Interference Among First-Price Pacing Equilibria: A Bias and Variance Analysis

## Abstract

Abstract. Online A/B testing is widely used in the internet industry to inform decisions on new feature roll-outs. For online marketplaces (such as advertising markets), standard approaches to A/B testing may lead to biased results when buyers operate under a budget constraint, as budget consumption in one arm of the experiment impacts performance of the other arm. To counteract this interference, one can use a budget-split design where the budget constraint operates on a per-arm basis and each arm receives an equal fraction of the budget, leading to “budget-controlled A/B testing.” Despite clear advantages of budget-controlled A/B testing, performance degrades when budget are split too small, limiting the overall throughput of such systems. In this paper, we propose a parallel budget-controlled A/B testing design where we use market segmentation to identify submarkets in the larger market, and we run parallel experiments on each submarket. Our contributions are as follows: First, we introduce and demonstrate the effectiveness of the parallel budget-controlled A/B test design with submarkets in a large online marketplace environment. Second, we formally define market interference in first-price auction markets leveraging the first price pacing equilibrium (FPPE) framework which has been proposed to model autobidding in repeated auctions. Third, we propose a debiased surrogate that eliminates the first-order bias of FPPE, drawing upon the principles of sensitivity analysis in mathematical programs. Fourth, we derive a plug-in estimator for the surrogate and establish its asymptotic normality. Fifth, we provide an estimation procedure for submarket parallel budget-controlled A/B tests. Finally, we present numerical examples on semi-synthetic data, confirming that the debiasing technique achieves the desired coverage properties.

## 1. Introduction

Online A/B testing, or *experimentation*, is widely used in the internet industry. Major tech companies conduct a large number of A/B tests annually to inform decisions related to ads delivery, online auction mechanisms, user interface components, new feature rollouts, infrastructure improvements, and many other aspects of their products and services.

In the context of online auctions in marketplaces, where buyers bid for sellers’ items or advertisers bid for users’ impressions, budget-controlled A/B testing has gained popularity and is supported by companies like LinkedIn, Google, and Meta (Basse et al., ; Liu et al., ). The primary advantage of budget control in A/B testing is that it eliminates interference between the A and B groups that could otherwise bias the results: In a traditional A/B test, the budget of a buyer is distributed across all auctions, involving users from both A and B groups. This couples the behavior of the buyer across the groups; for example, increased expenditure under B might decrease expenditure in A, causing interference. However, in a budget-controlled A/B test, each buyer spends half of their budget on users from the A group and the other half on users from the B group, creating two separate but mirrored markets, and thus removing the interference caused by a shared budget constraint. This experiment type, therefore, is commonly used to test various ads delivery changes, such as pacing algorithms (see Section 2.1).

Despite clear advantages of budget-controlled A/B testing, companies are extremely constrained by the number of such experiments they can run: buyers (advertisers) can only be charged once for the same item (impression), it is thus not feasible to create additional experimentation throughput by reshuffling users between the A and B groups. An alternative idea is to create more budget splits so more treatments can be tested at the same time. However, this will lower the budget-per-group substantially, which could lead to different equilibrium outcomes and may disproportionately affect smaller buyers. In addition, it poses challenges with respect to statistical power, in view of the fact that tech companies primarily focus on down-funnel conversion events which are difficult to measure effects on to begin with. In fact, the lack of bandwidth for budget-controlled A/B tests has become one of the major bottlenecks in experimentation for the internet industry.

A third approach to increase throughput is market segmentation. If the overall market can be divided into several relatively isolated submarkets, budget-controlled A/B tests can be conducted in parallel within these submarkets. However, this method also presents some challenges. First, submarkets are rarely completely isolated; certain items may attract buyers from multiple submarkets, resulting in interference across
submarkets when conducting tests in parallel. Second, submarkets differ in terms of buyer (and user) compositions, which might cause the local treatment effect estimates being not representative of the global treatment effect where all buyers are included in the market. The second challenge is relatively easy to address in practice by imposing balancing constraints in the clustering algorithm used to define submarkets, while the first challenge is more fundamentally important and requires deeper theoretical understanding.

As a motivating example, Fig 1(a) compares the results of submarket budget-controlled A/B tests (vertical axis) with those obtained from running budget-controlled A/B tests on the entire market (horizontal axis). The figure reports a normalized version of one of the real experiment metrics at Meta, with each submarket covering approximately 6-8% of advertisers and each experiment condition covering 30% of users. All submarkets are carefully created with limited imbalances in advertiser composition. We consider 16 different treatments, most of which are related to the pacing algorithm, each run for about 6 days, resulting in a total of 99 daily comparisons. The benchmark for sign-consistency at a 90% significance level (for neutral true effects) is 81.5%. The observed sign consistency rate between submarket and standard budget-controlled experiments is only slightly lower, at around 75%. Our focus on sign consistency in this context is intentional, as it directly relates to the frequency of production decision reversals, which we aim to minimize as much as possible.

Figure: (a) All datapoints.
Refer to caption: extracted/5403078/figs/metric2_before_guardrail.png

Table 1 shows the sign consistency rate of comparisons for daily and 6-day aggregated results, including for a second metric. Since submarkets are not perfectly isolated, we use a guardrail metric to measure the interference between submarkets, and use a threshold on the guardrail metric to improve reliability. As seen in Table 1, by eliminating experiments where the guardrail metric violates the threshold, we meaningfully improve the consistency rate for both metrics in daily and 6-day results. In practice, approximately 10% of submarket experiments violate the guardrail threshold. Fig 1(b) demonstrates the treatment effect comparisons for daily results after we remove such data points. The results are generally satisfactory, however the existence of biases is evident even after removing data points that violated the guardrail threshold.
Evidently, quantitatively understanding the interference bias to improve the accuracy of the estimates is conducive to better decisions, and that is what we set to do in this work.

**Table 1. Sign consistency between submarket and standard budget-controlled A/B test**
| Duration | Data | Num | Sign consistency (%) |  |
| --- | --- | --- | --- | --- |
| Metric 1 | Metric 2 |  |  |  |
| 6 days | All combos | 16 | 87.5 | 68.8 |
| 6 days | within guardrail | 14 | 100 | 78.6 |
| 1 day | All combos | 99 | 75.8 | 73.7 |
| 1 day | within guardrail | 84 | 81.0 | 76.2 |

The contributions of this paper are as follows. First, we introduce and demonstrate the effectiveness of the parallel budget-controlled A/B test design with submarkets in Meta’s production environment, which, to our best knowledge, is a novel approach. Second, we formally define market interference in first-price auction markets using the First Price Pacing Equilibrium (FPPE) framework (Conitzer et al., ). Third, we propose a debiased surrogate that eliminates the first-order bias of FPPE, drawing upon the principles of sensitivity analysis in mathematical programs (Shapiro et al., ). Fourth, we derive a plug-in estimator for the surrogate and establish its asymptotic normality. Fifth, we provide an estimation procedure for submarket parallel budget-controlled A/B tests. Finally, we present numerical examples on semi-synthetic data, confirming that the debiasing technique achieves the desired coverage properties.

Section 2 introduces the FPPE framework. The formal problem setup is given in Section 3. Asymptotic theories around FPPE are presented in Section 4. Section 5 formalizes the parallel A/B testing problem under interference and provides the estimation procedure. Numerical examples on synthetic and semi-synthetic data are presented in Section 6. We conclude in Section 7.

### 1.1. Parallel A/B Testing in Practice

In this section we describe the real-world problem of A/B testing with congestion that we wish to model, and our proposed solution of parallel A/B tests in carefully balanced submarkets; the formal model appears in Sec. 2 and Sec. 3. Along with the description of the real-world problem, we will also discuss which aspects of it will be or will not be captured by our theoretical model.

We start by describing the market environment. Assume that there is a set of $n$ advertisers, and each advertiser $i$ has a budget $b_{i}$. Whenever a user shows up on the platform an *impression opportunity* occurs, and an auction is conducted in order to determine which subset of ads will be shown to the user. Each advertiser $i$ has some stated value $v_{i}(\theta)$ of being shown to a particular impression opportunity $\theta$. This may not be their true valuation; in practice it may be their stated willingness to pay for a click, multiplied by an estimated click-through-rate for the pair ($i$,$\theta$). The advertiser submits a bid which is determined based on $v_{i}(\theta)$, as well as the expenditure of the advertiser so far. For example, in *multiplicative pacing* (Balseiro et al., ; Conitzer et al., , ), the platform adaptively learns a *pacing multiplier* $\beta_{i}\in[0,1]$ such that the bid is formed as $\beta_{i}v_{i}(\theta)$. The budget-management system then adaptively controls $\beta_{i}$ over time, in order to ensure the correct rate of budget expenditure on behalf of the advertiser.
The auction format for an impression opportunity is typically some generalization of the first or second-price auction, though the real-world auctions are not fully captured by any simple-to-describe auction model from the literature. For example, in the setting of inserting ads into a news feed, multiple ads may be allowed to be shown, but the exact number may depend on the size of each ad, the size of other content in the feed, and so forth.

In our theoretical model, we will simplify the auction format in three respects:
first, we will assume that each impression opportunity allows exactly one ad to be shown. This is a common assumption in the literature studying both online learning for budget management (Borgs et al., ; Balseiro and Gur, ) as well as in the literature on equilibrium outcomes under budget management (Balseiro et al., ; Conitzer et al., , ; Aggarwal et al., ; Balseiro et al., ).
Thirdly, we assume that each individual auction is a *first-price auction*, meaning that the highest bidder wins, and pays their bid. Internet advertising uses both first and second-price auctions, depending on the particular context (Balseiro et al., ; Goke et al., ), though generalizations of the second-price auction are arguably still more common, in particular for sponsored search and feed auctions. We focus on first-price auctions because the equilibrium outcomes under budget management satisfy two crucial properties from the perspective of statistical inference: they are unique, and the equilibrium is captured by a convex program (Conitzer et al., ).
In contrast, equilibria under budget management in second-price auctions may not be unique (Conitzer et al., ), and they are PPAD-hard to compute (Chen et al., ). It is an interesting future research direction to figure out how to develop results for the second-price setting, though we expect it to be much harder.
We will also abstract away the dynamic temporal aspects of budget management: instead of the pacing multipliers being learned over time, we will assume that what we observe is the set of pacing multipliers in equilibrium. Secondly, in the real world, advertisers may use a variety of budget management algorithms beyond multiplicative pacing. For example, Google also offers *probabilistic throttling*, where bids are not modified, and instead the expenditure is managed by independently flipping a coin for each auction, in order to determine whether to bid at all (Balseiro et al., ; Chen et al., ; Karande et al., ).
We assume that every advertiser uses multiplicative pacing in order to manage budgets.
Budget-management equilibria in first-price auctions with probabilistic throttling are known to also be unique and computable (Chen et al., ), so it is possible that similar results could be obtained for that setting, though it is not known whether those equilibria are captured by a convex mathematical program, which is a crucial fact for our technical approach.

Now that we have discussed budget management, we describe the A/B testing problem. Suppose that a platform wants to run $K$ A/B tests, which may affect, e.g., the valuations that advertisers have for impression slots, revenue, etc.
Naively performing the A/B test on a per-impression or per-user basis and then simply running the market as usual will not work, because the budget management system means that the A and B users will be affected by each other through the budget coupling.
As mentioned in the introduction, a possible option is to perform budget splitting: create $K+1$ copies of each advertiser (corresponding to each test, as well as a holdout “A” control copy), randomize users into one of the $K+1$ markets, and run separate budget management for each copy of the $K+1$ advertisers. As mentioned earlier, this approach runs into problems because small advertisers end up with prohibitively-small budgets, and there is a large negative impact on statistical power because some of the events measured by the A/B tests might be far downstream of an impression.

Instead, our paper considers the following parallel A/B test paradigm based on market segmentation.
We first define a bipartite graph between advertisers and users based on targeting criteria. Subsequently, we cluster the advertisers into $K$ clusters, where the objective is to minimize the sum of weighted edges between clusters subject to traffic balancing constraints to make the resulting clusters as similar to the whole market as possible. The edge weight between a pair of advertisers is the number of impressions (or users) where they are both within the top-$k$ bids. The choice of $k$ is a parameter that must be chosen based on experience with the specific application setting.
If the clustering achieves a small objective function value, then each cluster is a mostly isolated submarket, in the sense that each user will mostly receive bids from advertisers in a single cluster.
Then, we run an A/B test within each of the $K$ submarkets.
Every user is randomly assigned to either “A” or “B” in each submarket.
Note that the same user can be in “A” in submarket 1 and “B” in submarket 2. This means that all the A/B tests must be mutually compatible; in practice this means that our approach is best suited for A/B tests that apply to advertisers, as opposed to, e.g., user-side user interface changes.
If the interaction between different submarkets is not too strong, then we expect that hypothesis testing in a submarket is indicative of the true treatment effect within that submarket.
Secondly, if each submarket is representative of the overall market, then such a hypothesis test is also indicative of the true treatment effect across the whole market.

In our theoretical model we use a coarser notion of contamination between a pair of submarkets than the top-$k$ measure that we use in practice. In our theoretical model, we measure the number of items with *any* bids from buyers from separate submarkets.
In practice, this measure is too coarse (and often very large). It is an interesting follow-up question to come up with stronger theoretical results that allow for finer-grained measures of contamination.
Secondly, our theoretical model will assume that what we are interested in is the treatment effect within a submarket. This implicitly assumes that each submarket is representative of the overall market.

## 2. Theory overview

### 2.1. Related Work

Pacing equilibrium.
Pacing is a budget management strategy where bids are scaled down by a multiplicative factor in order to smooth out and control spending.
In the first-price setting,
Borgs et al. () study first price auctions with budget constraints in a perturbed model, where the limit prices converge to those of an FPPE.
Building on that work, Conitzer et al. () introduce the FPPE framework to model autobidding in repeated auctions,
and discover several properties of FPPE such as shill-proofness monotonicity properties, and a close relationship between FPPE and the quasilinear Fisher market equilibrium (Chen et al., ; Cole et al., ).
Gao and Kroer () propose an infinite-dimensional variant of the quasilinear Fisher market, which lays the probability foundations of the current paper.
Gao et al. () and Liao et al. () study online computation of the infinite-dimensional Fisher market equilibrium, and we utilize their methods to compute equilibria.
In the second-price setting, a variety of models have been proposed for modeling the outcome of budget management:
Balseiro et al. () study budget-management in second-price auctions through a fluid mean-field approximation;
Balseiro and Gur () investigate adaptive pacing strategy from buyers’ perspective in a stochastic continuous setting; and
Balseiro et al. () study several budget smoothing methods including multiplicative pacing in a stochastic context.
A second-price analogue of FPPE was explored by Conitzer et al. ().
Modeling statistical inference in the second-price setting is most likely harder, due to equilibrium multiplicity issues (Conitzer et al., ), and hardness of even computing equilibria (Chen et al., ).

Recently, a series of papers (e.g., Aggarwal et al. (); Deng et al. (); Balseiro et al. ()) investigated an “autobidding” model for value-maximizing bidders, who aim to maximize their value subject to budget or return-on-spend constraints, which may include buying individual ad impressions that have negative utility. Several extensions provide results for more general settings, such as the quality of equilibria for generalized rationality models for bidders (Babaioff et al., ), non-uniform bidding strategies (Deng et al., ), and liquid welfare guarantees when budget-constrained buyers fail to reach equilibrium (Gaitonde et al., ).
Statistical inference for parallel A/B testing under such models represent an interesting future direction of work.

Interference in A/B Testing for Auction Markets. Experimentation under interference has been extensively studied in the past decade within the context of social networks (Ugander et al., ; Aronow and Samii, ; Eckles et al., ; Athey et al., ; Li and Wager, ) and two-sided marketplaces (Zigler and Papadogeorgou, ; Wager and Xu, ). Under interference, the outcome of an experiment unit, either a user in a social network or a buyer in an online marketplace, can be affected by the treatment status of other units in the networks, violating the Stable Unit Treatment Values Assumption (SUTVA) assumption commonly employed in analyzing online A/B tests (Rosenbaum, ). To take into account this type of interference, novel designs and analysis approaches have been proposed, e.g., cluster randomization (Ugander et al., ; Karrer et al., ), regression-based estimators (Chin, ), and bipartite analysis (Harshaw et al., ). A/B testing in the presence of seller-side budget constraints has been discussed by Basse et al. () and Liu et al. () where the budget-splitting design spans the full market.
However, most of the above works do not consider the particular type of interference caused by competitive equilibrium effects.
Recent work has also addressed equilibrium effects (Wager and Xu, ; Liao et al., ; Liao and Kroer, ). Most relevant to our results, Liao and Kroer () consider A/B testing via budget splitting in Fisher markets and FPPE. However, none of these works consider the effects caused by parallel A/B tests that interfere with each other.
Our method builds on top of the budget-splitting design; our approach additionally segments the full market into submarkets, where different submarkets receive different treatment. In the formal sections of our paper we assume that the submarket structure is known. In practice, the submarkets may not be known; Rolnick et al. () and Viviano et al. () present methods to identify submarkets through clustering approaches that aim to limit the interference between clusters.

### 2.2. Review of FPPE

Notation.
For a measurable space $(\Theta,\mathop{}\!\mathrm{d}\theta)$, we let $L^{p}$ (and $L^{p}_{+}$, resp.) denote the set of (nonnegative, resp.) $L^{p}$ functions on $\Theta$ w.r.t the base measure $\mathop{}\!\mathrm{d}\theta$ for any $p\in[1,\infty]$ (including $p=\infty$).
Given $x\in L^{\infty}$ and $v\in L^{1}$, we let $\langle v,x\rangle=\int_{\Theta}v(\theta)x(\theta)\mathop{}\!\mathrm{d}\theta$.
We treat all functions that agree on all but a measure-zero set as the same.
For a sequence of random variables $\{X_{n}\}$, we say $X_{n}=O_{p}(1)$ if for any $\epsilon>0$ there exists a finite $M_{\epsilon}$ and a finite $N_{\epsilon}$ such that ${\mathbb{P}}(|X_{n}|>M_{\epsilon})<\epsilon$ for all $n\geq N_{\epsilon}$. We say $X_{n}=o_{p}(1)$ if $X_{n}$ converges to zero in probability.
For a subset $\Theta^{\prime}\subset\Theta$, let $1_{\Theta^{\prime}}({\cdot}):\Theta\to\{0,1\}$ be the indicator function of $\Theta^{\prime}$.
Convergence in distribution and probability is denoted by ${\,\overset{{d}}{\to}\,}$ and ${\,\overset{{p}}{\to}\,}$.

Given a vector $a=[a_{1},\dots,a_{n}]^{\scriptscriptstyle\mathsf{T}}$, let ${\operatorname*{Diag}}(a)$ denote the diagonal matrix with $(i,i)$-th entry being $a_{i}$; sometimes we write ${\operatorname*{Diag}}(a_{i})$ when it is convenient to define each $a_{i}$ inline.
Let $A^{\dagger}$ denote the Moore–Penrose inverse of the matrix $A$,
$e_{j}$ the $j$-th unit vector,
and $[n]=\{1,\ldots,n\}$.

Figure: Figure 2. Finite FPPE (left) and limit FPPE (right). In a finite FPPE, there are a finite number of items. In a limit FPPE, the item set is a continuum.
Refer to caption: x1.png

#### 2.2.1. Basic Definitions, Limit FPPE and Regularity Conditions

We introduce the definitions of limit FPPE and two regularity conditions on the market.
We have $n$ buyers and a possibly continuous set of items $\Theta$ with an integrating measure $\mathop{}\!\mathrm{d}\theta$.
For example, one could take $\Theta=[0,1]$ and $\mathop{}\!\mathrm{d}\theta=$ the Lebesgue measure on $[0,1]$.
An FPPE requires the following elements.

- •
The *budget* $b_{i}$ of buyer $i$. Let $b=(b_{1},\dots,b_{n})$.
- •
The *valuation* for buyer $i$ is a function $v_{i}\in L^{1}_{+}$, i.e., buyer $i$ has valuation $v_{i}(\theta)$ (value per unit supply) of item $\theta\in\Theta$. Let $v:\Theta\to{\mathbb{R}^{n}}$, $v(\theta)=[v_{1}(\theta),\dots,v_{n}(\theta)]$. We assume ${\bar{v}}=\max_{i}\sup_{\theta}v_{i}(\theta)<\infty$.
- •
The *supplies* of items are given by a function $s\in L^{\infty}_{+}$, i.e., item $\theta\in\Theta$ has $s(\theta)$ unit of supply.
Without loss of generality, we assume a unit total supply $\int_{\Theta}s\mathop{}\!\mathrm{d}\theta=1$.
Given $g:\Theta\to{\mathbb{R}}$, we let ${\mathbb{E}}[g]=\int g(\theta)s(\theta)\mathop{}\!\mathrm{d}\theta$ and ${\operatorname{Var}}[g]={\mathbb{E}}[g^{2}]-({\mathbb{E}}[g])^{2}$.
Given $t$ i.i.d. draws $\{\theta^{1},\dots,\theta^{t}\}$
from $s$, let $P_{t}g({\cdot})=\frac{1}{t}{\sum_{\tau=1}^{t}}g({\theta^{\tau}})$.

Next we introduce the market equilibrium concepts that are the foundation of our study. For that we leverage the *first-price pacing equilibrium* (FPPE) concept put forward by Conitzer et al. (). FPPE models equilibrium outcomes under budget-management systems employed in several practical settings. Each buyer is assigned a *pacing multiplier* $\beta_{i}$, which is used to control their budget expenditure. For each individual auction $\theta$, the buyer bids $\beta_{i}v_{i}(\theta)$, which can be seen as their adjusted valuation after factoring in their budget constraint (in practice, the valuation $v_{i}(\theta)$ may not be the buyer’s true valuation, but instead their unscaled bid reported to the platform).
The goal of the budget management system is to achieve an equilibrium, meaning that it must ensure that buyers spend their budget exactly by appropriately choosing $\beta_{i}$. If the budget cannot be fully spent, then no pacing must occur (i.e., $\beta_{i}=1$).

We now formally define the pacing equilibrium concepts in the case of infinite (below) and finite number of users (in the next section), as illustrated by Fig 2.

###### Definition 1 (Limit FPPE, Gao and Kroer ( 2022 ); Conitzer et al . ( 2022a ) ).

A limit FPPE, denoted as ${\small\mathsf{{FPPE}}}(b,v,s,\Theta)$, is the unique tuple $(\beta,p({\cdot}))\in[0,1]^{n}\times L^{1}_{+}(\Theta)$ such that there exist $x_{i}:\Theta\to[0,1]$ for all $i\in[n]$,

- (1)
(First-price)
Prices are determined by first price auctions:
for all item $\theta\in\Theta$, $p(\theta)=\max_{i}{\beta_{i}}v_{i}(\theta)$.
Only highest bidders obtain items:
for all $i$ and $\theta$, $x_{i}(\theta)>0$ implies ${\beta_{i}}{v_{i}(\theta)}=\max_{k}\beta_{k}v_{k}(\theta)$
- (2)
(Feasiblity, market clearing)
Buyers are budget constrained:
for all $i$, $\int x_{i}(\theta)p(\theta)s(\theta)\mathop{}\!\mathrm{d}\theta\leq b_{i}$.
There is no overselling:
for all $\theta$, ${\sum_{i=1}^{n}}x_{i}(\theta)\leq 1$.
All items are for which there is competition are fully allocated:
for all $\theta$, $p(\theta)>0$ implies ${\sum_{i=1}^{n}}x_{i}(\theta)=1$.
- (3)
(No unnecessary pacing) For all $i$, $\int x_{i}(\theta)p(\theta)s(\theta)\mathop{}\!\mathrm{d}\theta<b_{i}$ implies ${\beta_{i}}=1$.

Let $\beta^{*}$ and $p^{*}$ be the equilibrium pacing multipliers and prices.
Revenue in the limit FPPE is
${\mathsf{{REV}}}^{*}=\int p^{*}(\theta)s(\theta)\mathop{}\!\mathrm{d}\theta\;.$
It measures the profitability of the auction platform.
The leftover budgets for buyers are ${\delta^{*}_{i}}=b_{i}-\int p^{*}(\theta)s(\theta){x^{*}_{i}}(\theta)\mathop{}
\!\mathrm{d}\theta$.

The first two conditions simply describe the possible outcomes of an auction system that uses pacing as the budget management strategy.
The last condition, no unnecessary pacing, ensures that we only scale down a buyer’s bids in case their budget constraint is binding. FPPE has many nice properties, including that they are competitive equilibria and that they are revenue-maximizing among budget-feasible pacing multipliers (Conitzer et al., ).

In a pacing auction market ${\mathscr{M}}={\small\mathsf{{FPPE}}}(b,v,s,\Theta)$ the following two regularity conditions are important for the study of its statistical properties.

###### Definition 2 ( SMO ).

We say the smoothness condition holds if
the map $\beta\mapsto{\mathbb{E}}_{s}[\max_{i}{\beta_{i}}{v_{i}(\theta)}]$ is twice continuously differentiable in a neighborhood of $\beta^{*}$.

###### Definition 3 ( SCS ).

We say strict complementary slackness holds if,
whenever a buyer is unpaced (${\beta^{*}_{i}}=1$), then her leftover budget is strictly positive ($\delta^{*}_{i}>0$).

The condition Definition 2 (SMO). ensures that in the limit market, items that incur a tie are measure zero.
The condition Definition 3 (SCS). rules out degenerate buyers that spend their budget exactly at $\beta_{i}^{*}$. See Liao and Kroer () and Liao et al. () for an extensive discussion about these conditions.

###### Definition 1 (Limit FPPE, Gao and Kroer ( 2022 ); Conitzer et al . ( 2022a ) ).

###### Definition 2 ( SMO ).

###### Definition 3 ( SCS ).

#### 2.2.2. Finite FPPE

Next we introduce finite FPPE, which models the auction data we observe in practice.
Let $\gamma=(\theta^{1},\dots,\theta^{t})$ be a sequence of items.
Assume each item has the same supply of $\sigma\in{\mathbb{R}_{+}}$ units.
A finite FPPE is a limit FPPE where the supply is a discrete measure supported on the observed items $\gamma$.
Let ${v_{i}^{\tau}}={v_{i}(\theta^{\tau})}$.

###### Definition 4 (Finite FPPE, Conitzer et al . ( 2022a ) ).

Given $(b,v,\sigma,\gamma)$, the
observed FPPE, ${\small\widehat{\textsf{{FPPE}}}}(b,v,\sigma,\gamma)$, is the unique tuple
$(\beta,p)\in[0,1]^{n}\times{\mathbb{R}}^{t}_{+}$
such that there exist $x_{i}^{\tau}\in[0,1]$:

- •
(First-price) For all $\tau$, ${p^{\tau}}=\max_{i}{\beta_{i}}{v_{i}^{\tau}}$. For all $i$ and $\tau$, ${x_{i}^{\tau}}>0$ implies ${\beta_{i}}{v_{i}^{\tau}}=\max_{k}\beta_{k}v_{k}^{\tau}$.
- •
(Supply and budget feasible) For all $i$, $\sigma{\sum_{\tau=1}^{t}}{x_{i}^{\tau}}{p^{\tau}}\leq b_{i}$. For all $\tau$, ${\sum_{i=1}^{n}}{x_{i}^{\tau}}\leq 1$.
(Market clearing) For all $\tau$, ${p^{\tau}}>0$ implies ${\sum_{i=1}^{n}}{x_{i}^{\tau}}=1$.
- •
(No unnecessary pacing) For all $i$, $\sigma{\sum_{\tau=1}^{t}}{x_{i}^{\tau}}{p^{\tau}}<b_{i}$ implies ${\beta_{i}}=1$.

In Liao and Kroer (), it is shown that if $\gamma$ consists of $t$ i.i.d. draws from distribution $s$, and one takes $\sigma=1/t$, then the pacing multiplier in ${\small\widehat{\textsf{{FPPE}}}}(b,v,1/t,\gamma)$ converges to the pacing multiplier in ${\small\mathsf{{FPPE}}}(b,v,s,\Theta)$ in probability.
Also note that the ${\small\widehat{\textsf{{FPPE}}}}(tb,v,1,\gamma)$ converges to the same limit FPPE ${\small\mathsf{{FPPE}}}(b,v,s,\Theta)$ because the pacing multiplier of a finite FPPE does not change when budgets and supplies are multiplied by the same scalar.

###### Definition 4 (Finite FPPE, Conitzer et al . ( 2022a ) ).

#### 2.2.3. The Eisenberg-Gale Program

Both the limit FPPE and the finite FPPE have convex program characterizations (Chen et al., ; Conitzer et al., ; Gao and Kroer, ).
We define the dual Eisenberg-Gale (EG) objective for a single item $\theta$ as

$$ (1) $\displaystyle F(\theta,\beta)=f(\beta,\theta)-{\sum_{i=1}^{n}}b_{i}\log\beta_{ i}\;,\quad f(\beta,\theta)=\max_{i\in[n]}\beta_{i}v_{i}(\theta)\;.$ $$

The population and sample (dual) EG objectives are then defined as

$$ (2) $\displaystyle H(\beta)={\mathbb{E}}[F(\theta,\beta)]\;,\quad H_{t}(\beta)= \frac{1}{t}{\sum_{\tau=1}^{t}}F({\theta^{\tau}},\beta)\;.$ $$

We say ${\mathscr{H}}$ is the Hessian of market ${\mathscr{M}}$ when ${\mathscr{H}}=\nabla^{2}\int F(\theta,\beta)s\mathop{}\!\mathrm{d}\theta|_{
\beta=\beta^{*}}$.

The equilibrium pacing multipliers $\beta^{*}$ in ${\small\mathsf{{FPPE}}}(b,v,s,\Theta)$ can be recovered through
the population dual EG program

$$ (3) $\displaystyle\beta^{*}=\operatorname*{arg\,min}_{\beta\in(0,1]^{n}}H(\beta)\;.$ $$

The pacing multiplier $\beta^{*}$ is the unique solution to Eq. 3.
Let $\beta^{\gamma}$ be the equilibrium pacing multiplier in ${\small\widehat{\textsf{{FPPE}}}}(b,v,1/t,\gamma)$. Then $\beta^{\gamma}$ solves the sample analogue of Eq. 3:

$$ (4) $\displaystyle\beta^{\gamma}=\operatorname*{arg\,min}_{\beta\in(0,1]^{n}}H_{t}( \beta)\;.$ $$

Let us briefly comment on the differential structure of $f$, since it plays a role in later sections.
The function $f(\beta,\theta)$ is a convex function of $\beta$ and its subdifferential $\partial_{\beta}f(\beta,\theta)$ is the convex hull of $\{v_{i}e_{i}:\text{index $i$ such that ${\beta_{i}}{v_{i}(\theta)}=\max_{k}{
\beta_{k}}v_{k}(\theta)$}\}$, with $e_{i}$ being the base vector in ${\mathbb{R}^{n}}$.
When $\max_{i}{\beta_{i}}v_{i}(\theta)$ is attained by a unique $i^{*}$, the function $f(\cdot,\theta)$ is differentiable. In that case, all entries of $\nabla_{\beta}f(\beta,\theta)$ are zero expect that the $i^{*}$-th entry is filled with the value $v_{i^{*}}(\theta)$.

## 3. Setup: Interference as Contamination

In this section, we discuss how to estimate market equilibria when there is contamination in the supply. Then, we show that interference from other markets can be viewed as a form of contamination, and so the problem of removing interference bias can be analyzed via our contamination framework.

### 3.1. Contaminated First-Price Pacing Equilibrium

We assume that we are in the same FPPE setting as before:
there are $n$ buyers, each with budget $b_{i}$,
and an item set $\Theta$. However, now we assume that $\Theta$ is partitioned into $\Theta_{\mathsf{good}}$ and $\Theta_{\mathsf{bad}}$.
Let $s$ be supported on $\Theta_{\mathsf{good}}$ and $s^{\prime}$ on $\Theta_{\mathsf{bad}}$.
We define the $\alpha$-contaminated market and the uncontaminated market by

$$ $\displaystyle{\mathscr{M}}_{\alpha}={\small\mathsf{{FPPE}}}(b,v,s_{\alpha}, \Theta)\;,\quad{\mathscr{M}}_{0}={\small\mathsf{{FPPE}}}(b,v,s,\Theta)\;,$ $$

where $s_{\alpha}=\alpha s^{\prime}+(1-\alpha)s$.
Our goal is to perform inference about FPPE properties in the limit FPPE with the item set $\Theta_{\mathsf{good}}$ and supply $s$. However, we are given access to finite FPPEs sampled from $s_{\alpha}$ instead.
In particular, let $\gamma$ be $t$ i.i.d. draws from $s_{\alpha}$ and let

$$ $\displaystyle\widehat{\mathscr{M}}_{\alpha}={\small\widehat{\textsf{{FPPE}}}}( b,v,1/t,\gamma)\;.$ $$

We assume knowledge of $\alpha$ throughout the paper.
In practice, this can often be estimated from historical data; in the parallel A/B test setting, this can be estimated directly from the sampled set of items, since we know which items are good and bad in that setting.
Let $\beta^{*}$ and $\beta^{*}_{\alpha}$ be the limit pacing multipliers in ${\mathscr{M}}_{0}$ and ${\mathscr{M}}_{\alpha}$, respectively. Let $\beta^{\gamma}_{\alpha}$ be the pacing multipliers in the sampled market $\widehat{\mathscr{M}}_{\alpha}$ and let $H_{\alpha,t}(\beta)=\frac{1}{t}{\sum_{\tau=1}^{t}}F({\theta^{\tau}},\beta)$ be the sample EG objective.

Now, if we wanted to make inferences about ${\mathscr{M}}_{\alpha}$ then we could use existing statistical inference theory on how to use data in a finite FPPE $\widehat{\mathscr{M}}_{\alpha}$ to make inferences about the limit FPPE ${\mathscr{M}}_{\alpha}$.
In recent work of Liao and Kroer (), it is shown that for an *influence function* $D_{\alpha}:\Theta\to{\mathbb{R}}$ (to be defined in Thm. 3), the asymptotic distribution of $\beta_{\alpha}^{\gamma}$ around $\beta_{\alpha}^{*}$ satisfies

$$ $\displaystyle\sqrt{t}(\beta^{\gamma}_{\alpha}-\beta^{*}_{\alpha})=\frac{1}{ \sqrt{t}}{\sum_{\tau=1}^{t}}(D_{\alpha}({\theta^{\tau}})-{\mathbb{E}}_{\alpha} [D_{\alpha}(\theta)])+o_{p}(1)\;,$ $$

where $\{{\theta^{\tau}}\}$ is the set of items sampled in $\gamma$.
Our central research question can now be formulated as follows:

*How can we use the data in the finite contaminated market $\widehat{\mathscr{M}}_{\alpha}$ to make inferences about the uncontaminated limit market ${\mathscr{M}}_{0}$?*

### 3.2. Interference as Contamination

Now we show how the contamination model from the previous section can be used to model interference.
Consider $K$ separated auction markets, which together form a global market.
In the global market there are $n$ buyers, each with budget $b_{i}$,
and an item set $\Theta$, partitioned into $\Theta_{\mathsf{good}}$ and $\Theta_{\mathsf{bad}}$.
Let $C_{1},\dots,C_{K}$ be a partition of the buyers,
$\Theta_{1},\dots,\Theta_{K}$ be a partition of the good item set $\Theta_{\mathsf{good}}$,
and $s_{1},\dots s_{K}$ be a set of supply functions, supported on $\Theta_{1},\dots,\Theta_{K}$ respectively.
The $k$-th submarket
consists of buyers in $C_{k}$, the item set $\Theta_{k}$ and supply $s_{k}$.
Let $s=\frac{1}{K}\sum_{k}s_{k}$ be the average mixture and $s^{\prime}$ be a supply supported on $\Theta_{\mathsf{bad}}$.
Let the contaminated supply be $s_{\alpha}=\alpha s^{\prime}+(1-\alpha)s$.

By imposing structure on
$\Theta_{\mathsf{good}}$ and $\Theta_{\mathsf{bad}}$
the contamination model can capture interference among auction markets.
We assume that submarkets are separated, which models the ideal case where there is no interference.
A buyer $i\in C_{k}$ is only interested in items from the submarket he belongs to: $v_{i}(\theta)=0$ for $\theta\in\Theta_{k^{\prime}},k^{\prime}\neq k$.
Next, we let $\Theta_{\mathsf{bad}}$ represent items
that cause outbound edges from submarkets; see the green edges in Fig 3 left panel. An item is referred to as bad if it has positive values for buyers from at least two different submarkets. Formally,
$\theta\in\Theta_{\mathsf{bad}}$ if there exist $i\in C_{k}$, $j\in C_{k^{\prime}}$, $k\neq k^{\prime}$, such that $v_{i}(\theta)>0$ and $v_{j}(\theta)>0$.
Combining these assumptions, we have that a buyer $i$ from submarket $k$ has positive values *only* for items from the sets $\Theta_{k}$ and possibly $\Theta_{\mathsf{bad}}$.
Now we have fully specified a contaminated market setup: we wish to make inferences on the market consisting of only $\Theta_{\mathsf{good}}$ (which is really $K$ fully separate submarkets), but we observe an actual market containing items from $\Theta_{\mathsf{good}}\cup\Theta_{\mathsf{bad}}$.
With this setup, we can use the results developed in the following section to model interference in parallel submarkets.

In Fig 3 we present an example of interference among $K=3$ submarkets.
The market of interest is the perfectly separated market (right).
The reason is, in parallel A/B testing, submarkets are created in a way that each resembles the global market, and when a submarket receives a treatment, the observed quantities in that submarket, such as revenues and social welfare, are considered surrogates for the treatment effect in the global market.
However, in practice we only observe the interfered finite market (left), which converges to the interfered limit market (middle). In Sec. 5 we show how to analyze parallel A/B testing using this framework.

Figure: Figure 3. The model. Left ($\widehat{\mathscr{M}}_{\alpha}$): the observed market where interference is present among submarkets. Middle (${\mathscr{M}}_{\alpha}$): the limit market with interference from bad item set. Right (${\mathscr{M}}_{0}$): the limit market with perfectly separated submarkets. We use data from the left panel to make inferences about the market in the right panel.
Refer to caption: x2.png

## 4. Main Results

In this section, we develop a methodology for making inferences about, e.g., pacing multipliers or revenue in the uncontaminated limit FPPE.
We introduce a surrogate for pacing multipliers, based on the notion of directional derivatives, and establish its debiasing property in Sec. 4.1. Then, we focus on estimating this surrogate quantity in Sec. 4.2, and develop asymptotic normality results in Sec. 4.3.
Secondly, we consider estimating revenue. Revenue can be thought of as a smooth function of pacing multipliers, and we discuss debiased estimation and inference of revenue based on our pacing multiplier results in Sec. 4.4.

Before we proceed we briefly discuss two baseline approaches to the problem; see Table 2 for an overview.
The first method is doing nothing, i.e., treating quantities in the contaminated finite market $\widehat{\mathscr{M}}_{\alpha}$ as estimators of those in ${\mathscr{M}}_{0}$.
This approach suffers from a significant level of bias, roughly of order $\Theta(\alpha)$, and the bias does not vanish as the number of items increases.
The second method is that, since we know whether an item comes from
the contamination supply $s^{\prime}$, we could throw away the market equilibrium $\widehat{\mathscr{M}}_{\alpha}$,
collect the good items, and compute a finite market equilibrium formed by the good items via solving the EG program in Eq. 4. This approach has a vanishing bias but is computationally heavy, especially in real-world advertising platforms where there are millions of buyers.
Our proposed approach reduces bias to the order $o(\alpha)$ and does not require solving large-scale EG programs, by utilizing the equilibrium structure of the observed market $\widehat{\mathscr{M}}_{\alpha}$.

**Table 2. Comparison of Procedures**
| Procedure | Bias | Requires solving EG |
| --- | --- | --- |
| Ignore interference | $\Theta(\alpha)$ | No |
| Solve EG | $o(1)$ | Yes |
| Our debiasing procedure | $o(\alpha)$ | No |

### 4.1. A Debiased Surrogate for Pacing Multipliers

If we view $\beta^{*}_{\alpha}$ as a function of the level of contamination $\alpha$, then one can imagine that under sufficient regularity conditions, the pacing multipliers in the perfectly separated market, $\beta^{*}_{0}$, can be approximated by some form of Taylor expansion of $\alpha\mapsto\beta^{*}_{\alpha}$ at $\alpha$.
This can be made rigorous by the notion of directional derivatives. We define

$$ (5) $\displaystyle\mathop{}\!\mathrm{d}\beta^{*}(\alpha)=\lim_{\delta\to 0^{+}} \frac{\beta^{*}_{\alpha-\delta}-\beta^{*}_{\alpha}}{\delta}\;.$ $$

if the limit exists.
We will show that $\beta^{*}_{\alpha}+\alpha\mathop{}\!\mathrm{d}\beta^{*}(\alpha)$ serves as a good approximation to $\beta^{*}$.

We now show that the directional derivative $\mathop{}\!\mathrm{d}\beta^{*}(\alpha)$ has a closed-form expression under certain regularity conditions (the conditions are given in Thm. 1; the full proof is given in the appendix). We need a few notations for this expression.
Define
$\Delta_{\alpha}=\int\nabla f(\theta,\beta^{*}_{\alpha})(s-s^{\prime})\mathop{}
\!\mathrm{d}\theta.$
Let ${\mathscr{H}}_{\alpha}=\nabla_{\beta\beta}^{2}\int F(\theta,\beta^{*}_{\alpha}
)s_{\alpha}\mathop{}\!\mathrm{d}\theta$ be the Hessian matrix in the market ${\mathscr{M}}_{\alpha}$ and $P_{\alpha}={\operatorname*{Diag}}(1(\beta^{*}_{\alpha,i}<1))$. Then, under
regularity conditions,

$$ (6) $\displaystyle\mathop{}\!\mathrm{d}\beta^{*}(\alpha)=-(P_{\alpha}{\mathscr{H}}_ {\alpha}P_{\alpha})^{\dagger}\Delta_{\alpha}\;.$ $$

We present a heuristic argument for Eq. 6 here; for a rigorous treatment we need a perturbation theory result which is given in Lemma 1 in the appendix.
Define the contaminated EG objective $H_{\alpha}(\beta)=\int F(\beta,\theta)s_{\alpha}\mathop{}\!\mathrm{d}\theta$. Then ${\mathscr{H}}_{\alpha}=\nabla^{2}H_{\alpha}(\beta^{*}_{\alpha})$. The map $(\beta,\alpha)\mapsto H_{\alpha}(\beta)$ has gradient $[\nabla H(\beta^{*}),\int F(\beta^{*}_{\alpha},\theta)(s^{\prime}-s)\mathop{}
\!\mathrm{d}\theta]^{\scriptscriptstyle\mathsf{T}}$ and Hessian $[{\mathscr{H}}_{\alpha},-\Delta_{\alpha};-\Delta_{\alpha}^{\scriptscriptstyle
\mathsf{T}},0]$ at $(\beta^{*}_{\alpha},\alpha)$.
Then we have the following quadratic approximation of $H_{\alpha-\delta}(\beta)$

$$ $\displaystyle H_{\alpha-\delta}(\beta)\approx H_{\alpha}(\beta^{*}_{\alpha})+ \begin{pmatrix}\beta-\beta^{*}_{\alpha}\\ -\delta\end{pmatrix}^{\scriptscriptstyle\mathsf{T}}\begin{pmatrix}\nabla H_{ \alpha}(\beta^{*}_{\alpha})\\ \int F(\beta^{*}_{\alpha},\theta)(s^{\prime}-s)\mathop{}\!\mathrm{d}\theta\end {pmatrix}+\frac{1}{2}\begin{pmatrix}\beta-\beta^{*}\\ -\delta\end{pmatrix}^{\scriptscriptstyle\mathsf{T}}\begin{pmatrix}{\mathscr{H} }_{\alpha}&-\Delta_{\alpha}\\ -\Delta_{\alpha}^{\scriptscriptstyle\mathsf{T}}&0\end{pmatrix}\begin{pmatrix} \beta-\beta^{*}\\ -\delta\end{pmatrix}\;.$ $$

And so

$$ $\displaystyle\beta^{*}_{\alpha-\delta}=\operatorname*{arg\,min}_{\beta\in(0,1] ^{n}}H_{\alpha-\delta}(\beta)\approx\operatorname*{arg\,min}_{\beta\in(0,1]^{n }}(\nabla H_{\alpha}(\beta^{*}_{\alpha})+\delta\Delta_{\alpha})^{ \scriptscriptstyle\mathsf{T}}\beta+\tfrac{1}{2}(\beta-\beta^{*}_{\alpha})^{ \scriptscriptstyle\mathsf{T}}{\mathscr{H}}_{\alpha}(\beta-\beta^{*}_{\alpha})\;,$ $$

where we dropped terms that do not involve $\beta$.

Next we need two observations. For $\delta$ small enough, we have $\beta^{*}_{\alpha-\delta,i}=1$ if $\beta^{*}_{\alpha,i}=1$.
Intuitively, buyers in the market ${\mathscr{M}}_{\alpha}$ with $\beta^{*}_{\alpha,i}=1$ have some nonzero leftover budget, which acts as a buffer to small changes in supply, and thus their pacing multipliers remain one.
Next, we also note $\beta\mapsto\nabla H_{\alpha}(\beta^{*}_{\alpha})^{\scriptscriptstyle\mathsf{T
}}\beta$ is constant on the set $\{\beta:{\beta_{i}}=1\text{ if }\beta^{*}_{\alpha,i}=1\}$ by complementary slackness.
Then we further simplify and obtain

$$ $\displaystyle\beta^{*}_{\alpha-\delta}\approx\operatorname*{arg\,min}_{{\beta_ {i}}=1\text{ if }\beta^{*}_{\alpha,i}=1}\delta\Delta_{\alpha}^{ \scriptscriptstyle\mathsf{T}}\beta+\tfrac{1}{2}(\beta-\beta^{*}_{\alpha})^{ \scriptscriptstyle\mathsf{T}}{\mathscr{H}}_{\alpha}(\beta-\beta^{*}_{\alpha})\;.$ $$

The right-hand side is just a linearly constrained quadratic optimization problem, which admits a closed-form solution $\beta^{*}_{\alpha}-\delta(P_{\alpha}{\mathscr{H}}_{\alpha}P_{\alpha})^{\dagger
}\Delta_{\alpha}$.
Now we define a debiased pacing multiplier

$$ (7) $\displaystyle\widetilde{\beta}^{*}=\beta^{*}_{\alpha}+\alpha\cdot(-(P_{\alpha} {\mathscr{H}}_{\alpha}P_{\alpha})^{\dagger}\Delta_{\alpha})\;.$ $$

###### Theorem 1 (Analysis of Bias).

Suppose that in the market ${\mathscr{M}}_{0}$ conditions Definition 2 (SMO). and Definition 3 (SCS). hold, and assume that
$\beta\mapsto\nabla^{2}\int F(\beta,\theta)s^{\prime}\mathop{}\!\mathrm{d}\theta$ is twice continuously differentiable in a neighborhood of $\beta^{*}$.
Then the directional derivative $\mathop{}\!\mathrm{d}\beta^{*}({\cdot})$ is well-defined in a neighborhood of zero, and is given by Eq. 6.
Moreover, as $\alpha\downarrow 0$,

$$ $\displaystyle\|\widetilde{\beta}^{*}-\beta^{*}\|_{2}=o(\alpha)\;.$ $$

The proof of Thm. 1 is given in Sec. B.1.
Thm. 1 indicates that the debiased surrogate $\widetilde{\beta}^{*}$ removes first-order bias
caused by interference.
The limit pacing multipliers $\beta_{\alpha}^{*}$ of the contaminated market ${\mathscr{M}}_{\alpha}$ will have bias of order $\beta^{*}_{\alpha}-\beta^{*}=\Theta(\alpha)$. In contrast, Thm. 1 shows that
the debiased surrogate only incurs a bias of order $o(\alpha)$.

###### Theorem 1 (Analysis of Bias).

### 4.2. The Estimator

In this section we introduce a plug-in estimator for the debiased surrogate $\widetilde{\beta}^{*}$ and introduce a consistency theorem.
In next section we discuss constructing confidence intervals.

To estimate $\mathop{}\!\mathrm{d}\beta^{*}(\alpha)$ in Eq. 6 we need estimates of its three components: the Hessian ${\mathscr{H}}_{\alpha}=\nabla_{\beta\beta}^{2}\int F(\theta,\beta^{*}_{\alpha}
)s_{\alpha}\mathop{}\!\mathrm{d}\theta$, the diagonal matrix $P_{\alpha}$ and the vector $\Delta_{\alpha}=\int\nabla f(\theta,\beta^{*}_{\alpha})(s-s^{\prime})\mathop{}
\!\mathrm{d}\theta$.

- •
The Hessian. For simplicity in our theoretical results, we will simply assume a generic Hessian estimator $\widehat{\mathscr{H}}_{\alpha}$ such that for some $\eta_{t}\downarrow 0$ we have
$\widehat{\mathscr{H}}_{\alpha}-{\mathscr{H}}_{\alpha}=O_{p}(\eta_{t})$.
Hong et al. () discuss the estimation of the derivative in detail.
Different kinds of statistical guarantees require different rate conditions on $\eta_{t}$; see Theorems 2 and 3.
We then introduce two Hessian estimators: one is applicable for general FPPE, while the other requires an extra market regularity condition.
The first Hessian estimator is the finite difference method.
Let $e_{i},e_{j}$ be basis vectors and $\varepsilon_{t}$ be a step-size.
Then the estimator is
$\widehat{\mathscr{H}}_{\alpha}[i,j]=[H_{\alpha,t}(\beta^{\gamma}_{++})-H_{
\alpha,t}(\beta^{\gamma}_{+-})-H_{\alpha,t}(\beta^{\gamma}_{-+})+H_{\alpha,t}(
\beta^{\gamma}_{--})]/(4\varepsilon_{t}^{2})\;,$
where $\beta^{\gamma}_{\pm\pm}=\beta^{\gamma}_{\alpha}\pm e_{i}\varepsilon_{t}\pm e_{
j}\varepsilon_{t}$, and $H_{\alpha,t}(\beta)=\frac{1}{t}{\sum_{\tau=1}^{t}}F({\theta^{\tau}},\beta)$, with $\{{\theta^{\tau}}\}_{\tau}$ being the items in $\widehat{\mathscr{M}}_{\alpha}$.
In practice, a diagonal approximation of the Hessian suffices.
The second method relies on an additional regularity condition, in which case we derive a simplified formula for the Hessian, thereby enabling a simpler estimation procedure (see Thm. 3).
- •
The vector $\Delta_{\alpha}$.
Let $g$ be the Radon-Nikodym ratio $g(\theta)=(\mathop{}\!\mathrm{d}(s-s^{\prime})/\mathop{}\!\mathrm{d}s_{\alpha}
)(\theta)=\frac{1}{1-\alpha}1_{\Theta_{\mathsf{good}}}(\theta)-\frac{1}{\alpha
}1_{\Theta_{\mathsf{bad}}}(\theta)$.
With the ratio $g$, the true vector $\Delta_{\alpha}$ can be written as $\Delta_{\alpha}=\int g(\theta)\nabla f(\theta,\beta^{*}_{\alpha})s_{\alpha}(
\theta)\mathop{}\!\mathrm{d}\theta$, which is easy to estimate given i.i.d. draws from $s_{\alpha}$.
In particular, our estimator is then
$\displaystyle\widehat{\Delta}_{\alpha}=\frac{1}{t}{\sum_{\tau=1}^{t}}g({\theta
^{\tau}})\mu^{\tau}\;.$
Here $\mu^{\tau}=[x_{1}^{\tau}v_{1}^{\tau},\dots,x_{n}^{\tau}v_{n}^{\tau}]^{
\scriptscriptstyle\mathsf{T}}$
is a subgradient of $f({\theta^{\tau}},\beta^{\gamma}_{\alpha})$ w.r.t. $\beta$.
- •
The diagonal matrix $P_{\alpha}$. Recall $P_{\alpha}={\operatorname*{Diag}}(1(\beta^{*}_{\alpha,i}<1))$. So a natural estimator is
$\widehat{P}_{\alpha}={\operatorname*{Diag}}(1(\beta^{\gamma}_{\alpha,i}<1-
\iota_{t}))$,
where the slackness $\iota_{t}\asymp\frac{1}{\sqrt{t}}$.

With all three components estimated, using $\beta^{\gamma}_{\alpha}$ is the pacing multiplier in the market $\widehat{\mathscr{M}}_{\alpha}$ we define the plug-in estimator for $\mathop{}\!\mathrm{d}\beta^{*}(\alpha)$ in Eq. 6 as

$$ (8) $\displaystyle\widehat{\beta}=\beta^{\gamma}_{\alpha}-\alpha\,{\cdot}\,( \widehat{P}_{\alpha}\widehat{\mathscr{H}}_{\alpha}\widehat{P}_{\alpha})^{ \dagger}\widehat{\Delta}_{\alpha}\;.$ $$

###### Theorem 2 (Consistency).

If $\eta_{t}=o(1)$, then $\widehat{\beta}{\,\overset{{p}}{\to}\,}\widetilde{\beta}^{*}$. The proof is in Sec. B.2.

###### Theorem 2 (Consistency).

### 4.3. Asymptotic Normality and Inference

We present two asymptotic normality results.
In the first result, we require a
stronger condition on the Hessian error rate $\eta_{t}$.
In particular, as will be shown in Thm. 3, the rate condition $\eta_{t}=o(1/\sqrt{t})$ is sufficient for normality.
One could use a separate large historical dataset to obtain a good estimate of the Hessian matrix.
In the second result, we
impose an additional condition on market structure which simplifies the Hessian expression and facilitates efficient Hessian estimation.
To describe the additional market structure, we define the gap between the highest and the second-highest bid for an item $\theta$ under pacing $\beta$ by

$$ $\displaystyle{\mathsf{{bidgap}}}(\beta,\theta)=\max\{{\beta_{i}}{v_{i}(\theta) }\}-\operatorname{secondmax}\{{\beta_{i}}{v_{i}(\theta)}\}\;,$ $$

where $\operatorname{secondmax}$ is the second-highest entry
potentially equal to the highest; e.g., $\operatornamewithlimits{secondmax}([1,1,2])=1$.
When there is a tie for an item $\theta$ under pacing $\beta$, we have ${\mathsf{{bidgap}}}(\beta,\theta)=0$.
When there is no tie for an item $\theta$, the gap ${\mathsf{{bidgap}}}(\beta,\theta)$ is strictly positive.

For any $g:\Theta\to{\mathbb{R}}$, let ${\mathbb{E}}_{\alpha}[g]=\int gs_{\alpha}\mathop{}\!\mathrm{d}\theta$ and ${\operatorname{Cov}}_{\alpha}(g)={\mathbb{E}}_{\alpha}[(g-{\mathbb{E}}_{\alpha
}[g])(g-{\mathbb{E}}_{\alpha}[g])^{\scriptscriptstyle\mathsf{T}}]$.
Recall $\widetilde{\beta}^{*}$ is the debiased surrogate in Eq. 7 and $\widehat{\beta}$ is its estimator defined in Eq. 8.

We need to introduce a few more notations to describe the normality results. First, let

$$ $\displaystyle D_{\alpha}=-(P_{\alpha}{\mathscr{H}}_{\alpha}P_{\alpha})^{ \dagger}\nabla f({\cdot},\beta^{*}_{\alpha})\;,$ $$

As mentioned previously, the pacing multipliers in the contaminated market converge to the limit counterpart and have the representation

$$ $\displaystyle\sqrt{t}(\beta^{\gamma}_{\alpha}-\beta^{*}_{\alpha})=\frac{1}{ \sqrt{t}}{\sum_{\tau=1}^{t}}(D_{\alpha}({\theta^{\tau}})-{\mathbb{E}}_{\alpha} [D_{\alpha}({\theta^{\tau}})])+o_{p}(1)\;.$ $$

In the statistics literature, the function $D_{\alpha}({\cdot})-{\mathbb{E}}[D_{\alpha}]$ is called the influence function (van der Vaart, ).
For our debiased estimators, we need the following (uncentered) influence functions.

$$ $\displaystyle IF_{1}(\theta)=\big{(}\tfrac{1}{1-\alpha}1_{\Theta_{\mathsf{good }}}(\theta)\big{)}D_{\alpha}(\theta)\;,$ $\displaystyle IF_{2}(\theta)=IF_{1}(\theta)-2\alpha{\operatorname*{Diag}}( \beta^{*}_{\alpha,i}\Delta_{\alpha,i}b_{i}^{{-1}})D_{\alpha}(\theta)\;.$ $$

###### Theorem 3 (Asymptotic Normality).

Let Definition 3 (SCS). and Definition 2 (SMO). hold in ${\mathscr{M}}_{\alpha}$.

1.
It holds that
$\widehat{\beta}-\widetilde{\beta}^{*}={Z_{t}}+O_{p}(\eta_{t})+o_{p}(\tfrac{1}{
\sqrt{t}}),$
where

$$ $\displaystyle\sqrt{t}Z_{t}{\,\overset{{d}}{\to}\,}{\mathscr{N}}(0,\Sigma_{1})$ $$

with $\Sigma_{1}={\operatorname{Cov}}_{\alpha}(IF_{1})$.

2. If in addition ${\mathbb{E}}_{\alpha}[1/{\mathsf{{bidgap}}}(\beta^{*}_{\alpha},\theta)]<\infty$, then ${\mathscr{H}}_{\alpha}={\operatorname*{Diag}}(b_{i}/(\beta^{*}_{\alpha,i})^{2})$. Suppose we estimate ${\mathscr{H}}_{\alpha}$ with $\widehat{\mathscr{H}}_{\alpha}={\operatorname*{Diag}}(b_{i}/(\beta^{\gamma}_{
\alpha,i})^{2})$. Then

$$ $\displaystyle\sqrt{t}(\widehat{\beta}-\widetilde{\beta}^{*}){\,\overset{{d}}{ \to}\,}{\mathscr{N}}(0,\Sigma_{2})$ $$

where $\Sigma_{2}={\operatorname{Cov}}_{\alpha}(IF_{2})$.
The proof is in Sec. B.3.

Thm. 3 part 1 shows how the error of the Hessian estimate affects the distribution of the estimator $\widehat{\beta}$.
If $\eta_{t}=o(1/\sqrt{t})$, then
the decomposition becomes $\sqrt{t}(\widehat{\beta}-\widetilde{\beta}^{*})=\sqrt{t}Z_{t}+o_{p}(1)$, implying asymptotic normality, i.e.
$\sqrt{t}(\widehat{\beta}-\widetilde{\beta}^{*}){\,\overset{{d}}{\to}\,}{
\mathscr{N}}(0,\Sigma_{1})$, in which case one can construct an ellipsoidal confidence region for $\widetilde{\beta}^{*}$.
Thm. 3 part 2 shows direct
asymptotic normality under the extra condition, with a simpler Hessian estimator that avoids step-sizes.

To perform inference, we need to construct a consistent estimate of the covariance matrix.
Now we describe a plug-in estimate of $\Sigma_{1}$. Let the estimator $\widehat{IF_{1}}^{\tau}$ be

$$ $\displaystyle\widehat{IF_{1}}^{\tau}=-\big{(}\tfrac{1}{1-\alpha}1_{\Theta_{ \mathsf{good}}}({\theta^{\tau}})\big{)}(\widehat{P}_{\alpha}\widehat{\mathscr{ H}}_{\alpha}\widehat{P}_{\alpha})^{\dagger}\mu^{\tau}\;,$ $$

where $\mu^{\tau}=[x_{1}^{\tau}v_{1}^{\tau},\dots,x_{n}^{\tau}v_{n}^{\tau}]^{
\scriptscriptstyle\mathsf{T}}$, and $\widehat{P_{\alpha}},\widehat{\mathscr{H}}_{\alpha}$ have been defined in Sec. 4.2.
The plug-in estimator is

$$ $\displaystyle\widehat{\Sigma}_{1}=\frac{1}{t}{\sum_{\tau=1}^{t}}(\widehat{IF_{ 1}}^{\tau}-\overline{IF_{1}})(\widehat{IF_{1}}^{\tau}-\overline{IF_{1}})^{ \scriptscriptstyle\mathsf{T}}$ $$

with $\overline{IF_{1}}=\frac{1}{t}{\sum_{\tau=1}^{t}}\widehat{IF_{1}}^{\tau}$.
By similar arguments as in Liao et al. (), the plug-in estimates of $\Sigma_{1}$ and $\Sigma_{2}$ are consistent. Algorithm 1 summarizes the debiasing procedure.

Figure: ALGORITHM 1 Debiasing procedure

###### Theorem 3 (Asymptotic Normality).

### 4.4. Results for Revenue

The revenue in the good market is defined as ${\mathsf{{REV}}}^{*}=\int\max_{i}\{{v_{i}(\theta)}{\beta^{*}_{i}}\}s(\theta)
\mathop{}\!\mathrm{d}\theta$. We define the debiased revenue surrogate and its estimator

$$ (9) $\displaystyle\widetilde{{\mathsf{{REV}}}}^{*}=\int\max_{i}\{{v_{i}(\theta)} \widetilde{\beta}^{*}_{i}\}s(\theta)\mathop{}\!\mathrm{d}\theta\;,$ (10) $\displaystyle\widehat{{\mathsf{{REV}}}}=\frac{1}{1-\alpha}\frac{1}{t}{\sum_{ \tau=1}^{t}}1_{\Theta_{\mathsf{good}}}({\theta^{\tau}})\max_{i}\{{v_{i}(\theta ^{\tau})}\widehat{\beta}_{i}\}\;.$ $$

In words, we approximate ${\mathsf{{REV}}}^{*}$ with the revenue generated in the market where buyers bid according to the pacing profile $\widetilde{\beta}^{*}$, and then we estimate this via plug-in estimation on the subset of items that are good.

###### Theorem 4 ().

Let the conditions from Thm. 1 and Thm. 3 hold.
The debiased revenue removes first-order bias:
$\widetilde{{\mathsf{{REV}}}}^{*}={\mathsf{{REV}}}^{*}+o(\alpha)$.
Assume that $\sqrt{t}(\widehat{\beta}-\widetilde{\beta}^{*})$ has influence function $IF$,
i.e., $\sqrt{t}(\widehat{\beta}-\widetilde{\beta}^{*})=\frac{1}{\sqrt{t}}{\sum_{\tau=
1}^{t}}IF({\theta^{\tau}})+o_{p}(1)$,
then $\sqrt{t}(\widehat{{\mathsf{{REV}}}}-\widetilde{{\mathsf{{REV}}}}^{*}){\,
\overset{{d}}{\to}\,}{\mathscr{N}}(0,\sigma^{2}_{\mathsf{{REV}}})$ where
$\sigma^{2}_{\mathsf{{REV}}}={\operatorname{Cov}}_{\alpha}(IF_{\mathsf{{REV}}})$,
$IF_{\mathsf{{REV}}}({\cdot})=\frac{1}{1-\alpha}\big{(}1_{\Theta_{\mathsf{good}
}}({\cdot})f({\cdot},\widetilde{\beta}^{*})+{\mathbb{E}}_{\alpha}[\nabla f(
\theta,\widetilde{\beta}^{*})1_{\Theta_{\mathsf{good}}}(\theta)]^{
\scriptscriptstyle\mathsf{T}}IF({\cdot})\big{)}$.
The proof is in Sec. B.4.

Examples of influence functions compatible with Thm. 4 are the influence functions in Thm. 3.

We have two methods to construct CIs for the debiased revenue surrogate in Eq. 7.
The first method is based on the asymptotic normality results in Thm. 4.
In the second method, a revenue CI is constructed based on a CI for $\beta^{*}_{\alpha}$.
Suppose ${\small\mathbb{{CI}}}$ is some confidence region for $\beta^{*}_{\alpha}$.
Then $\{\frac{1}{1-\alpha}\frac{1}{t}{\sum_{\tau=1}^{t}}\allowbreak 1_{\Theta_{
\mathsf{good}}}({\theta^{\tau}})\max_{i}{v_{i}(\theta^{\tau})}{\beta_{i}}:
\beta\in{\small\mathbb{{CI}}}\}$ serves as a CI for revenue.
Another way to utilize a pacing multiplier CI is based on the observation that revenue is a monotone function of the pacing multiplier. Let
$\underline{{\beta_{i}}}=\min\{{\beta_{i}}:\beta\in{\small\mathbb{{CI}}}\}$ and $\bar{\beta}_{i}=\max\{{\beta_{i}}:\beta\in{\small\mathbb{{CI}}}\}$.
Then another natural (potentially wider) CI for $\widetilde{{\mathsf{{REV}}}}^{*}$ is $[\underline{{\mathsf{{REV}}}},\overline{{\mathsf{{REV}}}}]$, where

$$ (11) $\underline{{\mathsf{{REV}}}}=\frac{1}{1-\alpha}\frac{1}{t}{\sum_{\tau=1}^{t}}1 _{\Theta_{\mathsf{good}}}({\theta^{\tau}})\max_{i}{v_{i}(\theta^{\tau})} \underline{{\beta_{i}}}\;,\quad\overline{{\mathsf{{REV}}}}=\frac{1}{1-\alpha} \frac{1}{t}{\sum_{\tau=1}^{t}}1_{\Theta_{\mathsf{good}}}({\theta^{\tau}})\max_ {i}{v_{i}(\theta^{\tau})}\bar{\beta}_{i}\;.$ $$

###### Theorem 4 ().

## 5. Parallel A/B Test under Interference

In this section, we use the theory from the previous sections to formulate a statistical inference theory for parallel A/B tests under interference.

Consider $K$ auction markets in which we want to run parallel A/B tests.
There are $n$ buyers, each with budget $b_{i}$ corresponding to one unit supply of items,
and an item set $\Theta$, partitioned into $\Theta_{\mathsf{good}}$ and $\Theta_{\mathsf{bad}}$.
Let $C_{1},\dots,C_{K}$ be a partition of the buyers,
$\Theta_{1},\dots,\Theta_{K}$ be a partition of the good item set $\Theta_{\mathsf{good}}$,
and $s_{1},\dots s_{K}$ be a set of supply functions, supported on $\Theta_{1},\dots,\Theta_{K}$ respectively.
The $k$-th submarket
consists of buyers in $C_{k}$, the item set $\Theta_{k}$ and supply $s_{k}$.
Let $s=\frac{1}{K}\sum_{k}s_{k}$ be the average mixture and $s^{\prime}$ be a supply supported on $\Theta_{\mathsf{bad}}$.

To model treatment application we introduce the *potential value functions*

$$ $v(0)=(v_{1}(0,\cdot),\dots,v_{n}(0,\cdot)),\;v(1)=(v_{1}(1,\cdot),\dots,v_{n}( 1,\cdot))\;.$ $$

If item $\theta$ is exposed to treatment $w\in\{0,1\}$, then its value to buyer $i$ will be $v_{i}(w,\theta)$.

We extend the structure on $\Theta_{\mathsf{bad}}$ and $\Theta_{\mathsf{good}}$ introduced in Sec. 3.2 to A/B testing here.
We assume these submarkets are separated.
This is to model the ideal case where there is no interference.
A buyer $i\in C_{k}$ is only interested in items from the submarket he belongs to: $v_{i}(w,\theta)=0$ for $\theta\in C_{k^{\prime}},k^{\prime}\neq k$, and $w\in\{0,1\}$.
An item is called a bad item if it has positive values for buyers from different submarkets. Formally,
$\theta\in\Theta_{\mathsf{bad}}$ if there exist $w\in\{0,1\}$, $i\in C_{k}$, $j\in C_{k^{\prime}}$, $k\neq k^{\prime}$, so that $v_{i}(w,\theta)>0$ and $v_{j}(w,\theta)>0$.
A buyer $i$ from submarket $k$ has a (potentially) positive value only for items in $\Theta_{k}\cup\Theta_{\mathsf{bad}}$.

Step 1. The treatment effects. Let
${\mathscr{M}}(w,k)={\small\mathsf{{FPPE}}}(\{b_{i}\}_{i\in C_{k}},\{v_{i}({
\cdot},w)\}_{i\in C_{k}},s_{k},\Theta_{k})$be the $k$-th limit submarket under treatment $w\in\{0,1\}$, and let revenue be ${\mathsf{{REV}}}^{*}_{k}(w)$.
Then the treatment effect in the submarket $k$ is $\tau^{*}_{k}={\mathsf{{REV}}}^{*}_{k}(1)-{\mathsf{{REV}}}^{*}_{k}(0)$.
Equivalently, all $K$ submarkets can be formed simultaneously,
${\mathscr{M}}(w)={\small\mathsf{{FPPE}}}\left(\frac{1}{K}b,v(w),s,\Theta\right).$
The scaling $1/K$ in budgets ensures that a buyer in $C_{k}$ has budget $b_{i}/K$ to bid for items in $\Theta_{k}$,
whose supply in ${\mathscr{M}}(w)$ is $\frac{1}{K}s_{k}$.
The $k$-th component of ${\mathscr{M}}(w)$ corresponds to ${\mathscr{M}}(w,k)$.

Step 2. The experiment.
A practical parallel A/B test scheme is as follows.

*Step 2.1. Budget splitting.*
Decide on a budget split ratio $\pi_{0}$, $\pi_{1}$ satisfying $\pi_{0}+\pi_{1}=1$, and split budgets accordingly.
Choose A/B test size $t$, the total number of impressions in each submarket summed across treatments.
Then, the budget of buyer $i$ for treatment 0 0 will be $\pi_{0}tb_{i}$,
and $\pi_{1}tb_{i}$ for treatment 1.
This is because $b_{i}$ is normalized to correspond to one unit of total supply.

*Step 2.2. Observe markets with interference.*
Draw items/impressions from $\alpha s^{\prime}+(1-\alpha)s$.
Draw $\lfloor\pi_{0}t\rfloor$ items for each submarket under treatment 0 0 and $\lfloor\pi_{1}t\rfloor$ for treatment 1.
A total of $t_{w}=\lfloor\pi_{w}tK\rfloor$ items are drawn for the whole market under treatment $w$.
Since items come from the contaminated distribution, roughly a small fraction $\alpha$ of the $t_{w}$ items cause interference.

The data observed following the described A/B test scheme can be compactly represented by two finite FPPEs (one for each treatment) (Def. 4):
$\widehat{\mathscr{M}}_{\alpha}(w)={\small\widehat{\textsf{{FPPE}}}}(\pi_{w}tb,
v(w),1,\gamma)$
,
where $\gamma\subset\Theta$ consists of $t_{w}=\lfloor\pi_{w}tK\rfloor$ i.i.d. draws from
the mixture $\alpha s^{\prime}+(1-\alpha)s$.

It can be shown that
the observed market $\widehat{\mathscr{M}}_{\alpha}(w)$ converges to
${\mathscr{M}}_{\alpha}(w)={\small\mathsf{{FPPE}}}\ \left(\frac{1}{K}b,v(w),s_{
\alpha},\Theta\right).$

Step 3. Debiasing $\beta$.
Following previous section,
we can debias $\widehat{\mathscr{M}}_{\alpha}(w)$ to approximate ${\mathscr{M}}(w)$.
Let $\widehat{\beta}(w)$ be the debiased pacing multiplier.

Step 4. Revenue estimator. Estimate
$\widehat{\mathsf{{REV}}}_{k}(w)$ by

$$ $\displaystyle\frac{1}{\pi_{w}(1-\alpha)t}\sum_{\tau=1}^{t_{w}}1_{\Theta_{k}}({ \theta^{\tau}})\max_{i\in C_{k}}{v_{i}(\theta^{\tau})}\widehat{\beta_{i}}(w)\;.$ $$

Then $\widehat{\tau}_{k}=\widehat{\mathsf{{REV}}}_{k}(1)-\widehat{\mathsf{{REV}}}_{k
}(0)$ and
a variance estimate can be obtained by either
asymptotic normality, confidence region of $\beta$, or bootstrap.

## 6. Experiment

To evaluate our proposed framework and debiased estimator, we run both fully synthetic and semi-synthetic simulations to check if the proposed estimator for beta and revenue are indeed less biased as well as the coverage of the proposed estimator.

### 6.1. Fully synthetic experiments

In the fully synthetic experiments, we simulate 10 buyers and 1000 *good* items in two submarkets, with a varying number of *bad* items (up to 500) in order to study the effect of the contamination parameter $\alpha$. For each $\alpha$, we randomly sample a budget from a uniform distribution for each buyer, and compute $\beta^{*}$ and ${\mathsf{{REV}}}^{*}$ from the limit pure market ${\mathscr{M}}_{0}$ with a uniformly distributed value function in each submarket. To check if the debiased surrogate reduces bias, we compute $\beta^{*}_{\alpha}$ and ${\mathsf{{REV}}}^{*}_{\alpha}$ from the limit market with interference ${\mathscr{M}}_{\alpha}$ and their surrogates, $\widetilde{\beta}^{*}$ and $\widetilde{\mathsf{{REV}}}^{*}$. We look at the normalized bias for the surrogate, defined as $\|\widetilde{\beta}^{*}-\beta^{*}\|_{2}/\|\beta^{*}\|_{2}$ for pacing multipliers and as $|\widetilde{\mathsf{{REV}}}^{*}/{\mathsf{{REV}}}^{*}-1|$ for revenue, and similarly defined for the limit quantities. Fig 4 shows the normalized bias curves as a function of $\alpha$.
It can be seen that the magnitude of the bias increases with $\alpha$, for both the variables in the limit market with interference ${\mathscr{M}}_{\alpha}$ and their debiased surrogates. We see that the bias of the debiased surrogates is indeed much smaller than the contaminated limit quantities.

Figure: Figure 4. Normalized bias (in percent of true value) as a function of $\alpha$ in fully synthetic experiments. $\widetilde{\beta}^{*}$ and $\widetilde{\mathsf{{REV}}}^{*}$ are the debiased surrogates for pacing multiplier and revenue in the limit market with interference ${\mathscr{M}}_{\alpha}$.
Refer to caption: x3.png

Next, we check the coverage of the proposed variance estimator. For each $\alpha$ and each budget sample, we run 100 simulations in the following way: We sample items (or their values for each buyer) considering two submarkets and bad items. We then run the finite FPPE with bad items and obtain a baseline estimate for pacing multiplier and revenue without applying the debiasing procedure. Then, we apply the debiasing procedure to compute the debiased estimates. For each simulation, we check if the debiased surrogate is within the confidence interval of the debiased estimator. Finally, we aggregate them to compute the estimated coverage of the estimator. The results for both pacing multiplier and revenue are shown in Table 3. For the coverage of $\widehat{\beta}$, we first compute the coverage of each component and report only the average in the table. It is shown that $\widehat{\beta}$ is slightly over-covered. We found that the over-coverage is mainly due to the over-estimation of the estimator’s variance. For revenue, we construct the CI using the two approaches as mentioned in Sec. 4.4: one based on Eq. 11 and the other using parametric bootstrap based on the estimated asymptotic distribution of $\widehat{\beta}$ (with ”(b)” in the column names). First of all, we show in Fig 5 that both CIs converges to the true value from the limit market with interfrence ${\mathscr{M}}_{\alpha}$ as the number of items goes to infinity. Then, in  Table 3, we show that the bootstrap CI width is much smaller compared to that from Eq. 11. However, both CIs are over-covered, higher than the nominal level (95%). It is expected though that the CI from Eq. 11 is relatively large and hence conservative since it uses the min and max of each $\beta_{i}$, hence likely living outside of the confidence region ($n$-dimension ellipsoid) for $\widehat{\beta}$ and leading to over-coverage. For the bootstrap CI, it requires enough bootstrap samples to fully explore the whole boundary of the confidence region of $\widehat{\beta}$, as well as an accurate estimate of the asymptotic distribution of $\widehat{\beta}$. In our fully synthetic simulation, our hypothesis is that the over-coverage of the bootstrap CI is mainly due to the over-estimation of the variance for $\widehat{\beta}$.

**Table 3. Coverage of $\widehat{\beta}$ and revenue estimates in fully synthetic experiments. All quantities are averaged over 100 simulations for each $\alpha$. The coverage of $\widehat{\beta}$ is averaged over all components of $\widehat{\beta}$. For revenue estimates, the columns with ”(b)” represent the quantities from the bootstrap CI, while the columns without ”(b)” are for the CI from Eq. 11. The CI widths are normalized by the revenue from the limit market ${\mathscr{M}}_{\alpha}$. $\alpha$ is expressed as the ratio of the number of bad items and the total items.**
|  | $\widehat{\beta}$ | $\widehat{\mathsf{{REV}}}$ |  |  |  |
| --- | --- | --- | --- | --- | --- |
| $\alpha$ | coverage | CI width | CI width (b) | coverage | coverage (b) |
| 100/1100 | 0.993 | 0.132 | 0.043 | 1.0 | 1.0 |
| 200/1200 | 0.994 | 0.141 | 0.059 | 1.0 | 1.0 |
| 300/1300 | 0.980 | 0.107 | 0.059 | 1.0 | 1.0 |
| 400/1400 | 0.983 | 0.098 | 0.064 | 1.0 | 1.0 |
| 500/1500 | 0.971 | 0.108 | 0.069 | 1.0 | 1.0 |

Figure: Figure 5. Revenue confidence intervals as a function of the number of items. The analytic CI comes from Eq. 11. The true value is the debiased surrogates for revenue in the limit market with interference ${\mathscr{M}}_{\alpha}$.
Refer to caption: x4.png

### 6.2. Semi-synthetic experiments

We perform semi-synthetic experiments almost in the same way as with fully synthetic ones. The difference is that values are sampled from historical bidding data, making the budget and value distributions heavy-tailed. This matches real-world applications better, instead of the uniform distributions of the fully synthetic experiments. More specifically, we first sample a certain number of auctions. For each auction, we sample a given number of advertisers with their per-impression bids. Advertisers that are sampled across different auctions are treated as the same buyers and their budgets are determined by aggregating their values over auctions up to a scalar to calibrated
to get the percentage of budget-constrained buyers equal to what was observed in the real-world auction market, along the same lines as the experiments of Conitzer et al. ().

We observe similar results in terms of biases as in the fully synthetic experiments. The biases of the debiased surrogates are smaller than those from the limit market with interference ${\mathscr{M}}_{\alpha}$, with similar bias-$\alpha$ curves as shown in Fig 4. Coverage estimates for $\widehat{\beta}$ and $\widehat{\mathsf{{REV}}}$ are shown in Table 4. The CI from Eq. 11 is still very conservative as expected, achieving 100% coverage for all $\alpha$. However, unlike the fully synthetic experiment, the coverage for $\widehat{\beta}$ is slightly smaller than the nominal level, as well as the coverage of the bootstrap CI of revenue. The under-coverage for $\widehat{\beta}$ is mainly driven by the under-estimation of the variance of $\widehat{\beta}$, while the under-coverage of the bootstrap CI for revenue can also be partially attributed to the higher dimensionality (with 40 buyers), making the bootstrap resampling harder to explore the whole space. Although the proposed variance estimator has good asymptotic properties, the results from both the fully and semi- synthetic experiments suggest that it can perform badly, in either direction, for finite markets. Constructing more accurate variance estimators for our debiased estimator in finite settings would certainly mitigate the over- or under-coverage issues that we observe here and deserve more future research. One promising alternative is to construct the CI for $\widehat{\beta}$ and $\widehat{\mathsf{{REV}}}$ by directly bootstrapping the observed value matrix, though this might work best in a setting where the limit market has independent valuations across buyers.

**Table 4. Coverage of $\widehat{\beta}$ and revenue estimates in the semi-synthetic experiments. All quantities are averaged over 100 simulations for each $\alpha$ (the ratio of the number of bad items and the total items). The coverage of $\widehat{\beta}$ is averaged over all components of $\widehat{\beta}$. For revenue estimates, the columns with ”(b)” represent the quantities from the bootstrap CI, while the columns without ”(b)” are for the CI from Eq. 11. The CI widths are normalized by the revenue from the limit market ${\mathscr{M}}_{\alpha}$.**
|  | $\widehat{\beta}$ | $\widehat{\mathsf{{REV}}}$ |  |  |  |
| --- | --- | --- | --- | --- | --- |
| $\alpha$ | coverage | CI width | CI width (b) | coverage | coverage (b) |
| 1000/11000 | 0.877 | 0.244 | 0.044 | 1.0 | 0.95 |
| 2000/12000 | 0.849 | 0.225 | 0.043 | 1.0 | 0.87 |
| 3000/13000 | 0.852 | 0.210 | 0.041 | 1.0 | 0.81 |
| 4000/14000 | 0.828 | 0.200 | 0.040 | 1.0 | 0.90 |
| 5000/15000 | 0.826 | 0.191 | 0.039 | 1.0 | 0.90 |

## 7. Conclusion

We have proposed a practical experimental design for performing concurrent A/B tests in large-scale ad auction markets, using a submarket clustering approach, and showed that in production experiments, this submarket clustering approach leads to strong sign consistency performance, as compared to A/B testing on the full market, while allowing significantly-higher A/B test throughput.
In order to model the potential for interference between submarket A/B tests, we introduced a theoretical model of statistical inference in first-price pacing equilibrium problems, under settings with supply contamination. We showed how one can perform statistical inference in such a setting using a debiased estimation procedure, and studied the statistical properties of this procedure.
We then showed how our model of statistical inference in FPPE with contamination can be used to model the submarket clustering parallel A/B test design, and gave theoretical guarantees on performance.
Finally, we presented numerical experiments on fully synthetic and semi-synthetic data derived from Meta ad auctions. The experiments showed that our proposed debiased estimator achieves smaller biases and its statistical coverage on realistic data is generally in line with the predictions from our theory.

## Appendix A Technical Lemmas

### A.1. A perturbation result for constrained stochastic programs

We introduce a theorem from Shapiro ().
Theorem 1 in that paper handles the case where the constraints are also defined as expectations of some random functions, and constraints are also perturbed in the analysis. For simplicity we specialize the theorem to deterministic constraints and do not perturb constraints. Note that Theorem 7.27 from Shapiro et al. () can also be used to prove our Thm. 1.

Let $\Theta$ be a probability space equipped with an appropriate $\sigma$-algebra. Consider $F:\Theta\times{\mathbb{R}}^{n}\to{\mathbb{R}}$ and a set $B\subset{\mathbb{R}}^{n}$, given by

$$ $B=\left\{\beta:g_{i}(\beta)=0,i\in I;g_{i}(\beta)\leq 0,i\in J\right\}.$ $$

where $I$ and $J$ are finite index sets.
Let $P$ and $Q$ be two probability measures on $\Theta$.
Let $\phi(\beta,\alpha)=(P+\alpha(Q-P))F(\cdot,\beta)$ and $\phi(\beta)=\phi(\beta,0)$. Here $\min_{B}\phi$ is the main program of interest and $Q$ is a perturbation measure. The amount of perturbation is measured by $\alpha\in[0,1]$.

Let $\beta^{*}$ be the unique minimizer of $\phi(\beta)$ over $B$.
Let $J^{*}\subset J$ be the inequality constraints active at $\beta^{*}$, meaning $g_{i}(\beta^{*})=0$ for all $i\in J^{*}$.
Let $\beta^{*}(\alpha)$ be the unique optimal solution to $\min_{B}\phi({\cdot},\alpha)$.

###### Lemma 0 (Theorem 1 from Shapiro ( 1990 ) ).

Define the Lagrangian function by
$L(\beta,\lambda,\alpha)=\phi(\beta,\alpha)+\sum_{i\in I\cup J}\lambda_{i}g_{i}
(\beta)$ and $L_{0}(\beta,\lambda)=L(\beta,\lambda,0)$.
Let $\Lambda_{0}$ be the set of optimal Lagrangian multipliers (^1^11$\lambda\in\Lambda_{0}$ iff $\nabla_{x}L_{0}\left(\beta_{0},\lambda\right)=0$ and $\lambda_{i}\geq 0,i\in J^{*}$ and $\lambda_{i}=0$ for all $j\in J\setminus J^{*}$.). Define the critical cone $C=\{u:u^{\scriptscriptstyle\mathsf{T}}\nabla g_{i}(\beta^{*})=0,i\in I;u^{
\scriptscriptstyle\mathsf{T}}\nabla g_{i}(\beta^{*})\leq 0$ for $i\in J^{*};u^{\scriptscriptstyle\mathsf{T}}\nabla\phi(\beta^{*})\leq 0\}$. Define $\Lambda_{0}^{*}=\operatorname*{arg\,max}_{\lambda\in\Lambda_{0}}\sum_{i\in I
\cup J}\lambda_{i}g_{i}(\beta^{*})$.
Assume the following conditions.

- 1.
Differentiability. The functions $\phi({\cdot},0)$, $\phi({\cdot},1)$, and $g_{i},i\in I\cup J$, are continuously differentiable in a neighborhood of $\beta^{*}$.
- 2.
Constraint Qualification. The gradients $\nabla g_{i}(\beta^{*})$, $i\in I$ are linearly independent. And there exists $b\in{\mathbb{R}^{n}}$ such that $b^{\scriptscriptstyle\mathsf{T}}\nabla g_{i}(\beta^{*})=0$ for $i\in I$ and $b^{\scriptscriptstyle\mathsf{T}}\nabla g_{i}(\beta^{*})<0$ for $i\in J^{*}$.
- 3.
Differentiability. The functions $\phi({\cdot},0)$, $g_{i}$, $i\in I\cup J^{*}$ are twice continuously differentiable in a neighbourhood of $\beta^{*}$.
- 4.
Second-order sufficient condition. Assume for all nonzero $u\in C$, $\max_{\lambda\in\Lambda_{0}^{*}}u^{\scriptscriptstyle\mathsf{T}}\nabla^{2}_{xx
}L_{0}(\beta^{*},\lambda)u>0$.

Define the function $b$ and the set $\Sigma$ by

$$ (12) $\displaystyle b(u)$ $\displaystyle=\max_{\lambda\in\Lambda_{0}^{*}}u^{\scriptscriptstyle\mathsf{T}} \nabla^{2}_{xt}L_{0}(\beta^{*},\lambda)+\frac{1}{2}u^{\scriptscriptstyle \mathsf{T}}\nabla^{2}_{xx}L_{0}(\beta^{*},\lambda)u$ (13) $\displaystyle\Sigma$ $\displaystyle=\operatorname*{arg\,min}u^{\scriptscriptstyle\mathsf{T}}\nabla \phi(\beta^{*})$ $\displaystyle\mathrm{s.t.}\quad u^{\scriptscriptstyle\mathsf{T}}\nabla g_{i}( \beta^{*})=0,i\in I;u^{\scriptscriptstyle\mathsf{T}}\nabla g_{i}(\beta^{*}) \leq 0,i\in J^{*};$ $$

Then (i) there exists a positive $K$ such that $\|\beta^{*}(\alpha)-\beta^{*}\|\leq K\alpha$ for all positive $\alpha$ in a neighborhood of zero. (ii) If, in addition, the function $b$ has a unique minimizer $u$ over $\Sigma$, then the limit
$\lim_{\alpha\downarrow 0}(\beta^{*}(\alpha)-\beta^{*})/\alpha$ exists and equals to $u$.

###### Lemma 0 (Theorem 1 from Shapiro ( 1990 ) ).

## Appendix B Proofs

### B.1. Proof of Closed-Form Expression for d ⁢ β * ⁢ ( α ) d superscript 𝛽 𝛼 \mathop{}\!\mathrm{d}\beta^{*}(\alpha) roman_d italic_β start_POSTSUPERSCRIPT * end_POSTSUPERSCRIPT ( italic_α ) and Thm. 1

Now we prove Thm. 1.

To begin with, we define directional derivatives. For a probability measure $P$ on the item set $\Theta$, let $\beta(P)$ be the unique optimal solution to the Eisenberg-Gale program: $\beta(P)=\min_{\beta\in(0,1]^{n}}\int F(\theta,\beta)\mathop{}\!\mathrm{d}P(\theta)$.
Let $P$ and $Q$ be two measures.
When exists, the directional derivative is defined as

$$ $\displaystyle\mathop{}\!\mathrm{d}\beta(P;Q-P)=\lim_{t\downarrow 0}\frac{\beta (P+t(Q-P))-\beta(P)}{t}.$ $$

Step 1. Show there exist $K>0$ and $\bar{\alpha}_{1}>0$ such that $\|\beta^{*}_{\alpha}-\beta^{*}\|\leq K\alpha$ for $\alpha\in[0,\bar{\alpha}_{1}]$. And $\mathop{}\!\mathrm{d}\beta(s;s-s^{\prime})=-(P{\mathscr{H}}_{0}P)^{\dagger}
\Delta_{0}$.

We apply Lemma 1 with $F(\theta,\beta)=\max_{i}{v_{i}(\theta)}{\beta_{i}}-\sum_{i}b_{i}\log({\beta_{i
}})$, $g_{i}(\beta)={\beta_{i}}-1$, $B=(0,1]^{n}$, $\mathop{}\!\mathrm{d}P=s\mathop{}\!\mathrm{d}\theta$, $\mathop{}\!\mathrm{d}(Q-P)=(s-s^{\prime})\mathop{}\!\mathrm{d}\theta$.

By Definition 3 (SCS). in the market ${\mathscr{M}}_{0}$, the set $\Sigma$ (defined in Eq. 13) becomes the plane $\{u:u_{i}=0,i\in I_{+}\}$ where we recall $I_{+}=\{i:{\beta^{*}_{i}}=0\}$.
By Definition 2 (SMO). in ${\mathscr{M}}_{0}$, we know the Lagrangian multiplier of the EG program in ${\mathscr{M}}_{0}$ is unique, and thus $b({\cdot})$ defined in Eq. 12 becomes
$b(u)=u^{\scriptscriptstyle\mathsf{T}}\Delta_{0}+\frac{1}{2}u^{
\scriptscriptstyle\mathsf{T}}{\mathscr{H}}_{0}u$.
We conclude
the directional derivative is
$\mathop{}\!\mathrm{d}\beta(s;s-s^{\prime})=\lim_{\delta\to 0^{+}}(\beta^{*}_{0
}-\beta^{*}_{-\delta})/\delta=-(P{\mathscr{H}}_{0}P)^{\dagger}\Delta_{0}$, where
$P={\operatorname*{Diag}}(1({\beta^{*}_{i}}<1))$ and $\Delta_{0}=\int\nabla f({\cdot},\beta^{*})(s-s^{\prime})\mathop{}\!\mathrm{d}\theta$

Step 2. For all $\alpha\geq 0$ small enough, it holds $I_{\alpha}=I$, where we recall $I_{\alpha}=\{i:\beta^{*}_{\alpha,i}=1\}$ and $I=\{i:{\beta^{*}_{i}}=0\}$.

First we show $I\subseteq I_{\alpha}$, i.e., $\{i:{\beta^{*}_{i}}=1\}\subseteq\{i:\beta^{*}_{\alpha,i}=1\}$ for $\alpha$ small enough. This is saying, if a constraint ${\beta_{i}}\leq 1$ is strongly active in the EG program of ${\mathscr{M}}_{0}$, then it is also strongly active in the EG program of the market ${\mathscr{M}}_{\alpha}$.
This holds by Lemma 2.2 from Shapiro () and
the Definition 3 (SCS). condition in ${\mathscr{M}}_{0}$.
Next, we show, for small $\alpha$, $I_{\alpha}\subseteq I$ by showing $[n]\setminus I\subseteq[n]\setminus I_{\alpha}$, i.e., $\{i:{\beta^{*}_{i}}<1\}\subseteq\{i:\beta^{*}_{\alpha,i}<1\}$. This holds by $\|\beta^{*}_{\alpha}-\beta^{*}\|\leq K\alpha$ for all $\alpha\leq\bar{\alpha}_{1}$.

Step 3. For all $\alpha\geq 0$ small enough, the directional derivative $\mathop{}\!\mathrm{d}\beta(s_{\alpha};s-s^{\prime})=\lim_{\delta\to 0^{+}}(
\beta^{*}_{\alpha-\delta}-\beta^{*}_{\alpha})/\delta$ exists and $\alpha\mapsto\mathop{}\!\mathrm{d}\beta(s_{\alpha};s-s^{\prime})$ is continuous.

Applying Lemma 1 with $\mathop{}\!\mathrm{d}P=s_{\alpha}\mathop{}\!\mathrm{d}\theta$ and $\mathop{}\!\mathrm{d}(Q-P)=(s-s^{\prime})\mathop{}\!\mathrm{d}\theta$ shows that the directional derivative exists for all $\alpha$ small enough. Now we find an expression for $\mathop{}\!\mathrm{d}\beta(s_{\alpha};s-s^{\prime})$.
By Definition 2 (SMO). in ${\mathscr{M}}_{0}$, twice continuously differentiability of $\beta\mapsto\nabla^{2}\int F(\beta,\theta)s^{\prime}\mathop{}\!\mathrm{d}\theta$ in a neighborhood of $\beta^{*}$, and the Lipschitzness result from step 1,
we know $\beta\to\nabla^{2}\int F(\theta,\beta)s_{\alpha}\mathop{}\!\mathrm{d}\theta$ is twice continuously differentiable at $\beta^{*}_{\alpha}$, which implies
uniqueness of Lagrangian multiplier in the market ${\mathscr{M}}_{\alpha}$, for all $\alpha$ small enough (Lemma 2.2 from Shapiro ()). And thus $b({\cdot})$ defined in Eq. 12 becomes
$b(u)=u^{\scriptscriptstyle\mathsf{T}}\Delta_{\alpha}+\frac{1}{2}u^{
\scriptscriptstyle\mathsf{T}}{\mathscr{H}}_{\alpha}u$.
Next, the Definition 3 (SCS). condition in the market ${\mathscr{M}}_{0}$
implies SCS in ${\mathscr{M}}_{\alpha}$ for all $\alpha$ small enough.
The set $\Sigma$ (defined in Eq. 13) becomes the plane $\{u:u_{i}=0,i\in I_{\alpha}\}$.
The quadratic program $\min_{\Sigma}b$ has a closed-form solution: $u=-(P_{\alpha}{\mathscr{H}}_{\alpha}P_{\alpha})^{\dagger}\Delta_{\alpha}$ where $P_{\alpha}={\operatorname*{Diag}}(1(\beta^{*}_{\alpha,i}<1))$, $\Delta_{\alpha}=\int\nabla f({\cdot},\beta^{*}_{\alpha})(s-s^{\prime})\mathop{
}\!\mathrm{d}\theta$.
And we apply $I_{\alpha}=I$ for all $\alpha$ small enough from step 2, and so $P_{\alpha}=P$.
We conclude

$$ $\displaystyle\mathop{}\!\mathrm{d}\beta(s_{\alpha};s-s^{\prime})=-(P{\mathscr{ H}}_{\alpha}P)^{\dagger}\Delta_{\alpha}$ $$

for all $\alpha$ small enough.

Next, we show that the map $\alpha\mapsto(P{\mathscr{H}}_{\alpha}P)^{\dagger}\Delta_{\alpha}$ is continuous for all $\alpha$ small enough.
To see this, note ${\mathscr{H}}_{\alpha}=\nabla^{2}\int F(\theta,\beta^{*}_{\alpha})s_{\alpha}
\mathop{}\!\mathrm{d}\theta=(1-\alpha)\nabla^{2}\int F({\cdot},\beta^{*}_{
\alpha})s\mathop{}\!\mathrm{d}\theta+\alpha\nabla^{2}\int F({\cdot},\beta^{*}_
{\alpha})s^{\prime}\mathop{}\!\mathrm{d}\theta$ is continuous in $\alpha$. The $P$ matrix is fixed, and so $\alpha\mapsto P{\mathscr{H}}_{\alpha}P$ is continuous. Without loss of generality, suppose ${I^{c}}=\{i:{\beta^{*}_{i}}<0\}=[k]$ are the first $k$ buyers. Then $P{\mathscr{H}}_{\alpha}P$ creates a matrix with upper left $k$-by-$k$ block equal to the upper left $k$-by-$k$ block of ${\mathscr{H}}_{\alpha}$, denoted ${\mathscr{H}}_{\alpha,{I^{c}}{I^{c}}}$, and zeros everywhere else.
Then $(P{\mathscr{H}}_{\alpha}P)^{\dagger}$ is a matrix with upper left block being $({\mathscr{H}}_{\alpha,{I^{c}}{I^{c}}})^{{-1}}$ and zeros everywhere else.
Since ${\mathscr{H}}_{\alpha}$ is positive definite for all small $\alpha$, the submatrix ${\mathscr{H}}_{\alpha,{I^{c}}{I^{c}}}$ must also be positive definite. And so $\alpha\mapsto({\mathscr{H}}_{\alpha,{I^{c}}{I^{c}}})^{{-1}}$ is continuous, implying continuity of $\alpha\mapsto(P{\mathscr{H}}_{\alpha}P)^{\dagger}$.
Finally, $\Delta_{\alpha}=\int\nabla f({\cdot},\beta^{*}_{\alpha})(s-s^{\prime})\mathop{
}\!\mathrm{d}\theta$ is continuous in $\alpha$.

Step 4. The desired claim: $\widetilde{\beta}^{*}-\beta^{*}=o(\alpha)$.
Note $\mathop{}\!\mathrm{d}\beta^{*}(\alpha)$ defined in Eq. 6 is exactly $\mathop{}\!\mathrm{d}\beta(s_{\alpha};s-s^{\prime})$. And so

$$ $\displaystyle\widetilde{\beta}^{*}-\beta^{*}$ $\displaystyle=\beta^{*}_{\alpha}+\alpha\mathop{}\!\mathrm{d}\beta(s_{\alpha};s -s^{\prime})-\beta^{*}$ $\displaystyle=(\beta^{*}-\alpha\mathop{}\!\mathrm{d}\beta(s_{0};s-s^{\prime})+ o(\alpha))+\alpha\mathop{}\!\mathrm{d}\beta(s_{\alpha};s-s^{\prime})-\beta^{*}$ $\displaystyle=\alpha(\mathop{}\!\mathrm{d}\beta(s_{\alpha};s-s^{\prime})- \mathop{}\!\mathrm{d}\beta(s_{0};s-s^{\prime}))+o(\alpha)=o(\alpha).$ $$

where the last line uses continuity in $\alpha$ of the directional derivative.
This completes the proof of Thm. 1.

### B.2. Proof of Thm. 2

Recall the estimator $\widehat{\beta}=\beta^{\gamma}_{\alpha}-\alpha(\widehat{P}_{\alpha}\widehat{
\mathscr{H}}_{\alpha}\widehat{P}_{\alpha})^{\dagger}\widehat{\Delta}_{\alpha}$
and the debiased surrogate is $\widetilde{\beta}^{*}=\beta^{*}_{\alpha}-\alpha(P_{\alpha}{\mathscr{H}}_{
\alpha}P_{\alpha})^{\dagger}\Delta_{\alpha}$.
By results from Liao and Kroer () we know that Definition 3 (SCS). and Definition 2 (SMO). in the market ${\mathscr{M}}_{\alpha}$ and $\eta_{t}=o(1)$ imply $\beta^{\gamma}_{\alpha}{\,\overset{{p}}{\to}\,}\beta^{*}_{\alpha}$ and $(\widehat{P}_{\alpha}\widehat{\mathscr{H}}_{\alpha}\widehat{P}_{\alpha})^{
\dagger}{\,\overset{{p}}{\to}\,}(P_{\alpha}{\mathscr{H}}_{\alpha}P_{\alpha})^{\dagger}$.
Finally, $\widehat{\Delta}_{\alpha}{\,\overset{{p}}{\to}\,}\Delta_{\alpha}$ holds by the law of large numbers. We complete the proof of Thm. 2.

### B.3. Proof of Thm. 3

Recall the estimator $\widehat{\beta}=\beta^{\gamma}_{\alpha}-\alpha(\widehat{P}_{\alpha}\widehat{
\mathscr{H}}_{\alpha}\widehat{P}_{\alpha})^{\dagger}\widehat{\Delta}_{\alpha}$
and the debiased surrogate is $\widetilde{\beta}^{*}=\beta^{*}_{\alpha}-\alpha(P_{\alpha}{\mathscr{H}}_{
\alpha}P_{\alpha})^{\dagger}\Delta_{\alpha}$.
Define

$$ (14) $\displaystyle Z_{t}=\beta^{\gamma}_{\alpha}-\beta^{*}_{\alpha}-\alpha(P_{ \alpha}{\mathscr{H}}_{\alpha}P_{\alpha})^{\dagger}(\widehat{\Delta}_{\alpha}- \Delta_{\alpha})$ (15) $\displaystyle\mho_{t}=-\alpha((\widehat{P}_{\alpha}\widehat{\mathscr{H}}_{ \alpha}\widehat{P}_{\alpha})^{\dagger}-(P_{\alpha}{\mathscr{H}}_{\alpha}P_{ \alpha})^{\dagger})\Delta_{\alpha}$ (16) $\displaystyle\zeta_{t}=-\alpha((\widehat{P}_{\alpha}\widehat{\mathscr{H}}_{ \alpha}\widehat{P}_{\alpha})^{\dagger}-(P_{\alpha}{\mathscr{H}}_{\alpha}P_{ \alpha})^{\dagger})(\widehat{\Delta}_{\alpha}-\Delta_{\alpha})$ $$

Then clearly $\widehat{\beta}-\widetilde{\beta}^{*}=Z_{t}+\mho_{t}+\zeta_{t}$.

Step 1. We show $\sqrt{t}Z_{t}$ converges to a normal distribution.
By Liao and Kroer (), it holds

$$ (17) $\displaystyle\sqrt{t}(\beta^{\gamma}_{\alpha}-\beta^{*}_{\alpha})=\frac{1}{ \sqrt{t}}{\sum_{\tau=1}^{t}}D_{\alpha}({\theta^{\tau}})-{\mathbb{E}}[D_{\alpha }]+o_{p}(1)$ $$

where $D_{\alpha}(\theta)=-(P_{\alpha}{\mathscr{H}}_{\alpha}P_{\alpha})^{\dagger}
\nabla f(\theta,\beta^{*}_{\alpha})$ is the influence function.
Next, we define the likelihood ratio function $g(\theta)=(\mathop{}\!\mathrm{d}(s-s^{\prime})/\mathop{}\!\mathrm{d}s_{\alpha}
)(\theta)=-\frac{1}{\alpha}1_{\Theta_{\mathsf{bad}}}(\theta)+\frac{1}{1-\alpha
}1_{\Theta_{\mathsf{good}}}(\theta)$.
Then $\Delta_{\alpha}=\int\nabla f({\cdot},\beta^{*}_{\alpha})(s-s^{\prime})\mathop{
}\!\mathrm{d}\theta=\int g({\cdot})\nabla f({\cdot},\beta^{*}_{\alpha})s_{
\alpha}({\cdot})\mathop{}\!\mathrm{d}\theta$.
Then by definition $\widehat{\Delta}_{\alpha}$ can be written as $\widehat{\Delta}_{\alpha}=P_{t}(g({\cdot})\nabla f({\cdot},\beta^{\gamma}_{
\alpha}))$ where $P_{t}h({\cdot})=\frac{1}{t}{\sum_{\tau=1}^{t}}h({\theta^{\tau}})$ and ${\theta^{\tau}}$ are i.i.d. draws from $s_{\alpha}$.
Also let $\nu_{t}h=\sqrt{t}(P_{t}h-\int hs_{\alpha}\mathop{}\!\mathrm{d}\theta)$.
By the decomposition,

$$ (18) $\displaystyle\sqrt{t}(\widehat{\Delta}_{\alpha}-\Delta_{\alpha})$ (19) $\displaystyle=$ $\displaystyle\sqrt{t}\bigg{(}P_{t}\big{(}g({\cdot})\nabla f({\cdot},\beta^{ \gamma}_{\alpha})\big{)}-\int g({\cdot})\nabla f({\cdot},\beta^{*}_{\alpha})s_ {\alpha}\mathop{}\!\mathrm{d}\theta\bigg{)}$ (20) $\displaystyle=$ $\displaystyle\sqrt{t}\bigg{(}P_{t}\big{(}g({\cdot})\nabla f({\cdot},\beta^{*}_ {\alpha})\big{)}-\int g({\cdot})\nabla f({\cdot},\beta^{*}_{\alpha})s_{\alpha} \mathop{}\!\mathrm{d}\theta\bigg{)}+\nu_{t}\big{(}g({\cdot})(\nabla f({\cdot}, \beta^{\gamma}_{\alpha})-\nabla f({\cdot},\beta^{*}_{\alpha}))\big{)}$ (21) $\displaystyle=$ $\displaystyle\nu_{t}(g({\cdot})\nabla f({\cdot},\beta^{*}_{\alpha}))+o_{p}(1),$ $$

where the last line follows by a stochastic equicontinuity argument as in Liao and Kroer ().
Finally, $D_{\alpha}({\cdot})-\alpha g({\cdot})(P_{\alpha}{\mathscr{H}}_{\alpha}P_{
\alpha})^{\dagger}\nabla f({\cdot},\beta^{*}_{\alpha})$ is exactly the influence function $IF_{1}({\cdot})$ defined in the theorem.

Step 2. Show $\mho_{t}=O_{p}(\eta_{t})$.

We need a matrix perturbation result.

###### Lemma 0 (Theorem 2.2 from Stewart ( 1977 ) ).

Let $A$ and $B=A+E$ be nonsingular square matrices, and $\|A^{{-1}}\|\|E\|<1$.
Here $\|{\cdot}\|$ is the operator norm. Then

$$ (22) $\displaystyle\|B^{{-1}}-A^{{-1}}\|\leq\frac{\|A^{{-1}}\|^{2}}{1-\|E\|\|A^{{-1} }\|}\|E\|.$ $$

By Liao and Kroer (), we know ${\mathbb{P}}(\widehat{P}_{\alpha}=P_{\alpha})\to 1$.
Without loss of generality, suppose $I^{c}_{\alpha}=\{i:\beta^{*}_{\alpha,i}<0\}=[k]$ are the first $k$ buyers.
Then $(P_{\alpha}{\mathscr{H}}_{\alpha}P_{\alpha})$ creates a matrix with upper left $k$-by-$k$ block equal to the upper left $k$-by-$k$ block of ${\mathscr{H}}_{\alpha}$, denoted ${\mathscr{H}}_{\alpha,{I^{c}}{I^{c}}}$, and zeros everywhere else.
Under the event $\{\widehat{P}_{\alpha}=P_{\alpha}\}$,
the matrix $(\widehat{P}_{\alpha}\widehat{\mathscr{H}}_{\alpha}\widehat{P}_{\alpha})$ is one with upper left $k$-by-$k$ block being upper left $k$-by-$k$ block of $\widehat{\mathscr{H}}_{\alpha}$, denoted $\widehat{\mathscr{H}}_{\alpha,{I^{c}}{I^{c}}}$, and zeros everywhere else.
Now let $A=\widehat{\mathscr{H}}_{\alpha,{I^{c}}{I^{c}}}$ and $B={\mathscr{H}}_{\alpha,{I^{c}}{I^{c}}}$.
It is clear that $\|\mho_{t}\|_{2}\leq\alpha\|\Delta_{\alpha}\|_{2}\|A^{{-1}}-B^{{-1}}\|=O_{p}(
\eta_{t})$ by Lemma 1.

Step 3. Show $\zeta_{t}=o_{p}(\frac{1}{\sqrt{t}})$.
From previous derivation, $\widehat{\Delta}_{\alpha}-\Delta_{\alpha}=O_{p}(1/\sqrt{t})$ and $(\widehat{P}_{\alpha}\widehat{\mathscr{H}}_{\alpha}\widehat{P}_{\alpha})^{
\dagger}-(P_{\alpha}{\mathscr{H}}_{\alpha}P_{\alpha})^{\dagger}=O_{p}(\eta_{t}
)=o_{p}(1)$. We conclude $\zeta_{t}=O_{p}(1/\sqrt{t})o_{p}(1)=o_{p}(1/\sqrt{t})$.

We complete the proof of Thm. 3 part 1.

Now we prove part 2. The claim that ${\mathscr{H}}_{\alpha}={\operatorname*{Diag}}(b_{i}/(\beta^{*}_{\alpha,i})^{2})$ follows from Liao et al. (). Next we derive the asymptotic distribution.
We only need to handle the $\mho_{t}$ term defined as before. Under the event $\{\widehat{P}_{\alpha}=P_{\alpha}\}$, we have $\mho_{t}=-{\operatorname*{Diag}}(\alpha\Delta_{\alpha,i}1(\beta^{*}_{\alpha,i}
<1)b_{i}^{{-1}})(\beta^{\gamma}_{\alpha}\circ\beta^{\gamma}_{\alpha}-\beta^{*}
_{\alpha}\circ\beta^{*}_{\alpha})$.
By Liao and Kroer () we know $\sqrt{t}(\beta^{\gamma}_{\alpha}-\beta^{*}_{\alpha})=\frac{1}{\sqrt{t}}{\sum_{
\tau=1}^{t}}(D_{\alpha}({\theta^{\tau}})-{\mathbb{E}}_{\alpha}[D_{\alpha}])+o_
{p}(1)$. So by the delta method,

$$ $\displaystyle\sqrt{t}(\beta^{\gamma}_{\alpha}\circ\beta^{\gamma}_{\alpha}- \beta^{*}_{\alpha}\circ\beta^{*}_{\alpha})={\operatorname*{Diag}}(2\beta^{*}_{ \alpha})\frac{1}{\sqrt{t}}{\sum_{\tau=1}^{t}}(D_{\alpha}({\theta^{\tau}})-{ \mathbb{E}}_{\alpha}[D_{\alpha}])+o_{p}(1).$ $$

Summarizing,

$$ $\displaystyle\sqrt{t}\mho_{t}=-{\operatorname*{Diag}}(2\beta^{*}_{\alpha,i} \alpha\Delta_{\alpha,i}1(\beta^{*}_{\alpha,i}<1)b_{i}^{{-1}})\frac{1}{\sqrt{t} }{\sum_{\tau=1}^{t}}(D_{\alpha}({\theta^{\tau}})-{\mathbb{E}}_{\alpha}[D_{ \alpha}])+o_{p}(1)$ $$

Note that for $i$ such that $\beta^{*}_{\alpha,i}=1$, the $i$-th entry of $D_{\alpha}$ will be zero almost surely, so the indicator can be dropped in the diagonal matrix.
Now let $IF_{2}(\theta)=-{\operatorname*{Diag}}(2\alpha\beta^{*}_{\alpha,i}\Delta_{
\alpha,i}b_{i}^{{-1}})D_{\alpha}+IF_{1}$, where we recall $IF_{1}$ is defined in Thm. 3. Then under the condition that
${\mathbb{E}}_{\alpha}[\frac{1}{{\mathsf{{bidgap}}}(\beta^{*}_{\alpha},\theta)}
]<\infty$, we have $\sqrt{t}(\widehat{\beta}-\widetilde{\beta}^{*}){\,\overset{{d}}{\to}\,}{
\mathscr{N}}(0,{\mathbb{E}}_{\alpha}[(IF_{2}-{\mathbb{E}}_{\alpha}[IF_{2}])(IF
_{2}-{\mathbb{E}}_{\alpha}[IF_{2}])^{\scriptscriptstyle\mathsf{T}}])$.

###### Lemma 0 (Theorem 2.2 from Stewart ( 1977 ) ).

### B.4. Proof of Thm. 4

###### Proof.

Note

$$ $\displaystyle\bigg{|}\int\max_{i}({v_{i}(\theta)}\widetilde{\beta}^{*}_{i})s( \theta)\mathop{}\!\mathrm{d}\theta-\int\max_{i}({v_{i}(\theta)}\beta^{*}_{i})s (\theta)\mathop{}\!\mathrm{d}\theta\bigg{|}$ $\displaystyle\leq\int|\max_{i}({v_{i}(\theta)}\widetilde{\beta}^{*}_{i})s( \theta)-\max_{i}({v_{i}(\theta)}\beta^{*}_{i})s(\theta)|\mathop{}\!\mathrm{d}\theta$ $\displaystyle\leq\max_{i}\sup_{\theta}v_{i}(\theta)\cdot\|\beta^{*}-\widetilde {\beta}^{*}\|_{\infty}=o(\alpha)$ $$

as $\alpha\downarrow 0$.

Let $P_{t}h({\cdot})=\frac{1}{t}{\sum_{\tau=1}^{t}}h({\theta^{\tau}})$ and ${\theta^{\tau}}$ are i.i.d. draws from $s_{\alpha}$, and $Ph({\cdot})=\int hs_{\alpha}$.
Let $g({\cdot})=\mathop{}\!\mathrm{d}s/\mathop{}\!\mathrm{d}s_{\alpha}=\frac{1}{1-
\alpha}1_{\Theta_{\mathsf{good}}}$.
Also let $\nu_{t}h=\sqrt{t}(P_{t}h-\int hs_{\alpha}\mathop{}\!\mathrm{d}\theta)$.
We change the measure and write
$\widetilde{{\mathsf{{REV}}}}^{*}=\int f(\theta,\widetilde{\beta}^{*})s\mathop{
}\!\mathrm{d}\theta=\int f(\theta,\widetilde{\beta}^{*})g(\theta)s_{\alpha}
\mathop{}\!\mathrm{d}\theta$.
For the claim regarding asymptotic normality,
note

$$ $\displaystyle\sqrt{t}(\widehat{{\mathsf{{REV}}}}-\widetilde{{\mathsf{{REV}}}}^ {*})$ $\displaystyle=\sqrt{t}[P_{t}f({\cdot},\widehat{\beta})g({\cdot})-Pf({\cdot}, \widetilde{\beta}^{*})g({\cdot})]$ $\displaystyle=\sqrt{t}[P_{t}f({\cdot},\widetilde{\beta}^{*})g({\cdot})-Pf({ \cdot},\widetilde{\beta}^{*})g({\cdot})]+\sqrt{t}[Pf({\cdot},\widehat{\beta})g ({\cdot})-Pf({\cdot},\widetilde{\beta}^{*})g({\cdot})]$ $\displaystyle\quad+\sqrt{t}(P_{t}-P)(f({\cdot},\widehat{\beta})g({\cdot})-f({ \cdot},\widetilde{\beta}^{*})g({\cdot}))$ $\displaystyle=\nu_{t}(f({\cdot},\widetilde{\beta}^{*})g({\cdot}))+\nu_{t}({ \mathbb{E}}_{\alpha}[g({\cdot})\nabla f({\cdot},\widetilde{\beta}^{*})]^{ \scriptscriptstyle\mathsf{T}}IF({\cdot}))+o_{p}(1)$ $$

where the last line follows by the delta method and a stochastic equicontinuity argument. We complete the proof of the theorem.
∎

###### Proof.

## Peer Reviews

confidence: 3
contribution: 3
initial_rating: 8
presentation: 3
questions: 1. The theoretical guarantees heavily rely on the parameter \eta for the error in estimating the Hessian. Could the authors comment on the the rate \eta for the proposed finite differencing Hessian estimator proposed in the paper? 

2.  Does the guarantee on the revenue in Theorem 4 generalize to estimating other quantities that are a functional of the parameters \beta?

3. Providing a discussion on limitations and addressing my minor comment #4 above would be helpful.
soundness: 3
strengths: Interference is a well-known, important, and practical problem in A/B testing of online marketplaces. The paper meaningfully extends prior work on pacing equilibria by considering how interference affects the results. The paper provides rigorous theoretical guarantees, complemented by preliminary semi-synthetic experiments.
summary: This paper considers A/B testing in online marketplaces, where interference arises because items can be recommended to advertisers in both the control and the treatment groups. The paper adopts the first-price pacing equilibrium (FPPE) framework from prior work [Conitzer et al., 2022], and analyzes how the equilibrium (i.e., pacing parameters \beta and the total revenue) changes as a function of the level of contamination/interference. The paper proposes a first-order bias correction by Taylor expansion and proves consistency and asymptotic normality guarantees under certain regularity conditions. The paper conducts semi-synthetic experiments to demonstrate the effectiveness of the proposed estimators (where the distributions come from real data).
weaknesses: While I find the formulation and assumptions proposed in the paper reasonable, I think the paper can benefit from a discussion on limitations. For example, a natural alternative model is not to partition items into a good set and a bad set, but instead assume an item can be "good" for some groups but "bad" (interference) for other groups. The paper may also discuss other types of interference that the proposed approach does not cover, etc.

===
Additional minor comments:

1. I think the claim in the abstract on demonstrating "the effectiveness of this approach on real experiments on advertising markets at Meta" is an overstatement. I think the paper should make it clear that only semi-synthetic experiments are performed.

2. While I understand citing unpublished work is optional, I think the paper can benefit from citing the following paper:
Zhu, Cai, Zheng, and Si. "Seller-Side Experiments under Interference Induced by Feedback Loops in Two-Sided Platforms". arXiv, 2024.

3.  I find the term "buyer" confusing, as "buyers" can be understood as users who make purchases or sellers who bid for ads. I prefer the terminology of advertisers vs. users introduced in other parts of the paper.

4.  I don't understand details of the experiment in Fig 1. What're the axes? Are these ratios of the two experimental designs? How are two experiments paired? What is the guardrail metric? Also a 79% agreement compared to the 81.5% optimal agreement appears pretty good to me. Is the paper trying to address the remaining 2.5% of the cases?

5.  In Paragraph L87-118, two graphs are introduced. One is a bipartite graph between advertisers and users, from which a graph for advertisers is derived for clustering ("the edge weight between a pair of advertisers..."). It would be helpful to clarify they are not the same graph.

6.  Paragraph L54-63: "Sec" -> "Fig"

---

confidence: 3
contribution: 2
initial_rating: 5
presentation: 1
questions: NA
soundness: 3
strengths: 1. I can understand that A/B test design is indeed a very important issue in the industry. Since buyers will use strategic behaviors to counter the platform's strategy or mechanism, a/b testing is the most effective way to test the platform's strategy/mechanism. However, the vanilla a/b test setup will bring about the issue of mutual influence between a/b groups, which may interfere with the experimental results.

2. The theoretical proof in the paper is non-trivial and very solid. The authors have proved from many aspects that the estimator can indeed converge well to the limit FPPE in the uncontaminated market, which the platform hopes to observe.
summary: This paper studies how to estimate market equilibria in each submarket when there is interference across submarkets. The authors consider the first-price pacing equilibrium (FPPE) model where each buyer uses a single pacing multiplier to shade values. For each submarket, some items outside would also attract the buyers in this submarket, which is unavoidable since we are unable to find completely isolated submarkets. From the modeling point of view, the supply is contaminated by another distribution. The authors first propose a debiased surrogate, which can approximate the limit FPPE in the uncontaminated market based on the limit FPPE in the contaminated market, and then prove that it has only a small error due to the removal of the first-order bias. Then, since the finit FPPE will converge to the limit FPPE in probability, the above estimator can be applied to the actual dynamic market with a slight modification. The authors also present two asymptotic normality results to further demonstrate the superiority of the estimator. Experiments on semi-synthetic data show that the proposed estimator is indeed less biased in terms of pacing multipliers and revenue.
weaknesses: Overall, this paper is hard to parse. The flow of this paper is not clear enough to assist readers well. The abstract and introduction of this paper both start with a/b testing, but the contribution part does not mention it. It was not until page 6 that I began to realize what the authors are doing. After reading the paper several times, I finally understood the authors' real contribution. 

According to my understanding, the real question is how to study the impact of different strategies/mechanisms on market FPPE. The method proposed by the authors is to find relatively isolated submarkets as a/b groups, and then construct an estimator to eliminate cross-market interference. As far as I know, the former is a method that has been adopted in the industry (of course, the issue of cross-market interference remains), while the latter is tailor-made for the FPPE model. Therefore, the focus of this paper's contribution should actually be on FPPE rather than ab test design. In fact, even if all ab test parts are removed, this will still be a complete paper that studies how to approximate the limit FPPE of an uncontaminated market using contaminated data.

Since the estimator in this paper cannot be directly extended to other scenarios, I suggest that the authors should not emphasize the proposal of a novel ab test framework, but use the research on FPPE as the background to introduce challenges and solutions.

---

confidence: 3
contribution: 3
initial_rating: 8
presentation: 3
questions: - is the decomposition into submarkets part of the contribution or not? It appears as something important, but it's in the introduction, rather than being in the main paper. Why? Please clarify this. If this is a contribution, it should be explicitly evident and not appear in the introduction. If it is not, this should be explicit in the paper, maybe with proper citations.
- Please explain - why is the equilibrium framework applicable in practice, given the changing nature of pacing factors?
soundness: 3
strengths: - aims to solve an extremely important practical problem
- Introduces an interesting theoretical framework of modeling "budget interference" using a mixture of distributions.
- makes a good job explaining *why* such a modeling is reasonable.
summary: The paper proposes an estimation technique for A/B test revenue by modeling budget interference between two experiments as a mixture of distributions.

At first, the paper is hard to follow for readers unfamiliar with the FPPE formalism, such as myself. Even though I am familiar with advertising systems and budget pacing mechanisms. A lot of symbols and concepts to grasp. However, once I familiarized myself with the relevant cited background, the explanation of modeling of budget interference using a mixture of distribution makes sense, and I believe it's an important concept that this paper introduces.

The experimental section is quite shallow, but I believe that since the paper is focused on rigorous theory, extensive experiments are less important in such a paper.

However, there are many weaknesses of this paper, described below in the weaknesses section, that make me believe this paper requires additional work to make it ready for publication. The largest problem is, I believe, that the theoretical framework of an equilibrium does not appear to model real-world marketplaces with ever-changing pacing factors. However, the paper presents the technique as something applicable in practice, and no attempt is made to close this gap. More weaknesses appear in the weakness section.
weaknesses: - real budget pacing systems, from my experience, do not operate with constant pacing factors. But the FPPE theory assumes each buyer has one specific pacing factor defined by the equilibrium. There is no explanation in the paper for how this framework actually models the real world of dynamically changing pacing factors, and this limits the usefulness of the estimators in practice. If it is possible to apply the framework to such a dynamic system, there is no explanation in the paper for *how* to do it. So overall, there is no motivation explaining why the equilibrium framework is useful.
- the notation is extremely hard to follow. Typically there is a difference between how vectors, scalars, sets, random variables, and matrices are denoted. For example, vectors by boldface, matrices as upcase-bold, and so on.. It requires a lot of mental effort to follow all the symbols and concepts in the paper, which makes it practically unreadable to audience unfamiliar with the FPPE framework
- a paper aims to show us how to debias the revenue metrics in A/B tests, but it does *not* derive a debiased estimator for the revenue. only for the budget pacing factors. For such an important concept, I'd assume there should be at least a corollary for how the revenue estimator can be computed from the estimated pacing factors, and an explanation for why this estimator is also debiased (it's not obvious that a function of a de-biased estimator is also de-biased in the same sense).


## Update
Most of the weaknesses have been addressed by the rebuttal and the revised version.

Meta-review: This paper studies A/B testing with interference in online marketplaces. The authors considers first-price pacing equilibrium (FPPE) and analyzes how the equilibrium changes according to contamination/interference. The authors then proposes a debias surrogate against the first-order bias and an estimator. The reviewers recognized the following strengths: 
- The problem of interference in A/B testing is important and practical.
- Theoretical analysis is a major contribution. The results and proofs are non-trivial and sound.
- Semi-synthetic experiments demonstrated the effectiveness of the proposed estimators.

Weaknesses:
-  Presentation issue: there are concerns about unclear notations, confusing concepts, and missing details of experiments. The revision with clarifications addressed the issue.
- Limited generalization ability: the result is not generalizable to the problem outside of budget management settings. 

After discussion with authors and among reviewers, all reviewers agree to accept the paper considering the importance of the problem and the theoreical contribution. I agree with the reviewers and recommend acceptance.

Decision: Accept (Poster)