Prophet Secretary for Combinatorial Auctions and Matroids

The secretary and the prophet inequality problems are central to the field of Stopping Theory. Recently, there has been a lot of work in generalizing these models to multiple items because of their applications in mechanism design. The most important of these generalizations are to matroids and to combinatorial auctions (extends bipartite matching). Kleinberg-Weinberg KW-STOC12 and Feldman et al. feldman2015combinatorial show that for adversarial arrival order of random variables the optimal prophet inequalities give a 1/2-approximation. For many settings, however, it's conceivable that the arrival order is chosen uniformly at random, akin to the secretary problem. For such a random arrival model, we improve upon the 1/2-approximation and obtain (1-1/e)-approximation prophet inequalities for both matroids and combinatorial auctions. This also gives improvements to the results of Yan yan2011mechanism and Esfandiari et al. esfandiari2015prophet who worked in the special cases where we can fully control the arrival order or when there is only a single item. Our techniques are threshold based. We convert our discrete problem into a continuous setting and then give a generic template on how to dynamically adjust these thresholds to lower bound the expected total welfare.

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1 Introduction

Suppose there is a sequence of buyers arriving with different values to your single item. On arrival a buyer offers a take-it-or-leave-it value for your item. How should you decide which buyer to assign the item to in order to maximize the value. There are two popular models in the field of Stopping Theory to study this problem: the secretary and the prophet inequality models. In the secretary model we assume no prior knowledge about the buyer values but the buyers arrive in a uniformly random order [Dyn63]. Meanwhile, in the prophet inequality model we assume stochastic knowledge about the buyer values but the arrival order of the buyers is chosen by an adversary [KS78, KS77]

. Since the two models complement each other, both have been widely studied in the fields of mechanism design and combinatorial optimization (see related work).

These models assume that either the buyer values or the buyer arrival order is chosen by an adversary. In practice, however, it is often conceivable that there is no adversary acting against you. Can we design better strategies in such settings? The prophet secretary model introduced in [EHLM17] is a natural way to consider such a process where we assume both a stochastic knowledge about buyer values and that the buyers arrive in a uniformly random order. The goal is to design a strategy that maximizes expected accepted value, where the expectation is over the random arrival order, the stochastic buyer values, and also any internal randomness of the strategy.

In this paper, we consider generalizations of the above problem to combinatorial settings. Suppose the buyers correspond to elements of a matroid222A matroid consists of a ground set and a non-empty downward-closed set system satisfying the matroid exchange axiom: for all pairs of sets such that , there exists an element such that . Elements of are called independent sets. and we are allowed to accept any independent set in this matroid rather than only a single buyer. The buyers again arrive and offer take-it-or-leave-it value for being accepted. In the prophet inequality model, a surprising result of Kleinberg-Weinberg [KW12] gives a -approximation strategy to this problem, i.e., the value of their strategy, in expectation, is at least half of the value of the expected offline optimum that selects the best set of buyers in hindsight. Simple examples show that for adversarial arrival one cannot improve this factor. On the other hand, if we are also allowed to control the arrival order of the buyers, Yan [Yan11] gives a -approximation strategy. But what if the arrival order is neither adversarial and nor in your control. In particular, can we beat the -approximation for a uniformly random arrival order?

Matroid Prophet Secretary Problem (MPS): Given a matroid on

buyers (elements) and independent probability distributions on their values, suppose the outcome buyer values are revealed in a uniformly random order. Whenever a buyer value is revealed, the problem is to immediately and irrevocably decide whether to

select the buyer. The goal is to maximize the sum of values of the selected buyers, while ensuring that they are always feasible in .

Besides being a natural problem that relates two important Stopping Theory models, MPS is also interesting because of its applications in mechanism design. Often while designing mechanisms, we have to balance between maximizing revenue/welfare and the simplicity of the mechanism. While there exist optimal mechanisms such as VCG or Myerson’s mechanism, they are impractical in real markets [AM06, Rot07]. On the other hand, simple Sequentially Posted Pricing mechanisms, where we offer take-it-or-leave-it prices to buyers, are known to give good approximations to optimal mechanisms. This is because the problem gets reduced to designing a prophet inequality [CHMS10, Yan11, Ala14, KW12, FGL15].

Esfandiari et al. [EHLM17] study MPS in the special case of a rank  matroid and give a -approximation algorithm. For general matroids, as in the original models of [CHMS10, Yan11, KW12], it was unclear prior to the work of this paper whether beating the factor of is possible. In Section 4 we prove the following result.

Theorem 1.

There exists a -approximation algorithm to MPS.

Note that the approximation in this theorem as well as the following ones compare to the expected optimal offline solution for the particular outcomes of the distributions. That is, in the case of matroids, we have , where is the value of buyer 333It’s not known if is tight for MPS. In fact, it’s even open if one can beat for a single item [AEE17]..

Next, let us consider a combinatorial auctions setting. Suppose there are buyers that take combinatorial valuations (say, submodular) for indivisible items from independent probability distributions. The problem is to decide how to allocate the items to the buyers, while trying to maximize the welfare—the sum of valuations of all the buyers. Feldman et al. [FGL15] show that for XOS444A function is an XOS function if there exists a collection of additive functions such that for every we have . (a generalization of submodular) valuations there exist static prices for items that gets a -approximation for buyers arriving in an adversarial order. Since this factor cannot be improved for adversarial arrival, this leaves an important open question if we can design better algorithms when the arrival order can be controlled. Or ideally, we want to beat even when the arrival order cannot be controlled but is chosen uniformly at random.

Combinatorial Auctions Prophet Secretary Problem (CAPS): Suppose buyers take XOS valuations for items from independent probability distributions. The outcome buyer valuations are revealed in a uniformly random order. Whenever a buyer valuations is revealed, the problem is to immediately and irrevocably assign a subset of the remaining items to the buyer. The goal is maximize the sum of the valuations of all the buyers for their assigned subset of items.

In Section 3.2 we improve the online approximation result of [FGL15] for random order.

Theorem 2.

There exists a -approximation algorithm to CAPS.

Given access to demand and XOS oracles for stochastic utilities of different buyers, the algorithm in Theorem 2 can be made efficient. This is interesting because it matches the best possible -approximation for XOS-welfare maximization in the offline setting [DNS10, Fei09].

A desirable property in the design of an economically viable mechanism is incentive-compatibility. In particular, a buyer is more likely to make decisions about their allocations based on their own personal incentives rather than to accept a given allocation that might optimize the social welfare but not the individuals’ profit. For the important case of unit-demand buyers (aka bipartite matching), in Section 3.1 we extend Theorem 2 to additionally obtain this property.

Theorem 3.

For bipartite matchings, when buyers arrive in a uniformly random order, there exists an incentive-compatible mechanism based on dynamic prices that gives a -approximation to the optimal welfare.

For this result, we require unit-demand buyers. This is because for general XOS functions shifting buyers to earlier arrivals can change the availability of items arbitrarily. For unit-demand functions, we show that this effect is bounded.

Finally, in Section 5 we conclude by showing that for the single-item case one can obtain a -approximation even by using static prices, and that nothing better is possible.

1.1 Our Techniques

In this section we discuss our three main ideas for a combinatorial auction. In this setting, our algorithm is threshold based, which means that we set dynamic prices to the items and allow a buyer to purchase a set of items only if her value is more than the price of that set. This allows us to view total value as the sum of utility of the buyers and the total generated revenue. Although powerful, dynamic prices often lead to involved calculations and become difficult to analyze beyond a single item setting [EHLM17, AEE17]. To overcome this issue, we convert our discrete problem into a continuous setting. This is possible because a random permutation of buyers can be viewed as each buyer arriving at a time chosen uniformly at random between and . The benefit of such a transformation is that the arrival times are independent, which keeps correlations managable. Besides, it allow us to use tools from integral calculus such as integration by parts.

Our algorithm for combinatorial auctions sets a base price for every item based on its contribution to the expected offline optimum . Our approach is to define two time varying continuous functions: discount and residual. The discount function is chosen such that the price of an unsold item at time is exactly . We define a residual function that intuitively denotes the expected value remaining in the instance at time . Hence, and . Computing is difficult for a combinatorial auction since it depends on several random variables. However, assuming that we know , we use application specific techniques to compute lower bounds on both the expected revenue and the expected utility in terms of the functions and .

Finally, although we do not know , we can choose the function in a way that allows us to simplify the sum of expected revenue and utility, without ever computing explicitly. This step exploits properties of the exponential function for integration (see Lemma 6).

1.2 Related Work

Starting with the works of Krengel-Sucheston [KS78, KS77] and Dynkin [Dyn63], there has been a long line of research on both prophet inequalities and secretary problems. One of the first generalizations is the multiple-choice prophet inequalities [Ken87, K85, Ker86] in which we are allowed to pick items and the goal is to maximize their sum. Alaei [Ala14] gives an almost tight ()-approximation algorithm for this problem (the lower bound is due to [HKS07]). Similarly, the multiple-choice secretary problem was first studied by Hajiaghayi et al. [HKP04], and Kleinberg [Kle05] gives a -approximation algorithm.

The research investigating the relation between prophet inequalities and online auctions is initiated in [HKS07, CHMS10]. This lead to several interesting follow up works for matroids [Yan11, KW12] and matchings [AHL12]. Meanwhile, the connection between secretary problems and online auctions is first explored in Hajiaghayi et al. [HKP04]. Its generalization to matroids is considered in [BIK07, Lac14, FSZ15] and to matchings in [GM08, KP09, MY11, KMT11, KRTV13, GS17].

Secretary problems and prophet inequalities have also been studied beyond a matroid/matching. For the intersection of matroids, Kleinberg and Weinberg [KW12] give an -approximation prophet inequality. Dütting and Kleinberg [DK15] extend this result to polymatroids. Feldman et al. [FGL15] study the generalizations to combinatorial auctions. Later, Dütting et al. [DFKL17] give a general framework to prove such prophet inequalities. Submodular variants of the secretary problem have been considered in [BHZ10, GRST10, FZ15, KMZ15]. Prophet and secretary problems have also been studied for many classical combinatorial problems (see e.g., [Mey01, GGLS08, GHK14, DEH15, DEH17]). Rubinstein [Rub16] and Rubinstein-Singla [RS17] consider these problems for arbitrary downward-closed constraints.

In the prophet secretary model, Esfandiari et al. [EHLM17] give a -approximation in the special case of a single item. Going beyond has been challenging. Only recently, Abolhasani et al. [AEE17] and Correa et al. [CFH17] improve this factor for the single item i.i.d. setting. Extending this result to non-identical items or to matroids are interesting open problems.

2 Our Approach using a Residual

In this section, we define a residual and discuss how it can be used to design an approximation algorithm for a prophet secretary problem. Suppose there are requests that arrive at times

drawn i.i.d. from the uniform distribution in

. These requests correspond to buyers of a combinatorial auction or to elements of a matroid.

Whenever a request arrives, we have to decide if and how to serve it. Depending on how we serve request , say , we gain a certain value . Our task is to maximize the sum of values over all requests . Our algorithm Alg includes a time-dependent payment component. The payment that request has to make is the product of a time-dependent discount function and a base price . The base price depends on the allocation up to this point and how much the new choice limits other allocations in the future. However, it does not depend on , the time that has passed up to this point. If request has to pay for our decision , then its utility is given by . We write for the sum of utilities and for the sum of payments. The value achieved by Alg equals .

Next we define a residual function that has the interpretation of “expected remaining value in the instance at time ”. In Lemma 6 we show that the existence of a residual function for Alg suffices to give a -approximation prophet secretary.

Definition 4 (Residual).

Consider a prophet secretary problem with expected offline value . For any algorithm Alg based on a differentiable discount function , a differentiable function is called a residual if it satisfies the following three conditions for every choice of .

(1a)
(1b)
(1c)

We would like to remark here that this definition is similar in spirit to balanced thresholds [KW12] and balanced prices [DFKL17]. However, it is different because we have to take into account the random arrivals.

As an illustration of Definition 4, consider the case of a single item. That is, we are presented a sequence of real numbers and may select only up to one of them (previously studied in [EHLM17]).

Example 5 (Single Item).

Suppose buyer arrives with random value at time chosen uniformly at random between and . Define as the base price of the single item. A buyer arriving at time is offered the item at price , and she accepts the offer if and only if . We show that is a residual function.

By definition, (1a) holds trivially. To see that (1b) holds, observe that the increase in revenue from time to time is approximately if the item is allocated during this time, and is otherwise. That is, the expected increase in revenue is approximately . Taking the limit for then implies (1b), i.e., .

For (1c), we consider the expected utility of a buyer conditioning on her arriving at time

Here we use that the event the item is sold before does not depend on because buyer only arrives at time . The expectation in turn only depends on . It is also important to observe that . Next, we take the sum over all buyers and use that to get

This implies

We now use the properties of a residual function to design a -approximation algorithm. To this end, we choose in a manner that makes the sum of the expected revenue and buyers’ utilities independent of . This allows us to compute expected welfare, even though we cannot compute directly.

Lemma 6.

For a prophet secretary problem, if there exists a residual function for algorithm Alg as defined in Definition 4, then setting gives a -approximation.

Proof.

To further simplify Eq. (1b), we observe that applying integration by parts gives

So in combination

(2)

Now adding (2) and (1c) gives,

Although we do not know and computing seems difficult, we have the liberty of selecting the function . By choosing satisfying for all , this integral becomes independent of and simplifies to . In particular, let . This gives,

3 Prophet Secretary for Combinatorial Auctions

Let denote a set of buyers and denote the set of indivisible items. Suppose buyer arrives at a time chosen uniformly at random between and . Let (similarly denote the random combinatorial valuation function of buyer . In order to ensure polynomial running times, we assume that the distribution of has a polynomial support , where . Note that this assumption only simplifies notation. If we only have sample access to the distributions, then we can replace by an appropriate number of samples. Within our proofs, we will use to denote an independent, fresh sample from the distribution.

By T and v (similarly

) we denote the vector of all the buyer arrival times and valuations, respectively. Also, let

(similarly ) denote valuations of all buyers except buyer . For the special case of single items, we let denote . Let denote the probability that item has not been sold before time , where the probability is over valuations v, arrival times T, and any randomness of the algorithm.

3.1 Bipartite Matching

In the bipartite matching setting all buyers are unit-demand, i.e. . We can therefore assume that no buyer buys more than one item. We restate our result.

See 3

To define prices of items, let base price denote the expected value of the buyer that buys item in the offline welfare maximizing allocation (maximum weight matching). Now consider an algorithm that prices item at at time and allows the incoming buyer to pick any of the unsold items; here is a continuous differentiable discount function.

Consider the function . Clearly, . Using the following Lemma 7 and Claim 8, we prove that is a residual function for our algorithm. Since the algorithm is clearly incentive-compatible, Lemma 6 implies Theorem 3.

Lemma 7.

We can lower bound the total expected utility by

(3)
Proof.

Since buyer arriving at time can pick any of the unsold items, we have

One particular choice of buyer is to choose item if it is still available, and no item otherwise. This gives us a lower bound of

Note that in the product, the fact whether is sold before only depends on and the arrival times of the other buyers. It does not depend on or . The remaining terms, in contrast, only depend on and . Therefore, we can use independence to split up the expectation and get

Next, we use that by Lemma 9 and that and are identically distributed. Therefore, we can swap their roles inside the expectation. Overall, this gives us

(4)

Next, observe that by the definition of . Therefore, using linearity of expectation, summing up (4) over all buyers gives us

Now, taking the expectation over , we get

We next give a bound on the revenue generated by our algorithm.

Claim 8.

We can bound the total expected revenue by

(5)
Proof.

Since is the probability that item is bought between and (note is decreasing in ), we have

Finally, we prove the missing lemma that removes the conditioning on the arrival time.

Lemma 9.

We have

The idea is that if buyers arrive earlier in the process, this only reduces the available items. It can never happen that such a change makes an item available at a later point. For a single item this is trivial, for multiple items and other combinatorial valuations it does not necessarily hold.

Proof.

Consider the execution of our algorithm on two sequences that only differ in the arrival time of buyer . To this end, let v be arbitrary values and T be arbitrary arrival times. Let be the set of items that are sold before time on the sequence defined by v and T. Furthermore, let be the set of items sold before time if we replace by . Ties are broken in the same way in both sequences.

We claim that for all .

To this end, we observe that by definition for because the two sequences are identical before . This already shows the claim for . Otherwise, assume that there is some for which . Let be the infimum among these . It has to hold that some buyer arrives at time and buys item in the original sequence and in the modified sequence. Furthermore, we now have to have because was defined to be the infimum of all for which is not fulfilled. Therefore, . Additionally, . The reason is that for any before the next arrival .

Overall this means that in both sequences at time buyer has the choice between and . As his values are identical and ties are broken the same way, it has to hold that , which then contradicts that .

Taking the expectation over both v and T, we get

This implies the Lemma 9 because

3.2 XOS Combinatorial Auctions

In this section we prove our main result (restated below) for combinatorial auctions. See 2

Recollect that the random valuation of every buyer has a polynomial support. We can therefore write the following expectation-version of the configuration LP, which gives us an upper bound on the expected offline social welfare.

s.t. for all
for all

The above configuration LP can be solved with a polynomial number of calls to demand oracles of buyer valuations (see [DNS10]). Since all functions are XOS, there exist additive supporting valuations; that is, there exist numbers s.t. for , , and for all . Before describing our algorithm, we define a base price for every item.

Definition 10.

The base price of every item is .

Since , consider an algorithm that on arrival of buyer with valuation draws an independent random set with probability . Let denote this drawn set. This distribution also satisfies that for every item ,

(6)

Now consider the supporting additive valuation for in the XOS valuation function of buyer . This can be found using the XOS oracle for  [DNS10]. Our algorithm assigns her every item that has not been allocated so far and for which , where is a continuous differentiable function of . Note that since we do not allow buyer to choose items outside set , the mechanism defined by this algorithm need not be incentive compatible.

Consider the function , where again denotes the probability that item has not been sold before time . Clearly, . Using the following Lemma 11 and Claim 12, we prove that is a residual function for our algorithm. Hence, Lemma 6 implies Theorem 2.

Lemma 11.

The expected utility of the above algorithm is lower bounded by

(7)
Proof.

Given that buyer arrives at and only buys item if , her utility is

Using the fact that whether is sold before only depends on and T, and not on or ,

Now, observe that in our algorithm every buyer independently decides which set of items it will attempt to buy. Crucially, the probability of an item being sold by time can only increase if more buyers arrive before . Therefore,

Thus, we get

Finally, recollect from Eq. (6) that . Moreover,

Hence, by linearity of expectation

We next give a bound on the revenue generated by our algorithm.

Claim 12.

We can bound the total expected revenue by

(8)
Proof.

Since is the probability that item is bought between and (note is decreasing in ), we have

4 Prophet Secretary for Matroids

Let denote the random value of the ’th buyer (element) and let denote another independent draw from the value distribution of the ’th buyer. The problem is to select a subset of the buyers that form a feasible set in matroid , while trying to maximize . We restate our main result for the matroid setting. See 1

We need the following notation to describe our algorithm.

Definition 13.

For a given vector of values of items and , we define the following:

  • Let denote the optimal solution set in the contracted matroid .

  • Let denote the remaining value after selecting set .

We next define a base price of for every buyer .

Definition 14.

Let denote the independent set of buyers that have been accepted till now.

  • Let denote a threshold for buyer .

  • Let denote the base price for buyer .

Starting with , let denote the set of accepted buyers before time . This is a random variable that depends on the values v and arrival times T. Suppose a buyer arrives at time , then our algorithm selects iff both and selecting is feasible in .

Consider the function , where is a function of v and T. Clearly, . Using the following Lemma 16 and Claim 15, we prove that is a residual function. Hence, Lemma 6 implies Theorem 1.

Claim 15.
Proof.

Consider the time from to for some , . Let us fix the arrival times T and values v of all elements. This also fixes the sets . Let be the arrivals between and that get accepted in this order. Note that it is also possible that . The revenue obtained between and is now given as

Taking the expectation over v and T, we get by linearity of expectation

By the same argument, we also have

In combination, we get that

which implies the claim. ∎

Lemma 16.
Proof.

The utility of buyer arriving at time is given by

Observe that does not depend on if because it includes only the acceptances before . It does not depend on either, as is only used for analysis purposes and not known to the algorithm. Since and are identically distributed, we can also write

(9)

Now observe that buyer can belong to only if it’s not already in , which implies . Using this and removing non-negativity, we get