On the Power of Randomization in Algorithmic Mechanism Design

On the Power of Randomization in Algorithmic Mechanism Design

Abstract

In many settings the power of truthful mechanisms is severely bounded. In this paper we use randomization to overcome this problem. In particular, we construct an FPTAS for multi-unit auctions that is truthful in expectation, whereas there is evidence that no polynomial-time truthful deterministic mechanism provides an approximation ratio better than .

We also show for the first time that truthful in expectation polynomial-time mechanisms are provably stronger than polynomial-time universally truthful mechanisms. Specifically, we show that there is a setting in which: (1) there is a non-polynomial time truthful mechanism that always outputs the optimal solution, and that (2) no universally truthful randomized mechanism can provide an approximation ratio better than in polynomial time, but (3) an FPTAS that is truthful in expectation exists.

1 Introduction

Background and Our Results

The last few years have been quite disappointing for researchers in Algorithmic Mechanism Design. Indeed, much progress has been made, but in the “wrong” direction: several papers proved that the power of polynomial time truthful mechanisms is severely bounded, either comparing to polynomial time non-truthful algorithms, or to non-polynomial time truthful mechanisms [13, 7, 8, 10, 21, 16]. This paper brings some good news: randomization might help in breaking these lower bounds to achieve better approximation guarantees.

Before studying the power of randomization in the context of mechanism design, we recall the three common notions of truthfulness:

  • Deterministic Truthfulness: A bidder always maximizes his utility by bidding truthfully. No randomization is allowed.

  • Universal Truthfulness: A universally-truthful mechanism is a probability distribution over deterministic truthful mechanisms. The mechanism is truthful even when the realization of the random coins is known.

  • Truthfulness in Expectation: A mechanism is truthful in expectation if a bidder always maximizes his expected profit by bidding truthfully. The expectation is taken over the internal random coins of the mechanism.

It is well known that, in some settings (e.g. online), randomized algorithms are strictly more powerful than determinic algorithms. In their seminal paper introducing algorithmic mechanism design, Nisan and Ronen [19] showed that universally truthful mechanisms can be more powerful than their deterministic counterparts. In this paper we show that truthfulness in expectation yields more power than universal truthfulness.

For the well-studied problem of multi-unit auctions (e.g., [17, 4, 12, 20, 14, 8, 3]), we show a polynomial time truthful in expectation mechanism with the best ratio possible from a pure algorithmic point of view:

Theorem: There exists a truthful in expectation FPTAS for multi-unit auctions1.

There are many truthful in expectation mechanisms in the literature [2, 1, 14, 5]. However, somewhat surprisingly, many truthful in expectation mechanisms were followed by universally truthful mechanisms with the same performance [4, 6, 9, 8]. As a result, there are few settings in which the best known truthful in expectation algorithms provide better approximation ratios than the best known universally truthful mechanisms. Furthermore, [15] shows that for digital good auctions the two notions of randomization are equivalent in power. In contrast, our second main result shows that truthful in expectation mechanisms are strictly more powerful than universally truthful ones. We study a variant of multi-unit auctions that we term restricted multi-unit auctions, and show a first-of-a-kind separation result:

Theorem: If is a universally truthful mechanism for restricted multi-unit auctions that achieves an approximation ratio of for some constant , then has exponential communication complexity. However, there exists a truthful in expectation FPTAS for restricted multi-unit auctions.

We note that there exists a deterministic truthful mechanism that optimally solves restricted multi-unit auctions in exponential time.

Multi-unit Auctions

In a multi-unit auction a set of identical items is to be allocated to bidders. Each bidder has a valuation function , where is non-decreasing, and normalized: . The goal is the usual one of finding an allocation of the items that maximizes the social welfare: . All items are identical, so algorithms should run in time polynomial in the number of bits needed to represent the number , and the number of bidders: .

It is not hard to see that in the “single minded” case, where each is a single-step function (i.e., for some , if and otherwise), the problem is just a reformulation of the NP-hard Knapsack problem. The standard FPTAS for knapsack generalizes easily to multi-unit auctions.

The study of truthfulness in multi-unit auctions has a long history, starting with Vickrey’s 1961 paper [22]. The VCG mechanism is truthful and solves the problem optimally, but is not computationally-efficient (see, e.g., [18]). For the single-minded case, Mu’alem and Nisan provided a truthful polynomial-time -approximation algorithm, followed by an FPTAS by Briest et al [4]. In the general case, a truthful in expectation -approximation mechanism was presented by [14], followed by a deterministic -approximation mechanism in [8]. Can polynomial-time truthful mechanisms guarantee an approximation ratio better than ? This is one of the major open questions in algorithmic mechanism design, and only a partial answer is known: a deterministic truthful mechanism with an approximation ratio better than that always allocates all items must run in exponential time2 [13, 8]. As mentioned before, this paper provides an FPTAS for multi-unit auctions that is truthful in expectation.

A word is in order on how the valuations are accessed. The valuation functions are objects of size , whereas we are interested in algorithms that run in time polynomial in (the running time of the non-truthful FPTAS for multi-unit auctions). Hence, we assume that each valuation is given by a black box. For our upper bounds, the black box corresponding to needs to answer only the weak “value queries”: given , what is the value of . Our lower bound for the power of universally truthful mechanisms assumes a black box that can answer any query based on (the “communication model”).

Main Result I: A Truthful FPTAS for Multi-unit Auctions

We now give a short description of our truthful FPTAS for multi-unit auctions. The only technique known for designing truthful deterministic mechanisms for “rich” problems like multi-unit auctions is by designing maximal-in-range (henceforth MIR) algorithms. An algorithm is called maximal-in-range if there is a set of allocations (the “range”) that does not depend on the input, such that always outputs the allocation in that maximizes the welfare: . Using the VCG payment scheme together with an MIR mechanism results in a truthful mechanism. See [7] for a more formal discussion.

Therefore, one way to obtain truthful mechanisms is to identify a range that, on one hand, is “rich” enough to provide a good approximation ratio, and, on the other hand, is “simple” enough so that exact optimization over this range is computationally feasible. For multi-unit auctions, there is an MIR mechanism that provides an approximation ratio of in polynomial time, but unfortunately no MIR algorithm can provide a better ratio in polynomial time [8].

In this paper we let the range consist of distributions over allocations. An algorithm that always selects the allocation that maximizes the expected welfare and uses VCG payments is truthful in expectation. We term these mechanisms maximal in distributional range (MIDR).

Before discussing our truthful FPTAS, recall the spirit of the standard non-truthful FPTAS for multi-unit auctions: each valuation is simplified by rounding down each value to the nearest power of . Then, the best solution that uses only the ”breakpoints” of the rounded valuations is found. This solution has a value close to the optimal unrestricted solution. However, to guarantee truthfulness via a maximal-in-range algorithm we must find the solution with the optimal welfare in the range. Thus, we give a “weight” to each allocation : when is selected by the algorithm, the output will be with probability , and with probability no bidder will receive any items, effectively reducing the expected welfare of the allocation by a factor of . The weight of a less “structured” allocation is smaller than the weight of a more “structured” one. We show that in the optimal solution among these “weighted allocations” each bidder is assigned a bundle that is a breakpoint, or very close to a breakpoint of his valuation. Thus the optimal solution can be found efficiently using exhaustive search, when the number of bidders is a constant, since the number of breakpoints of each bidder is polynomial in the number of bits.

Efficiently handling any number of bidders is more involved. Given two bidders we define a “meta-bidder” by merging their valuations: the value of the meta bidder for items is equal to the value of the optimal solution that allocates items between the two bidders, using the weighted allocations mentioned above. We recursively define new meta-bidders given the previous ones, until we are left with only two meta bidders. Now the optimal solution can be found efficiently.

We note that MIDR mechanisms were used in [14], although only implicitly. The beautiful construction of [14] is general and applies to many settings. However, its strength and weakness is its generality: for some settings, a specifically-tailored mechanism might have more power than a mechanism obtained from the general construction of [14]. Indeed, for some of the most important settings discussed in [14] better constructions have been found [9, 6, 8]. Moreover, the LP-based techniques of [14] cannot guarantee an approximation ratio better than the integrality gap for multi-unit auctions, which is (see [14]). The techniques in this paper may help in designing better mechanisms in settings where [14] performs poorly, like combinatorial auctions with submodular bidders (see below).

Main Result II: Truthful in Expectation Mechanisms are more Powerful

Next we prove that truthful in expectation mechanisms are strictly more powerful than universally truthful ones. Ideally, we would like to prove that universally truthful mechanisms for multi-unit auctions cannot provide an approximation ratio better than in polynomial time. However, this remains open question even for deterministic mechanisms. Hence, we study a close variant of multi-unit auctions, called restricted multi-unit auctions. The FPTAS for multi-unit auctions extends almost immediately to restricted multi-unit auctions. It is more involved to show that universally truthful polynomial time mechanisms cannot provide an approximation ratio better than .

Roughly speaking, we first show that a polynomial-time universally truthful mechanism with an approximation ratio of must yield a polynomial-time deterministic truthful mechanism with an approximation ratio of on “many” instances. We then show that this deterministic mechanism must be an affine maximizer (a slight generalization of MIR algorithms). Finally, we prove that deterministic affine maximizers cannot provide an approximation ratio better than for restricted multi-unit auctions in polynomial time for “many” instances, hence no polynomial-time universally truthful mechanism for restricted multi-unit auctions with an approximation ratio better than exists.

Open Questions

The obvious open question is whether the techniques of this paper can be extended to improve the approximation guarantees of other questions in algorithmic mechanism design, such as variants of combinatorial auctions, or combinatorial public projects [21]. In particular:

Main Open Question: Is there a truthful mechanism for combinatorial auctions with subadditive (or submodular) bidders that provides a constant approximation ratio?

There are non-truthful algorithms with constant approximation ratios for combinatorial auctions with subadditive bidders[11, 23]. However, the best known polynomial-time truthful mechanisms provide a ratio that is no better than logarithmic [6, 9]. It seems that our current techniques reached a dead end: random sampling methods do not seem amenable to an approximation factor that is better than logarithmic, deterministic MIR mechanisms cannot provide an approximation ratio better than in polynomial time [7], and the techniques of [14] are not applicable to this setting. The methods of this paper might help in constructing truthful in expectation mechanisms. Alternatively, lower bounds on the power of MIDR mechanisms would also be extremely interesting.

2 Preliminaries

2.1 The Setting

In a multi-unit auction there is a set of identical items, and a set of bidders. Each bidder has a valuation function , which is normalized () and non-decreasing. Denote by the set of possible valuations. An allocation of the items to is a vector of non-negative integers with . Denote the set of allocations by . The goal is to find an allocation that maximizes the welfare: .

The valuations are given to us as black boxes. For algorithms, the black box will only answer the weak value queries: given , what is the value of . For the impossibility result, we assume that the black box can answer any query that is based on (the “communication model”). Our algorithms run in time , while our impossibility result gives a lower bound on the number of bits transferred, and holds even if the mechanism is computationally unbounded.

2.2 Truthfulness

An -bidder mechanism for multi-unit auctions is a pair where and , where . might be either randomized or deterministic.

Definition 2.1

Let be a deterministic mechanism. is truthful if for all , all and all we have that .

Definition 2.2

is universally truthful if it is a probability distribution over truthful deterministic mechanisms.

Definition 2.3

is truthful in expectation if for all , all and all we have that , where the expectation is over the internal random coins of the algorithm.

2.3 Maximal in Range, Maximal in Distributional range, and Affine Maximizers

Definition 2.4

is an affine maximizer if there exist a set of allocations , a constant for , and a constant for each , such that . is called maximal-in-range (MIR) if for , and for each .

Definition 2.5

is a distributional affine maximizer if there exist a set of distributions over allocations , a constant for , and a constant for each ) such that: . We say is maximal in distributional range (MIDR) if for all in , and for each .

The following proposition is standard:

Proposition 2.6

The following statements are true:

  1. Let be an affine maximizer (in particular, maximal in range). There are payments such that is a truthful mechanism.

  2. Let be a distributional affine maximizer (in particular, maximal in distributional range). There are payments such that is a truthful in expectation mechanism.

3 A Truthful FPTAS for Multi-Unit Auctions

This section provides a -approximation algorithm for multi-unit auctions that is truthful in expectation by providing a maximal-in-distributional-range mechanism. The running time is polynomial in , and . The construction has several stages. We start by defining a certain family of ranges, structured ranges. The main property of this family is that finding the optimal solution in a structured range is computationally feasible when the number of bidders is fixed. To handle any number of bidders, we use a more involved tree-like construction, where structured ranges serve as the basic building block. We start by defining structured ranges.

3.1 Structured Ranges

Definition

Towards constructing an FPTAS, fix the approximation parameter such that . It will be useful to let . We construct ranges that consist only of weighted allocations:

Definition 3.1

A weighted allocation is a distribution over allocations of the following form: with some probability the allocation is chosen, and with probability the empty allocation (where each bidder is allocated an empty set) is chosen. is called the weight of the weighted allocation.

The weight of a weighted allocation of items to bidders is determined using the following function , where is the set of all allocations of items to bidders. Let be the maximal (non-negative) integer such that for each , is a multiple of . Let if for bidders, let otherwise. Now define . Observe that the number of different outputs of is , where increasing by makes the output value increase by a factor of .

A structured range consists only of weighted allocations whose weight is determined by . Informally, in a structured range “structurally simpler” allocations have greater weight.

Definition 3.2

An -structured range for bidders and items is the following set of weighted allocations: , where is the set of allocations for these bidders. We sometimes call an -structured range a structured range when is clear from context.

The optimal solution in a structured range provides a good approximation to the (unweighted) optimal solution: every allocation is in the range with weight at least . In particular, the optimal (unweighted) allocation is in the range with a weight of at least .

Lemma 3.3

The expected approximation ratio of an algorithm that optimizes over an -structured range is at least .

The Optimal Solution is in Neighborhoods of Breakpoints

The important property of structured ranges is that finding an optimal weighted allocation is computationally easy. We show that the optimal solution consists of bundles that are near breakpoints, where breakpoints are bundles at which the valuation first exceeds a power of .

Definition 3.4

A bundle is called a -breakpoint of a valuation if it is the smallest multiple of s.t. , for some integer . The bundle is also considered a -breakpoint of , for all .

Definition 3.5

Let be a -breakpoint of . The neighborhood of consists of the bundles .

Lemma 3.6

Let be a weighted allocation that maximizes the welfare in a structured range among all allocations with , for some . Let be the maximal non-negative integer s.t. each is a multiple of . Then, each is in the neighborhood of a -breakpoint.

Proof:   Suppose towards a contradiction that there is some bidder , where is not in the neighborhood of a -breakpoint. Round down each (that is not a -breakpoint) to the nearest -breakpoint of denoted . Let be such that . For each , define as follows:

For each , let . We will show that the expected welfare of is greater than that of the optimal solution . Note that : on one hand by our assumption, . On the other hand, , since for each , if is odd, else . (In a sense, we “take” items from bidder and spread them among the other bidders.)

We now calculate the expected welfare of . All ’s are even, thus each is a multiple of . In addition, the number of bidders that receive no items is larger in . Hence , by the definition of . Since for each , , the monotonicity of the valuations implies that:

Using the definition of a breakpoint, the value of each is close to the value of :

I.e., the expected welfare of the distribution is greater than that of the optimal one: . A contradiction.

3.2 Warmup: An FPTAS for a Fixed Number of Bidders

As a warm-up we provide an MIDR -approximation algorithm for a fixed number of bidders (Figure 1). The idea is to fully optimize over a structured range. Specifically, the number of value queries the algorithm makes is . This algorithm has two drawbacks. The first major one is that while the number of value queries is small, the running time of the algorithm is exponential in : exhaustive search is used to find the best allocation that consists only of bundles in the neighborhood of breakpoints. However, for a fixed number of bidders the exhaustive search (and hence the algorithm) is also computationally efficient3.

The second drawback is that the running time and the number of value queries depends (polynomially) on (in a sense, the algorithm is only “weakly polynomial”). This can be fixed (using “significant breakpoints” – see the next section), but to keep the presentation of this warmup simple we do not address this here. We stress that the FPTAS for a general number of bidders we present in the next subsection is “strongly polynomial”. I.e., runs in time .

  1. For each :

    • Find the -breakpoints of each bidder .

    • Let be the allocation maximizing welfare among all allocations where for each , is in the neighborhood of a -breakpoint of .

  2. Let be the welfare maximizing distribution in . Return .

Figure 1: FPTAS for Fixed Number of Bidders
Lemma 3.7

The number of value queries the algorithm makes is .

Proof:   For each one of the possible values of , we find all breakpoints of each bidder , and then query the bundles in the neighborhood of the breakpoints. Binary search finds a single breakpoint in queries, thus the total number of queries is .

The following is an immediate corollary of Lemma 3.6 (setting in the statement of the lemma):

Lemma 3.8

The algorithm finds a welfare maximizing distribution in an -structured range.

Using Lemma 3.3 we now have:

Theorem 3.9

There is a -approximation algorithm for multi-unit auctions that is truthful in expectation and makes value queries.

3.3 The Main Result: An FPTAS for a General Number of Bidders

This section presents a computationally-efficient FPTAS for a general number of bidders. We use structured ranges as a basic building block.

Let and be valuations of two bidders. Informally speaking, define a “meta bidder” with valuation as follows: the value is set to the optimal expected welfare of allocating at most items in a structured range for items and the two bidders. Note that, by lemma 3.6 and a straightforward modification of the algorithm in the previous section, each query to can be calculated in time . Apply this construction recursively, combining two meta bidders into one “larger” meta bidder (see Figure 2). We end up with two meta-bidders at the top level, a case solved in the previous section. We will show that this algorithm runs in time , and that the solution obtained well-approximates the optimal welfare. We will also show that this algorithm optimizes over some range of distributions, yielding truthfulness in expectation. Throughout this section we assume, without loss of generality, that is a power of (as we can always add bidders that have a value of for each bundle).

The Range

Before defining the range of the algorithm, we define a binary hierarchal division of the bidders, illustrated in Figure 2. We think of each node in the hierarchy as a meta-bidder representing a subset of the bidders. The root meta-bidder, representing , is composed of two meta-bidders representing and , respectively. Applying the division recursively, meta-bidder has two children: is the left child and is the right child. The leaves correspond to single bidders. Number the tree levels from to , where the leaves are in level , and the top vertex is in level . The following definitions will prove helpful in defining the range of the algorithm .

Definition 3.10

Let and be two disjoint set of bidders. Let and be two distributions over allocations to bidders in and , respectively. Let . A weighted composition is the following distribution over allocations to bidders in : with probability , allocate to bidders in according to and to bidders in according to , and with probability allocate no items to all bidders in .

Definition 3.11

The set of components of a weighted composition distribution includes and . If and are weighted composition distributions then the components of and are also contained in the set of components of .

Let . In the remainder of this section, every reference to a structured range refers to an -structured range on items and two bidders, though we often refer to a subset of the structured range that allocates at most items, for some . For each vertex in the hierarchical tree with two leaf children and , let be the set of all weighted allocations of at most items in a -structured range of bidders and . For a vertex with two non-leaf children and , let

Now let the range of the algorithm be .

Consider some distribution (for some and ). Let be a descendant of in the hierarchical tree. There is a unique such that some is a component of . The distribution is said to be induced by on , and is said to be the number of items associated with in . Let be the number of items associated with bidder in . The (feasible) allocation is called the base allocation of . Notice that for every allocation there is at least one allocation such that is the base allocation of . Moreover, distribution always allocates either items or items to bidder .

Figure 2: Meta-Bidder Hierarchy

An algorithm that optimizes over is truthful in expectation. It also provides a good approximation:

Lemma 3.12

An algorithm that optimizes over provides an approximation ratio of at least .

Proof:   The lemma follows from the following claim:

Claim 3.13

Let . Let be its base allocation. .

Proof:   If the claim is true. Otherwise, for a non-leaf vertex with children and , let denote the weighted composition distribution induced by on . is the probability that the child vertices of receive a non-empty bundle. By definition, . The probability that a bidder receives is the probability that all non-leaf vertices containing “allocate” items to their children. Let be the set of all non-leaf vertices on the path in the hierarchal tree from the leaf to the root . We can bound this probability as follows:

Observe that the optimal (unweighted) allocation is induced by some . Hence, each bidder receives with probability at least in . The expected welfare of is therefore at least times the welfare of the optimal unweighted allocation. The algorithm optimizes over all distributions in the the range , and thus returns an allocation with at least that expected welfare.

We now make the meta-bidder intuition formal.

Definition 3.14

Let be a range of distributions of items to a set of bidders. Let be the set of all distributions in that never allocate more than items. Let be the set of all allocations that allocate at most items to bidders in . Define to be the valuation of a meta-bidder over using as follows:

Note that a valuation of a meta bidder is monotone and normalized. We now define a meta-bidder for each vertex . The definition is recursive. For a vertex with two leaf children and , let be the valuation of the meta bidder defined over using the range of distributions (i.e., an -structured range). For a vertex with two meta-bidder children and , let be the valuation of the meta-bidder defined over using the the -structured range of allocations to (meta-) bidders and . The meta-bidder idea is powerful for precisely the following reason:

Lemma 3.15

Let be a welfare maximizing distribution in . Let be the distribution of allocations to bidders in induced by on , where is a descendant of in the hierarchical tree or . Denote the expected welfare of by . Let be the bundle that is associated with in . Then, .

Proof:   The proof is by induction on the level of . For a vertex with two leaf children and , equals the value of the welfare maximizing solution that allocates at most items in a structured range. On the other hand, observe that induces a weighted allocation that allocates at most items in a structured range, and that is welfare maximizing. Hence, the expected welfare of equals .

Consider a vertex with two leaf children and , and let . The expected welfare of is by induction , i.e., is the welfare maximizing weighted allocation in a structured range with bidders and that allocates at most items, which is by definition the value of .

Fix . The important property of distributions in the range is that, similar to structured ranges, the bundle that is associated with a vertex is in the neighborhood of a breakpoint of the corresponding (meta-) bidder.

Lemma 3.16

Let be a welfare maximizing distribution in . Let and be two sibling vertices that are descendants of . Let and be the bundles associated with and in , respectively. Let be the maximal non-negative integer such that and are multiples of . Then, and are in the neighborhood of a -breakpoint of and , respectively.

Proof:   Let be the parent of and , let its associated bundle in be . By Lemma 3.15 and the definition of meta bidders, is equal to the expected welfare of a distribution of the form – the welfare maximizing distribution that allocates at most items to bidders and in a structured range. By Lemma 3.6, and are in neighborhoods of -breakpoints of and , respectively.

The FPTAS

The FPTAS we construct is the obvious one: find the optimal allocation to meta-bidders and in a structured range, then proceed recursively to find the best allocation between bidders and , and so on. In a naive implementation of this algorithm every value query to the valuations of the two meta-bidders is calculated recursively “on the fly”. However, a short calculation shows that in this implementation the number of value queries is polynomial in . To improve the running time, we use the fact that to calculate , for any bundle , we only need to know the value of a relatively small number of bundles in the neighborhood of breakpoints of the two (meta-) bidders that are ’s children.

We are also interested in a “strongly polynomial” algorithm. I.e., the running time should be independent of . Towards this end, define:

Definition 3.17

A bundle is an -significant -breakpoint of if it is a -breakpoint of and if either or .

By the next lemma, we can “ignore” insignificant breakpoints:

Lemma 3.18

Let be a welfare maximizing distribution in . Denote its expected welfare by . Let be two sibling descendants of in level . Let be the bundles associated with in , respectively. Let be the largest integer s.t. both and are multiples of . Then, and are in the neighborhood of an -significant -breakpoint of and , respectively, for .

Proof:   From Lemma 3.16 we know that and are in the neighborhood of -breakpoints of and . It remains to show that they are -significant. We make use of the following claim:

Claim 3.19

Let be a welfare maximizing distribution of items in an -structured range for bidders. Denote its expected welfare by . For each with we have that .

Proof:   The proof is almost identical to the proof of Lemma 3.6, and therefore it is only sketched. Suppose for contradiction that there is some bidder with and . Let be the maximal integer s.t. each is a multiple of . Define the following allocation : , and for the other bidder , let (notice that ). All bidders but one receive no items, thus , by definition of . We have that . However, now , a contradiction.

We prove the lemma using the claim by induction on the level of the tree, starting with children of to the leaves. By Lemma 3.15, . is a meta bidder, thus is also equal to the value of a welfare maximizing allocation in a structured range: . By the claim . Similarly, .

Assume correctness for level and prove for . For and in level with parent , from arguments similar to those above it follows that , where , and are the bundles associated with and in , respectively. By induction we get that: , as needed.

  1. For each vertex in level evaluate (recursively, from bottom to top) all bundles that are in the neighborhood of an -significant -breakpoint of (for , and )

  2. Evaluate (recursively, from top to bottom starting from the top vertex and ) for each vertex with children and a welfare maximizing weighted distribution of (meta-) bidders and , where . Let .

  3. Each bidder receives items with probability where is the set that includes all the proper ancestors of in the hierarchal tree.

Figure 3: An FPTAS for a General Number of Bidders
Theorem 3.20

The algorithm in Figure 3 is a truthful-in-expectation FPTAS for multi-unit auctions.

Before proving the theorem, we comment on the definition of in the algorithm. Let be the expected welfare of . On one hand, observe that (allocating all items to bidder that maximizes is an allocation that is induced by some . has an expected welfare of at least ), and on the other hand . In other words, is on one hand small enough so we can calculate the significant breakpoints of the valuations according to Lemma 3.18, and on the other hand is not too small, so we do not have too many significant points to evaluate. The proofs below make this discussion formal.

Lemma 3.21

The algorithm finds a welfare maximizing distribution .

Proof:   We prove the following claim first:

Claim 3.22

Let be a vertex in level . The -significant -breakpoint are evaluated correctly (for ).

Proof:   The proof is by induction on the level , starting from the leaves to the top. For with two leaf children and , the claim is trivially true, since and by Lemma 3.18.

We assume correctness for and prove for . Consider vertex in level with meta-bidder children and . By induction the values of bundles in the neighborhood of -significant breakpoints of and are computed correctly. Consider some bundle in the neighborhood of an -significant breakpoint of . Denote by the welfare maximizing weighted allocation to and that allocates at most items. By induction, the values of are calculated correctly, since they are -significant breakpoints of and (Lemma 3.18). Hence is calculated correctly, as needed.

Now we can prove the lemma by induction from the top vertex to the leaves. The welfare of the distribution induced by on the top vertex is, by Lemma 3.15, equal to a welfare maximizing weighted allocation to bidders and in a structured range . By the same arguments as above, the algorithm finds a welfare maximizing weighted allocation to bidders and of at most items. Proceeding similarly until we reach the leaves, we end up with a distribution in with an expected welfare equal to that of .

Lemma 3.23

The algorithm runs in time .

Proof:   Fix some vertex with children and . Observe that , for any , can be computed using only bundles in the neighborhood of -significant -breakpoints of and (for ).

To see that these values can be obtained efficiently, notice that each valuation in level has only -significant -breakpoints (by our choice of – since ). Recall that via binary search a -breakpoint of can be can be found in value queries to . For each breakpoint we also need to query all bundles in its neighborhood. Hence we need to query each valuation (of a bidder or a meta bidder) only times. The number of valuations of bidders and meta-bidders in the tree is . and thus the total number of queries is still . Notice that the computational overhead is polynomial in the number of value queries: for each value query we compute a welfare maximizing allocation of the previous level in a structured range of bidders, which can be done in time . In total, the running time of the algorithm is .

This completes the proof of Theorem 3.20

4 Truthful in Expectation Mechanisms Have More Power

This section shows that polynomial-time truthful-in-expectation mechanisms are strictly more powerful than universally-truthful polynomial-time mechanisms. Ideally, we would like to prove this for multi-unit auctions. However, finding a non-trivial lower bound on the power of polynomial time mechanisms for multi-unit auctions, even deterministic ones, is a big open question. Hence we study a variant of multi-unit auctions that is artificially restricted, with the sole purpose of proving such a separation for the first time. We prove that universally truthful polynomial time mechanisms cannot provide an approximation ratio better than for this variant, while we provide a truthful in expectation FPTAS. As in multi-unit auctions, the (deterministic) VCG mechanism solves this problem optimally, but in exponential time.

In a restricted multi-unit auction, a set of items, where is a power of , is to be allocated to two bidders. The set of feasible allocations is restricted as follows: either no items are allocated, or all items are allocated with at least one item per bidder. Each bidder has a valuation function , given as a black box, specifying the bidder’s value for each number of items. We restrict to be normalized () and strictly increasing. The objective is to maximize social welfare, the sum of the values of the bidders. Our algorithms should run in time polynomial in .

Theorem 4.1

The following statements are true:

  1. For every constant , every universally truthful -approximation mechanism for restricted multi-unit auction requires communication.

  2. There exists a -approximation algorithm for restricted multi-unit auctions that is truthful in expectation and runs in time (an FPTAS).

Before proceeding with the proof, we discuss the differences between restricted multi-unit auctions and the standard multi-unit auctions discussed in previous sections, and the role these differences play in the proof.

  • Strictly Increasing Valuations, is a power of : These restrictions are only there to simplify the proof. They can be removed without changing the statement of the theorem.

  • Allocate all Items with at least One Item per Bidder or Allocate Nothing: The all-items-are-allocated constraint fulfills the conditions of the characterization of deterministic truthful mechanisms for multi-unit auctions [10]. The restriction that each bidder must receive at least one item, and the relaxation allowing the empty allocation, will prove useful in arguing about universally truthful randomized mechanisms which were not considered in [10].

To prove Theorem 4.1, first we bound the power of universally truthful polynomial time mechanisms for restricted multi-unit auctions. Then, we show that there exists a truthful in expectation FPTAS for restricted multi-unit auctions.

4.1 A Lower Bound on Universally Truthful Mechanisms

From Universally Truthful to Deterministic Mechanisms

It is inconvenient to study randomized mechanisms directly. Therefore, we start by showing that the existence of a universally truthful mechanism with a good approximation ratio implies the existence of a deterministic mechanism that provides a good approximation on “many” instances. The following definition and propositions are adapted from [7]. We repeat the proof here for completeness.

Definition 4.2

Fix , , and a finite set of instances of restricted multi-unit auctions. A deterministic algorithm for restricted multi-unit auctions is -good on if returns an -approximate solution for at least a -fraction of the instances in .

Proposition 4.3 (essentially [7])

Let be some finite set of instances of restricted multi-unit auctions, and let . Let be a universally truthful mechanism for restricted multi-unit auctions that provides an expected welfare of for every instance with expected communication complexity . Then, there is a -time (deterministic) algorithm in the support of that is -good on and has a communication complexity of .

Proof:   For each mechanism in the support of , Let denote its expected communication complexity. Let be the mechanism obtained from by replacing in the support of each mechanism for which by a mechanism that never allocates any items. Notice that the expected approximation ratio of is at least .

Fix some instance . The expected welfare of is at least . The probability that has welfare at least is lower-bounded by : this is achieved if is optimal with probability and has welfare otherwise. Hence, there exists a deterministic algorithm in the support of that returns an -approximate solution for at least a -fraction of the instances in . Notice that the communication complexity of is .

Let be a universally truthful mechanism for restricted multi-unit auctions with communication complexity and an approximation ratio of , for a constant . By the claim, there must exist a -good deterministic mechanism in its support with communication complexity of (for some constant and some set of instances ). We will show that the communication complexity of this “good” mechanism for that is , thus the communication complexity of is .

A Lower Bound on “Good” Deterministic Mechanisms

To start, we recall the following characterization result for deterministic mechanisms for multi-unit auctions:

Theorem 4.4 ([10])

Let be a truthful deterministic mechanism for multi-unit auctions with strictly monotone valuations that always allocates all items with range at least . Then, is an affine maximizer.

Fix some universally truthful mechanism for restricted multi-unit auctions. The support of contains three possible (non-disjoint) types of deterministic mechanisms:

  1. Imperfect Mechanisms: Mechanisms where there exist valuations such .

  2. Tiny-Range Mechanisms: Mechanisms that have a range of size at most .

  3. Affine Maximizers: Mechanisms that are affine maximizers.

Notice that there are no other mechanisms in the support of : a mechanism for restricted multi-unit auctions that always allocates some items must always allocate all items. If its range is of size at least , then by Theorem 4.4 it must be an affine maximizer.

Claim 4.5

Let be a deterministic imperfect mechanism for restricted multi-unit auctions. Then, there is a constant such that whenever .

Proof:   Let be such that . Let . Since valuations are normalized and strictly increasing, . Consider some as in the statement of the claim. Observe that : by weak monotonicity (see [18]) bidder should be allocated no items, hence bidder is allocated no items (recall that every non-empty feasible allocation assigns at least one item to each bidder). Similarly, .

Let be a universally truthful mechanism for restricted multi-unit auctions that provides an approximation ratio of . Define , where the minimum is taken over all imperfect mechanisms in ’s support. Fix a constant such that . For each integer such that , define the instance as follows: for all , and for all ; for all , and for all . Let . Notice that the optimal welfare for