![1 Introduction
Many products like software, electronics, or automobiles evolve over time. When
a consumer considers buying such a product, he faces a tradeoff between buying
a 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 purchase
the latest Windows operating system. By buying early, the consumer takes full
advantage of all the new features. However, operating systems may have more
bugs and security holes at the beginning, and hence a consumer may prefer to wait
with the rationale that, if more people already own the operating system, then more
bugs will have already been uncovered and corrected. The key observation is, the
more people that use the operating system, or any product for that matter, the more
inherent value it accrues. In other words, the product exhibits a particular type of
externality, 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 the
company extract greater revenue from future customers. On the other hand, too
low a price will result in significant revenue loss from the initial sales. Often, when
faced with such a dilemma, a company will offer an initial promotional price at the
product’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 on
July 11th, 2009. Additionally, beta testers, who can be interpreted as consumers
who “bought” the product even prior to release, received the release version of
Windows 7 for free (as is often the case with software beta-testers).2
We study this phenomenon in the following stylized model: a monopolistic
seller wishes to derive a pricing and marketing plan for a product with historical
externalities. To this end, she commits to a price trajectory. Potential consumers
observe the price trajectory of the seller and make simultaneous decisions regarding
the day on which they will buy the product (and whether to buy at all). The payoff
of a consumer is a function of the day on which he bought the product, the price on
that 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 computer
science literature where a product (e.g., a cell phone) accrues value as more consumers buy it simply
because the product is used in conjunction with other consumers.
2
Historical prices, announced upon the press release, can be found in archived versions of various
technology news websites such as Ars Technia [22] and the Microsoft blog [16]. The current prices
were accessed on Microsoft’s website at the time of submission.
2](https://image.slidesharecdn.com/wine2010-130210223253-phpapp01/85/Wine-2010-2-320.jpg)
![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 are
observed 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 the
economics literature on pricing as well as in other games in the form of Stackel-
berg strategies (see Section 1.2). Further, commitment increases revenue: clearly
a seller who commits to price trajectories can extract at least as much revenue as a
seller who does not (or can not). We further observe via example in Section 2 that
in fact commitment enables a seller to extract unboundedly higher revenue than
in settings without commitment. Second, we assume a consumer’s payoff is only
a 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 in
proportion to usage rates. Of course, strictly speaking, consumers of Windows
7 benefit from future purchases as well via software updates and the like. How-
ever, this forward-looking benefit is substantially dampened in comparison to past
benefits by safety and security risks, and time commitments involved in updates.
Another justification for our payoff model comes from consumers’ uncertainties
regarding products. In many settings, consumers have signals regarding the value
of 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 ensuing
online reviews may help consumers improve their estimates of their values prior to
purchase, an especially important factor for risk-adverse buyers.
1.1 Our Results
We focus on the non-atomic setting in which we have a continuum of consumers
so that each consumer is infinitesimally small and therefore his own action has a
negligible effect on the actions of others. Consumers are drawn from a (possi-
bly infinite) set of types. These types capture varying behavior among consumer
groups. 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 in
the 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 a
paper 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 purchase
the product (thus indifferent consumers purchase the product). In general multi-
ple such equilibria may exist. However, in an aggregate model in which the value
function of each consumer type depends only on the aggregate behavior of the pop-
3](https://image.slidesharecdn.com/wine2010-130210223253-phpapp01/85/Wine-2010-3-320.jpg)
![ulation (i.e. the total fraction of potential consumers that have bought the product
and not the total fraction of various types), then we are able to show that when
they exist well-behaved equilibria of this game are unique in the sense that the
fraction of purchases per-type-per-day is fixed among all equilibria. This enables
us to search for the revenue-maximizing price trajectory. In Section 5, we address
this 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 are
special 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 a
monopolistic 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 set
of rectangles that fit underneath a given curve. We study the rectanglular covering
problem 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-unique
and 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-behaved
equilibrium 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 of
products 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 our
own work. Bensaid and Lesne [5] analyzed the two and infinite period pricing
problems in the presence of linear historical externalities and the study equilib-
ria of the induced games both with and without commitment. They observe, as
we do, that optimal price trajectories are increasing. The historical externalities
that we study generalize the externalities of Bensaid and Lesne [5], and in this
more general model, we solve for the optimal price sequence for any fixed number
of price periods. Most of the remaining externalities literature studies externali-
ties in which consumers care about the total population of users of a product and
hence 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; here
we assume that for any price trajectory announced by the seller, the consumer respond by playing a
well-behaved equilibrium.
4](https://image.slidesharecdn.com/wine2010-130210223253-phpapp01/85/Wine-2010-4-320.jpg)
![phenomenon studied is different from ours, some of the modeling assumptions in
these 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 and
then observe that the revenue-maximizing price sequence with fully rational con-
sumers (playing a Bayesian equilibrium) is increasing. Similar to our model, they
study 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 maximization
over social networks for products with externalities. However, their models assume
naive behavior for consumers. Namely, they assume consumers act myopically,
buying the product on the first day in which it offers them positive utility without
reasoning about future prices and sales that could affect optimal buying behavior
and 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 agents
that 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 the
correct model of pricing and consumer behavior probably lies somewhere between
these two extremes, we believe studying fully rational consumers is an important
first step in relaxing myopic assumptions. Bimpikis et al. [6] study such a model in
which players act strategically. They study a two stage pricing problem in which
the 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 model
and 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 the
time in which consumption happens. They assume that each player’s utility is a
non-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 the
structural properties of the optimal prices, as well as the algorithmic problem of
finding the optimal price sequence.
Our work can be viewed as a analysis of the optimal commitment strategy
of a seller in the presence of historical externalities. Stackelberg [24] first stud-
ied optimal commitment strategies, called Stackelberg strategies, in the Cournots
duopoly [9] model, proving that a firm who can commit has strategic advantage.
The computational aspects of Stackelberg strategies have been studied in generic
normal form [8, 21] and extensive form games [17]. Letchford and Conitzer [17]
suggested polynomial-time algorithms for various models of perfect information
extensive form games and proved NP-hardness of the problem with imperfect in-
formation even with two players. A perfect information extensive form game is
one in which at every stage of the game, every player is exactly aware of what has
happened earlier in the game. This does not happen in games with simultaneous
5](https://image.slidesharecdn.com/wine2010-130210223253-phpapp01/85/Wine-2010-5-320.jpg)
![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 of
potential consumers or buyers. We model our setting as a sequential game whose
players consist of the monopolistic seller and a continuum of potential consumers
or buyers b ∈ [0, 1]. In our game, the seller moves first, selecting a price trajectory
p = (p1 , . . . , pk ) where pi ∈ ℜ assigning a (possibly negative) price pi to each day
i. The buyers move next, selecting a day on which to buy the product given the
complete 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 the
product is bought (∅ is used to indicate that the product was not purchased). Hence
the 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 buyers
that are of type t and buy the product before day i, and we define X1,t = 0 for all
t. Note that by normalization t Xk+1,t ≤ 1 and 1 − t Xk+1,t is the fraction of
buyers that don’t buy the product at any time. Corresponding to this matrix X we
also define the marginal strategy profile matrix x = {xi,t }i=1,...,k;t=1,...,n where
xi,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 the
fraction of buyers who bought before day i and xi as a scalar to denote the fraction
of buyers who buy on day i.
Given a strategy profile X, we define the value of buyers of type t buying on
day i by a value function Fit (Xi ) where Xi is the i’th row of X (hence buyers
are indifferent to future buying decisions). Note the explicit dependence of F on
time, 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 of
particular interest as the β factor models settings in which the value degrades over
time 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 is
defined to be Fit (Xi ) − pi . We additionally allow buyers to have a discount factor
α such that their payoff is (1 − α)i (Fit (Xi ) − pi ). Thus α represents the way in
which agents discount future payoffs with respect to present payoffs. We say that
a strategy profile X is a Nash equilibrium of the induced subgame given by price
trajectory 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](https://image.slidesharecdn.com/wine2010-130210223253-phpapp01/85/Wine-2010-6-320.jpg)
![day i we have i ∈ arg maxj (Fjt (Xj ) − pj )(1 − α)j , and the strategy is ∅ whenever
the maximum is not positive (in which case the buyer’s payoff is zero). We call an
equilibrium well-behaved if all indifferent buyers buy, i.e. a buyer does not buy if
and only if his payoff (1 − α)i (Fit (Xi ) − pi ) is negative on all days 1 ≤ i ≤ k. We
say 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 marginal
strategy profile x of a stragety profile X is a (well-behaved) Nash equilibrium
for the subgame of price trajectory p if the profile X is a (well-behaved) Nash
equilibrium for the subgame of price trajectory p.
Given a price trajectory p and a marginal strategy profile x that arises in the
subgame induced by p, we define the payoff of the seller to be the revenue of x for
p, which is R(p, x) = k i=1
n i
t=1 xi,t pi (1 − α) . A subgame perfect equilibrium
of the sequential game is then a price trajectory p∗ and a set of marginal strategy
profiles 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 outcome
of 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 subgame
perfect 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 the
best response of the seller given the strategies {xp } of the buyers. We solve this
problem for special settings in which there exist revenue-maximizing well-behaved
equilibria 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 each
buyer to have a unique type and hence there are infinitely many types. We will use
b ∈ [0, 1] to denote the type of buyer b.
Definition 1 The Aggregate Model: The value function of each type in this model
is a function of the aggregate behavior of the population and is invariant with
respect to the behavior of each separate type. That is, the value function of buyer
b is a function of Xi only, where Xi is a scalar indicating the total fraction of all
buyers who buy before day i. In this instance, we overload the notation for the
value function and let Fib (Xi ) indicate the value of buyer b (hence Fib (·) now maps
the unit interval to the non-negative reals).
Definition 2 The Linear Model: This is a special case of the aggregate model
which is defined by a function Fi , an initial bias I, and a function C so that the
value 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 buyers
b with C(b) ≤ c∗ .
7](https://image.slidesharecdn.com/wine2010-130210223253-phpapp01/85/Wine-2010-7-320.jpg)
![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 sequential
game with multiple stages where in each day i the seller selects a price pi and then
buyers simultaneously choose whether to buy or not. Such a model is appropriate
when 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 analyze
the resulting revenue. Clearly the revenue with commitment is at least as high
as that without commitment. The below example shows that the revenue without
commitment can actually be unboundedly less.
Example. We consider buyers b ∈ [0, 1] with utility function F b (X) = (1+a|X|)b
for a non-negative constant a, and focus on a setting with k = 2 days. In a strategy
profile, 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 1
given any possible announcement p1 of the seller, and also whether to buy on day 2
given 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 best
response given the strategies of others. In particular, this implies that the seller
must 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 are
p1 and p2 , and subsets S1 , S2 ⊆ 2[0,1] buy at days 1 and 2, respectively. We claim
p1 ≥ p2 . Suppose for contradiction that p1 < p2 . First note that if |S1 | = 0, then
buying 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 lower
the 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-negative
utility 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 that
b(1 + 0) − p1 > b(1 + a|S1 |) − p2 , or equivalantly b < pa|S1 |1 (remember that we
2 −p
assumed |S1 | > 0). Let b∗ = pa|S1 |1 . The set of buyers buying on day 1 is equal to
2 −p
the buyers with b satisfying p1 ≤ b and also b < b∗ . Our assumption that |S1 | > 0
now implies that p1 < b∗ . We observe that p2 ∈ (p1 (1 + a|S1 |), b∗ (1 + a|S1 |)), as
can be seen from the equality p2 = a|S1 |b∗ + p1 and the inequality b∗ > p1 . By
the properties of subgame-perfect equilibria, the seller has to play his best response
in the subgame induced at day 2. In day 2 the utilities of players are outside the
interval (p1 (1 + a|S1 |), b∗ (1 + a|S1 |)). In fact player b utility at day 2 is in the
interval (p1 (1 + a|S1 |), b∗ (1 + a|S1 |)) if and only if p1 < b and b < b∗ which
implies the player b bought in day 1. So p2 ∈ (p1 (1 + a|S1 |), b∗ (1 + a|S1 |))
8](https://image.slidesharecdn.com/wine2010-130210223253-phpapp01/85/Wine-2010-8-320.jpg)
![can not be a revenue-maximizing price. We conclude that in any subgame perfect
equilibrium, the seller has to set the first price at least equal to second price, and
no player will buy in the first day. The highest revenue for the seller is to set the
price of the second day equal to the monopoly price (and any weakly higher price
for the first day), which gives revenue 1/4.
Now consider the game with commitment in which the seller selects a price
trajectory (p1 , p2 ) and then buyers choose their actions. It is easy to check that
p1 = 1/2, p2 = 8+3a , S1 = [1/2, 3/4], S2 = [3/4, 1] is an equilibrium, and its
16
revenue is 1/4 + 3a/64. We conclude by observing that the ratio of the revenue
with commitment to that without goes to infinity as a grows.
An important observation is that without externalities (when a = 0), both equilibria
would have been the same, with revenue equal to the optimal revenue of the 1-stage
game. 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-continuous
valuation function in which no equilibria exist.
Note that while the above theorems prove existence of equilibria, they do not
guarantee that the equilibria are necessarily well-behaved (i.e. all indifferent buy-
ers buy at some point). Well-behaved equilibria may not exist. Furthermore, even
when they exist they do not necessarily maximize revenue. However, for the linear
and 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 in
Section 3.1; the results in Section 5 imply the revenue-maximizing property.
We prove our pricing game has an equilibrium whenever buyers have finitely
many types and continuous valuation functions Fit , 1 ≤ t ≤ n, 1 ≤ i ≤ k. To
do so, we define a set-valued function on the space of marginal strategy profile
matrices whose fixed point is an equilibrium of our game and show the existence
of a fixed point using Kakutani’s fixed point theorem (KFPT). Let φ : X → 2Y be
a set-valued function, i.e. a function from X to the power set of Y . We say that φ
has a closed graph if the set {(x, y)|y ∈ φ(x)} is a closed subset of X × Y in the
product 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](https://image.slidesharecdn.com/wine2010-130210223253-phpapp01/85/Wine-2010-9-320.jpg)
![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 matrices
and then use KFPT to show that this mapping has a fixed point.
Theorem 2 If valuation functions are increasing and continuous, then our game
has an equilibrium.
Proof : Let S be a subset of the Euclidean space Rk×n consisting of all valid
marginal profile matrices x. Also let µ(t) be the length of Tt for each type t (recall
that Tt is a subinterval of [0, 1]). Each x ∈ S is a marginal strategy profile matrix
x = (x1,1 , · · · , xk,n ), where xi,t is the fraction of buyers who are of type t and
choose 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 ∈ S
the set of all marginal strategy profile matrices y ∈ φ(x) which are simultaneous
best-responses to the profile x. Formally, φ(x) consists of all y satisfying the
following 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 point
x 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 the
conditions of KFPT. Set S can be defined as the set of points x ∈ Rk×n satisfying
the 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 is
closed 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 intersection
of S, and a set of (possibly open) half-spaces defined by linear inequalities listed
in the conditions above. Thus, φ(x) is a convex set. It is also nonempty as each
buyer of each type t has a well-defined set of best-responses to X, the cumulative
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![Example 8 Assume there are two types and Fit (Y ) = Yt′ + 2 where t = 1, 2 and
t′ = t. In other words a social valuation function is only depends on fraction of
buyers of another type. The population of type 1 buyers are 0.3 and the population
of type 2 buyers are 0.7. Suppose the seller wants to sell the product in two days
and 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 the
product 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). The
revenue 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 is
unique. We show this for an infinite number of types in the aggregate model which
generalizes both the linear and symmetric models.
Recall that we allow for each buyer b ∈ [0, 1] to have a unique type in the
aggregate model such that the valuation function of buyer b is Fib . We will show
that in all of the well-behaved equilibrium points the fraction of people buying
on each day is the same. In turn, it implies that the revenue of all well-behaved
equilibrium 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. We
start with a definition: Consider two well-behaved equilibria x and y. Partition the
set of k days to two sets as follows: We call a day i a level 1 day, and denote it by
i ∈ D1 (x, y), if Xi < Yi . Otherwise, if Xi ≥ Yi , we call i a level 2 day and denote
it 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) and
whose strategy in y is j ∈ D2 (x, y).
Proof : Since x = y, assume WLOG that there exists a day i so that Xi < Yi . Let
S1 be the set of buyers who have bought on a level 1 day in x and S2 be the set of
buyers who have bought on a level 2 day in y. Further define S = S1 ∪ S2 . We
show that |S1 | + |S2 | > |S|. Therefore S1 and S2 have a nonempty intersection
whose 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](https://image.slidesharecdn.com/wine2010-130210223253-phpapp01/85/Wine-2010-14-320.jpg)
![by solving the Rectangular Covering Problem for a specific curve. This curve is
obtained 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 i
if and only if i ≤ k′ (just remove the days with no purchase to the end). So we
can assume that for all i < j ≤ k′ , we have Xi < Xj . The utility of a buyer
b for buying on day i is Ib + Cb F (Xi ) − pi . In order to concentrate on network
externalities, for now we restrict the model and assume that Ib is always equal to a
fixed 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
ˆ
be equal to (pi − pj )/(F (Xi ) − F (Xj )), which is the slope of the line between
points qi and qj . Also let s(i) = s(i, i + 1). In Lemma 13, we prove that s is non-
ˆ
decreasing in i. Lemma 14 then shows that the utility of buyer b will be maximized
on day i if and only if Cb ∈ [s(i−1), s(i)]. Finally, we use these lemmas in Lemma
15 to show how to find a price vector given x. We use these properties in the next
section 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 −p
that we know Xi > Xj , and therefore F (Xi ) > F (Xj )):Cb ≥ F (Xi )−Fj(Xj ) . The
converse is also true. If Cb is less than (pi − pj )/(F (Xi ) − F (Xj )), then day j
will 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 who
chooses 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). 2
Lemma 14 If Cb ∈ [s(i − 1), s(i)] then b will have the maximum utility buying on
day i.
Proof : Assume that Cb ∈ [s(i − 1), s(i)]. For each j > i, the utility of day i
is at least as good as that of day j, because Cb ≤ s(i) ≤ s(i, j). Similarly for
ˆ
each 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), ∞)
ˆ
are dealt with the same arguments. 2
This lemma enables us to find the key relations between prices and vector x.
18](https://image.slidesharecdn.com/wine2010-130210223253-phpapp01/85/Wine-2010-18-320.jpg)
![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 ’s
lie 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))} have
identical differences of consecutive terms. They also have identical initial elements
X1 = 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
(X
we 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 in
the revenue-maximizing equilibrium, the first price p1 will be equal to I in Lemma
16. 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 Lemma
18, we prove that the revenue maximization problem can be reduced to Rectangu-
lar Covering Problem, and therefore there exists an FPTAS to solve the revenue
maximization 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 all
elements of p by I − p1 to get p′ . It’s easy to check that x, p′ still define an
equilibrium. The new equilibrium’s revenue is greater than the original one. Hence
p1 = I in the revenue-maximizing equilibrium. 2
Note that by calculating all Xi with respect to vector x, we know the values
pi − pi−1 using Lemma 15. On the other hand, p1 = I in the revenue-maximizing
equilibrium. So we would know all pi ’s. So x uniquely determines prices. Let
Price(x) be price vector which has been determined by vector x. It is easy to
verify that vectors x and Price(x) make an equilibrium. Hence it suffices to view
everything as functions of the free variable x. Now we express the revenue in a
candidate equilibrium in terms of vector x.
Lemma 17 If x and p correspond to the revenue-maximizing equilibrium, the total
revenue 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](https://image.slidesharecdn.com/wine2010-130210223253-phpapp01/85/Wine-2010-19-320.jpg)
![Proof : Since the utility of buying on the first day is nonnegative for everybody, all
buyers 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=i
To interpret the above formula, note that the price change from day i − 1 to day i
is exerted on all buyers who have bought on day i or later. Since k xj = 1,
j=1
we can rewrite the above formula as R(p, x) = p1 + k (pi − pi−1 )(1 − Xi ).
i=2
Now we can substitute for p1 and pi − pi−1 from Lemmas 16 and 15 and write
revenue as a function of vector x. Therefore we haveR(p, x) = I + k (1 − i=2
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 Rectangular
Covering Problem. Note that I is a constant in Lemma 17 which does not affect
revenue maximization.
Lemma 18 The problem of maximizing k (1−Xi )C −1 (Xi )(F (Xi )−F (Xi−1 ))
i=2
can be reduced to the Rectangular Covering Problem.
Proof : Since F is a monotone and hence bijective function, we can write the
revenue as
k
R= (1 − F −1 (F (Xi )))C −1 (F −1 (F (Xi ))(F (Xi ) − F (Xi−1 ))
i=2
The 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 define
H : [Fmin , Fmax ] → R≥0 as follows : H(x) = (1 − F −1 (x))C −1 (F −1 (x)). Note
that the revenue is exactly R = k H(F (Xi ))(F (Xi ) − F (Xi−1 )). Therefore
i=2
we want to find values ti = F (Xi )’s in order to maximize R. Revenue R can be
shown 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 Covering
Problem. If we define H ′ (x) = H(−x) and t′ = −tk−i then we can write the
i
revenue 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 j
So we should solve Rectangular Covering Problem for function H ′ . 2
We have proved that the revenue maximization problem can be solved using
Rectangular Covering Problem. Therefore, there exists an FPTAS to solve this
problem (Theorem 22 in Section 6.2).
20](https://image.slidesharecdn.com/wine2010-130210223253-phpapp01/85/Wine-2010-20-320.jpg)
![H
H(ti )
ti−1 ti
Figure 3: Revenue with respect to non-monotone function H
.
6 Rectangular Covering Problem
Definition 5 Rectangular Covering Problem: A function F : [0, 1] → R is
given. We want to find k indices x1 , x2 , · · · , xk in order to maximize the goal
function 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 the
curve F , and maximizing the discounted covering area (see Figure 2). We propose
an FPTAS algorithm for this problem for concave function F in Section 6.1 and
for bounded slope function F in Section 6.2. Algorithm described in Lemma 19
can be used to solve the rectangular covering problem for general function F with
F (0) > 0. The running-time of the algorithm is poly(k, log1+ǫ F (1)/F (0)) in
this case. In the remaining part of paper we assume that F is non-decreasing and
F (1) = 1. We prove that these assumptions do not hurt the generality in Section
6.3.
Now we define some restricted versions of the rectangular covering problem
which are useful in the paper.
Definition 6 δ-Rectangular Covering Problem: it is an instance of rectangular
covering 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 F
In this section we propose an FPTAS algorithm for the rectangular covering prob-
lem when the function F is concave.
21](https://image.slidesharecdn.com/wine2010-130210223253-phpapp01/85/Wine-2010-21-320.jpg)
![F (x)
F (oi )γ i
F (xi )γ i
xi oi
x
Figure 4: The Region with Area A
Lemma 19 For every ǫ > 0 and δ ≥ 0, the δ-rectangular covering problem can
be 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 that
if we choose indices x1 , x2 , · · · , xk from sequence S, we could approximate the
optimum. Let si = F −1 ((1 + ǫ)i−1 δ). In fact F (si+1 ) = (1 + ǫ)F (si ) for every
i < 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=0
solution. 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 assume
x0 = 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 less
than or equal to k (xi+1 − xi )F (xi )γ i . So if we choose indices x1 , x2 , · · · , xk
i=0
from sequence S we can approximate the optimum with factor 1 + ǫ.
Now we design a dynamic programming algorithm to find indices x1 , x2 , · · · , xk
from S. Let A[n, r] is equal to the best solution when we want to select r indices
from 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](https://image.slidesharecdn.com/wine2010-130210223253-phpapp01/85/Wine-2010-22-320.jpg)
![For every i < k′ , we have (1 − oi ) ≤ (1 + ǫ)(1 − xi ). On the other hand for every
i ≥ k′ , we have F (oi ) − δ2 ≤ (1 + ǫ)(F (xi ) − δ2 ). Because F (ok′ −1 ) ≤ δ2 we
have F (oi ) − F (ok′ −1 ) ≤ (1 + ǫ)(F (xi ) − F (ok′ −1 )), for every i ≥ k′ . With
respect 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 less
than or equal to k (xi+1 − xi )F (xi ). So if we choose indices x1 , x2 , · · · , xk
i=1
from 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. Let
A[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 goal
OP T . Construct k2 + 1 instances of (δ2 , δ1 )-Rectangular Covering Problem from
Rectangular Covering Problem instance. In the i-th instance we set δ2 = ki−1 and
2 +1
δ1 = k2i and assume the value of goal function in the optimum solution of this
+1
instance 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 such
that every oi is out of interval ( kj−1 , k2j ). Therefore we have OP T = OP Tj .
2 +1 +1
Assume we solve all of the k2 + 1 instances of (δ2 , δ1 )-Rectangular Covering
Problem 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 Ai
as the output for the Rectangular Covering Problem instance, we can approx-
imate the optimum with factor 1 + ǫ. Now we prove that this algorithm run
in polynomial time. In order to prove this fact we should prove that if we set
δ2 = 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−δ2
mial. 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](https://image.slidesharecdn.com/wine2010-130210223253-phpapp01/85/Wine-2010-25-320.jpg)
![clude that 1 − F −1 (δ2 ) ≥ L(k1+ 1) . So 1−F −1 (δ2 ) is at most L(k2 + 1) which is
1
polynomial 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) ≤ L
can be solved in poly(k, log1+ǫ k, log1+ǫ L) time, with approximation factor 1 + ǫ.
6.3 Assumptions about function F
In this section, we prove that some restrictions can be assumed about function F
in 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](https://image.slidesharecdn.com/wine2010-130210223253-phpapp01/85/Wine-2010-26-320.jpg)
![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 Conclusion
In this paper, we studied the problem of item pricing in the presence of historical
network externalities and strategic buyers. We provided necessary and sufficient
conditions on the valuation functions for the existence and the uniqueness of the
equilibria. We defined the revenue maximization problem in settings in which the
equilibrium exists and is unique. We provided nearly optimal algorithms for the
two special cases of linear and symmetric valuation functions. The problem of
providing algorithms or proving inapproximability for the more general aggregate
model 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 another
interesting 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 of
duopoly from algorithmic perspective.
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Wine 2010
- 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,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 1
- 2. 1 Introduction Many products like software, electronics, or automobiles evolve over time. When a consumer considers buying such a product, he faces a tradeoff between buying a 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 purchase the latest Windows operating system. By buying early, the consumer takes full advantage of all the new features. However, operating systems may have more bugs and security holes at the beginning, and hence a consumer may prefer to wait with the rationale that, if more people already own the operating system, then more bugs will have already been uncovered and corrected. The key observation is, the more people that use the operating system, or any product for that matter, the more inherent value it accrues. In other words, the product exhibits a particular type of externality, 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 the company extract greater revenue from future customers. On the other hand, too low a price will result in significant revenue loss from the initial sales. Often, when faced with such a dilemma, a company will offer an initial promotional price at the product’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 on July 11th, 2009. Additionally, beta testers, who can be interpreted as consumers who “bought” the product even prior to release, received the release version of Windows 7 for free (as is often the case with software beta-testers).2 We study this phenomenon in the following stylized model: a monopolistic seller wishes to derive a pricing and marketing plan for a product with historical externalities. To this end, she commits to a price trajectory. Potential consumers observe the price trajectory of the seller and make simultaneous decisions regarding the day on which they will buy the product (and whether to buy at all). The payoff of a consumer is a function of the day on which he bought the product, the price on that 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 computer science literature where a product (e.g., a cell phone) accrues value as more consumers buy it simply because the product is used in conjunction with other consumers. 2 Historical prices, announced upon the press release, can be found in archived versions of various technology news websites such as Ars Technia [22] and the Microsoft blog [16]. The current prices were accessed on Microsoft’s website at the time of submission. 2
- 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 which the seller has the ability to commit to a price trajectory. Such commitments are observed 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 the economics literature on pricing as well as in other games in the form of Stackel- berg strategies (see Section 1.2). Further, commitment increases revenue: clearly a seller who commits to price trajectories can extract at least as much revenue as a seller who does not (or can not). We further observe via example in Section 2 that in fact commitment enables a seller to extract unboundedly higher revenue than in settings without commitment. Second, we assume a consumer’s payoff is only a 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 in proportion to usage rates. Of course, strictly speaking, consumers of Windows 7 benefit from future purchases as well via software updates and the like. How- ever, this forward-looking benefit is substantially dampened in comparison to past benefits by safety and security risks, and time commitments involved in updates. Another justification for our payoff model comes from consumers’ uncertainties regarding products. In many settings, consumers have signals regarding the value of 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 ensuing online reviews may help consumers improve their estimates of their values prior to purchase, an especially important factor for risk-adverse buyers. 1.1 Our Results We focus on the non-atomic setting in which we have a continuum of consumers so that each consumer is infinitesimally small and therefore his own action has a negligible effect on the actions of others. Consumers are drawn from a (possi- bly infinite) set of types. These types capture varying behavior among consumer groups. 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 in the 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 a paper 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 purchase the product (thus indifferent consumers purchase the product). In general multi- ple such equilibria may exist. However, in an aggregate model in which the value function of each consumer type depends only on the aggregate behavior of the pop- 3
- 4. ulation (i.e. the total fraction of potential consumers that have bought the product and not the total fraction of various types), then we are able to show that when they exist well-behaved equilibria of this game are unique in the sense that the fraction of purchases per-type-per-day is fixed among all equilibria. This enables us to search for the revenue-maximizing price trajectory. In Section 5, we address this 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 are special 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 a monopolistic 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 set of rectangles that fit underneath a given curve. We study the rectanglular covering problem 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-unique and 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-behaved equilibrium 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 of products 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 our own work. Bensaid and Lesne [5] analyzed the two and infinite period pricing problems in the presence of linear historical externalities and the study equilib- ria of the induced games both with and without commitment. They observe, as we do, that optimal price trajectories are increasing. The historical externalities that we study generalize the externalities of Bensaid and Lesne [5], and in this more general model, we solve for the optimal price sequence for any fixed number of price periods. Most of the remaining externalities literature studies externali- ties in which consumers care about the total population of users of a product and hence 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; here we assume that for any price trajectory announced by the seller, the consumer respond by playing a well-behaved equilibrium. 4
- 5. phenomenon studied is different from ours, some of the modeling assumptions in these 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 and then observe that the revenue-maximizing price sequence with fully rational con- sumers (playing a Bayesian equilibrium) is increasing. Similar to our model, they study 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 maximization over social networks for products with externalities. However, their models assume naive behavior for consumers. Namely, they assume consumers act myopically, buying the product on the first day in which it offers them positive utility without reasoning about future prices and sales that could affect optimal buying behavior and 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 agents that 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 the correct model of pricing and consumer behavior probably lies somewhere between these two extremes, we believe studying fully rational consumers is an important first step in relaxing myopic assumptions. Bimpikis et al. [6] study such a model in which players act strategically. They study a two stage pricing problem in which the 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 model and 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 the time in which consumption happens. They assume that each player’s utility is a non-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 the structural properties of the optimal prices, as well as the algorithmic problem of finding the optimal price sequence. Our work can be viewed as a analysis of the optimal commitment strategy of a seller in the presence of historical externalities. Stackelberg [24] first stud- ied optimal commitment strategies, called Stackelberg strategies, in the Cournots duopoly [9] model, proving that a firm who can commit has strategic advantage. The computational aspects of Stackelberg strategies have been studied in generic normal form [8, 21] and extensive form games [17]. Letchford and Conitzer [17] suggested polynomial-time algorithms for various models of perfect information extensive form games and proved NP-hardness of the problem with imperfect in- formation even with two players. A perfect information extensive form game is one in which at every stage of the game, every player is exactly aware of what has happened earlier in the game. This does not happen in games with simultaneous 5
- 6. 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 of potential consumers or buyers. We model our setting as a sequential game whose players consist of the monopolistic seller and a continuum of potential consumers or buyers b ∈ [0, 1]. In our game, the seller moves first, selecting a price trajectory p = (p1 , . . . , pk ) where pi ∈ ℜ assigning a (possibly negative) price pi to each day i. The buyers move next, selecting a day on which to buy the product given the complete 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 the product is bought (∅ is used to indicate that the product was not purchased). Hence the 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 buyers that are of type t and buy the product before day i, and we define X1,t = 0 for all t. Note that by normalization t Xk+1,t ≤ 1 and 1 − t Xk+1,t is the fraction of buyers that don’t buy the product at any time. Corresponding to this matrix X we also define the marginal strategy profile matrix x = {xi,t }i=1,...,k;t=1,...,n where xi,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 the fraction of buyers who bought before day i and xi as a scalar to denote the fraction of buyers who buy on day i. Given a strategy profile X, we define the value of buyers of type t buying on day i by a value function Fit (Xi ) where Xi is the i’th row of X (hence buyers are indifferent to future buying decisions). Note the explicit dependence of F on time, 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 of particular interest as the β factor models settings in which the value degrades over time 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 is defined to be Fit (Xi ) − pi . We additionally allow buyers to have a discount factor α such that their payoff is (1 − α)i (Fit (Xi ) − pi ). Thus α represents the way in which agents discount future payoffs with respect to present payoffs. We say that a strategy profile X is a Nash equilibrium of the induced subgame given by price trajectory 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. day i we have i ∈ arg maxj (Fjt (Xj ) − pj )(1 − α)j , and the strategy is ∅ whenever the maximum is not positive (in which case the buyer’s payoff is zero). We call an equilibrium well-behaved if all indifferent buyers buy, i.e. a buyer does not buy if and only if his payoff (1 − α)i (Fit (Xi ) − pi ) is negative on all days 1 ≤ i ≤ k. We say 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 marginal strategy profile x of a stragety profile X is a (well-behaved) Nash equilibrium for the subgame of price trajectory p if the profile X is a (well-behaved) Nash equilibrium for the subgame of price trajectory p. Given a price trajectory p and a marginal strategy profile x that arises in the subgame induced by p, we define the payoff of the seller to be the revenue of x for p, which is R(p, x) = k i=1 n i t=1 xi,t pi (1 − α) . A subgame perfect equilibrium of the sequential game is then a price trajectory p∗ and a set of marginal strategy profiles 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 outcome of 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 subgame perfect 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 the best response of the seller given the strategies {xp } of the buyers. We solve this problem for special settings in which there exist revenue-maximizing well-behaved equilibria 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 each buyer to have a unique type and hence there are infinitely many types. We will use b ∈ [0, 1] to denote the type of buyer b. Definition 1 The Aggregate Model: The value function of each type in this model is a function of the aggregate behavior of the population and is invariant with respect to the behavior of each separate type. That is, the value function of buyer b is a function of Xi only, where Xi is a scalar indicating the total fraction of all buyers who buy before day i. In this instance, we overload the notation for the value function and let Fib (Xi ) indicate the value of buyer b (hence Fib (·) now maps the unit interval to the non-negative reals). Definition 2 The Linear Model: This is a special case of the aggregate model which is defined by a function Fi , an initial bias I, and a function C so that the value 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 buyers b with C(b) ≤ c∗ . 7
- 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 sequential game with multiple stages where in each day i the seller selects a price pi and then buyers simultaneously choose whether to buy or not. Such a model is appropriate when 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 analyze the resulting revenue. Clearly the revenue with commitment is at least as high as that without commitment. The below example shows that the revenue without commitment can actually be unboundedly less. Example. We consider buyers b ∈ [0, 1] with utility function F b (X) = (1+a|X|)b for a non-negative constant a, and focus on a setting with k = 2 days. In a strategy profile, 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 1 given any possible announcement p1 of the seller, and also whether to buy on day 2 given 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 best response given the strategies of others. In particular, this implies that the seller must 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 are p1 and p2 , and subsets S1 , S2 ⊆ 2[0,1] buy at days 1 and 2, respectively. We claim p1 ≥ p2 . Suppose for contradiction that p1 < p2 . First note that if |S1 | = 0, then buying 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 lower the 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-negative utility 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 that b(1 + 0) − p1 > b(1 + a|S1 |) − p2 , or equivalantly b < pa|S1 |1 (remember that we 2 −p assumed |S1 | > 0). Let b∗ = pa|S1 |1 . The set of buyers buying on day 1 is equal to 2 −p the buyers with b satisfying p1 ≤ b and also b < b∗ . Our assumption that |S1 | > 0 now implies that p1 < b∗ . We observe that p2 ∈ (p1 (1 + a|S1 |), b∗ (1 + a|S1 |)), as can be seen from the equality p2 = a|S1 |b∗ + p1 and the inequality b∗ > p1 . By the properties of subgame-perfect equilibria, the seller has to play his best response in the subgame induced at day 2. In day 2 the utilities of players are outside the interval (p1 (1 + a|S1 |), b∗ (1 + a|S1 |)). In fact player b utility at day 2 is in the interval (p1 (1 + a|S1 |), b∗ (1 + a|S1 |)) if and only if p1 < b and b < b∗ which implies the player b bought in day 1. So p2 ∈ (p1 (1 + a|S1 |), b∗ (1 + a|S1 |)) 8
- 9. can not be a revenue-maximizing price. We conclude that in any subgame perfect equilibrium, the seller has to set the first price at least equal to second price, and no player will buy in the first day. The highest revenue for the seller is to set the price of the second day equal to the monopoly price (and any weakly higher price for the first day), which gives revenue 1/4. Now consider the game with commitment in which the seller selects a price trajectory (p1 , p2 ) and then buyers choose their actions. It is easy to check that p1 = 1/2, p2 = 8+3a , S1 = [1/2, 3/4], S2 = [3/4, 1] is an equilibrium, and its 16 revenue is 1/4 + 3a/64. We conclude by observing that the ratio of the revenue with commitment to that without goes to infinity as a grows. An important observation is that without externalities (when a = 0), both equilibria would have been the same, with revenue equal to the optimal revenue of the 1-stage game. 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-continuous valuation function in which no equilibria exist. Note that while the above theorems prove existence of equilibria, they do not guarantee that the equilibria are necessarily well-behaved (i.e. all indifferent buy- ers buy at some point). Well-behaved equilibria may not exist. Furthermore, even when they exist they do not necessarily maximize revenue. However, for the linear and 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 in Section 3.1; the results in Section 5 imply the revenue-maximizing property. We prove our pricing game has an equilibrium whenever buyers have finitely many types and continuous valuation functions Fit , 1 ≤ t ≤ n, 1 ≤ i ≤ k. To do so, we define a set-valued function on the space of marginal strategy profile matrices whose fixed point is an equilibrium of our game and show the existence of a fixed point using Kakutani’s fixed point theorem (KFPT). Let φ : X → 2Y be a set-valued function, i.e. a function from X to the power set of Y . We say that φ has a closed graph if the set {(x, y)|y ∈ φ(x)} is a closed subset of X × Y in the product 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. 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 matrices and then use KFPT to show that this mapping has a fixed point. Theorem 2 If valuation functions are increasing and continuous, then our game has an equilibrium. Proof : Let S be a subset of the Euclidean space Rk×n consisting of all valid marginal profile matrices x. Also let µ(t) be the length of Tt for each type t (recall that Tt is a subinterval of [0, 1]). Each x ∈ S is a marginal strategy profile matrix x = (x1,1 , · · · , xk,n ), where xi,t is the fraction of buyers who are of type t and choose 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 ∈ S the set of all marginal strategy profile matrices y ∈ φ(x) which are simultaneous best-responses to the profile x. Formally, φ(x) consists of all y satisfying the following 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 point x 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 the conditions of KFPT. Set S can be defined as the set of points x ∈ Rk×n satisfying the 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 is closed 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 intersection of S, and a set of (possibly open) half-spaces defined by linear inequalities listed in the conditions above. Thus, φ(x) is a convex set. It is also nonempty as each buyer of each type t has a well-defined set of best-responses to X, the cumulative 10
- 11. corresponding to marginal profile x, which is either some day j if Fjt (Xj )−pj ≥ 0 or the empty strategy (not buying) otherwise. It only remains to show that the graph of φ is a closed subset of R2(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 form A × B, where A is an open neighborhood of x and B is an open neighborhood of y. Since (x, y) is not in the graph, either (x, y) is not in S × S or y does not satisfy one of the conditions defining φ(x). In the former case, since S × S is closed, we can find a suitable neighborhood of (x, y) having no intersection with S × S and by 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 buyer of type t. Assuming the valuation functions are continuous and increasing, U and hence U −1 is continuous and invertible. As y ∈ φ(x), there is some type t such that 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 continuous and 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 of φ 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 when valuation 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 2 price 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’t buy in day one are not playing a best-response so X is not an equilibrium. On the other hand, if x1 > 1/2 then the payoff on day 1 is still 1 but the payoff on day 2 is now greater than 3/2 + 1/2 − 1 ≥ 1. Hence the buyers who buy in day one are not playing a best-response and so X is not an equilibrium. 11
- 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 changing the function to Fi (X) = 1 if X < 1 and Fit (X) = 3 + X otherwise, for i = 1, 2. 2 2 In this case x = (1/2, 1/2) is an equilibrium for price trajectory p = (0, 1). The following 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 3 valuation function Fit (X) = 2 if Xt′ < 1 and Fit (X) = 4, where t′ = 2, 3, 1 3 respectively for t = 1, 2, 3. Then similar reasoning shows price trajectory p = (1, 2) has no equilibrium. 3.1 Existence of Well-Behaved Equilibria Our revenue results assume the existence of revenue-maximizing well-behaved equilibria for all price sequences. Unfortunately, this does not hold even for the aggregate model, as the following example shows: Example 5 Suppose there are three days k = 3 and two types in the market, each of which contain half the total population. Let Fi1 (Xi ) and Fi2 (Xi ) be as shown in 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 utility of buying is 0, then we have an equilibrium point in this example. The vector x = ((0, 0.25), (0.35, 0), (0, 0.25)) is an equilibrium point and a 0.15 fraction of type 1 will not buy the product. However a simple case analysis shows there are no well-behaved equilibria (i.e. equilibria in which a buyer buys when the utility of 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. we derive a condition on the utilities that is sufficient to guarantee existence of well-behaved equilibria in which either all or no buyers buy. We will present the proof 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 form Fit (0)−pi which is negative. Hence everyone has a negative utility and the strategy profile 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 buying on 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 same well-behaved equilibrium is also an equilibrium for the original price trajectory. When prices are all decreased by the same amount all utilities are also decreased by that same amount. Hence the relative ordering of utilities is the same for both price trajectories. So after changing the prices back to the original ones, everyone is still buying an optimal day. The only equilibrium condition that might not hold anymore is the one asserting that buyers only buy when the optimum utility is non-negative. However, since mint,i Fit (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 in which 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 i one of the conditions of Theorem 6 holds and we can conclude Corollary 7. Note for 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-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 all the valuation functions are continuous. 13
- 14. Example 8 Assume there are two types and Fit (Y ) = Yt′ + 2 where t = 1, 2 and t′ = t. In other words a social valuation function is only depends on fraction of buyers of another type. The population of type 1 buyers are 0.3 and the population of type 2 buyers are 0.7. Suppose the seller wants to sell the product in two days and 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 the product 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). The revenue 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 is unique. We show this for an infinite number of types in the aggregate model which generalizes both the linear and symmetric models. Recall that we allow for each buyer b ∈ [0, 1] to have a unique type in the aggregate model such that the valuation function of buyer b is Fib . We will show that in all of the well-behaved equilibrium points the fraction of people buying on each day is the same. In turn, it implies that the revenue of all well-behaved equilibrium 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. We start with a definition: Consider two well-behaved equilibria x and y. Partition the set of k days to two sets as follows: We call a day i a level 1 day, and denote it by i ∈ D1 (x, y), if Xi < Yi . Otherwise, if Xi ≥ Yi , we call i a level 2 day and denote it 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) and whose strategy in y is j ∈ D2 (x, y). Proof : Since x = y, assume WLOG that there exists a day i so that Xi < Yi . Let S1 be the set of buyers who have bought on a level 1 day in x and S2 be the set of buyers who have bought on a level 2 day in y. Further define S = S1 ∪ S2 . We show that |S1 | + |S2 | > |S|. Therefore S1 and S2 have a nonempty intersection whose 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. • 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).5 Note that |S1 | + |S2 | = i∈D1 (x,y) xi + i∈D2 (x,y) yi = k zi . Thus it suffices i=1 to show that k zi > |S|. Let Zi = z1 + . . . + zi−1 . So we must argue that i=1 Zk+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 have Zi+1 ≥ (respectively >) min(Xi+1 , Yi+1 ). Proof : For day i, there are four possibilities: If i and i − 1 are level 1 days, then Zi+1 = Zi + zi = Zi + xi ≥ (>)Xi + xi = Xi+1 . If i is a level 1 day and i − 1 is a 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, then Zi+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 all i we have Zi ≥ min(Xi , Yi ). Second, there exists a day of level 1, and since day 1 is level 2, there must be a day i that falls in the second case of the claim for which we must have Zi > min(Xi , Yi ). Then by induction we will have Zk+1 > min(Xk+1 , Yk+1 ). 2 Theorem 11 Let Fib (X) be a strictly increasing function for each buyer b and day i. For a price sequence p and two well-behaved equilibrium points x and y, we have Xi = Yi , i.e. the fraction of buyers who have bought the product before day i is unique. Proof : Assume for contradiction that we have two well-behaved equilibrium points 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 a level 1 day in x and buys on a level 2 day in y. Assume that b buys on day i in x and on 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 monotonicity Fib (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. 5 Revenue Maximization In this section, we solve the revenue-maximizing problem in two special cases: the discounted version of the symmetric model, and the general linear model without discount 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 marginal strategy profile matrix will reduce to the vector x = (x1 , . . . , xk ). Also, fixing p 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 renaming qi = pi (1 − α) i and γ = β(1 − α), the utility of buyer b for the item on day i will 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. For convenience, we use p for the discounted prices q. Since we only have one type in this model, we know that the utility of buying in day i is equal among all players. We use the term utility of a day i, denoted by ui , for ui = F (Xi )γ i − pi . Define u(p, x) = maxi ui . Consider a price sequence and its equilibrium strategy profile x. We get the following properties immediately from the facts that players are utility maximizing: (i) players are allowed to choose inaction and have utility zero, (ii) they choose to buy if there is a day with a strictly positive 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- ˆ rium x. Then u(ˆ, x) = 0. ˆ p ˆ Proof : Assume for contradiction that u(ˆ, x) = w > 0. Let w = (w, . . . , w) p ˆ ˆ be a vector of length k with all of its elements equal to w, and consider the price sequence p + w and vector x. The utility of each day decreases by the same amount ˆ ˆ ˆ w, and the set of maximizers have positive utility, that is, u(ˆ + w, x) ≥ 0. There- ˆ p ˆ ˆ fore, each day i with xi > 0 is still a maximizer with ui ≥ 0. We conclude that ˆ x is an equilibrium for p + w. Since i xi = 1, this price sequence has strictly ˆ ˆ ˆ ˆ greater revenue, contradicting the optimality of p. ˆ 2 Assume that there is a price sequence p with equilibrium x and u(p, x) = 0 such that for some day i, we have xi = 0 and xi+1 > 0. Then we can define a new 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. price F (x) p′ t pt x F −1 (p′ ) F −1 (p′ ) 1 t t+1 Figure 2: The discounted RCP problem. The blue area is the total area covered by rectangles, discounted by the index of each rectangle). revenue. So we can assume WLOG that for a revenue maximizing price sequence p associated with x, there exists a k′ ≤ k such that xi = 0 if and only if i ≤ k′ . ˆ ˆ For such a price sequence, Lemma 12 shows that F (Xi )γ i − pi = 0 for each 1 ≤ i ≤ k′ . As a result, we have Xi = F −1 (pi /γ i ), which is well-defined as F is increasing. Now set p′ = pt /γ t . The fraction of people buying on day i t and 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 i areas of a number of rectangles, discounted by γ, that are fit under the graph of F (see Figure 2). So the revenue maximization problem reduces to the following rectangular 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 the discounted 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 problem for 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 different versions of the RCP. 5.2 Linear Version To solve the problem in the linear version, we first study the properties of any equilibrium in Section 5.2.1. We also show how to derive prices and the total revenue from vector x. Then we study the revenue maximization problem in linear version in Section 5.2.2. We prove that we can approximate the maximum revenue 17
- 18. by solving the Rectangular Covering Problem for a specific curve. This curve is obtained 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 i if and only if i ≤ k′ (just remove the days with no purchase to the end). So we can assume that for all i < j ≤ k′ , we have Xi < Xj . The utility of a buyer b for buying on day i is Ib + Cb F (Xi ) − pi . In order to concentrate on network externalities, for now we restrict the model and assume that Ib is always equal to a fixed 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 ˆ be equal to (pi − pj )/(F (Xi ) − F (Xj )), which is the slope of the line between points qi and qj . Also let s(i) = s(i, i + 1). In Lemma 13, we prove that s is non- ˆ decreasing in i. Lemma 14 then shows that the utility of buyer b will be maximized on day i if and only if Cb ∈ [s(i−1), s(i)]. Finally, we use these lemmas in Lemma 15 to show how to find a price vector given x. We use these properties in the next section 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 −p that we know Xi > Xj , and therefore F (Xi ) > F (Xj )):Cb ≥ F (Xi )−Fj(Xj ) . The converse is also true. If Cb is less than (pi − pj )/(F (Xi ) − F (Xj )), then day j will 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 who chooses 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). 2 Lemma 14 If Cb ∈ [s(i − 1), s(i)] then b will have the maximum utility buying on day i. Proof : Assume that Cb ∈ [s(i − 1), s(i)]. For each j > i, the utility of day i is at least as good as that of day j, because Cb ≤ s(i) ≤ s(i, j). Similarly for ˆ each 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), ∞) ˆ are dealt with the same arguments. 2 This lemma enables us to find the key relations between prices and vector x. 18
- 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 ’s lie 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))} have identical differences of consecutive terms. They also have identical initial elements X1 = 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 (X we 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 in the revenue-maximizing equilibrium, the first price p1 will be equal to I in Lemma 16. 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 Lemma 18, we prove that the revenue maximization problem can be reduced to Rectangu- lar Covering Problem, and therefore there exists an FPTAS to solve the revenue maximization 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 all elements of p by I − p1 to get p′ . It’s easy to check that x, p′ still define an equilibrium. The new equilibrium’s revenue is greater than the original one. Hence p1 = I in the revenue-maximizing equilibrium. 2 Note that by calculating all Xi with respect to vector x, we know the values pi − pi−1 using Lemma 15. On the other hand, p1 = I in the revenue-maximizing equilibrium. So we would know all pi ’s. So x uniquely determines prices. Let Price(x) be price vector which has been determined by vector x. It is easy to verify that vectors x and Price(x) make an equilibrium. Hence it suffices to view everything as functions of the free variable x. Now we express the revenue in a candidate equilibrium in terms of vector x. Lemma 17 If x and p correspond to the revenue-maximizing equilibrium, the total revenue 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. Proof : Since the utility of buying on the first day is nonnegative for everybody, all buyers 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=i To interpret the above formula, note that the price change from day i − 1 to day i is exerted on all buyers who have bought on day i or later. Since k xj = 1, j=1 we can rewrite the above formula as R(p, x) = p1 + k (pi − pi−1 )(1 − Xi ). i=2 Now we can substitute for p1 and pi − pi−1 from Lemmas 16 and 15 and write revenue as a function of vector x. Therefore we haveR(p, x) = I + k (1 − i=2 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 Rectangular Covering Problem. Note that I is a constant in Lemma 17 which does not affect revenue maximization. Lemma 18 The problem of maximizing k (1−Xi )C −1 (Xi )(F (Xi )−F (Xi−1 )) i=2 can be reduced to the Rectangular Covering Problem. Proof : Since F is a monotone and hence bijective function, we can write the revenue as k R= (1 − F −1 (F (Xi )))C −1 (F −1 (F (Xi ))(F (Xi ) − F (Xi−1 )) i=2 The 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 define H : [Fmin , Fmax ] → R≥0 as follows : H(x) = (1 − F −1 (x))C −1 (F −1 (x)). Note that the revenue is exactly R = k H(F (Xi ))(F (Xi ) − F (Xi−1 )). Therefore i=2 we want to find values ti = F (Xi )’s in order to maximize R. Revenue R can be shown 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 Covering Problem. If we define H ′ (x) = H(−x) and t′ = −tk−i then we can write the i revenue 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 j So we should solve Rectangular Covering Problem for function H ′ . 2 We have proved that the revenue maximization problem can be solved using Rectangular Covering Problem. Therefore, there exists an FPTAS to solve this problem (Theorem 22 in Section 6.2). 20
- 21. H H(ti ) ti−1 ti Figure 3: Revenue with respect to non-monotone function H . 6 Rectangular Covering Problem Definition 5 Rectangular Covering Problem: A function F : [0, 1] → R is given. We want to find k indices x1 , x2 , · · · , xk in order to maximize the goal function 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 the curve F , and maximizing the discounted covering area (see Figure 2). We propose an FPTAS algorithm for this problem for concave function F in Section 6.1 and for bounded slope function F in Section 6.2. Algorithm described in Lemma 19 can be used to solve the rectangular covering problem for general function F with F (0) > 0. The running-time of the algorithm is poly(k, log1+ǫ F (1)/F (0)) in this case. In the remaining part of paper we assume that F is non-decreasing and F (1) = 1. We prove that these assumptions do not hurt the generality in Section 6.3. Now we define some restricted versions of the rectangular covering problem which are useful in the paper. Definition 6 δ-Rectangular Covering Problem: it is an instance of rectangular covering 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 F In this section we propose an FPTAS algorithm for the rectangular covering prob- lem when the function F is concave. 21
- 22. F (x) F (oi )γ i F (xi )γ i xi oi x Figure 4: The Region with Area A Lemma 19 For every ǫ > 0 and δ ≥ 0, the δ-rectangular covering problem can be 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 that if we choose indices x1 , x2 , · · · , xk from sequence S, we could approximate the optimum. Let si = F −1 ((1 + ǫ)i−1 δ). In fact F (si+1 ) = (1 + ǫ)F (si ) for every i < 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=0 solution. 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 assume x0 = 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 less than or equal to k (xi+1 − xi )F (xi )γ i . So if we choose indices x1 , x2 , · · · , xk i=0 from sequence S we can approximate the optimum with factor 1 + ǫ. Now we design a dynamic programming algorithm to find indices x1 , x2 , · · · , xk from S. Let A[n, r] is equal to the best solution when we want to select r indices from 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. Now we are ready to propose an FPTAS algorithm for the rectangular covering problem with concave function F . Theorem 20 For every ǫ > 0, the rectangular covering problem with concave function 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 at least F (1)/4. Let the optimum for the δ-rectangular covering problem be OP Tδ and the solution found by Lemma 19 be Aδ . We have proved that Aδ ≥ OP T′δ . 1+ǫ On the other hand, the optimum solution for the rectangular covering problem is at most 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) ). 2 6.2 Function F With Bounded Slope In this section we propose an FPTAS algorithm to solve the rectangular covering problem for function F when γ = 1 and F ′ (x) ≤ L. Lemma 21 For every δ1 , δ2 , ǫ > 0, the (δ2 , δ1 )-Rectangular Covering Problem can be solved in poly(k, log1+ǫ ( 1−F −1 (δ2 ) ), log1+ǫ ( δ1−δ22 )) time, with approxi- 1 1 −δ mation factor 1 + ǫ Proof : At first we define a sequence S = (s1 , s2 , · · · , sm ) and then we prove that if 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 been constructed 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. 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=1 optimum 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 not less 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 equations assume 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=0 And 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. For every i < k′ , we have (1 − oi ) ≤ (1 + ǫ)(1 − xi ). On the other hand for every i ≥ k′ , we have F (oi ) − δ2 ≤ (1 + ǫ)(F (xi ) − δ2 ). Because F (ok′ −1 ) ≤ δ2 we have F (oi ) − F (ok′ −1 ) ≤ (1 + ǫ)(F (xi ) − F (ok′ −1 )), for every i ≥ k′ . With respect 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 less than or equal to k (xi+1 − xi )F (xi ). So if we choose indices x1 , x2 , · · · , xk i=1 from 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. Let A[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 goal OP T . Construct k2 + 1 instances of (δ2 , δ1 )-Rectangular Covering Problem from Rectangular Covering Problem instance. In the i-th instance we set δ2 = ki−1 and 2 +1 δ1 = k2i and assume the value of goal function in the optimum solution of this +1 instance 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 such that every oi is out of interval ( kj−1 , k2j ). Therefore we have OP T = OP Tj . 2 +1 +1 Assume we solve all of the k2 + 1 instances of (δ2 , δ1 )-Rectangular Covering Problem 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 Ai as the output for the Rectangular Covering Problem instance, we can approx- imate the optimum with factor 1 + ǫ. Now we prove that this algorithm run in polynomial time. In order to prove this fact we should prove that if we set δ2 = 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−δ2 mial. 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. clude that 1 − F −1 (δ2 ) ≥ L(k1+ 1) . So 1−F −1 (δ2 ) is at most L(k2 + 1) which is 1 polynomial 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) ≤ L can be solved in poly(k, log1+ǫ k, log1+ǫ L) time, with approximation factor 1 + ǫ. 6.3 Assumptions about function F In this section, we prove that some restrictions can be assumed about function F in 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. 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 Conclusion In this paper, we studied the problem of item pricing in the presence of historical network externalities and strategic buyers. We provided necessary and sufficient conditions on the valuation functions for the existence and the uniqueness of the equilibria. We defined the revenue maximization problem in settings in which the equilibrium exists and is unique. We provided nearly optimal algorithms for the two special cases of linear and symmetric valuation functions. The problem of providing algorithms or proving inapproximability for the more general aggregate model 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 another interesting 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 of duopoly 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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