Optimal Budget-Feasible Mechanisms for Additive Valuations

02/12/2019 ∙ by Nick Gravin, et al. ∙ Shanghai University of Finance & Economics Northwestern University The Hong Kong University of Science and Technology 0

In this paper, we obtain tight approximation guarantee for budget-feasible mechanisms with an additive buyer. We propose a new simple randomized mechanism with approximation ratio of 2, improving the previous best known result of 3. Our bound is tight with respect to either the optimal offline benchmark, or its fractional relaxation. We also present a simple deterministic mechanism with the tight approximation guarantee of 3 against the fractional optimum, improving the best known result of √(2) + 2 for the weaker integral benchmark.

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1 Introduction

In a typical procurement setting, a buyer wants to purchase items from a set of agents. Each agent can supply an item (or provide a service) at an incurred cost of to himself, and the buyer wants to optimize his valuation for the set of acquired items taking into account the costs of items. Since agents may strategically report their costs, the setting is usually considered as a truthful mechanism design problem.

These problems have been extensively studied by the AGT community. The earlier work analyzed the case where the buyer’s valuation takes - values (see, e.g., [AT02]) in the frugality framework, with the objective of payment minimization. A more recent line of work on the budget-feasible mechanism design (see, e.g., [Sin10]) studies more general valuation functions with a budget constraint on the buyer’s total payment. Our work belongs to the latter category.

The research in the budget-feasible framework focuses on different classes of complement-free valuations (ranging from the class of additive valuations to the most general class of subadditive valuations), and has many applications such as procurement in crowdsourcing markets [SM13], experimental design [HIM14], advertising in social networks [Sin12]. For example, consider an online crowdsourcing platform (like Amazon’s Mechanical Turk), where a buyer may hire multiple workers to perform certain tasks like image labeling, text translations, or writing consumer surveys. The central problem for these online labor markets is to properly price each task, since the buyer is usually budget constrained, and would run out of budget once he pays too high to the hired workers.

This setting corresponds to the most basic additive valuation of the buyer, which is the topic of our paper. That is, we assume that every hired worker generates value to the buyer, whose value from all hired workers equals to . Without any incentive constraints, this naturally defines the Knapsack optimization problem:

Find workers :   , subject to .

In the budget-feasible framework, the goal is to design truthful direct-revelation mechanism that decides (1) which workers to select, and (2) how much to pay them under the budget constraint. A mechanism is evaluated against the benchmark of the optimal solution to the Knapsack problem.

For the above problem with the additive buyer, Singer [Sin10] gave the first -approximation mechanism. Later, the result was improved by Chen et al. [CGL11] with a -approximation deterministic mechanism and a -approximation randomized mechanism, which still remain the best known upper bounds to the problem for nearly a decade. The best known lower bounds are for the deterministic and for the randomized mechanisms [CGL11]111There are gaps for both deterministic and randomized mechanisms. As these two intervals intersect, it is not even clear whether the best randomized mechanism is necessarily better than the deterministic one.. On the other hand, Anari et al. [AGN14] studied an important special case of large markets, i.e., the setting where each worker has vanishingly small cost (compared to the buyer’s budget), and acquired tight bound of .

Fractional Knapsack.

Interestingly, all previous work on budget-feasible mechanisms for the additive buyer actually obtains results against the stronger benchmark of the optimal solution to Fractional Knapsack222However, the lower bounds apply to the Knapsack benchmark instead of the Fractional Knapsack benchmark., i.e., the fractional relaxation of the Knapsack problem. Indeed, Knapsack is a well-known NP-hard problem, while its fractional relaxation admits efficient solution by a simple greedy algorithm, and generally has much better behavior than the integral optimum. We also compare performance of our mechanisms to the Fractional Knapsack benchmark.

Our Results.

We propose two natural mechanisms that both achieve tight guarantees against the Fractional Knapsack benchmark. Namely, we show -approximation guarantee for the deterministic and -approximation guarantee for the randomized mechanisms. Given the matching lower bound of even against the weaker Knapsack benchmark, the guarantee from our randomized mechanism is also tight against the standard benchmark. Our results establish clear separation between the powers of randomized and deterministic mechanisms: no deterministic mechanism has approximation guarantee better than , while our randomized mechanism already achieves -approximation.

We propose a new natural design principle of two-stage mechanisms: in the first stage, we greedily exclude the items with low value-per-cost ratios333This is essentially the main approach used in the previous work, had we continued until the set of the remaining items becomes budget-feasible.; in the second stage, we use the simple posted-pricing schemes, based on the values of the remaining items. Both of our randomized and deterministic mechanisms share the first stage, which stops earlier than its analogues from the previous work. A remarkable property of the first stage, which we call pruning (similar to the pruning approach in the frugality literature [CEGP10, KSM10]), is that it can be composed (in the sense of [AH06]) with any truthful follow-up mechanism that runs on the items left to the second stage. The difference between our randomized and deterministic mechanisms lies in the follow-up posted-pricing schemes: the randomized mechanism uses non-adaptive posted prices with the total sum below the budget, while our deterministic mechanism employees adaptive pricing that depends on whether previous agents accepted or rejected their posted-pricing offers.

1.1 Related Work

A complementary concept of budget-feasible mechanism design is frugality with the objective of payment minimization under feasibility constraint on the set of winning agents. In that framework, there is a rich literature studying different systems of feasible sets, including matroid set systems [KKT05], path and -paths auctions [AT02, Tal03, ESS04, CK07, CEGP10], vertex cover and -vertex cover [EGG07, KSM10, HKS18].

The framework of budget-feasible mechanism design was proposed by Singer [Sin10]. Besides additive valuations, other more general classes of complement-free valuations also have been studied in the literature: submodular, fractionally subadditive, and subadditive valuations. Singer provided an -approximation mechanism for submodular valuations [Sin10]. This bound was improved to and for randomized and deterministic mechanisms by Chen et al. in [CGL11], and afterward it was improved to and by Jalaly and Tardos in [KT18], respectively. For fractionally subadditive valuations, Bei et al. [BCGL17] gave a -approximation randomized mechanism. For the most general class of subadditive valuations, Dobzinski et al. [DPS11] first designed -approximation randomized and -approximation deterministic mechanisms. Later, Bei et al. [BCGL17] proved the existence of -approximation budget-feasible mechanism for the subadditive buyer; however, any explicit description of such a mechanism is still unknown.

There also have been many interesting and practically motivated adjustments to the original budget feasibility model. In particular, Anari et al. [AGN14] considered the variant with the additional large-market assumption, i.e., that the maximal cost of each agent is small compared to the budget, and attained tight result of for the additive buyer. Badanidiyuru et al. [BKS12] studied the family of online pricing mechanisms in the budget feasibility model, motivated by practical restrictions imposed by existing platforms. Balkanski and Hartline [BH16] obtained improved guarantees in the Bayesian framework. Goel et al. [GNS14] considered more complex scenarios on a crowdsourcing platform, where the buyer hires workers to complete more than one tasks. Balkanski and Singer  [BS15] considered fair mechanisms (rather than incentive compatible mechanisms) in the budget feasibility model.

2 Preliminaries

In the procurement auction, there are items for sale, each held by a single agent with a privately known cost and a publicly known value for the buyer. The buyer has an additive valuation function for purchasing a subset of items. By the revelation principle, we only consider direct-revelation mechanisms. Upon receiving bids of the claimed costs from the agents, a mechanism determines a set of winning agents and the payments to them. Any mechanism can be described by its allocation function and payment function . A truthful (deterministic) mechanism satisfies the following properties, for all bids and costs .

  • (Individual rationality and no positive transfer): if , and if . I.e., each winning agent gets a non-negative utility of , and each loosing agent gets a utility of .

  • (Budget feasibility): , for a given budget .

  • (Truthfulness): bidding the true cost maximizes the utility of each agent .

It is well-known (see [Mye81]) that truthfulness holds if and only if: (1) allocation function is monotone, i.e., decreasing bid unilaterally cannot alter the status of agent

from loosing to winning (in case of a randomized mechanism, the allocation probability

never increases); and (2) payment is the threshold/maximum bid for an agent to keep winning. A universally truthful randomized

mechanism is a probability distribution over truthful deterministic mechanisms.

We denote by alg the value derived from a deterministic mechanism, or the expected value in case of a randomized mechanism. For each agent , we assume without loss of generality that , in that agent can never win when (due to individual rationality and budget feasibility constraints).

If the buyer knows the private costs of the items, he would simply select the subset of items with the maximal total value, under the budget constraint. Let opt denote the optimal solution to such Knapsack problem:

We also consider the fractional relaxation of the problem, and define its optimum as

While opt is NP-hard to calculate, finding fopt is easy: one greedily takes the items in the decreasing order of value-per-cost ratios , until the budget is exhausted or no item is left. Under our assumption that for each item 444Without this assumption, the gap between the two optima can be arbitrary large., .

We say that a mechanism is an -approximation (with respect to the benchmark opt) if the value of alg is always at least -fraction of the value of opt corresponding to the Knapsack problem. In this work, we mostly compare the performance of a mechanism to the stronger benchmark fopt, i.e., the solution to the Fractional Knapsack problem.

3 Composition of Mechanisms: Pruning

Each of our mechanisms consists of two stages, and can be described as a composition of two consecutive mechanisms. The common first stage is called Pruning-Mechanism, and the purpose of this is to exclude the items with low value-per-cost ratios.

Pruning-Mechanism [label = (0), itemindent=-0.5em] Let and While do: [label = (), itemindent=-0.5em] Continuously increase ratio If , then discard555If there are multiple such items, we discard them one by one in lexicographical order; we stop discarding items, whenever Stop-Condition of While-Loop holds. item : Return pair

Pruning-Mechanism possesses a remarkable composability property: the combination of it with any truthful follow-up mechanism running on the output set of Pruning-Mechanism is still a truthful mechanism. More concretely, the composition mechanism of Pruning-Mechanism with a mechanism works as follows:

[label = (0), itemindent=-0.5em] Receive pair from Pruning-Mechanism. Run mechanism on agent set : [label = (), itemindent=-0.5em] Select the winning set from according to . Cap payment to , i.e., , for each winning agent .

Lemma 1.

If a mechanism is individually rational, budget-feasible, and truthful, then so is the composition mechanism .

Proof.

By Step (2b) in Pruning-Mechanism, each item has a value-per-cost ratio at least , which means . Hence, capping a payment to does not break individual rationality. Mechanism is already budget-feasible, and mechanism can only decrease the payment to each agent. We are only left to show that is truthful.

First, we claim that any agent cannot alter the output of Pruning-Mechanism by manipulating his bid, unless agent bids too high to be included in set . Formally, denote by the output under another bid , then whenever . Indeed, agent ’s bid may affect any step in Pruning-Mechanism only when the mechanism is excluding him from the candidate set; given that agent belongs to the output set, his bid cannot change the output.

As mechanism has a monotone allocation rule, so does the composition mechanism . Indeed, if a loosing agent claims a higher cost: (1) the increased bid does not help agent to pass the first stage of Pruning-Mechanism; (2) given that agent always gets selected in set before and after increasing the bid, he would always loose in truthful mechanism , in that the pair remains the same.

The payment of is exactly the threshold bid for an agent to keep winning: the bid of is the threshold bid to pass the first stage of Pruning-Mechanism; the bid of is the threshold bid to win in the second stage of mechanism . ∎

We show now several useful properties of the output of Pruning-Mechanism.

Lemma 2.

Let denote the highest-value item666When there are multiple highest-value items, we break ties lexicographically., and let . Then,

  1. [label=()]

  2. for each item .

  3. .

  4. .

Proof.

In (a): the first inequality is due to Step (2b); the second inequality holds, since ratio is initialized to be , and never decreases during While-Loop.

To prove (b), we observe that the first inequality is a reformulation of Stop-Condition of While-Loop. To prove the second inequality, we note that there are two possibilities that may lead to the termination of While-Loop, and holds in both cases.

  • [Increase of ratio ]. Continuous increase of implies .

  • [Discard of an item ]. Value-per-cost ratio is fixed before and after the discard. Before the discard, in that Stop-Condition has not been invoked,

The second inequality in (c) directly follows from (b). We would verify the first inequality in (c) via case analysis. Let denote the solution to the Fractional Knapsack problem. We have either or . This claim comes as: (1) Pruning-Mechanism removes items in the increasing order of value-per-cost ratios; while (2) the greedy algorithm takes items in the decreasing order of value-per-cost ratios; and (3) in both processes, we break ties lexicographically.

  • [When ]. Observe that , and for each item . Hence, the total value of the items beyond set is .

  • [When ]. Clearly, , and for each item . Then, .

This concludes the proof of all points (a), (b), and (c) in Lemma 2. ∎

Mechanisms in the Second Stage.

Based on Lemma 1, Pruning-Mechanism can be composed with any follow-up truthful mechanism. In fact, we focus on the class of simple posted-pricing mechanisms, with each mechanism parametrized777To obtain our -approximation deterministic mechanism in Section 4, we actually use an adaptive pricing scheme. I.e., the take-it-or-leave price offered to each item may change, depending on whether the items that had made decisions before accepted or rejected their posted prices. by a set of prices subject to . Such a mechanism naturally satisfies individual rationality, no positive transfer, budget feasibility, and truthfulness.

To illustrate how to analyze the approximability of a two-stage pricing scheme, and as a warm-up exercise, we discuss below two simple mechanisms.

Warm-Up.

In our first mechanism, we choose the higher-value subset of either or as the winning set , by offering price to each item or . As a result, we know from Lemma 2 (c) that .

-Approx-Mechanism [label = (0), itemindent=-0.5em] If888 Item will accept the offer , by Lemma 2 (a) that . , get item by offering price Else999 Each item will accept the offer , by Lemma 2 (a) that ., get items by offering price to each item

Our next pricing mechanism recovers the best known ratio of by Chen et al. [CGL11]. We justify its approximability via case analysis.

-Approx-Mechanism [label = (0), itemindent=-0.5em] If , get item by offering price Else, [label = (), itemindent=-0.5em] Get items by offering price to each item Offer price101010 Notice from Lemma 2 (b) that . to item

[Case (1): ]. Item is the only winner, hence . By Lemma 2 (c),

[Case (2): ]. There are two possibilities. First, when , all items in set together form the winning set , i.e., . By Lemma 2 (c), . Second, when , only the items in set are chosen as the winners, i.e., . Consequently,

fopt (by Lemma 2 (c))
(since )
(since )

When and , the last mechanism admits a -approximation. One might ask a natural question: is it possible to achieve a better trade-off between this -approximation case and the -approximation cases? In the next section, we affirm this guess by presenting a slightly more complicated adaptive pricing scheme, resulting in a -approximation deterministic mechanism.

4 Deterministic Mechanism

The warm-up mechanisms have merely a few possible outcomes, and do not adapt to the decisions of the items: either the highest-value item , or the remaining items , or rarely both of item and items win; the posted prices are almost all equal to the maximum possible values . Such rigid structure hinders those pricing mechanisms from achieving better performance guarantees than -approximation. In this section, we introduce an improved adaptive pricing mechanism.

Deterministic-Mechanism [label = (0), itemindent=-0.5em] If , get items by offering price to each item Else if , get item by offering price to Else, i.e., when : [label = (), itemindent=-0.5em] Offer price to item If111111 When , item will accept offer . , offer to each item Else121212 When , item will reject offer , and then each item will accept offer ., get items by offering price to each item

We also call by the same name Deterministic-Mechanism the composition of two mechanisms: Pruning-Mechanism with Deterministic-Mechanism.

Theorem 1.

Deterministic-Mechanism is a -approximation mechanism (individually rational, budget-feasible, and truthful) against the Fractional Knapsack benchmark.

Proof.

Individual rationality, no positive transfer, and truthfulness are due to the pricing nature of Deterministic-mechanism, Lemma 1, and Lemma 2 (a). To show budget feasibility, we consider either Case (3b) or Case (3c) in the mechanism:

  • [Case (3b)]. .

  • [Case (3c)]. Since , we know from Lemma 2 (b) that .

In the reminder of the proof, we focus on the approximation guarantee. Both of Case (1) and Case (2) in the mechanism, in which either or , are easy to analyze. In either case, and thus

where the first inequality comes from Lemma 2 (c), and the second inequality holds since either or .

From now on, we safely assume that . Conditioned on either or , we are only left to deal with Case (3b) and Case (3c).

[Case (3b) that ]. Let be the set of winners in set , then . We have for each item . Note that for each item , since otherwise item would accept price and would actually get in set . Thus, for each item and

(1)

By Lemma 2 (c), . We plug inequality (1) into it and get

(by Lemma 2 (b): )
(since )
(since )

[Case (3c) that ]. By Lemma 2 (a), , and for all item . Because , we have . The winning set , which means . Hence,

fopt (by Lemma 2 (c))
(since )
(by Lemma 2 (b): )
(since )

Thus, holds in all cases, which concludes the proof of Theorem 1. ∎

4.1 Matching Lower Bound

It turns out that our Deterministic-Mechanism reaches the best possible approximation ratio among all deterministic mechanisms, against the Fractional Knapsack benchmark. Here, we give an instance similar in spirit to the example of Singer [Sin10, Proposition 5.2] with a matching lower bound of .

Theorem 2.

No deterministic mechanism (truthful, individually rational and budget-feasible) has an approximation ratio less than against the Fractional Knapsack benchmark, even if there are only three items.

Proof.

For the sake of contradiction, assume that there is a -approximation deterministic mechanism, for some constant . Consider the following two scenarios with three items having values . Let .

  • [With costs ]. Based on individual rationality, each winning item gains a payment of at least . To guarantee the promised approximation ratio of under budget feasibility, there is exactly one winning item; without loss of generality, we assume that the winner is the first item.

  • [With costs ]. By truthfulness, item wins once again, getting the same payment of at least . Thus, the budget left is at most . Due to budget feasibility and individual rationality, neither item nor item can win.

In the later scenario, the mechanism generates value , yet the Fractional Knapsack benchmark achieves value . This contradicts our assumption that the mechanism is -approximation, concluding the proof of the theorem. ∎

5 Main Result: Randomized Mechanism

Next, we present the main result of our work, a randomized mechanism that achieves a -approximation to the Fractional Knapsack benchmark. Given the matching lower bound by Chen et al. [CGL11, Theorem 4.2] against the weaker Knapsack benchmark, this approximation guarantee is tight for both benchmarks. Our mechanism described below and called Randomized-Mechanism receives pair from Pruning-Mechanism as input, and runs in the second stage of the composition. The mechanism simply randomizes among non-adaptive pricing schemes.

Randomized-Mechanism [label = (0), itemindent=-0.5em] Let If , let and Else, let and Offer price131313 For all item , price is well defined in range , by Lemma 2 (a). to item , where is defined as follows: [label = (), itemindent=-0.5em] With probability , let With probability , let With probability , draw Offer price to each item

We first verify that all quantities in Randomized-Mechanism are well defined.

Lemma 3.

and .

Proof.

The first inequality is due to Lemma 2 (b) that . Lemma 2 further implies and , i.e., . Now, the second inequality in Lemma 3 follows, as . Finally, rearranging leads to the last two inequalities. ∎

Similar to Deterministic-Mechanism in Section 4, we also slightly abuse notations and also refer to Randomized-Mechanism as the composition of two mechanisms: Pruning-Mechanism with Randomized-Mechanism.

Theorem 3.

Randomized-Mechanism is a -approximation mechanism (individually rational, budget-feasible, and universally truthful) against the Fractional Knapsack benchmark.

Proof.

Since Randomized-Mechanism is a pricing scheme, it is individually rational; since each random realization of the prices is budget-feasible, i.e., by construction, the mechanism is also budget-feasible. Note that (1) all random choices in Randomized-Mechanism, i.e., the prices , can be made before execution of the mechanism; and (2) for each such choice, the resulting pricing mechanism is obviously truthful. Due to Lemma 1, all desired properties extend to the composition mechanism, hence being individually rational, budget-feasible, and universally truthful.

In the rest of the proof, we show that Randomized-Mechanism is a -approximation to fopt. Let denote the allocation probabilities, then the mechanism generates an expected value of . In order to prove the approximation guarantee, we need the following equation (2), inequality (3), and inequality (4), which will be proved later.

(2)
(3)
(4)

Indeed, these mathematical facts together with Lemma 2 (c) imply that .

fopt

Now, we are only left to prove equation (2), inequality (3) and inequality (4).

[Equality (2)]. By the definitions of and , in either case of Step (2) or Step (3),

(by the definition of )

[Inequality (3)]. It is equivalent to showing that .

  • [When ]. Item always accepts price , i.e., , which gives us the desired bound of , because .

  • [When ]. By Lemma 2 (a), . We consider the random events in Step (4a) that and in Step (4c) that . Since and , putting everything together gives

    (since )
    (by the definition of )
    (since )

[Inequality (4)]. The argument is similar to the above. For each item , we claim that .

  • [When ]. Item always accepts price , i.e., , which gives us the desired bound of , in that .

  • [When ]. By Step (5), if and only if . We consider the random events in Step (4b) that