Optimal Competitive Auctions

# Optimal Competitive Auctions

## Abstract

We study the design of truthful auctions for selling identical items in unlimited supply (e.g., digital goods) to unit demand buyers. This classic problem stands out from profit-maximizing auction design literature as it requires no probabilistic assumptions on buyers’ valuations and employs the framework of competitive analysis. Our objective is to optimize the worst-case performance of an auction, measured by the ratio between a given benchmark and revenue generated by the auction.

We establish a sufficient and necessary condition that characterizes competitive ratios for all monotone benchmarks. The characterization identifies the worst-case distribution of instances and reveals intrinsic relations between competitive ratios and benchmarks in the competitive analysis. With the characterization at hand, we show optimal competitive auctions for two natural benchmarks.

The most well-studied benchmark measures the envy-free optimal revenue where at least two buyers win. Goldberg et al. [13] showed a sequence of lower bounds on the competitive ratio for each number of buyers . They conjectured that all these bounds are tight. We show that optimal competitive auctions match these bounds. Thus, we confirm the conjecture and settle a central open problem in the design of digital goods auctions. As one more application we examine another economically meaningful benchmark, which measures the optimal revenue across all limited-supply Vickrey auctions. We identify the optimal competitive ratios to be for each number of buyers , that is as approaches infinity.

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## 1 Introduction

A central question in auction theory is to design optimal truthful auctions that maximize the revenue of the auctioneer. A truthful auction must be incentive compatible with the selfish behavior of bidders and encourage them to reveal their private information truthfully. In the classic single-parameter setting, the auctioneer sells multiple copies of an item to unit-demand bidders, each bidder having a private value for the item. The revenue maximization problem in economics is traditionally analyzed in the Bayesian framework. The seminal Myerson’s auction [23] provides an optimal design that extracts the maximum expected revenue for a given distribution of values.

While Myerson’s auction gives an optimal design in the Bayesian framework, in many scenarios determining or estimating the prior distribution in advance is impossible. Without prior information on the distribution, how should an auction be designed? The auctioneer not being certain about the prior distribution is likely to resort to truthful auctions that generate a good revenue on every possible instance of bidders’ valuations. We therefore employ the worst-case competitive analysis, which characterizes the worst-case performance of an auction across all possible bidders’ valuations. We follow the framework summarized in [14] where a truthful auction is required to generate a profit comparable to the value of a certain economically meaningful benchmark on every input of values. The motivation of this framework comes from the analysis of on-line algorithms, where the performance of an on-line algorithm, which is unaware of the future, is measured in terms of the performance of the optimal off-line algorithm that knows the future. The assumption that an on-line algorithm does not know the future in advance corresponds to the assumption that an auction does not know the biddersâ valuations in advance. How well an auction can perform in the worst-case? We will answer this question in the present paper.

A particularly interesting single-parameter setting is digital goods auctions in which the number of units for sale is unlimited or is greater than or equal to the number of bidders. Digital goods auctions are motivated by the applications of selling digital goods like downloadable software on the Internet or pay-per-view television where there is a negligible cost for producing a copy of the item. The design of the optimal Bayesian auction for digital goods is trivial: given a prior distribution of values, one may independently offer each bidder a fixed price tailored to the value’s distribution so as to maximize the expected revenue . From the worst-case perspective, digital goods auctions attracted considerable attention over the last decade [15, 10, 12, 13, 19, 9, 2, 21]. However, the optimal design was still unknown. The question about designing optimal digital goods auctions remained widely open and is fundamental to our understanding of the optimal competitive auction design in the general single-parameter setting.

Formally, in the competitive analysis framework, given a benchmark function , the competitive ratio (with respect to ) of a truthful auction is defined as , where is the expected revenue of the auction on valuation vector . The objective is to design an auction that minimizes the competitive ratio with respect to a given benchmark.

We note that a number of functions can serve as meaningful benchmarks and that they may have different optimal competitive ratios. Such flexibility in the choice of a target benchmark function may prove to be helpful for modeling different objectives of the auctioneer. Conversely, it may allow us to make meaningful conclusions and prescriptions about target benchmarks. However, most of the work done along the line of competitive analysis in algorithmic game theory [24] is devoted to the competitive analysis of a well-motivated but fixed concrete benchmark function. Namely, while there are basic guidelines for choosing a good benchmark function (e.g., the benchmark should have a strong economic motivation and should match the performance of the optimal auction as closely as possible), there are no formal criteria for distinguishing different benchmarks. In this paper, instead of justifying what a good benchmark is, we focus on the intrinsic relation between benchmarks and their corresponding competitive ratios. We give a complete characterization for almost all possible benchmark functions.

Theorem 1. (Characterization) For any non-negative and monotonically increasing function , there is a truthful digital goods auction that achieves a competitive ratio of with respect to if and only if

 ∫Sf(v)⋅w(v)dv≤λ⋅n∑i=1 ∫ S↓iw(v-i)dv-i, (1)

where is any upward closed set in the support of valuation vectors, is the projection of along the -th coordinate, , and .

The fact that the weight function appears in the above theorem is not a coincidence. The corresponding single-parameter distribution with the density function for is a common tool to provide bounds on the performance of auctions. It is called equal-revenue distribution, as it enjoys a remarkable property that if the value of a bidder is drawn from this distribution, then any fixed price offered to generates the same expected revenue . In particular, the product of independent equal-revenue distributions was used in [13] to obtain the best known lower bounds on the competitive ratios of digital goods auctions; it has also been used in the auction analysis of other models, e.g., in [17].

As we know from Yao’s minimax principle, for any benchmark function , there is a distribution of instances such that on average for that distribution no truthful auction can beat the worst-case competitive ratio with respect to . The inequality (1) in the theorem states that the equal revenue distribution is indeed the worst-case distribution to estimate the competitive ratio of an auction. In other words, in the context of digital goods auctions, the worst-case distribution can be described only by its support, and the actual density of the distribution is given by the equal-revenue distribution. The theorem gives a sufficient and necessary condition for a benchmark function to admit a competitive ratio of . It implies that the optimal competitive ratio with respect to is the smallest value of for which the inequality (1) holds. The theorem indicates that all benchmarks and their competitive ratios are tied to the equal revenue distribution: it is the worst distribution not only for the benchmark considered in [13] but also for all monotone benchmarks.

We note that our characterization is provided by the set of inequalities (1) that only involves a function and a ratio , but does not describe an actual auction. This is similar in spirit to the characterization of truthfully implementable allocation functions, where an allocation function is truthfully implementable if there exists a payment function that makes the allocation function truthful. To characterize a truthfully implementable allocation function, one uses a set of inequalities such as weak-monotonicity [25, 6, 3] that only specifies an allocation function but says nothing about payments. Hence, without describing any payment function, one can determine whether there is a truthful auction with a specified allocation. Our characterization of benchmarks with a given competitive ratio shares a similar philosophy: the condition determines whether there exists a truthful auction with a certain completive ratio with respect to , but does not explicitly describe the auction.

The characterization theorem provides us with a powerful tool to analyze the optimal competitive ratios of auctions for different benchmarks. We first consider the most well-studied benchmark introduced in [15], which is defined as the maximum revenue achieved in an envy-free allocation provided that at least two bidders receive the item. We next study another natural benchmark, denoted by maxV, which is defined as the maximal revenue of the -item Vickrey auction across all possible values of . We have the following results on the optimal competitive ratios for these two benchmarks.

Theorem 2. There are truthful digital goods auctions that achieve the optimal competitive ratios of and for any with respect to the benchmarks and maxV, respectively, where

 λn=1−n∑i=2(−1n)i−1ii−1(n−1i−1) \ \ and \ \ γn=(nn−1)n−1−1.

It was shown in [13] that gives a lower bound on the competitive ratio of any auction with respect to for any . These lower bounds were obtained by calculating the expected value of the benchmark when is drawn from the equal revenue distribution. In particular, , , and in general, is an increasing sequence with a limit of roughly . Goldberg et al. [13] conjectured that the lower bounds given by are tight. Indeed, for bidders, the second price auction gives a matching competitive ratio of ; for bidders, [19] gave a sophisticated auction with a competitive ratio that matches the lower bound of . These are the only cases for which optimal competitive auctions were known. Our result confirms the conjecture of [13] and settles the long standing open problem of designing optimal digital goods auctions with respect to the benchmark .

For the benchmark maxV, we calculate the expected value of when is drawn from the equal revenue distribution, and derive a sequence of optimal competitive ratios for each number of bidders in the auction. We note that , , and is an increasing sequence with the limit of as approaches infinity.

Finally, as another application of the characterization theorem, we consider optimal competitive auctions for the multi-unit limited supply setting where there is an item with units of supply for sale to unit-demand bidders. It was observed in [14] that there is a competitive ratio preserving reduction from unlimited supply to limited supply. Namely, given a digital goods auction with a competitive ratio with respect to for bidders, one can construct a truthful auction for bidders with the same competitive ratio with respect to (where is the optimal fixed price revenue provided that at least and at most bidders receive the item). Our analysis continues to hold for those benchmarks that only depend on the highest values, and thus, gives optimal competitive auctions in the competitive analysis framework.

### 1.1 Related Work

The study of competitive digital goods auctions was coined by Goldberg et al. [15], where the authors introduced the random sampling optimal price auction and showed that it has a constant competitive ratio with respect to . Later on the competitive ratio of the auction was shown to be 15 and 4.68, by Feige et al. [9] and Alaei et al. [2], respectively. Since the pioneer work of [15], a sequence of work has been devoted to design auctions with improved competitive ratios: the random sampling cost sharing auction [10] with ratio 4; the consensus revenue estimate auction [12] with ratio 3.39; the aggregation auction [19] with ratio 3.25; the best known ratio 3.12 is attained by the averaging auction [21].

Fiat et al. [10] formulated the prior-free analysis framework for digital goods auctions. The framework was further developed to general symmetric auction problems and connected with the Bayesian framework in [20]. The relation between envy-freedom and prior-free mechanism design was further investigated in [18]. Aggarwal et al. [1] showed that every randomized auction in the digital goods environment can be derandomized in polynomial time with an extra additive error that depends on the maximal range of values.

Leonardi and Roughgarden [26] introduced another benchmark, namely monotone-price benchmark . Later it was shown in [5] that digital goods auctions have a constant competitive ratio with respect to .

A number of variants of digital goods auctions have been investigated, including, e.g., online auctions [4, 7], limited supply (-unit auctions) [8], online auctions with unknown limited supply [22], externalities between bidders [16], and matroid permutations and position environments [18].

## 2 Preliminaries

In a digital goods auction, an auctioneer sells multiple copies of an item in unlimited supply to bidders. Each bidder is interested in a single unit of the item and values it at a privately known value . We consider a single-round auction, where each bidder submits a sealed bid to the auctioneer. Upon receiving submitted bids from all bidders, the auctioneer decides on whether each bidder receives an item and the amount that pays. If bidder wins an item, his utility is the difference between his value and his payment; otherwise, the bidder pays and his utility is . The auctioneer’s revenue is the total payment of the bidders.

We assume that all bidders are self-motivated and aim to maximize their own utility. We say that an auction is truthful or incentive compatible if it is a dominant strategy for every bidder to submit his private value, i.e., , no matter how other bidders behave. A randomized auction is (universally) truthful if it is randomly distributed over deterministic truthful auctions.

An auction is called bid-independent if, for each bidder , the auctioneer computes a threshold price according to the bids of the rest bidders . In other words, there is a function such that . It was shown in [15] that an auction is truthful if and only if it is bid-independent. Thus, it is sufficient to consider bid-independent auctions in order to design truthful auctions.

To evaluate the performance of an auction, we need to have a reasonable benchmark function , where measures our target revenue for the bid vector . Given a benchmark function , we say that an auction has a competitive ratio of with respect to if

 f(v)E[A(v)]≤λ,∀v=(v1,…,vn)

where is the expected revenue of auction on the bid vector . The focus of our paper is to design truthful auctions that minimize the competitive ratio with respect to different benchmarks.

In this paper, we assume that a benchmark function is non-negative and monotone. These are natural conditions for a function to serve as a reasonable benchmark. Specifically, we will focus on the following benchmark functions. (Given a bid vector we reorder bids so that .)

• . That is, gives the largest possible revenue obtained in a fixed price auction given that there are at least two winners. was denoted sometimes as in the previous literature and provides the optimal envy-free revenue conditioned on that at least two bidders receive the item.

• . We note that is the revenue of the -item Vickrey auction with a fixed supply of items. Hence, maxV gives the largest revenue obtained in the Vickrey auction for selling items for all possible values of the limited supply .

###### Theorem 1 (Goldberg et al. [13]).

The competitive ratio with respect to of any truthful randomized auction is at least

 λn=1−n∑i=2(−1n)i−1ii−1(n−1i−1).

We note that , , and is an increasing sequence with a limit of when approaches infinity. In the proof of the above theorem, the authors of [13] constructed a so-called equal revenue distribution where all values are drawn identically and independently with probability for any . A remarkable property of this distribution is that any truthful auction has the same expected revenue . It was shown that the expected value of is . Thus, the theorem follows since

 maxvF(2)(v)E[A(v)]≥E[F(2)(v)][F(2)(v)]E[A(v)]=n×λnn=λn.

## 3 Characterization of Benchmarks

We introduce the following central definition for a Benchmark function.

###### Definition 1 (Attainability).

A benchmark function is -attainable if there exists a truthful auction that has a competitive ratio of with respect to .

We shall give a sufficient and necessary condition of attainability of a benchmark in this section with a set of inequalities which only involves the function and ratio but not any auction. After having the characterization, we analyze the attainability of two well-studied benchmarks in the next section.

For technical simplicity, we consider a discrete and bounded domain for all bids: for any bidder , we assume that , where is any fixed small constant. Thus, all bids are between 1 and , and are multiples of . Let

 Λ={(1+δ)t | t=0,1,2,…,N}

denote the support of a single bidder’s bids and denote the support of bid vectors of all bidders. We note that such a multiplicative discretization is not critical for our characterization. (Alternatively, we may consider an additive discretization with bids being integer multiples of .) We will discuss how to generalize our analysis to continuous and unbounded domains at the end of this section.

For the domain we assume without loss of generality that for any , the price offered to bidder is also from , which is of the form . For and , let denote the probability that the auctioneer offers price to bidder when observing others’ bids . There exists a truthful auction that is -competitive with respect to the benchmark if and only if the following linear system is feasible:

 {\sc LS}1: ⎧⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪⎨⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪⎩λ⋅∑ibi∑pi=1pi⋅zi(b-i,pi)≥f(b),∀b(1+δ)N∑pi=1zi(b-i,pi)≤1,∀i,b-izi(b-i,pi)≥0,∀i,b-i,pi

Note that the summation over in is taken in the domain .

We remark that the correspondence between and truthful auctions is not one-to-one but one-to-many. For any auction, there is a corresponding probability profile that satisfies . On the other hand, different auctions may have the same probability profile (thus, they have the same expected revenue). Note that for any given , we can construct at least one corresponding truthful auction, which independently offers the threshold price to each bidder with probability .

We define

 xi(b)=xi(b-i,bi)=bi∑pi=1pi⋅zi(b-i,pi).

Intuitively, gives the expected revenue obtained from bidder when the bid vector is . We further define

 w((1+δ)t)=⎧⎪⎨⎪⎩δ(1+δ)t+1,t=0,1,2,…,N−11(1+δ)N,t=N

We note that can be viewed as a equal revenue distribution over , which satisfies the following nice property:

 N∑t=kw((1+δ)t)=1(1+δ)k.

Let and .

Given these definitions, the aforementioned linear system can be rewritten as follows.

 {\sc LS}2: ⎧⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪⎨⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪⎩λ⋅∑ixi(b-i,bi)≥f(b),∀bN∑t=0w((1+δ)t)⋅xi(b-i,(1+δ)t)≤1,∀i,b-ixi(b-i,(1+δ)t)≤xi(b-i,(1+δ)t+1),∀i,b% -i,t=0,1,2,…,N−1xi(b-i,bi)≥0,∀b

The third constraint in requires monotonicity of , which is a necessary condition for the equivalence of the two linear systems. Indeed, must be non-negative, where we denote . To summarize, we have the following claim.

###### Proposition \theproposition.

A monotone function is -attainable if and only if the linear system has a feasible solution (or equivalently, has a feasible solution ).

For any set , let denote the projections of along the -th coordinate. Formally,

 S↓i={b-i | ∃bi s.t. (b-i,bi)∈S}.

For a non-negative and monotone function , we have the following characterization.

###### Theorem 2.

Over a domain , a non-negative and monotone function is -attainable if and only if for any upward closed set ,

 ∑b∈Sw(b)⋅f(b)≤λ⋅∑i∑b-i∈S↓iw(b-i). (2)
###### Proof.

Only if (necessity). If is -attainable, then there exists a solution which satisfies all constraints in . Thus,

 ∑b∈Sw(b)⋅f(b) ≤λ⋅∑b∈Sw(b)∑ixi(b-i,bi) =λ⋅∑i∑b-i∈S↓iw(b-i)∑bi:b∈Sw(bi)⋅xi(b-i,bi) ≤λ⋅∑i∑b-i∈S↓iw(b-i)∑bi∈Λw(bi)⋅xi(b-i,bi) ≤λ⋅∑i∑b-i∈S↓iw(b-i).

If (sufficiency). Our goal is to find a feasible solution to the above linear system , given the system of inequalities (2). Our proof is constructive: we provide a procedure of continuously increasing , starting from 0, to a point where all constraints of are satisfied. In order to do that, we write a slightly more general linear system of the following form.

 {\sc LS}3: ⎧⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪⎨⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪⎩λ⋅∑ixi(b-i,bi)≥f(b),∀bN∑t=0w((1+δ)t)⋅xi(b-i,(1+δ)t)≤gi(b-i),∀i,b-ixi(b-i,(1+δ)t)≤xi(b-i,(1+δ)t+1),∀i,b% -i,t=0,1,2,…,N−1xi(b-i,bi)≥0,∀b

The only difference between and is in the second constraint. Intuitively, represents a total mass along direction at the point . Initially, , for all and , and in which case, the two linear systems and are identical. We will however decrease the values of ’s in the process of the proof while maintaining the following condition, derived from (2), for all upward closed sets .

 ∑b∈Sw(b)⋅f(b)≤λ⋅∑i∑b-i∈S↓igi(b-i)w(b-i). (3)

Let be the collection of upward closed sets for which the above inequality (3) is tight. It turns out that has a nice structure summarized in the following claim.

If , then .

###### Proof.

We note that both and are upward closed sets, then so are and . This implies that

 ∑b∈S1∪S2w(b)⋅f(b) ≤ λ⋅∑i∑b-i∈(S1∪S2)↓igi(b-i)⋅w(b-i) (4) ∑b∈S1∩S2w(b)⋅f(b) ≤ λ⋅∑i∑b-i∈(S1∩S2)↓igi(b-i)⋅w(b-i) (5)

As both , we have

 ∑b∈S1w(b)⋅f(b) = λ⋅∑i∑b-i∈S1↓igi(b-i)⋅w(b-i) ∑b∈S2w(b)⋅f(b) = λ⋅∑i∑b-i∈S2↓igi(b-i)⋅w(b-i)

For each , if is a projection of some , then either or ; if is a projection of some , then and . Hence,

 ∑b∈S1∪S2w(b)⋅f(b)+∑b∈S1∩S2w(b)⋅f(b) = ∑b∈S1w(b)⋅f(b)+∑b∈S2w(b)⋅f(b) = λ⋅∑i⎛⎜⎝∑b% -i∈S1↓igi(b-i)⋅w(b-i)+∑b-i∈S2↓igi(b-i)⋅w(b-i)⎞⎟⎠ ≥ λ⋅∑i⎛⎜⎝∑b% -i∈(S1∪S2)↓igi(b-i)⋅w(b-i)+λ⋅∑b-i∈(S1∩S2)↓igi(b-i)⋅w(b-i)⎞⎟⎠

Therefore, both inequalities (4) and (5) are tight. Thus, . ∎

The high-level idea of our proof is to continuously increase , starting from 0, to a point where all constraints of are satisfied. To implement the idea, we identify two special sets and , where , and a special set for a specific coordinate . Then for each , we find a threshold point and increase by for all . (In the proof, turns out to be the boundary of .) However, in the process, instead of increasing ’s, we decrease the values of and to simplify our analysis (thus, the values of ’s do not change): For each , we decrease by for to have an equivalent effect on the first constraint of ; further, we subtract from to balance the update of ’s in the second constraint of . The process continues until all ’s become 0, from which point we get an equivalent solution to the original problem. We next describe the formal proof.

Let be the support of . Since is a monotone function, we know that is an upward closed set. If , then we are done; thus, we assume that . For any , the following chain of inequalities

 ∑b∈Sw(b)⋅f(b)=∑b∈S∩Rw(b)⋅f(b)≤λ⋅∑i∑b-i∈(S∩R)↓iw(b-i)≤λ⋅∑i∑b-i∈S↓iw(b-i)

is in fact an equality. Hence, the above two inequalities are tight and . Hence, in the following, we will only consider those tight sets in that are contained in .

Let . We will maintain a chain of upward closed sets with the following structure:

 R=S0⊋S1⊋S2⊋⋯⊋Sm=∅.

During our process, we preserve the following key properties ():

Invariant properties () Inequality (3) holds for all upward closed sets. All sets in the chain remain tight. is nonnegative and monotonically increasing. is nonnegative and monotonically decreasing on for every and .

We note that all these properties hold at the beginning of the process (e.g., we may simply choose and ). Since , we have

 ∑b∈Rw(b)⋅f(b)>∑b∈S1w(b)⋅f(b).

Since satisfies condition (3) and , we have

 ∑b∈Rw(b)⋅f(b) ≤λ⋅∑i∑b-i∈R↓igi(b-i)⋅w(b-i) ∑b∈S1w(b)⋅f(b) =λ⋅∑i∑b-i∈S1↓igi(b-i)⋅w(b-i).

Hence,

 ∑i∑b-i∈R↓igi(b-i)⋅w(b-i)>∑i∑b-i∈S1↓igi(b-i)⋅w(b-i). (6)

Thus, we can find and such that and From now on to the end of the proof, for notational simplicity we use to denote this particular index rather than a generic one. Let ; note that

For each we consider the smallest such that We note that is well defined for each since . For the fixed index , we intend to update as follows:

 xi(b-i,b)←xi(b-i,b)+ε, for all b-i∈Ti and b≥ci(b-i),

for some fixed . In our process, instead of increasing we decrease the values of and as follows:

 (◊)  {f(b-i,b)←f(b-i,b)−λ⋅ε, for all b%−i∈Ti and b≥ci(b-i),gi(b-i)←gi(b-i)−εci(b-i), for all b-i∈Ti.

The decrements are with respect to the fixed index only1 and are implemented by continuously increasing from 0 until the value of one of and drops down to , or one more inequality (3) becomes tight for a new upward closed set.

Before describing how to proceed with the process, we establish some observations for the above updates .

###### Claim \theclaim.

For any and upward closed set , the two sides of the inequality (3) for remain unchanged. In particular, the inequality is still tight for all sets in the chain .

###### Proof.

The claim follows trivially since for any , none of and changes for every and . ∎

###### Claim \theclaim.

For any upward closed set and , the condition (3) is still satisfied.

###### Proof.

For the considered set , the changes of the left-hand side (LHS) and right-hand side (RHS) of (3) are as follows.

 LHS ← LHS−ελ⋅∑b-i∈Tiw(b-i)⋅(1+δ)N∑bi=ci(b-i)w(bi)=LHS−ελ⋅∑b-i∈Tiw(b-i)⋅1ci(b-i) RHS ← RHS−λ⋅∑b-i∈Tiw(b-i)⋅εci(b-i)

Hence, two sides of (3) decrease by exactly the same amount. Therefore, inequality (3) remains true for . ∎

###### Claim \theclaim.

The function remains monotonically increasing.

###### Proof.

Let us assume to the contrary that becomes non-monotone after one update. We note that all the four key properties hold before the update. Then there must exist a pair of vectors such that after the update. We have the following observations, where all variables denote their values before the update.

• Every value of either remains the same or decreases by . Thus, decreases and remains the same.

• , since must decrease. Thus, .

• , as .

• (otherwise, we would decrease by , since ).

• , since .

• , as .

• , as and is decreasing on .

Therefore, we obtain that and , a contradiction. ∎

###### Claim \theclaim.

The function remains monotonically decreasing on for each set in the chain .

###### Proof.

We note that does not change on for every . Thus, we only need to verify the claim for . Assume to the contrary that there exists a pair of vectors such that and after the update. We have the following observations.

• must decrease.

• .

• There exists , since .

• Let , then , as and is upward closed.

• since