# LP-based Approximation for Personalized Reserve Prices

We study the problem of computing revenue-optimal personalize (non-anonymous) reserve prices in eager second price auction in a single-item environment without having any assumption on valuation distributions. Here, the input is a dataset that contains the submitted bids of n buyers in a set of auctions and the goal is to return personalized reserve prices ṟ that maximize the revenue earned on these auctions by running eager second price auctions with reserve ṟ. We present a novel LP formulation to this problem and a rounding procedure which achievesOur main result in a polynomial-time LP-based algorithm that achieves a (1+2(√(2)-1)e^√(2)-2)^-1≈-approximation. This improves over the 12-approximation algorithm due to Roughgarden and Wang. We show that our analysis is tight for this rounding procedure. We also bound the integrality gap of the LP, which bounds the performance of any algorithm based on this LP.and obtains the best known approximation ratio even for the special case of independent distributions.

## Authors

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07/24/2020

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## 1 Introduction

Second price (Vickrey) auctions with reserves have been prevalent in many marketplaces such as online advertising markets [GLMN17, PLPV16, CS14]. A key parameter of this auction format is its reserve price, which is the minimum price at which the seller is willing to sell an item. While we have a full understanding of the optimal reserve prices when the buyers’ valuation distributions are, for example, i.i.d. and regular [Mye81]111The seminal work of [Mye81] shows that when the buyers’ valuation distributions are i.i.d. and regular, the monopoly price defined as for being the buyers’ valuation distribution is optimal., there are many practical applications including online advertising markets in which these assumptions fail to hold [GLMN17, CLMN14]. Furthermore, there are empirical and theoretical evidence that highlight the significance of setting personalized reserve prices for the buyers in order to maximize the revenue [EOS07, OS11, BGL18].

We study the problem of optimizing personalized reserve prices in second price auctions when the buyer valuations can be correlated. There are two different ways that personalized reserve prices can be applied in the second price auctions: lazy and eager [DRY15]. In the lazy version, we first determine the potential winner and then apply the reserve prices. In the eager version, we first apply the reserve prices and then determine the winner. In this work, we focus on optimizing eager reserve prices because (i) while the optimal lazy reserve prices can be computed exactly in polynomial time, they have worse revenue performance both in theory and practice, and (ii) eager reserves perform better in terms of social efficiency for similar revenue levels [PLPV16].

To optimize the eager reserve prices, we take a data-driven approach as suggested in the literature [PLPV16, RW16]. The input in this setting is a history of the buyers’ submitted bids/valuations over multiple runs of an auction and the goal, roughly speaking, is to set a personalized reserve price for each buyer such that the total revenue obtained on the same data set according to these reserve prices is maximized (see Section 2 for the formal definition). While the problem is APX-hard [RW16], the state-of-the-art algorithm of Roughgarden and Wang [RW16] achieves a -approximation which itself improves over an earlier -approximation algorithm by Paes Leme, Pál and Vassilvitskii [PLPV16]. Our main result is an algorithm with a significantly improved approximation factor:

[width=enhanced, frame hidden, boxsep=6pt, left=1pt, right=1pt, top=4pt, boxrule=1pt, arc=0pt, colback=mylightgray, colframe=black, breakable ]

###### Theorem (formally as Theorem 1).

There exists a randomized polynomial time algorithm that given a dataset, outputs a vector of reserve prices whose expected revenue is a

-approximation of that of the optimal value.

The known algorithms of the literature are all greedy and only take into account the two highest bids in each auction. Another limitation of these algorithms is that the reserve price for each buyer is computed in isolation. That is, the reserve price for a buyer only depends on the bids of the auctions in which the buyer submits the highest bid. In fact, [RW16] argue that these limitations are precisely what prevent their algorithm from obtaining any guarantee better than

. We bypass this bound by a careful analysis of a rounding technique for a natural linear programming formulation of the problem proposed in this work.

The optimal data-driven reserve prices solve an offline optimization problem, i.e., given a table of bid data, it computes the optimal reserve prices in retrospect. Such an approach, which is inspired by practice, does not need the knowledge of valuations/bids distributions. Suppose that there is a distribution over buyers’ valuations/bids and the goal is compute the optimal prices by having access to samples from that distribution [MM14, HMR18]. This setting is called batch learning in [RW16]. Using the machinery developed by Morgenstern and Roughgarden [MR15], by solving the data-driven offline optimization problem on the dataset with auctions, we can obtain fraction of the maximum revenue of any eager second price auction that one could have hoped to obtain by knowing the valuation distribution. This implies that the data-driven approach leads to approximately optimal reserve prices in the batch learning setting.

If the value distributions are independent, an improved approximation to personalized reserves are known via techniques like the correlation gap [Yan11, CHMS10] and prophet inequalities [KS78, HK81, ACK18, EHLM17, BGL18, CSZ19] (to cite a few). The latest result is -approximation by Correa et al [CSZ19]. Although those results are typically states as an approximation ratio with respect to the (stronger) Myerson revenue benchmark, those are also the best-known approximation ratios with respect to the optimal reserve prices for independent distributions.

#### Our result and techniques

Our main contribution is a polynomial-time -approximation algorithm for the data-driven reserve prices problem with correlated distributions, improving over the -approximation of [RW16]. This implies a -algorithm for the batch learning version of the problem using the reduction in [MR15]. It also implies a polynomial time -algorithm for independent distributions, which beats the best approximation known via prophet techniques222While we provide a better guarantee against the optimal reserves, our technique does not provide approximation guarantees with respect to the optimal auction as prophet inequalities do..

To overcome the limitation of algorithms by [PLPV16] and [RW16], we present an algorithm called “Profile-based LP-Rounding”, Pro-LPR for short, that takes advantage of a concise representation of the solution space. This representation, that we call profile space, is inspired by how revenue is computed in the eager auctions. Working with the profile space enables us to consider all the bids in an auction, not only the highest and second highest bids, to set the reserve prices. It further allows us to describe the optimal solution by a polynomial-size integer program. By relaxing the integrality constraints on the variables of the integer program, we construct a linear program (LP). The fractional solution of the LP is then rounded to obtain the reserve prices. The final reserve price of the algorithm is the best of the zero reserves and the reserves obtained from rounding the solution of the LP. The most technically challenging step in the analysis is to bound the approximation ratio. This is done via careful probabilistic analysis of the rounding procedure which leads to a non-linear mathematical program bounding the ratio. Our last step is to use techniques from non-linear optimization to bound the solution of the mathematical program. We would like to emphasize that our analysis of our algorithm is tight in a sense that there is an example for which our algorithm cannot get an approximation factor better than .

Finally, we point out that the performance of our algorithm is evaluated against the optimal value of the LP, which is an upper bound on the maximum revenue. By analyzing the integrality gap of the LP, we show that no algorithm can obtain more a 0.828 fraction of the optimal value of the LP; see Theorem 3. This highlights that our algorithm is evaluated against a powerful benchmark and despite that, it obtains   fraction of this powerful benchmark.

#### Other related work

Our work relates and contributes to the broad literature on revenue-maximizing mechanisms in a single-item environment. Within that line of work, our paper is in the intersection of two major sub-streams: (i) reserve price optimization and (ii) auction design for correlated valuations. Most of the reserve price optimization literature has been devoted to the case where valuations are independent, see Hartline and Roughgarden [HR09], Yan [Yan11], Dhangwatnotai et al [DRY15] and more recently, a very fruitful line of work on posted-price and reserve-price optimization via prophet inequalities [KS78, HK81, ACK18, EHLM17, BGL18, CSZ19].

This work on auction design for correlated distributions pioneered by Ronen [Ron01] and Ronen and Saberi [RS02]. The positive and negative results were later improved by Dobzinski et al [DFK11] and Papadimitriou and Pierrakos [PP11]. Our paper departs from this line work in the sense that we do not try to approximate the optimal incentive-compatible auction, but instead, we try to approximate the best auction in the subclass of second price auctions with reserves. Note that this is the auction format adopted by most online marketplaces, including online display advertising markets.

The rest of the paper is organized as follows. In Section 2, we define the model. Section 3 presents a high level view of the results and techniques. In Section 4, we provide our LP, which will be used as our benchmark. In Section 5, we present the LP-base algorithm and show its performance guarantee. Section 7 provides the proof of the integrality gap and Section 6 shows that our analysis is tight. We conclude in Section 8.

## 2 Preliminaries and Problem Statement

There are buyers participating in a set of single-item eager second price auctions. Let and respectively denote the set of auctions and buyers. For any buyer , and for any auction , we are given a non-negative number which indicates the bid of buyer in auction . Let be the personalized reserve price of buyer . Then, given the bids in auction and reserve prices , the eager second price (ESP) auction works as follows.

• First, any buyer with is eliminated. Let be the set of buyers who clear their reserve prices in auction .

• When set is nonempty, the item is allocated to buyer who has the highest bid among all the buyers in set and is charged

 \sf{Rev}a(r):=max{rb⋆a, maxb∈Sa,b≠b⋆a {βa,b}}.

Note that and are implicitly depend on reserve prices . Any other buyers , are not charged. Further, when set is empty, the item is not allocated and .

Note that the reserve prices are the same across all the auctions . However, each buyer is assigned a personalized reserve price . Given the dataset of bids , our goal here is to find personalized reserve prices that maximize revenue of the auctioneer. See the introduction section for a discussion on the nice properties of this data-driven optimization. Formally, we would like to solve the following optimization problem:

 \sf{ESP}⋆ = maxr∈Rn  \sf{Rev}(r):=∑a∈A\sf{Rev}a(r). (ESP-OPT)

Note that, without loss of generality, we assume that the optimal reserve price for buyer is equal to one of his submitted bids . Let . Then, Problem ESP-OPT can be rewritten as , which leads to a search space of size .

## 3 Results and Techniques

The main result of the paper is a randomized algorithm that returns an -approximation solution for Problem ESP-OPT.

[width=enhanced, frame hidden, boxsep=6pt, left=1pt, right=1pt, top=4pt, boxrule=1pt, arc=0pt, colback=mylightgray, colframe=black, breakable ]

###### Theorem 1 (Main Theorem).

There exists a randomized polynomial time algorithm that given a dataset , outputs a vector of eager reserve prices whose expected revenue is a -approximation of that of the optimal value of Problem ESP-OPT, denoted by .

To find an approximate solution, the overall idea is to construct an LP whose objective function at its optimal solution provides an upper bound for . The LP that takes advantage of a concise representation of the solution space, has a polynomial number of variables and constraints. Then, we use a rounding technique to transform the optimal solution of the LP to a vector of reserve prices. We show that if we consider the reserve prices obtained from the rounding technique and the vector of all-zero reserve prices and choose the one with the maximum revenue, we obtain the desired approximation factor. In Theorem 2, we further show that our analysis of our approximation factor is tight. That is, we provide an example for which our algorithm obtains exactly fraction of the optimal value of the LP, i.e., the upper bound on for . Finally, in Theorem 3, we bound the integrality gap of the LP. This characterization shows that no algorithm can obtain more than fraction of the LP.

## 4 Linear Program

The main challenge in designing an LP formulation for this problem is to find a concise representation of the solution space. Instead of considering all possible assignments of reserves to buyers, we will consider only partial assignments in which we only specify the reserve prices of two buyers. We will call such partial assignment a profile. Formally, a profile is a tuple , which represents an assignment of reserve to buyer and reserve to buyer . If it is the case that the reserves are below the corresponding bids in an auction , i.e. and , then no matter how the assignment of the remaining reserves, the revenue of this partial assignment is at least for . We also note that given any vector of reserve prices , the revenue that can be obtained from only depends on the reserve price of the highest and second highest bidders that clear the reserve prices.

Next, we formally define the notion of valid profile and show that the ESP-OPT problem can be relaxed to finding the best consistent distribution over valid profiles in each auction. To define valid profiles, we assume that the data has two additional buyers and who always bid zero which means . We further elaborate on this.

###### Definition 4.1 (Valid Profiles).

We define the set of valid profiles for auction as the set consisting of all tuples satisfies the following conditions:

1. Bid of buyer is greater than or equal to that of buyer ; that is, .

2. Buyer clears his reserve; that is, .

3. Buyer clears his reserve; that is, .

For any given , we define .

By adding buyers and to , we can define valid profiles to represent the cases in which less than two buyers cleared their reserve prices. We present the cases with one (respectively zero) cleared buyer with valid profile of (respectively ).

Note that we abuse notation and use for both revenue from reserves and revenue from profiles. The following lemma (which follows from the preceding discussion) states that reserve price vectors can always be mapped to a profile with the same revenue.

###### Lemma 4.2.

Given a vector of reserve prices and an auction , there is a valid profile such that . Such a profile is called the profile corresponding to reserve price vector .

We are now ready to describe our LP.

#### Decision variables of the LP:

The LP will have two sets of variables:

1. For any auction and any valid profile , define a variable such that

. This variable represents a probability distribution over valid profiles in auction

. We refer to as a profile-solution.

2. For any buyer and reserve price ,define a variable such that . This variable represents be the probability that buyer is assigned a reserve price of .

Finally, we add constraints relating and which will ensure the consistency of probability distributions across all profiles. To define this set of constraints, for every , , and , we define set

 Qb,r,a:={p=(b1,b,r1,r):p∈Pa}∪{p=(b,b2,r,r2):p∈Pa}, (1)

which corresponds to all valid profiles of auction that assign reserve to buyer . A natural constraint to add is that the total probability assigned to profiles in is at most the probability that buyer is assigned to reserve price . That is,

 ∑p∈Qb,r,asa,p≤qb,r.

Finally, we can put it all together in the following LP:

 maxq,s ∑a∈A∑p∈Pasa,p⋅\sf{Rev}a(p) s.t. ∑p∈Pasa,p≤1 ∀a:a∈A ∑p∈Qb,r,asa,p≤qb,r ∀a,b,r:b∈B,r∈\sf{R},a∈A ∑r∈Rqb,r=1 ∀b:b∈B sa,p≥0 ∀a,p:a∈A, p∈Pa (Profile-LP)

We start by noting that the LP is a relaxation of the ESP-OPT problem:

###### Lemma 4.3 (Upper bound on Revenue).

The solution of Profile-LP is an upper bound to , i.e., the optimal value of Problem ESP-OPT.

###### Proof.

Given reserve prices such that , we construct a feasible solution to the LP as follows. For each , we let for the profile corresponding to (according to lemma 4.2) and for all remaining profiles. Further, we let and for all remaining reserves. It is straightforward to verify that it is a feasible solution to the Profile-LP and that . ∎

## 5 Profile-based LP-rounding (Pro-LPR) Algorithm

In this section, we present an algorithm, called Profile-based LP-rounding (Pro-LPR), that uses the optimal solution of (Profile-LP), , to devise reserve prices. Our rounding procedure is as follows:

• Construct reserve prices . To do so, for each buyer , independently sample reserve price with probability proportional to .

• Let be the vector of all zero reserves. Output the best of and , i.e.,

 rout←argmaxr∈{z,rR}\sf{Rev}(r).

Now we analyze the rounding procedure and show that is at least a fraction of the solution of the Profile-LP and hence at most . One of the biggest strengths of our LP formulation is that it allows the analysis to decouple the effect of rounding for each individual auction.

###### Lemma 5.1 (Two Conditions).

Let and be the optimal solution of (Profile-LP) and be a random reserve price obtained from the rounding procedure. If there exists a constant such that for any and any auctions , we have

 ∑{p:p∈Pa,Rev(p)≥t}s⋆a,p−Pr[Reva(rR)≥t] ≤ 0 for t>β(2)a (2) ∑{p:p∈Pa,Rev(p)≥t}s⋆a,p−Pr[Reva(rR)≥t] ≤ c for t≤β(2)a, (3)

then Pro-LPR algorithm is a -approximation. That is, it obtains at least fraction of the optimal value of Problem ESP-OPT. Here, is the second highest bid in auction and is the revenue in auction under reserve prices .

###### Proof of Lemma 5.1.

By integrating over in Equations (2) and (3) and adding them up, we get

 ∫∞β(2)a(∑{p:p∈PaRev(p)≥t}s⋆a,p−Pr[Reva(rR)≥t])dt

This is simplified as follows

 ∑p∈Pas⋆a,pReva(p)−E[Reva(rR)]≤c⋅β(2)a. (4)

Then, if , Equation (4) leads to

 ∑a∈A∑p∈Pas⋆a,pReva(p)−∑a∈AE[Reva(rR)]≤c⋅∑a∈Aβ(2)a=c⋅x⋅\sf{ESP}⋆ (5)

Here . Note that by Lemma 4.3, the optimal value of Problem ESP-OPT, denoted by , is upper bounded by . That is,

 \sf{ESP}⋆≤\sf{Rev}(s⋆)=∑a∈A∑p∈Pas⋆a,pRev(p). (6)

Further, the revenue of Pro-LPR algorithm, i.e, , is lower bounded by

 E[Rev(rout)]≥max(∑a∈Aβ(2)a,E[Rev(rR)]). (7)

To see why this holds note that Pro-LPR algorithm returns the best of reserve price and all zero prices, where the revenue under all zero prices is the sum of the second highest highest bids . By using Equations (6) and (7) in (5), we have

 \sf{ESP}⋆−E[Rev(rout)]≤c⋅x⋅%ESP⋆.

Invoking Equation (7) again, we have

 E[Rev(rout)]≥∑a∈Aβ(2)a=x⋅\sf{ESP}⋆.

Putting these together, we have

 E[Rev(rout)]≥max(x,1−c⋅x)⋅% \sf{ESP}⋆≥11+c⋅\sf{ESP}⋆,

which is the desired result. ∎

By Lemma 5.1, to complete the proof of the main theorem it suffices to prove that Equation (2) holds for any auction , and find a constant that satisfies Equation (3). The following lemma shows that Equations (2) holds.

###### Lemma 5.2 (First Condition Holds).

Let denote an optimal solution of (Profile-LP) and let be the profile-solution associated with the vector of reserve prices , defined in Pro-LPR Algorithm. For any auction , we have

 ∑{p:p∈Pa,Rev(p)≥t}s⋆a,p−Pr[Reva(rR)≥t] ≤ 0 for t>β(2)a. (8)
###### Proof.

The first term in the l.h.s. of (8) can be written as

 ∑{p:p∈Pa,Rev(p)≥t}s⋆a,p=∑{p:p∈Pa,p=(b(1)a,b2,r,r2), r≥t}s⋆a,p ≤ ∑r≥tqb(1)a,r=Pr[Reva(rR)≥t], (9)

where the first equation holds because revenue of a profile is if and only if the bidder with the highest bid in auction , i.e., , is assigned a reserve price and the bid of this bidder is greater than . The second equation holds because of the second set of constraints of (Profile-LP). The last equation follows from the construction of reserve prices . Note that Equation (9) verifies condition (8). ∎

### 5.1 Bounding the Constant in the Second Condition

We start by noting that the second condition in Lemma 5.1 holds trivially for , which recovers the same approximation factor of of [RW16]. For the rest of the paper, we will improve past by constructing a non-linear mathematical program to optimize and then applying the first order conditions in non-linear programming to bound the optimal solution. In Lemma 5.3, we show that

 c=maxθ∈[0,1]\sf OPT(θ),

where for any real number , is defined as follows

 \sf OPT(θ) = maxx≥0 eθ−1⎡⎣∏i∈[n](1−xi)+∑i∈[n]xi∏j∈[n],j≠i(1−xj)⎤⎦ s.t. 12∑i∈[n]xi=θ xi≤θ,∀i∈[n], (10)

where is the number of buyers. Characterizing is technically involved and because of that its details is postponed to Section 5.2. There, we show that for any number of buyers and any real number ,

 \sf OPT(θ)≤2(√2−1)e√2−2≈0.4612.

Then, invoking Lemmas 5.1 and 5.2, this leads to the approximation factor of , which is the desired result.

In the next lemma, we formally state the relationship between and the approximation factor of our algorithm.

###### Lemma 5.3 (Second Condition).

Let denote an optimal solution of Profile-LP and be the vector of reserve prices, defined in Pro-LPR Algorithm. Let

 c=maxθ∈[0,1]\sf OPT(θ).

Then, for any auction , the following equation holds.

 ∑{p:p∈Pa,Rev(p)≥t}s⋆a,p−Pr[Reva(rR)≥t] ≤ c for t≤β(2)a.

To show Lemma 5.3, we make use of the following lemma.

###### Lemma 5.4.

Given fixed with and , the following inequality holds:

 ∏b∈B(1−x1,b−x2,b)+∑b∈Bx2,b∏b′≠b(1−x1,b′−x2,b′)≤∏b∈B(1−x1,b)⎡⎣∏b∈B(1−x2,b)+∑b∈Bx2,b∏b′≠b(1−x2,b′)⎤⎦

The proof of Lemma 5.4 is provided at the end of this section. Before it, we use Lemma 5.4 to prove Lemma 5.3.

###### Proof of Lemma 5.3.

We start with a few definitions. Consider a certain auction and all of its valid profiles . Fix some threshold and an optimal solution of (Profile-LP), denoted by . Consider a buyer . Then, define

 X′1,b={p=(b,b2,r1,r2):p∈Pa, r1≥t}
 X′′1,b={p=(b1,b,r1,r2):p∈Pa, r1
 X2,b={p=(b1,b2,r1,r2):p∈Pa,b∈{b1,b2},r1,r2

and then set:

 x1,b=∑p∈X′1,b∪X′′1,bs⋆a,pandx2,b=∑p∈X2,bs⋆a,p.

We note that is the set of all valid profiles in which reserve of buyer is at least . is the set of all valid profiles in which reserve of buyer is less than and reserve of buyer is greater than or equal to . Observe that for all the profiles in , reserve of buyer is at least . This implies that for all of these profiles, . We also note that is the set of all valid profiles such that buyer and bid of buyer is at least . Again, it is easy to see that for any valid profile , we have . Finally, we point that while any profile in and has , by construction, and are disjoint. Therefore, we have

 ∑{p: p∈Pa, Rev(p)≥t}s⋆a,p=∑b∈Bx1,b+12∑b∈Bx2,b, (11)

where the coefficient accounts for double-counting. That is, while any profile in contributes to once, any profile in contributes to twice.

Define as the probability that the sampled reserve of buyer , i.e., , is in and as the probability that the sampled reserve is in . By the sampling procedure we know that:

 y1,b≥x1,bandy2,b≥x2,b.

Observe that iff at least one of the two following events happen.

1. Event : There exists a buyer with a reserve of at least whose bid is cleared.

2. Event : There are at least two buyers with cleared bids greater than or equal to .

Precisely,

 Pr[Reva(rR)≥t]=Pr[E1 or E2]=Pr[E1]+Pr[E2 and ¯E1]=Pr[E1]+Pr[¯E1]Pr[E2|¯E1], (12)

where

 Pr[E1]=1−∏b∈B(1−y1,b)

and

 Pr[E2|¯E1]=1−∏b∈B(1−~y2,b)−∑b∈B~y2,b∏b′≠b(1−~y2,b′) for ~y2,b=y2,b1−y1,b

This gives us

 Pr[E2 and ¯E1]=Pr[¯E1]Pr[E2|¯E1]=Pr[¯E1]−∏b∈B(1−y1,b−y2,b)−∑b∈By2,b∏b′≠b(1−y1,b′−y2,b′).

Thus, by Equation (12), we get

 Pr[E2 or E1]=1−∏b∈B(1−y1,b−y2,b)−∑b∈By2,b∏b′≠b(1−y1,b′−y2,b′).

Now observe that the expression above, i.e., , is increasing in both and , . To see why is increasing in , note that

 ∂(Pr[E2 or E1])∂y2,b =∑b′∈B,b′≠by2,b′∏b′′≠b,b′(1−y1,b′′−y2,b′′)≥0.

This implies that:

 Pr[E2 or E1]≥1−∏b∈B(1−x1,b−x2,b)−∑b∈Bx2,b∏b′≠b(1−x1,b′−x2,b′).

We now invoke Lemma 5.4, stated earlier, to get

 Pr[E2 or E1]≥1−∏b∈B(1−x1,b)⎡⎣∏b∈B(1−x2,b)+∑b∈Bx2,b∏b′≠b(1−x2,b′)⎤⎦. (13)

Using Equations (11), (13), and (12), we get

 ∑{p:p∈Pa,Rev(p)≥t}s⋆a,p−Pr[Reva(rR)≥t]≤ ∑b∈Bx1,b+12∑b∈Bx2,b−⎛⎝1−∏b∈B(1−x1,b)⎡⎣∏b∈B(1−x2,b)+∑b∈Bx2,b∏b′≠b(1−x2,b′)⎤⎦⎞⎠ (14)

It easy to check that for any , the above expression is non-decreasing in . This allows that to assume without loss of generality333To see why, suppose that , where . Then, by replacing with , the r.h.s. of Equation (14) can only increase. Therefore, without loss of generality, we assume that . that .

As a result, we have

 ∑{p:p∈Pa,Rev(p)≥t}s⋆a,p−Pr[Reva(rR)≥t]≤∏b∈B(1−x1,b)⎡⎣∏b∈B(1−x2,b)+∑b∈Bx2,b∏b′≠b(1−x2,b′)⎤⎦,

where , . Here, . To complete the proof, we simply use that: . Given how we constructed the variables , we also need . Hence,

 ∑{p:p∈Pa,Rev(p)≥t}s⋆a,p−Pr[Reva(rR)≥t]≤eθ−1⎡⎣∏b∈B(1−x2,b)+∑b∈Bx2,b∏b′≠b(<