nonempty intersectionwhose elements are the buyers we are looking for. We do this by showing that|S1 | + |S2 | > min(Xk+1 , Yk+1 ) ≥ |S|. First observe that |S| ≤ min(Xk+1 , Yk+1 ) since • If b ∈ S1 then b bought on some level 1 day i ∈ D1 (x, y) in x. Therefore his utility on day i in y is non-negative. As y is a well-behaved equilibrium, this means that b must buy on some day in y. Thus b buys in both x and y. 14
  15. 15. • If b ∈ S2 then b bought on some level 2 day in y and so similar reasoning shows that as x is a well-behaved equilibrium, b must also buy on some day in x. Thus again b buys in both x and y.Hence |S| ≤ min(Xk+1 , Yk+1 ) as claimed. Next let zi be equal to xi for i ∈ D1 (x, y), and equal to yi for i ∈ D2 (x, y).5Note that |S1 | + |S2 | = i∈D1 (x,y) xi + i∈D2 (x,y) yi = k zi . Thus it suffices i=1to show that k zi > |S|. Let Zi = z1 + . . . + zi−1 . So we must argue that i=1Zk+1 > min(Xk+1 , Yk+1 ). To do so we use the following claim:Claim 10 For each day i if Zi ≥ (respectively >) min(Xi , Yi ), then we haveZi+1 ≥ (respectively >) min(Xi+1 , Yi+1 ).Proof : For day i, there are four possibilities: If i and i − 1 are level 1 days, thenZi+1 = Zi + zi = Zi + xi ≥ (>)Xi + xi = Xi+1 . If i is a level 1 day and i − 1 isa level 2 day, then Zi+1 = Zi + zi = Zi + xi ≥ (>)Yi + xi > Xi + xi = Xi+1 .Note that in this case we get strict inequality even assuming weak inequality for i.If i is a level 2 day and i − 1 is a level 1 day, then Zi+1 = Zi + zi = Zi + yi ≥(>)Xi + yi ≥ Yi + yi = Yi+1 . Finally, if i and i − 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 two observa-tions: First, for i = 1, we have X1 = Y1 = Z1 = 0 so by induction for alli we have Zi ≥ min(Xi , Yi ). Second, there exists a day of level 1, and sinceday 1 is level 2, there must be a day i that falls in the second case of the claimfor which we must have Zi > min(Xi , Yi ). Then by induction we will haveZk+1 > min(Xk+1 , Yk+1 ). 2Theorem 11 Let Fib (X) be a strictly increasing function for each buyer b and dayi. For a price sequence p and two well-behaved equilibrium points x and y, wehave Xi = Yi , i.e. the fraction of buyers who have bought the product before day iis unique.Proof : Assume for contradiction that we have two well-behaved equilibriumpoints x and y and a day i for which Xi = Yi . Again assume without loss of gener-ality that Xi < Yi . By Lemma 9 we know that there exists a buyer b who buys on alevel 1 day in x and buys on a level 2 day in y. Assume that b buys on day i in x andon day j in y. Then Fib (Xi ) − pi ≥ Fjb (Xj ) − pj and Fjb (Yj ) − pj ≥ Fib (Yi ) − pi .Adding the two inequalities we get: Fib (Xi ) + Fjb (Yj ) ≥ Fjb (Xj ) + Fib (Yi ).On the other hand since i is a level 1 day, Xi < Yi ; hence by monotonicityFib (Xi ) < Fib (Yi ). Since j is a level 2 day, Xj ≥ Yj ; hence Fjb (Yj ) ≤ Fjb (Xj ).The addition of these two inequalities contradicts the previous one. 2 5 Strictly speaking z is not a valid marginal strategy profile as some buyers will buy in two differ-ent days in z. 15
  16. 16. 5 Revenue MaximizationIn this section, we solve the revenue-maximizing problem in two 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 SettingSince all players in this model have the same valuation function F , the marginalstrategy profile matrix will reduce to the vector x = (x1 , . . . , xk ). Also, fixingp and x, the utility of buyer b for the item on day i is Fib (Xi ) = F (Xi )β i (1 −α)i − pi (1 − α)i , and the revenue R(p, x) = i i xi pi (1 − α) . By renamingqi = pi (1 − α) i and γ = β(1 − α), the utility of buyer b for the item on day iwill be F (Xi )γ i − qi , and the revenue becomes i xi qi . Using this new notation,we may assume without loss of generality that the only discount factor is γ. Forconvenience, we use p for the discounted prices q. 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 term utility of a day i, denoted byui , for ui = F (Xi )γ i − pi . Define u(p, x) = maxi ui . Consider a price sequenceand its equilibrium strategy profile x. 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 an i with xi > 0, then u(p, x) ≥ 0 and ui =u(p, x). Second, if there is a day i with xi > 0, then k xi = 1. i=1 First, we observe the following lemma:Lemma 12 Let p be the revenue-maximizing price vector that results in equilib- ˆrium x. Then u(ˆ, x) = 0. ˆ p ˆProof : Assume for contradiction that u(ˆ, x) = w > 0. Let w = (w, . . . , w) p ˆ ˆbe a vector of length k with all of its elements equal to w, and consider the pricesequence p + w and vector x. The utility of each day decreases by the same amount ˆ ˆ ˆw, and the set of maximizers have positive utility, that is, u(ˆ + w, x) ≥ 0. There- ˆ p ˆ ˆfore, each day i with xi > 0 is still a maximizer with ui ≥ 0. We conclude that ˆx is an equilibrium for p + w. Since i xi = 1, this price sequence has strictlyˆ ˆ ˆ ˆgreater revenue, contradicting the optimality of p. ˆ 2 Assume that there is a price sequence p with equilibrium x and u(p, x) = 0such that for some day i, we have xi = 0 and xi+1 > 0. Then we can define anew price sequence p which is equal to p except that pj = pj+1 /γ for each j ≥ i. ˜ ˜Also define the vector x to be equal to x except that xj = xj+1 for each j ≥ i, ˜ ˜and xk = 0. One can observe that the pair (˜, x) is an equilibrium with no less ˜ p ˜ 16
  17. 17. price F (x) p′ t pt x F −1 (p′ ) F −1 (p′ ) 1 t t+1Figure 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 with x, there exists a k′ ≤ k such that xi = 0 if and only if i ≤ k′ .ˆ ˆFor such a price sequence, Lemma 12 shows that F (Xi )γ i − pi = 0 for each1 ≤ i ≤ k′ . As a result, we have Xi = F −1 (pi /γ i ), which is well-defined asF is increasing. Now set p′ = pt /γ t . The fraction of people buying on day i tand paying price pi is equal to xi = F −1 (p′ ) − F −1 (p′ ). So the revenue is i+1 i −1 (p′ −1 (p′ ))p′ γ i . This sum is equal to the sum of the i xi pi = i (F i+1 ) − F i iareas of a number of rectangles, discounted by γ, that are fit under the graph ofF (see Figure 2). So the revenue maximization problem reduces to the followingrectangular covering problem.Definition 4 Rectangular Covering Problem (RCP) Given an increasing func-tion F and an integer k, find a sequence p of size at most k that maximizes thediscounted total area of the rectangles fit under the graph of F , that is, p ∈arg maxp′ t (F −1 (p′ ) − F −1 (p′ ))p′ γ t . t+1 t 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 general treatment of differentversions of the RCP.5.2 Linear VersionTo 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 vector x. Then we study the revenue maximization problem in linearversion in Section 5.2.2. We prove that we can approximate the maximum revenue 17
  18. 18. by solving the Rectangular Covering Problem for a specific curve. This curve isobtained from the function F and the distribution function C.5.2.1 Equilibrium Properties:Fix a price sequence p with an equilibrium x. Since we do not have discounts,we can assume without loss that for some k′ ≤ k, a purchase happens in day iif and only if i ≤ k′ (just remove the days with no purchase to the end). So wecan assume that for all i < j ≤ k′ , we have Xi < Xj . The utility of a buyerb for buying on day i is Ib + Cb F (Xi ) − pi . In order to concentrate on networkexternalities, for now we restrict the model and assume that Ib is always equal to afixed constant I for all buyers. Then the utility can be written as I + Cb F (Xi ) − pi . Consider the set of points qi = (F (Xi ), pi ), 1 ≤ i ≤ k′ . Define s(i, j) to ˆbe equal to (pi − pj )/(F (Xi ) − F (Xj )), which is the slope of the line betweenpoints qi and qj . Also let s(i) = s(i, i + 1). In Lemma 13, we prove that s is non- ˆdecreasing in i. Lemma 14 then shows that the utility of buyer b will be maximizedon day i if and only if Cb ∈ [s(i−1), s(i)]. Finally, we use these lemmas in Lemma15 to show how to find a price vector given x. We use these properties in the nextsection in order to find the desirable equilibrium. For a fixed b, let i > j be two distinct days. The player prefers day i to j if I +Cb F (Xi ) − pi ≥ I + Cb F (Xj ) − pj . The above inequality can be written as (recall pi −pthat we know Xi > Xj , and therefore F (Xi ) > F (Xj )):Cb ≥ F (Xi )−Fj(Xj ) . Theconverse is also true. If Cb is less than (pi − pj )/(F (Xi ) − F (Xj )), then day jwill be preferred to day i.Lemma 13 For the function s as defined above, s(i) is non-decreasing in i.Proof : Let i, i + 1, i + 2 be three consecutive days, and let b be a buyer whochooses to buy on day i + 1. For b, day i + 1 is at least as good as days i, i + 2.Hence Cb must be greater than or equal to s(i) and less than or equal to s(i + 1).We conclude that s(i + 1) ≥ s(i). 2Lemma 14 If Cb ∈ [s(i − 1), s(i)] then b will have the maximum utility buying onday i.Proof : Assume that Cb ∈ [s(i − 1), s(i)]. For each j > i, the utility of day iis at least as good as that of day j, because Cb ≤ s(i) ≤ s(i, j). Similarly for ˆeach j < i, the utility of day i is at least as good as that of day j, because Cb ≥s(i − 1) ≥ s(j, i). The two special cases Cb ∈ [0, s(1)] and Cb ∈ [s(k − 1), ∞) ˆare dealt with the same arguments. 2 This lemma enables us to find the key relations between prices and vector x. 18
  19. 19. Lemma 15 In an equilibrium, the following holds for each 2 ≤ i ≤ k: pi −pi−1 =(F (Xi ) − F (Xi−1 ))C −1 (Xi ).Proof : The fraction of people who buy on day i is exactly the fraction whose Cb ’slie inside interval [s(i−1), s(i)]. So we have xi = 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 have identical initial elementsX1 = C(s(0)) = 0. Hence they are identical and we have Xi = C(s(i − 1)). On the other hand s(i − 1) is equal to F (Xp)−Fi−1 i−1 ) by definition. Therefore i i −p (Xwe can conclude the desirable result. 25.2.2 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 price p1 will be equal to I in Lemma16. Using this result, we express total revenue in the revenue-maximizing equi-librium as a function of vector x. 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 to solve 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. Obviously p1 ≤ I,because people who buy on the first day, have nonnegative utility. Now raise allelements of p by I − p1 to get p′ . It’s easy to check that x, p′ still define anequilibrium. The new equilibrium’s revenue is greater than the original one. Hencep1 = I in the revenue-maximizing equilibrium. 2 Note that by calculating all Xi with respect to vector x, we know the valuespi − pi−1 using Lemma 15. On the other hand, p1 = I in the revenue-maximizingequilibrium. So we would know all pi ’s. So x uniquely determines prices. LetPrice(x) be price vector which has been determined by vector x. It is easy toverify that vectors x and Price(x) make an equilibrium. Hence it suffices to vieweverything as functions of the free variable x. Now we express the revenue in acandidate equilibrium in terms of vector x.Lemma 17 If x and p correspond to the revenue-maximizing equilibrium, the totalrevenue can be expressed by the following formula: k R(p, x) = I + (1 − Xi )C −1 (Xi )(F (Xi ) − F (Xi−1 )) i=2 19
  20. 20. Proof : Since the utility of buying on the first day is nonnegative for everybody, allbuyers would choose to buy. i.e. Xk+1 = 1. The total revenue can be written as:     k k k k R(p, x) = pi xi = p1  xj  + (pi − pi−1 )  xj  i=1 j=1 i=2 j=iTo interpret the above formula, note that the price change from day i − 1 to day iis exerted on all buyers who have bought on day i or later. Since k xj = 1, j=1we can rewrite the above formula as R(p, x) = p1 + k (pi − pi−1 )(1 − Xi ). i=2Now we can substitute for p1 and pi − pi−1 from Lemmas 16 and 15 and writerevenue as a function of vector x. Therefore we haveR(p, x) = I + k (1 − i=2Xi )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 that I is a constant in Lemma 17 which does not affectrevenue maximization.Lemma 18 The problem of maximizing k (1−Xi )C −1 (Xi )(F (Xi )−F (Xi−1 )) i=2can be reduced to the Rectangular Covering Problem.Proof : Since F is a monotone and hence bijective function, we can write therevenue as k R= (1 − F −1 (F (Xi )))C −1 (F −1 (F (Xi ))(F (Xi ) − F (Xi−1 )) i=2The above formula which only depends on F (Xi )’s can be interpreted as a Rect-angular Covering Problem instance. Let Fmin = 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 exactly R = k H(F (Xi ))(F (Xi ) − F (Xi−1 )). Therefore i=2we want to find values ti = F (Xi )’s in order to maximize R. Revenue R can beshown as total area of some rectangles with one corner on curve H (See Figure 3). The problem of finding these rectangles is very similar to Rectangular CoveringProblem. If we define H ′ (x) = H(−x) and t′ = −tk−i then we can write the irevenue as: R = H(ti )(ti − ti−1 ) = H ′ (−ti )(ti − ti−1 ) = H ′ (t′ )(t′ k−i ′ k−i+1 − tk−i ) = H ′ (t′ )(t′ − t′ )) j j+1 jSo 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
  21. 21. H H(ti ) ti−1 ti Figure 3: Revenue with respect to non-monotone function H .6 Rectangular Covering ProblemDefinition 5 Rectangular Covering Problem: A function F : [0, 1] → R isgiven. We want to find k indices x1 , x2 , · · · , xk in order to maximize the goalfunction k (xi+1 − xi )F (xi )γ i , where x0 = 0 and xk+1 = 1. i=0 In the rectangular covering problem we want to place k rectangles under thecurve F , and maximizing the discounted covering area (see Figure 2). We proposean FPTAS algorithm for this problem for concave function F in Section 6.1 andfor bounded slope function F in Section 6.2. Algorithm described in Lemma 19can be used to solve the rectangular covering problem for general function F withF (0) > 0. The running-time of the algorithm is poly(k, log1+ǫ F (1)/F (0)) inthis case. In the remaining part of paper we assume that F 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 every F (xi ) should be greater than or equal to δ.Definition 7 (δ2 , δ1 )-Rectangular Covering Problem: it is an instance of rect-angular covering problem in which every F (xi ) should be out of a given interval(δ2 , δ1 ).6.1 Concave Function FIn this section we propose an FPTAS algorithm for the rectangular covering prob-lem when the function F is concave. 21
  22. 22. F (x) F (oi )γ i F (xi )γ i xi oi x Figure 4: The Region with Area ALemma 19 For every ǫ > 0 and δ ≥ 0, the δ-rectangular covering problem canbe solved in poly(k, log1+ǫ (F (1)/δ)) time, with approximation factor 1 + ǫ.Proof : At first we define a sequence S = (s1 , s2 , · · · , sm ) and then we prove thatif we choose indices x1 , x2 , · · · , xk from sequence S, we could approximate theoptimum. Let si = F −1 ((1 + ǫ)i−1 δ). In fact F (si+1 ) = (1 + ǫ)F (si ) for everyi < m, s0 = 0 and sm ≤ 1 < sm+1 . Assume optimum indices are o1 , o2 , · · · , ok in the rectangular covering prob-lem. The goal function is equal to O = k (oi+1 − oi )F (oi )γ i in the optimum i=0solution. Let xi be the maximum index of sequence S with value no more than oi .So F (oi ) ≤ F (xi )(1 + ǫ). We can bound O as follows: (In all equations assumex0 = o0 = 0 and ok+1 = xk+1 = 1) k k i O= (oi+1 − oi )F (oi )γ ≤ (oi+1 − oi )(1 + ǫ)F (xi )γ i = (1 + ǫ)A i=0 i=0 (1) The region with area A has been shown in Figure 4. It is clear that A is lessthan or equal to k (xi+1 − xi )F (xi )γ i . So if we choose indices x1 , x2 , · · · , xk i=0from sequence S we can approximate the optimum with factor 1 + ǫ. Now we design a dynamic programming algorithm to find indices x1 , x2 , · · · , xkfrom S. Let A[n, r] is equal to the best solution when we want to select r indicesfrom sequence (s1 , s2 , · · · , sn ) and also sn has been selected. We have: A[n, r] = maxn′ <n {A[n′ , r − 1] + (sn − sn′ )F (sn′ )γ r } The running time of the algorithm is Θ(poly(m, k)), where m = Θ(log1+ǫ (F (1)/δ))2 22
  23. 23. Now we are ready to propose an FPTAS algorithm for the rectangular coveringproblem with concave function F .Theorem 20 For every ǫ > 0, the rectangular covering problem with concavefunction F can be solved in poly(k, 1/ log(1 + ǫ), log(1/ǫ)) time with approxima-tion factor 1 + ǫ.Proof : Let ǫ′ = ǫ/2. Assume we want to solve the δ-rectangular covering problem ′for function F with δ = F (1)( 1+ǫ′ )n , where n = log1+ǫ′ 4(1+2ǫ ) . Because F is 1 ǫ′concave the optimum solution for the rectangular covering problem (OP T ) is atleast F (1)/4. Let the optimum for the δ-rectangular covering problem be OP Tδand the solution found by Lemma 19 be Aδ . We have proved that Aδ ≥ OP T′δ . 1+ǫOn the other hand, the optimum solution for the rectangular covering problem is atmost OP Tδ + δ. So Aδ ≥ OP T −δ = OP T − (1+ǫ(1) ≥ OP T − (1+ǫ′ )n+1 . If we 1+ǫ′ 1+ǫ′ F ′ )n+1 1+ǫ′ 4OP T ′replace (1 + ǫ′ )n by 4(1+2ǫ ) , we can conclude Aδ ≥ OP T . ǫ′ 1+ǫ 1 We have used Lemma 19 with ǫ′ and δ = F (1)( 1+ǫ′ )n . So the algorithm runs log(1/ǫ)in Θ(k, log1+ǫ′ F (1)/δ) = Θ(k, n) time, where n = Θ( log(1+ǫ) ). 26.2 Function F With Bounded SlopeIn this section we propose an FPTAS algorithm to solve the rectangular coveringproblem for function F when γ = 1 and F ′ (x) ≤ L.Lemma 21 For every δ1 , δ2 , ǫ > 0, the (δ2 , δ1 )-Rectangular Covering Problemcan be solved in poly(k, log1+ǫ ( 1−F −1 (δ2 ) ), log1+ǫ ( δ1−δ22 )) time, with approxi- 1 1 −δmation factor 1 + ǫProof : At first we define a sequence S = (s1 , s2 , · · · , sm ) and then we prove thatif we choose indices x1 , x2 , · · · , xk from sequence S, we could approximate the ′′optimum. The sequence S consists of two sequences S ′ and S , which have beenconstructed as follows: • Sequence S ′ = (s′ , s′ , · · · , s′ ′ ) consist of indices with F (s′ ) ≥ δ1 . Let 1 2 m i s′ = F −1 (δ1 ) and s′ = F −1 ((1 + ǫ)(F (s′ ) − δ2 ) + δ2 ) and s′ ′ = 1. In 1 i+1 i m fact the we have F (s′ ) − δ2 ≤ (1 + ǫ)(F (s′ ) − δ2 ) for every i < m′ . The i+1 i length of sequence S ′ is m′ = log1+ǫ ( δ1−δ22 ). 1 −δ ′′ ′′ ′′ ′′ ′′ • Sequence S = (s1 , s2 , · · · , sm′′ ) consist of indices with F (si ) ≤ δ2 . Let ′′ ′′ 1 ′′ ′′ s1 = 0 and 1−si = ( 1+ǫ )i−1 and sm′′ = F −1 (δ2 ). In fact we have 1−si ≤ ′′ 1 (1 + ǫ)(1 − si+1 ). The length of sequence S ′′ is m′′ = log1+ǫ ( 1−F −1 (δ2 ) ). 23
  24. 24. F (x) δ1 δ2 oj xj xi oi x Figure 5: The Region with Area A Assume optimum indices are o1 , o2 , · · · , ok in the (δ2 , δ1 )-Rectangular Cov-ering Problem. And let the goal function be O = k (oi+1 − oi )F (oi ) in the i=1optimum solution. We know that every F (oi ) is out of interval (δ2 , δ1 ). For every ′′oi ≤ F − 1(δ2 ), Let xi be the minimum index in sequence S (S ) with value notless than oi , and for every oi ≥ F − 1(δ1 ), Let xi be the maximum index in se- ′′quence S (S ′ ) with value no more than oi . Assume xi ∈ S for every index i < k′and xi ∈ S ′ for every index i ≥ k′ . We can bound O as follows: (In all equationsassume x0 = o0 = 0 and ok+1 = xk+1 = 1) k k ′ −1 k O= (oi+1 − oi )F (oi ) = (oi+1 − oi )F (oi ) + (oi+1 − oi )F (oi ) (2) i=0 i=0 i=k ′The right hand side of above equation can be written as:k ′ −1 k ′ −1 (oi+1 − oi )F (oi ) = (1 − oi )(F (oi ) − F (oi−1 )) − F (ok′ −1 )(1 − ok′ ) (3)i=0 i=0And the left hand side can be written as: k k (oi+1 − oi )F (oi ) = (oi+1 − oi )(F (oi ) − F (ok′ −1 )) + F (ok′ −1 )(1 − ok′ ) (4)i=k ′ i=k ′Rewrite equality 2 with respect to equality 3 and 4. k ′ −1 k O= (1 − oi )(F (oi ) − F (oi−1 )) + (oi+1 − oi )(F (oi ) − F (ok′ −1 )) (5) i=0 i=k ′ 24
  25. 25. For every i < k′ , we have (1 − oi ) ≤ (1 + ǫ)(1 − xi ). On the other hand for everyi ≥ k′ , we have F (oi ) − δ2 ≤ (1 + ǫ)(F (xi ) − δ2 ). Because F (ok′ −1 ) ≤ δ2 wehave F (oi ) − F (ok′ −1 ) ≤ (1 + ǫ)(F (xi ) − F (ok′ −1 )), for every i ≥ k′ . Withrespect to these facts we can bound the optimum as follows: k ′ −1 O ≤ (1 + ǫ) (1 − xi )(F (oi ) − F (oi−1 )) i=0 k + (1 + ǫ) (oi+1 − oi )(F (xi ) − F (ok′ −1 )) i=k ′ = (1 + ǫ)A (6) The region with area A has been shown in Figure 5. It is clear that A is lessthan or equal to k (xi+1 − xi )F (xi ). So if we choose indices x1 , x2 , · · · , xk i=1from sequence S we can approximate the optimum with factor 1 + ǫ. Now we de-sign a dynamic programming algorithm to find indices x1 , 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 also sn 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 goalOP T . Construct k2 + 1 instances of (δ2 , δ1 )-Rectangular Covering Problem fromRectangular Covering Problem instance. In the i-th instance we set δ2 = ki−1 and 2 +1δ1 = k2i and assume the value of goal function in the optimum solution of this +1instance is OP Ti . Assume o1 , o2 , ..., ok are the optimum indices in the Rectangu-lar Covering Problem instance. It is clear that there exist 1 ≤ j ≤ k2 + 1 suchthat every oi is out of interval ( kj−1 , k2j ). Therefore we have OP T = OP Tj . 2 +1 +1Assume we solve all of the k2 + 1 instances of (δ2 , δ1 )-Rectangular CoveringProblem using Lemma 21. Let Ai be the output of the algorithm for i-th in-stance. We proved in Lemma 21 that OP Ti ≤ (1 + ǫ)Ai . So if return maxi Aias the output for the Rectangular Covering Problem instance, we can approx-imate the optimum with factor 1 + ǫ. Now we prove that this algorithm runin polynomial time. In order to prove this fact we should prove that if we setδ2 = ki−1 and δ1 = k2i then log1+ǫ ( 1−F −1 (δ2 ) ), log1+ǫ ( δ1−δ22 ) are polyno- 2 +1 +1 1 1 −δ 1−δ2mial. First we have δ1 −δ2 = k2 + 2 − i which is polynomial with respect to 1 ′k. On the other hand we have δ2 + F −1 (δ2 ) F (x)dx = F (1) = 1. Therefore 1 ′ 1 F −1 (δ2 ) F (x)dx = 1 − δ2 ≥ k 2 +1 . If we assume that F ′ (x) ≤ L we can con- 25
  26. 26. clude that 1 − F −1 (δ2 ) ≥ L(k1+ 1) . So 1−F −1 (δ2 ) is at most L(k2 + 1) which is 1polynomial with respect to k and L. Now we can conclude Theorem 22.Theorem 22 For every ǫ > 0, the Rectangular Covering Problem with F ′ (x) ≤ Lcan be solved in poly(k, log1+ǫ k, log1+ǫ L) time, with approximation factor 1 + ǫ.6.3 Assumptions about function FIn this section, we prove that some restrictions can be assumed about function Fin the rectangular covering problemWLOG. These restrictions are: • F is non-decreasing: Let G(x) = max0≤y≤x F (y). We prove that the best rectangular covering of G and F are exactly the same. We prove this state- ment by showing every optimum solution of rectangular covering problem for F is a solution for G with same objective value and vice versa. First as- sume x1 , x2 , · · · , xk are optimum indices corresponding to some rectangular covering for curve F . We prove that F (xi ) = G(xi ), for every 1 ≤ i ≤ k. If it is not the case, let j be the smallest index which F (xj ) < G(xj ). Let xj ′ be the smallest index which G(xj ′ ) = G(xj ). It is clear that F (xj ′ ) is the biggest value in interval [0, xj ]. Now if we replace xj by xj ′ in the optimum indices, the change in the objective function will be: (xj ′ − xj−1 )F (xj−1 )γ j−1 + (xj+1 − xj ′ )F (xj ′ )γ j − (xj − xj−1 )F (xj−1 )γ j−1 + (xj+1 − xj )F (xj )γ j Note that F (xj ′ )γ j is greater that F (xj )γ j and F (xj−1 )γ j−1 . Therefore we can conclude that the amount of change in the objective function is positive, which is a contradiction with optimality of indices x1 , x2 , ·, xk . So indices x1 , x2 , ·, xk is a solution for curve G with the same objective value. On the other hand assume x1 , x2 , ·, xk are optimum indices corresponding to some rectangular coveringfor curve G. Note that G is non-decreasing. If there are two indices xj ′ and xj such that xj ′ < xj and F (xj ′ ) = F (xj ), then xj will not appear in any optimum solution of G. If remove indices with this property from the search space only indices xi with G(xi ) = F (xi ) remain. So indices x1 , x2 , ·, xk is a solution for curve F with the same ob- jective value. • F (1) = 1: Let F (1) = c = 1. Define G(x) = F (x) and solve the problem c for G. Assume x1 , x2 , · · · , xk are indices corresponding to some rectangular 26
  27. 27. coveringfor curve G. It is clear that: k k 1 (xi+1 − xi )G(xi )γ i = (xi+1 − xi )F (xi )γ i c i=0 i=0 So we can convert every solution for covering area under curve G to a so- lution for covering area under curve F and vice versa. So by solving the rectangular covering problemfor curve G we can solve the problem for curve F.7 ConclusionIn 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 and the 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 the more 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 without the presence of com-peting sellers. It will be interesting to study the extensions of classical models ofduopoly from algorithmic perspective.References [1] N. AhmadiPourAnari, S. Ehsani, M. Ghodsi, N. Haghpanah, N. Immorlica, H. Mahini, and V. S. Mirrokni. Equilibrium pricing with positive externali- ties. In WINE, 2010. [2] H. Akhlaghpour, M. Ghodsi, N. Haghpanah, H. Mahini, V. S. Mirrokni, and A. Nikzad. Optimal iterative pricing over social networks. In WINE, 2010. 27
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