1. Introduction
In the ongoing discussion of what it means for automated decisionmaking systems to be fair, the topic of online advertising has merited particular interest. In the United States, segregated employment ads for men and women proved to be a flashpoint in the 1960s, and the introduction of evermore finelytuned advertising online has renewed concerns about discrimination in ads for critical categories such as employment, housing and credit. Although individual advertisers certainly have opportunities to use finegrained targeting (in some cases, targeting individual users) to implement biased advertising strategies, there is both empirical evidence and theoretical support for the idea that the troubling trends in skewed advertisement between different demographic groups do not occur solely because of bad actors among advertisers. In fact, recent work has shown that
even when the advertisers all act fairly in isolation, revenueoptimized platform mechanisms can result in unfairness. In particular, competition between advertisers in firstprice auctions, particularly across categories, can introduce a significant skew in the types of ads people see (e.g., (LT2016)). For example, two equally qualified software engineers may see wildly different numbers of employment ads depending on competition for their attention from other categories. Competition from lucrative categories like children’s products can be difficult for an individual advertiser to correct for, particularly in categories like employment where parental status is considered very sensitive. While the idea of parents being excluded from employment ads due to competition is inherently troubling, blunt policies like equalizing the number of ads across users within each category can also result in poor outcomes for individuals. For example, suppose Bob is highly qualified for a new credit card, and he is also searching for daycare options in his area. Alice is equally qualified for the credit card, but she is not interested in daycare. Bob would not want to see fewer ads for daycare services to ensure that he sees as many ads for credit as Alice, and likewise Alice would not want good credit offers suppressed in order to ensure that she doesn’t see more credit ads than Bob who doesn’t even care about credit cards anyway! Competition across ads within a single category can likewise result in skew. Even if two equally qualified individuals see the same number of ads in a category, these ads may not be equally relevant or valuable to them. For example, these individuals could see the same total number of job ads, but with one receiving ads for jobs with significantly higher salaries than the other.^{1}^{1}1 Subtle issues, like willingness to relocate, company size and type, and other policies may also significantly influence the usefulness of the ads to each user. How can we formalize these intuitive desiderata? Are these fairness desiderata compatible with consumer, platform, and advertiser utility?Our Contributions
Model and Definitions
In this work, we examine the problem of fairness in advertising across multiple categories from a theoretical perspective. We propose a stylized model and fairness requirements that match the intuitive fairness desiderata of inter and intracategory guarantees.^{2}^{2}2Our aim is to propose a model and examine its properties to gain insight into how certain notions of fairness interact with platform utility as well as our intuition about what is fair rather than to propose a specific notion of fairness for practical implementation. Moreover, our fairness requirements incorporate the qualifications of users as well as their preferences across different ad categories. Our definitions are inspired by and combine the complementary notions of envyfreeness and individual fairness. Envyfreeness (caragiannis2012efficiency) requires that every user should prefer their own allocation to that of everyone else; it ignores users’ qualifications and considers preferences as paramount. Individual or metric fairness (DHPRZ2012), on the other hand, ignores preferences and essentially requires that similar users should be treated similarly. Its extension to multidimensional allocations as introduced in (DI2018), multiple task fairness
, requires that individual fairness is satisfied separately and simultaneously for all categories. For intercategory competition, we note that multiple task fairness is at odds with consumer and platform utility. Consider the example of Alice and Bob discussed above. Taking this example to the extreme suppose that a user, whom we will call Jack, is a “jack of all trades,” i.e., qualified or interested in many categories. Per multiple task fairness, another user who is as qualified as Jack in a single category but unqualified in all other categories, would have their allocation within the single category of interest limited to match Jack’s allocation within that category. Multiple task fairness would then either (1) enforce a minimum amount of exposure for Jack to ads they may not care about or (2) dictate a maximum amount of exposure to relevant ads for other users who are qualified for a smaller number of categories. In effect, by ignoring users’ own preferences, this stringent notion of fairness helps no one. We thus introduce
intercategory envyfreeness, which allows users to specify a set of categories that they “care” about, and guarantees that they see at least as many ads that they care about as any other individual.^{3}^{3}3Kim et al. (KKRY2019) has similar motivation, but takes a different approach to blending userpreferences and individual fairness. For intracategory competition, we show that multipletask fairness can be too weak to avoid discriminatory allocations and is therefore vulnerable to certain subversion attacks. For example, if Alice sees every highpaying job ad slightly less often than Bob and every lowpaying job ad slightly more often than Bob, then for any single ad the two users obtain similar allocations, but across the set of all highpaying jobs, Alice may receive a far smaller share than Bob. We thus introduce total variation fairness, which offers protection against these subversion attacks and captures fairness over all possible sets of substitutes.^{4}^{4}4Although “highpaying” or “lowpaying” are welldefined attributes in the employment setting, there are many subtle properties which may be harder to articulate, e.g., “commute time.” By protecting all sets of substitutes, we obviate the need to enumerate all possible attributes. Combining these fairness aims yields a definition giving fairness both across categories and within categories. More specifically, we demonstrate how to combine intercategory envyfreeness and totalvariation fairness into a hybrid fairness notion that we call compositional fairness. For example, suppose two individuals, Alice and Bob, are equally qualified and interested in a particular type of job. Alice is also interested in the latest movie releases, while Bob is interested in buying a new car. We provide the guarantee that Alice and Bob see the same mixof ads for jobs, but are permitted to see job ads overall with different probabilities, so long as each sees their preferred categories in total at least as often as the other.
Fair mechanisms achieving high utility
In practice, fairness is obviously not the only consideration in mechanism design. Platforms typically aim to optimize for (1) shortterm revenue, i.e. the sum of payments made by advertisers for displaying their ads, or (2) allocative efficiency a.k.a. social welfare, i.e. the sum of the perceived values of the userad matches, or a combination of these objectives. As a proxy for these complex objectives, we use the sum of the bids of the ads displayed as a measure of the quality of an allocation. We call this objective platform utility. High platform utility correlates both with advertiser happiness (ads get matched with targeted “high bid” users) as well as with platform revenue (advertisers often pay their bid or an increasing function of their bid). While our goal is to design fair mechanisms that achieve high platform utility, we compare the performance of our mechanisms to the unfair optimum, following the intuition that platforms and advertisers are unlikely to adopt mechanisms which deviate wildly in utility from the status quo. In this respect, we show that our fairness desiderata are compatible with high platform utility. Apart from these fairness and utility objectives, from a mechanism design perspective, it is desirable for our mechanisms to satisfy certain implementation properties. For example, the platform may like to have some ability to outsource auditing of fairness instead of taking on the responsibility of determining what constitutes fairness in each category and enforcing those determinations.^{5}^{5}5Particularly in cases in which determining fairness requires sensitive information (or sensitive information may be leaked by the determination procedure) the platform is likely to want to avoid collecting or possessing such information. In this context, the platform would likely prefer to be oblivious to the particulars of fairness requirements, and instead make a guarantee that if each advertiser behaves “fairly”, then the platform does not introduce any additional unfairness. With this guarantee, the platform would thus be able to leave auditing of advertiser behavior to a neutral third party or governing body. Our main result is a family of mechanisms that achieve compositional fairness and utility close to optimal within the class of fair mechanisms, while also being oblivious to fairness requirements. The nature of the fairness definitions enables us to compose a mechanism satisfying totalvariation fairness within each category with a mechanism satisfying intercategory envyfreeness across categories to obtain a mechanism satisfying compositional fairness. Thus, we can separately design intercategory mechanisms and intracategory mechanisms. The intercategory mechanisms that we propose are always envyfree regardless of the bids, and the intracategory mechanisms that we propose are oblivious to fairness requirements: our intracategory mechanisms implicitly use fairness of advertiser bids to infer the necessary fairness requirements. Unsurprisingly, if advertisers are allowed to bid very differently on similar individuals, it is impossible to simultaneously achieve fairness and high platform utility. As part of our study, we also explore upper and lower bounds on the strength of a bid fairness condition needed to achieve a high platform utility.
Outline for the rest of the paper
The remainder of this work is organized as follows: related work is discussed in Section 2; definitions and our formal model are introduced in Section 3; mechanisms achieving intracategory fairness are discussed in Section 4, and mechanisms achieving intercategory envyfreeness are discussed in Section 5; composing these mechanisms is addressed in Section 6; finally future work is discussed in Section 7. We defer proofs and some discussion to the Appendix.
2. Related Work
Fairness in advertising is a topic of particular interest in the popular press, the empirical and theoretical computer science literature, as well as economics. There are number of compelling empirical studies and popular press articles demonstrating “skew” in advertisement delivery between groups, either due to advertiser targeting choices, platform delivery issues, or competition between advertisers (AP2016; BTI2019; AST2017; LT2016; ASBKMR2019). In the theoretical computer science literature, Dwork et al. (DI2018) proposed the notion of individual fairness as a fairness concept for settings including advertising. Dwork and Ilvento (DI2018) consider individual fairness (and group fairness) in advertising as a composition problem, and poses a limited set of fair composition mechanisms. Similar to this work, Kim et al. (KKRY2019) propose a notion of “Preference Informed Individual Fairness” (PIIF) to capture the idea that deviation from individual fairness is acceptable as long as it is based on individuals’ preferences. However, while PIIF is intended to expand the definition of fairness within the context of taskspecific metrics, our preferencesbased envyfreeness concept applies across multiple categories independent of the metric for each category. Furthermore, whereas Kim et al. focus primarily on optimizing user utility via offline optimization across all fair outcomes, we study online mechanisms and measure utility relative to the unfair optimum. We view our work as complementary to (KKRY2019), and we anticipate that combining the insights from their perspective and ours will be useful for proposing alternative mechanisms and fairness relaxations.Other works on fair advertising and related problems employ different notions of fairness. With respect to group or statistical notions of fairness, Mehrotra et al. (celis2019toward) propose a mechanism for ad delivery while maintaining certain group level statistics. A variety of fairness notions have also been considered in related problems such as ranking (celis2017ranking; evidencerankings2019), recommender systems (antikacioglu2018new), and news search engines (Kapoor2018). Finally, mostly disjoint from the work referenced above, there is an extensive literature in fair division concerning other notions of fairness (caragiannis2012efficiency). For one of these fairness notions, called “envyfreeness”, the goal is to partition a limited shared resource among multiple agents with heterogenous preferences. A major difference between this literature and the fairness literature referenced above is that envyfreeness is defined exclusively based on agents’ preferences and not on their traits or qualifications. Two recent works, Balcan et al. (balcan2018envy) and Zafar et al. (zafar2017parity)
seek to combine notions of envyfreeness with paritybased fairness in machine learning in the context of classification.
3. Advertising and Fairness Models
We model the online advertising problem as follows. A universe of users arrive in an online fashion. There are advertisers indexed by . When a user arrives, each advertiser places a bid on the user. The allocation algorithm then assigns allocation probabilities to the advertisers with . The allocation mechanism is an online algorithm, i.e. it assigns allocation probabilities without observing bids on users that arrive in the future or the ordering of future user arrivals. Moreover, we assume for simplicity that each user arrives at most once. We use to denote the allocation rule output by the allocation algorithm. The goal of the allocation mechanism is to maximize the sum of the bids of the ads displayed. Formally, this is given by:
We measure the utility of our mechanisms against the best achievable in the absence of any fairness constraints. The utility is easy to maximize in the absence of any constraints on how allocations vary across users: the mechanism can simply assign a probability mass of to the highest bidder for every user.^{6}^{6}6This is equivalent to running a firstprice auction. We call the corresponding utility the unfair optimum:
The Fair Value of an allocation mechanism is the ratio of its utility to the UnfairOPT. Note that this ratio is always less than 1; the larger the fair value the better the utility of the allocation is. Why compare with the unfair optimal utility instead of the fair optimal (as do other related works (KKRY2019; celis2019toward))? First, the utility of the fair optimal mechanism can be difficult to analyze due to the fairness requirements being revealed in an online fashion. Second, if there is a large gap between the utility of the mechanisms we propose and the utility of the unfairoptimal mechanism, the platform and advertisers may be unwilling to adopt the mechanism, so it is critical that we compare with the platform’s status quo.
3.1. Fairness of allocation
Our fairness guarantees are based on the concept of individual fairness defined by (DHPRZ2012) and the wellstudied concept of envyfreeness^{7}^{7}7See (caragiannis2012efficiency).. At a high level, individual fairness guarantees that similar individuals are treated similarly. Similarity between individuals is captured through a fairness metric over and similarity between outcomes is captured by defining a metric over distributions over outcomes.
Definition 3.1 (Individual Fairness (Dhprz2012)).
A function assigning users to distributions over outcomes is said to be individually fair with respect to distance metrics over and over , if for all we have .
Dwork and Ilvento (DI2018) proposed extending the notion of individual fairness to settings involving multidimensional allocations by ensuring fairness separately within each dimension or “task”. This gives rise to the notion of multipletask fairness, which we now define in the context of online advertising. Let denote a partition of the set of advertisers into categories. For , let denote a pseudometric over the users relevant to all advertisers in category ; . In our setting, the outcome assigned to each user corresponds to the advertiser who is assigned to the slot for user . Our mechanism maps users to distributions over outcomes, i.e. to the allocation probabilities . We use the absolute difference between these probabilities to capture the similarity of allocations, and multipletask fairness becomes the following condition:^{8}^{8}8We can also view the assignment as a fractional allocation. In this case, the distance corresponds to the difference in the portion of allocation for each advertiser.
Definition 3.2 (MultipleTask Fairness (Di2018)).
An allocation function satisfies multipletask fairness with respect to distance metrics if for all , , and , we have .
We will demonstrate that multipletask fairness is too weak for fairness within a single category, and it results in suboptimal allocations across categories from the perspective of envyfreeness. We propose two new multidimensional fairness notions, one applying across different categories and the other to multiple advertisers within the same category, and combine these into the notion of compositional fairness that overcomes the shortcomings of multipletask fairness.
Intracategory fairness
First, we consider a setting in which all of the advertisers belong to a single category (i.e. ) with a single metric that we denote by (e.g., a tech jobsearch website containing advertisements only from tech employers). We observe that multipletask fairness is insufficient to protect against two broad classes of problems.

[leftmargin=*]

Intentional unfairness: It is vulnerable to subversion by malicious advertisers. In particular, consider an advertiser that submits multiple different ads (pretending to be distinct advertisers) for the same job and bids separately on each user for each of those ads. The advertiser effectively poses as multiple subadvertisers; let denote the set of these subadvertisers. In this case, the multipletask fairness constraint only ensures for each , and so it is possible that . As a result, the advertisers may be able to amplify the difference in probabilities of allocation between two users to an arbitrarily large extent.

Unintentional unfairness: Multipletask fairness can also interact in undesirable ways with wellintentioned advertisers. Suppose that the set consists of all highpaying job ads.^{9}^{9}9The set can also be job ads from a certain geographical area. Suppose highpaying advertisers all bid higher on user than on user , and the mechanism sets for all (to maximize utility). Then, it would again be the case that , so the total allocation on highpaying job ads (i.e. advertisers in ) can be vastly different for and .
To rectify these issues, we propose total variation fairness
which requires that the allocation vectors
and are not only close componentwise, but are also close in terms of distance or total variation distance.Definition 3.3 (Total Variation Fairness).
An allocation function satisfies total variation fairness with respect to a metric if for all and all , we have . Equivalently, for all , .
This stronger definition, which provides guarantees on all subsets of advertisers , effectively mitigates the issues outlined above. First, it prevents the multiple bid attack by ensuring that . Second, it provides nice guarantees over substitutes in the following sense. Consider a user who regards some arbitrary subset of advertisers to be substitutes. In that case, the probability that the user observes an ad from this subset is and total variation fairness ensures that this sum is close to the corresponding sum for similar users.^{10}^{10}10This definition can be viewed as combination of the multipletask fairness and ORfairness definitions put forth in (DI2018). The definition essentially provides ORfairness over all possible subsets of advertisers.
Intercategory fairness
Next consider a setting where every advertiser belongs to a different category, i.e. where . As the “jack of all trades” example in the introduction (formalized in Section 5) shows, multipletask fairness can lead to outcomes that, although technically fair, are undesirable from every stakeholder’s perspective – (1) users get low allocations in their desired categories, (2) advertisers reach far fewer qualified users and (3) the platform gets low utility owing to the poor quality of the matching produced. In this example, allocations available to the jack of all trades are constrained by the limited attention of this single user (a single ad slot in our model). Multipletask fairness combined with this constraint limits the allocations of all of the other users, thereby hurting their utility, without in turn providing any benefit to the jack of all trades. Within this context, we view the multiple task fairness objective to be unduly skewed in favor of a single individual over the collective good. Is it possible to achieve a better balance? It is if we slightly shift our viewpoint. Consider a platform where each user is allowed to choose the category they are most interested in with the guarantee that they see at least as many ads in this category as any other user. In effect, no user is envious of other users given their own choice of preferred category. A few remarks are in order. First, in the special case where every category has a single advertiser,^{11}^{11}11We discuss the case of multiple advertisers per category and allocation between those advertisers in the subsection on compositional fairness. within the chosen category, the fairness guarantee provided to the user is much stronger than that guaranteed by individual fairness – the user does not just obtain an allocation close to that of other similar users, but rather obtains one that is as good or better than that of everyone else.^{12}^{12}12The reader may worry that a user who is “unqualified” in their chosen category could obtain a large allocation at the expense of advertisers that don’t want to target such a user. Note that advertisers can bid on users that are not targeted and thereby pay nothing for those users. Furthermore, it is rare for a user to be wholly unqualified for a category from the perspective of advertising. For example, an individual who is unqualified for a job is likely a good candidate for a job training ad. Second, this is balanced by the lack of any fairness guarantee on nonchosen categories. However, note that from the user’s perspective those other categories are anyway not important. To take an example, suppose that Alice and Bob are identical in terms of their creditworthiness as well as job qualifications. Suppose Alice is looking for a job and Bob for a credit card. Then a platform that shows a job ad to Alice and a credit card ad to Bob makes both users happy and envyfree. In contrast, a platform that shows each user one of the two ads uniformly at random is multipletask fair but makes both users worse off. Third, while our definitions focus on a single arrival of each user, one may envision a system where a user interacting with the platform multiple times can change their preferred category at each interaction and thereby obtain a fair allocation within the chosen category at each individual interaction. Finally, observe that the envyfreeness guarantee is directional and entirely independent of distance metrics. More generally, we allow users to select multiple preferred categories and provide an envyfreeness guarantee with respect to the total probability of seeing an ad within their set of preferred categories. Formally, each user picks a preferred set of categories.^{13}^{13}13From an implementation perspective, we might imagine that users specify these categories on a user profile. We then ensure that is at least as large as the corresponding sum for any other user .
Definition 3.4 (InterCategory EnvyFreeness).
An allocation function satisfies intercategory envyfreeness with respect to preferred sets if for all , we have .
Compositional fairness
Now we consider the general setting where there can be multiple categories and multiple advertisers in each category. We discuss how to combine the two definitions above to provide hybrid fairness guarantees. We have two goals: (1) each user should be envyfree with respect to the categories of ads they see and (2) within each category, the mix of ads presented to each user satisfies our strengthened notion of individual fairness. For example, suppose that two similarly qualified users Alice and Bob both select jobs as their preferred ad category. Not only should the two users then see job ads with the same total likelihood, but they should also see a similar mix of highpaying and lowpaying job ads. This composition of definitions becomes subtle when users select multiple preferred categories. Suppose that Alice continues to choose jobs as her preferred category, but Bob chooses both jobs and household product ads. Suppose, further, that Alice is allocated a jobs ad with probability and Bob sees an ad in each of the two categories with probability each. This allocation satisfies intercategory envyfreeness. However, within the job ads category there is no way to assign probabilities that satisfy unconditional total variation fairness simply because of the fact that we have different total probabilities to distribute. Intuitively, we want Bob to be able to see the same mix of ads as Alice even though Alice may see job ads more frequently overall. Accordingly, we enforce total variation fairness on the conditional distribution of allocation within each category. Formally, we define compositional fairness as follows. We use to denote the metric specific to category and to denote the total allocation within category for user .
Definition 3.5 (Compositional Fairness).
An allocation function satisfies compositional fairness with respect to distance metrics if the assignments satisfy intercategory envyfreeness, and for each such that , the conditional probabilities satisfy total variation fairness with respect to .
Multiplicative Relaxations
We can further refine each of the above notions by defining multiplicative relaxations parameterized by :

[leftmargin=*]

total variation fairness: for all we have .

intercategory envyfreeness: for all we have .

compositional fairness: satisfies intercategory envyfreeness, and for each such that , satisfies total variation fairness.
4. Intracategory fairness
In this section, we focus on the case where advertisers are in a single category , i.e. where advertisers face the same metric over users, in particular, . In Section 4.1, we describe the need for a fairness condition on advertiser bids, that we call a bid ratio condition. In Section 4.2, we investigate the special case of uniform metrics and establish impossibility results, i.e. upper bounds on the fair value of any allocation mechanism that satisfies multipletask fairness as a function of the bid ratio condition. In Section 4.3, we consider settings with arbitrary distance metrics and exhibit an allocation mechanism that is metricoblivious, historyoblivious, and achieves total variation fairness with respect to the given metric with an appropriate bid ratio condition. We then bound the fair value achieved by this mechanism as a function of (the number of advertisers) and a parameter defining the bid ratio constraint. Moreover, we show that this mechanism achieves the nearoptimal tradeoff between bid ratio condition and fair value within a restricted class of mechanisms. In Section 4.4, we show that the fair value achieved by this mechanism is close to the upper bound established in Section 4.2 for allocation mechanisms with a uniform metric. We emphasize that while our negative result in Section 4.2 applies to general online algorithms which satisfy multipletask fairness with access to the underlying metric, our positive result applies to a mechanism that is metricoblivious and satisfies the stronger notion of total variation fairness. All of the proofs can be found in the Appendix.
4.1. Fairness in bids
Our goal is to develop an allocation mechanism that simultaneously satisfies total variation fairness and achieves large fair value with respect to the UnfairOPT. As one might expect, it is impossible to achieve this if advertisers are allowed to place arbitrary bids on users without regard to the relevant similarity metric over users.^{14}^{14}14One could consider platform mechanisms which alter or decrease the bids of advertisers as needed to achieve fairness constraints but this has the downside of (1) making the platform responsible for advertiser behavior and (2) making it difficult for advertisers to optimize their bids. The question thus becomes: what kind of fairness constraint on bids enables a reasonable fair value? The following example illustrates the need for a fairness constraint that requires an advertiser’s bids on pairs of similar users to be close in ratio, even when allocations are merely required to satisfy the weaker condition of multipletask fairness. In particular, being close in terms of their absolute difference is not sufficient to achieve a good fair value.
Example 4.1 ().
Suppose that there are advertisers and exactly users. Suppose that the metric is uniform: for some parameter , every pair of users is a distance of apart. For each , advertiser bids on user and on all other users . Observe that UnfairOPT . On the other hand, due the symmetry of this instance and the fact that a fair allocation requires that , for each user, it turns out the optimal fair allocation assigns an allocation probability of to all advertisers with the low bid and a probability of to the advertiser with the high bid. The fair value of this allocation turns out to be^{15}^{15}15We prove this explicitly in Lemma A.3 in Appendix A.1.1 as a corollary of a more general result about the optimal utility achieved by a fair offline mechanism on a uniform metric. . Observe that a fair value of is trivial to achieve via a fair allocation.^{16}^{16}16In particular, for each user, assigning an allocation probability of to the highest bidder and to all other advertisers achieves this bound. When is very small, this trivial bound is tiny. If , then, in this example, no fair allocation can perform much better than the trivial algorithm. In order to be able to do better, must be bounded.
Motivated by the example above, we require that for every advertiser and every pair of users, the ratio of the bids the advertiser places on the users is bounded by a function of the distance between the users according to . The closer the two users, the closer this ratio bound should be to ; on the other hand, the ratio bound should be large between far apart users. We formally define this constraint as follows.
Definition 4.2 ().
A bid ratio constraint is a function . We say that the bid function of advertiser satisfies the bid ratio constraint with respect to metric if we have for all : .
How should we choose ? On the one hand, the bid ratio constraint needs to be sufficiently strict to provide meaningful fairness and utility guarantees and on the other hand it cannot be so overly restrictive as to prohibit reasonably expressive bidding strategies. We characterize by boundary conditions for identical users and maximally distant users with the requirement that is weakly increasing and in the intermediate range. In the case of identical users, i.e. , the advertiser is required to place identical bids, i.e., . For maximally distant users, i.e. , we choose , allowing advertisers to bid arbitrarily differently on this pair. For ease of analysis, we also make continuous.
In this work, we show that a specific parameterized class of bid ratio constraints that satisfy the above properties performs quite well in terms of fair value. The family is parameterized by , and is defined as: . Figure 1 displays some functions in this family. Note that as the parameter increases, the bid ratio condition becomes more and more strict. In Appendix I, we discuss implications of certain structural properties of . We emphasize that bid ratios are not required to satisfy the constraint exactly, but rather should lie at or below the imposed curve. From an algorithmic viewpoint, this means that if we design an algorithm based on the polynomial family described above but the actual bid ratio constraint imposed on bids, say , does not belong to this family, it nevertheless suffices for the algorithm to find a value of the parameter for which for all and use the function in making allocations.
4.2. Lower bounds on fair value using uniform metrics
We prove upper bounds on the fair value of any allocation mechanism that satisfies multipletask fairness. Observe that these upper bounds apply also to total variation fairness, which is a stronger requirement. We use uniform metrics, i.e. metrics of the form for all . We use Example 4.1 to show an upper bound on the fair value as a function of the bid ratio constraint.^{17}^{17}17In Appendix D, we give an explicit formula for the optimal revenue that can be achieved in the offline setting for a uniform metrics as a function of the bids and . This formula yields the bound in Lemma 4.3 on Example 4.1.
Lemma 4.3 ().
Given and , there exists an instance of the online advertising problem for which the fair value of every offline allocation mechanism satisfying multipletask fairness with respect to the uniform metric is at most .
Observe that as increases, the upper bound on the fair value increases, due to a weaker fairness constraint on setting allocation probabilities. On the other hand, for any fixed and , the upper bound decreases as a function of : as increases, weakening the fairness constraint on bids, the algorithm’s performance becomes worse. In the online setting it is possible to prove even stronger bounds on the fair value.^{18}^{18}18To be explicit, the online lower bound applies in the limit as , where the adversary is given access to the probabilities output by the mechanism when designing the next set of bids. More specifically, we can tighten the term in the fair value to . In Appendix D, we develop online allocation mechanisms that are tailored to uniform metrics and achieve a fair value that nearly matches these lower bounds, demonstrating that stronger lower bounds for general online algorithms cannot be obtained through uniform metrics.
Lemma 4.4 ().
Given and , there exists an instance of the online advertising problem for which no online allocation mechanism satisfying multipletask fairness with respect to the uniform metric can obtain fair value better than .
The main idea of the proof of Lemma 4.4 is to make the bids on the first user equal. Then for the next user, the adversary maximally increases some bids and decreases other bids so that the advertiser receiving the lowest probability on the first user has the highest bid on the second user. The distance metric limits the extent to which the mechanism can increase the probability placed on this advertiser for the second user due to the low probability placed on the first user.
4.3. Mechanisms for the general metric case
We construct an allocation mechanism for a general fairness metric that achieves total variation fairness and a large fair value. From a mechanism design standpoint, it is desirable for our mechanism to satisfy certain other properties. Most important is metricobliviousness, i.e. where the mechanism doesn’t require direct knowledge of . In addition, the following other properties are desirable:

[leftmargin=*]

Identityobliviousness: that is, the mechanism treats advertisers in a symmetric manner in each individual iteration — the allocation to an advertiser does not depend on their identity.

Monotonicity: that is, the allocation probabilities increase monotonically as functions of the advertisers’ bids, which means that we can make this mechanism truthful by setting the payoffs appropriately by Myerson’s lemma.

Historyobliviousness, which has the nice implication that the memory required by the mechanism is independent of the number of users and the solution is independent of the ordering of the users so we don’t need to worry about impact of the order in which the users are given on advertiser strategies and/or utility.

Protecting against advertiser splitting: We would like the mechanism to disincentivize an advertiser from splitting up into subadvertisers (or submitting multiple ads) in an attempt to obtain a higher probability allocation.
We now describe a class of mechanisms that satisfy these properties. The key intuition is to convert the bids on a user into probabilities using a function that places higher probabilities on higher bids. Note that the first three properties (identityoblivious, monotonic, and historyoblivious) mean that the mechanism must be defined by a symmetric, coordinatewise increasing function that maps bids into probabilities. We observe that the overall optimal solution (UnfairOPT) can be placed in this framework: the mechanism that distributes the probability mass equally among the highest bidders for each user corresponds to the function that places the full mass on the highest bids. This function can be viewed as assigning allocation probabilities in proportion to their contribution to the norm over the bids.
Proportional allocation mechanisms
One rich class of mechanisms of this form are mechanisms where the probabilities are proportional to some deterministic function of the bids, i.e. where , with appropriate normalization to make . We call these mechanisms proportional allocation mechanisms, defined as follows:^{19}^{19}19The special case of the proportional allocation mechanism with was considered in a different context in (AZM2018).
Mechanism 1 ().
Let be a continuous, superadditive (i.e. ), increasing function. The proportional allocation mechanism with parameter assigns for every user and advertiser .
It is straightforward to verify that proportional allocation mechanisms are identityoblivious, monotonic, historyoblivious, and protect against submitting multiple ads (this last property follows from the superadditivity of ). These mechanisms bear similarity to position auctions^{20}^{20}20Roughly speaking, a position auction (Varian2006; EOS2005) only uses the ordering of the bids (and not the bid values or advertiser identities) in determining the allocation. See Appendix H for a discussion of why position auctions cannot achieve both fairness and a high fair value.: while proportional allocation mechanisms are not position auctions since they can still rely heavily on the bids, they are “close” to position auctions in that the highest bidder is always assigned the highest probability regardless of their identity.
A proportional allocation mechanism with high fair value
We construct a family of functions where the fair value of the proportional allocation mechanisms is high. More specifically, we show that for can achieve fair value approaching as . This mechanism can be viewed as assigning allocations in proportion to each bid’s contribution to the norm of the bid vector. We emphasize that our bound on the fair value does not require bids to satisfy the bid ratio constraint.
Theorem 4.5 ().
Let be a proportional allocation mechanism with parameter for a positive integer . If and , then the fair value of is at least .
Observe that fair value is an increasing function of , and as (where places the entire mass on the highest bid), the bound equals as expected. This means that fair value can be made arbitrarily close to by sufficiently strengthening the bid ratio constraint. To achieve a fair value of , we can set . We show that with the bid ratio condition shown in Figure 1, this mechanism satisfies total variation fairness.
Theorem 4.6 ().
Let be a proportional allocation mechanism with parameter for a positive integer . If all advertisers in a category satisfy the bid ratio condition , satisfies total variation fairness in that category.
Observe that the bid ratio condition becomes stronger as increases. Thus, for a fixed number of advertisers, a higher fair value is accompanied by a stronger bid ratio condition. Moreover, observe that to maintain a fair value of , a stronger bid ratio condition is needed as the number of advertisers increases, since grows with . In Appendix E we discuss relaxations of the fairness constraint and analyze the bid ratio condition needed for Mechanism 1 with parameter under those relaxations.
Nearoptimality within proportional allocation mechanisms
A natural question is to ask is whether Mechanism 1 with a different function can achieve a a much better fair value, potentially using a differently shaped bid ratio condition. In fact, we show that Mechanism 1 is nearly optimal within the family of proportional allocation mechanisms. We specifically prove that any proportional allocation mechanism achieving a certain fair value will have a corresponding bid ratio condition that is pointwise stronger than , where within a constant factor of what is achieved by Mechanism 1 with . This result demonstrates that changing the shape of the bid ratio condition will not significantly improve the fair value. We show the following lower bound:
Lemma 4.7 ().
Suppose that is a proportional allocation mechanism achieves fair value and achieves total variation fairness with a bid ratio condition of . For , we have that for infinitely many points on (more specifically, for ).
How does the lower bound in Lemma 4.7 compare to the proportional allocation mechanisms with parameter ? Lemma 4.7 shows that any proportional allocation mechanism will satisfy where for infinitely many points on the curve. Meanwhile, Theorem 4.5 and Theorem 4.6 show that to achieve a fair value , it suffices to take with . Thus, there is essentially a constant factor difference in the lower and upper bounds on . A consequence of Lemma 4.7 is in the family of proportional allocation mechanisms, the fair value must necessarily degrade with the number of advertisers if the bid ratio condition is fixed. Or equivalently, to maintain a fair value of , a stronger bid ratio condition is needed as the number of advertisers increases. In fact, Lemma 4.7 implies that that the proportional mechanism with parameter achieves the optimal rate of change of the bid ratio condition as a function of : within the family of proportional allocation mechanisms, a better dependence on is not possible.
4.4. Discussion
In Section 4.3, we showed that Mechanism 1 with is nearly optimal in the class of proportional allocation mechanisms. Now, we compare Mechanism 1 with to general online mechanisms, using our negative results in Section 4.2. It is a little tricky to directly compare the fair value lower bounds achieved by Mechanism 1 with with the upper bounds in Section 4.2, because the bounds depend on different parameters. In particular, the lower bounds hold for arbitrary metrics whereas the upper bounds are designed only for the uniform metric. To perform an applestoapples comparison, we fix parameters , , and some number , and set . Figure 2 displays the ratio of the upper bound and lower bound for various parameter settings. Observe that the ratio is bounded by a reasonably small constant except when is very small. This indicates that Mechanism 1 with parameter in general obtains a large fraction of the utility that can be obtained by any online mechanism satisfying multipletask fairness, despite satisfying a stronger form of fairness and nice mechanism design properties.
5. Intercategory fairness
In this section, we consider the setting where different product categories correspond to very different metrics. We specifically consider the setting where every category has exactly one advertiser (i.e. where ). We first show it is not possible to achieve the same tradeoff between multipletask fairness and utility as in the case of identical metrics. We then show that with intercategory envyfreeness, significantly better tradeoffs are possible. In Section 5.1, we focus on the UnfairOPT benchmark considered in the previous section. First, we show that multipletask fairness is suboptimal for users, advertisers, and the platform. We then consider the weaker notion of intercategory envyfreeness introduced in Section 3, and we present tight upper and lower bounds on the fair value achievable as a function of upper bounds on the sizes of the preferred sets. In Section 5.2, we argue in favor of a weaker benchmark to evaluate the performance of fair allocation mechanisms, and show that this benchmark can be exactly met by mechanisms that satisfy intercategory envyfreeness. Proofs are in the Appendix.
5.1. Intercategory envyfreeness and fair value relative to UnfairOPT
First, we show a upper bound that demonstrates that multipletask fairness is in conflict with utility, even in the offline setting, if the metrics are permitted to be different. The result is based on the “jack of all trades” example in the introduction. Formally:
Example 5.1 (Jackofalltrades).
Suppose that the universe has users and there are categories (with one advertiser per category). The metric is defined so that and all other distances are . Suppose that advertiser bids on and , and on everybody else. Observe that the bids are fair.^{21}^{21}21We assume that .
Consider any allocation mechanism and suppose that this mechanism chooses an ad from distribution to display for user . Then, in order to respect multiple task fairness as defined above, the mechanism cannot allocate ad to user with probability greater than . As a result, most of the “specialists” necessarily obtain a low allocation within their desired categories: since the “jack of all trades” can only be served a single ad, the “specialists” are penalized. Multipletask fairness thus limits the allocation of almost all of the users. Moreover, the utility of any fair allocation is bounded by some constant, whereas an unfair allocation can achieve utility . We obtain the following bound on fair value:
Proposition 5.2 ().
Suppose that bids satisfy the bid ratio constraint , then no offline mechanism that satisfies multipletask fairness across categories can obtain a fair value more than .
We now consider the weaker fairness notion of intercategory envyfreeness defined in Section 3 that limits the number of fairness constraints imposed by any given user. We assume that the platform obtains from each user a preferred set of categories as the user arrives (i.e. the preferences are not known to the allocation mechanism in advance). Intercategory envyfreeness requires that the total allocation of ads in to the user should be at least as large as the total allocation of ads in to any other user. Observe that our definition of fairness doesn’t involve a metric over the users. Moreover, obtaining good utility requires that we place a reasonable amount of mass on the highest bid(s) for the user . We develop an allocation mechanism that reserves some allocation mass for the category with the highest bid on each user , and distributes the remaining allocation probability across categories in . In doing so, we must ensure that no subset of categories gets too much mass in total. This is because if such a set exists, and a future user sets , then the mechanism is forced to give a high allocation to this set for user . Unless is small, this may then leave little probability mass for the highest bidder for . We now formally define our mechanism and bound the fair value achieved by it.
Mechanism 2 ().
The equalspread mechanism with parameters and is defined as follows. We assume that every user specifies a subset with . Let be a category with the highest bid. The mechanism assigns an allocation probability of to and to categories in , where and .
The parameter allows the mechanism to tradeoff between fairness and utility. By allowing the mechanism to achieve intercategory envyfreeness for larger values, we obtain a better approximation to the UnfairOPT.
Theorem 5.3 ().
If every user’s preferred set contains categories, then for any , the equalspread mechanism with parameters and (Mechanism 2) satisfies intercategory envyfreeness and achieves a fair value of .
We show a matching upper bound on the fair value, thus showing that Mechanism 2 is optimal. Our proof boils down to bounding the amount of mass that can be placed on the highest bid. We construct a sequence of users with the property that bids in are always and each user has a category that bids . We adaptively construct the sets and to minimize the fair value.
Lemma 5.4 ().
Suppose that , and every user’s preferred set can contain any number of categories . Then, regardless of the bid ratio constraint imposed on the advertisers (but assuming ), any online mechanism that satisfies intercategory envyfreeness obtains a fair value of at most .
In the above construction, we consider a uniform metric with distance (where the bid ratio conditions do not implicitly provide any guarantees). Nonetheless, even though the users are maximally distant, the intercategory envyfreeness still provides strong unidimensional guarantees between these users. A natural question to ask is: can we achieve a higher fair value by considering a relaxed version of intercategory envyfreeness that reintroduces the metric? However, we show in Appendix F that even with these relaxations, it is not possible to achieve a much higher fair value than that achieved by Mechanism 2. We plot the tight bound on fair value obtained for intercategory envyfreeness as a function of and in Figure 3. For the dependence on , observe that as increases, the weakened fairness guarantees cause the fair value to increase. For the dependence on , observe that the mechanism must balance between allocating to a category with the highest bid to achieve high utility, and allocating to categories in to achieve intercategory envyfreeness. As increases, the mechanism has a greater number of categories to consider for each user (and the highest bid can still be outside of ), thus causing the optimal fair value to decrease. Let’s now consider the strongest setting of , so the fair value becomes . The fair value thus becomes small when users are permitted to specify a large number of categories, and when (and ), the fair value is .
5.2. Relaxing the utility benchmark
The results in the previous section, for the strongest form of fairness, place an upper bound of (or lower) on the fair value that can achieved by intercategory envyfreeness. In this section, we consider the setting where we restrict to mechanisms that receive utility only for allocations in : that is, the utility is . In this case, we assume that userspecified categories are aligned with interest, and in a clickthroughrate based revenue/utility model, the platform only obtains benefit from allocations within the userspecified sets. It is straightforward to see that the best possible utility achieved by any (potentially unfair) mechanism in this restricted class is , which may be much smaller than unrestricted optimal utility of . Thus, when considering the utility for this setting, we compare against , the best possible userpreferencecompatible utility for a mechanism in this setting. This motivates considering the relaxed fair value given by: . Note that when there can be more one advertiser per category, this quantity naturally generalizes to . Returning to the case where there is at most one advertiser per category, we show a strong positive result in this setting. More specifically, we remark that a simple highestbidderwins mechanism achieves intercategory envyfreeness and a relaxed fair value of . This mechanisms does not assume that all users specify the same number of categories and permits users to specify any number of categories.
Mechanism 3 ().
For each , the mechanism allocates a probability of to the category in with the highest bid. If there are multiple categories tied for the highest bid, then the mechanism splits the probability equally between these categories.
Theorem 5.5 ().
Mechanism 3 achieves intercategory envyfreeness with and achieves a relaxed fair value of .
Given the simplicity of Mechanism 3, a natural question is to ask is: can we obtain better fairness guarantees while still maintaining a high relaxed fair value? However, in Appendix G, we show that it is not possible to have multipletask fairness guarantees and a high fair value in this setting even against the relaxed benchmark.
6. Compositional fairness
We now discuss how to combine our mechanisms to handle the setting where there can be multiple categories and multiple advertisers in each category. Let denote a partition of the set of advertisers into categories. We can compose our mechanisms from Section 4 and Section 5 by running mechanisms from Section 5 to allocate probability between categories, and mechanisms from Section 4 to distribute the probability assigned to a category between advertisers in that category. The key intuition for fairness is that compositional fairness is implied by intercategory envyfreeness and intracategory total variation fairness.^{22}^{22}22In order to conclude that is actually the probability that the first mechanism assigned to category , we require that the second mechanism assigns the full probability mass to each user. This is true of proportional allocation mechanisms (Mechanism 1). Moreover, we show that these combined constructions achieve a high fair value, since the fair value of the composed mechanism is the product of the fair values of the individual mechanisms. Note that the two choices of category selection mechanisms in Section 5 result in combined constructions that differ in terms of both utility guarantees (i.e. choice of benchmark) and category distribution properties (i.e., concentrating or spreading probability between categories), which are important for practical considerations. Proofs are in the Appendix.
High fair value compared to the relaxed benchmark
To achieve a high fair value compared to the relaxed utility benchmark (taking user preferences as an indicator of click probability discussed in Section 5.2), we can compose Mechanism 3 with Mechanism 1. The idea is that we run Mechanism 3 to identify the category in with the highest bid, and then we run Mechanism 1 to divide the probability mass between advertisers in this category.
Mechanism 4 ().
For each user , the mechanism runs Mechanism 3 to allocate probabilities between categories, where for , the bid is taken to be .^{23}^{23}23This sets the bid for a category to be the highest bid of any advertiser within that category. The mechanism then determines the conditional probabilities for advertisers within each category using Mechanism 1 with parameter .
Theorem 6.1 ().
Let be the maximum number of advertisers in any category. If all advertisers satisfy the bid ratio condition , then Mechanism 4 achieves compositional fairness and a relaxed fair value of at least .
High fair value compared to UnfairOPT
It is likewise possible to combine mechanisms 1 and 2 to obtain bounds on the fair value against UnfairOPT.
Mechanism 5 ().
For each user , the mechanism runs Mechanism 2 with parameters and to allocate probabilities between categories, where for , the bid is taken to be .^{24}^{24}24Again, this sets the bid for a category to be the highest bid of any advertiser within that category. The mechanism then determines the conditional probabilities for advertisers within each category using Mechanism 1 with parameter .
Theorem 6.2 ().
Suppose that every user’s preferred set contains at most categories. Let be the maximum number of advertisers in any category. If all advertisers satisfy the bid ratio condition , then Mechanism 5 achieves compositional fairness and a fair value of at least .
7. Future work
In this work, we give an initial framework for understanding the utility and fairness properties of a combination of individual fairness and envyfreeness. We highlight areas for future work below.

[leftmargin=*]

Incentivizing and auditing fair bidding. A significant benefit of the online mechanism presented for intracategory competition is that it is metric oblivious which frees the platform from explicitly checking whether bids placed are in accordance with the relevant fairness metric. Although total variation fairness does mitigate some malicious advertiser behavior, there is still an open question of how best to incentivize, audit, and enforce fair bidding. We anticipate a mechanism which can impose penalties (either monetary penalties or reduced platform access) on advertisers who do not behave fairly may provide sufficient incentive to follow the rules.

Closing the gaps and tighter lower bounds. Although uniform metrics provide a good starting point for lower bounds, and we have provided nearly matching bounds for proportional allocation mechanisms, we anticipate that it may be possible to show more general lower bounds in certain oblivious settings, i.e., requiring metric or identity obliviousness.

Multiple slot auctions. In this work, we have focused on the case of single slot auctions. However, in practice multiple slot auctions, or auctions where each user appears at several times, are more common. In addition to technical implementation questions, there are several important considerations in defining fairness for the setting. For instance, the ordering of advertisements may be important when there is significant visual distinction (e.g. the user must scroll to see ads lower in the slate).

Different combinations of inter and intracategory fairness. We have presented one combination of definitions of inter and intracategory fairness, but it’s likely that others (e.g., intracategory total variation fairness and intercategory PIIF) may have interesting properties.
References
Appendix A Proofs for Section 4
a.1. Proofs for Section 4.2
In Section A.1.1, we prove bounds for offline mechanisms. In Section A.1.2, we prove bounds for online mechanisms.
a.1.1. Offline mechanisms
First, we compute the optimal offline mechanism revenue for uniform metrics.
Lemma A.1 ().
Suppose that is an advertiser such that
If the metric is uniform with distance , then the optimal offline, multipletask fair mechanism achieves a revenue of exactly:
where if is the first price auction revenue, and otherwise, is the maximum integer in such that the mth price auction revenue is bigger than .
The proof for this lemma boils down to solving the following linear program. In the offline setting for multipletask fairness, an optimalrevenue mechanism that achieves multipletask fairness is solving the following linear program, where the solutions are
, and the rest of the variables are inputs.We first show the following property of the LP. In order to better study this LP, we also consider the dual with variables and :
The variable represents the condition . The variable represents the condition .
Proposition A.2 ().
At an optimal solution, suppose that
Then there exists some such that .
Proof.
We prove the contrapositive. We use complementary slackness to analyze the structure of this LP. Suppose that we are at an optimal solution and for all for some . The corresponding condition in the dual is that for all . This means that
Moreover, we know that for any , it holds that for all . This means that
Thus, we know that for any . ∎
Now, we prove Lemma A.1.
Proof of Lemma a.1.
We first show this result when there is a unique advertiser that achieves the top bid sum. For , we know that for some by Proposition A.2. Thus, for all by the fairness constraint. Now, we know that the bids of advertiser are in the interval for some by the fairness constraints. Any set of bids such that for and satisfies the fairness constraints. Let’s examine an individual item . We see that the optimal strategy is to assign mass to and mass to all other advertisers to start, and then distribute the remaining mass by assigning mass to the top bidder, then mass to the next highest bidder, until all of the mass runs out (assigning potentially mass to the last bidder considered). Let’s suppose that . Let where . We see that the utility achieved at is
This is equal to
Thus, the maximum is achieved (not necessarily uniquely) at if and achieved at if . Now, let be the maximum value such that is bigger than and . The above argument shows that the optimal revenue can be achieved by setting . This achieves the revenue in the lemma statement (for the case where there is a unique advertiser with the top bid sum). Now, we obtain the general case result (where multiple advertisers can attain the top bid sum). Consider all advertisers who achieve the top bid sum. If is the first price auction revenue, then the statement follows by considering the mechanism that picks an arbitrary advertiser and assigns every user to advertiser . Otherwise, we obtain the result via a limiting argument. If , let’s also pick an advertiser arbitrarily. For the other advertisers , we change the bid to for some where advertiser is not the unique top bid (this exists – otherwise advertiser would be the unique advertiser with the top bid sum). We keep all of the other bids the same. Now, we can apply the result to these bids, since advertiser is the unique advertiser having the top bid sum on these new bids. It suffices to show , (a) the optimal offline mechanism utility for the modified bids approaches the optimal offline mechanism utility for the original bids, and (b) the utility in the formula for the modified bids approaches the utility in the formula for the original bids (In combination, (a) and (b) imply that the utility in the formula for the original bids is the optimal offline mechanism utility for the original bids.) First, we show (a). Note that when bids are modified by , the primal LP constraints remain the same, and the objective is changed by at most . Thus, the optimal offline revenue changes by at most . As , this goes to , so we know that the optimal offline revenue is a continuous function of the bids. Now, we show (b). The only term in the formula that changes is the ith price revenue. We see that the ith price revenue will be dampened by at most , leading to at most a reduction in the expression. Moreover, if is sufficiently small, then will not be affected (since we have a strict inequality in its definition since ). Thus, the revenue expression converges to its limit as for this sequence of bids. ∎
We use this lemma to compute the fair value in the offline case on Example 4.1.
Lemma A.3 ().
The fair value of the offline revenue in Example 4.1 is
Proof of Lemma a.3.
We see by Lemma A.1 that since while the 1st price auction revenue is and the ith price auction revenue for is . ∎
Proof of Lemma 4.3.
We use that to obtain the desired result. ∎
a.1.2. Online mechanism
We now consider the online setting.
Proof of Lemma 4.4.
Suppose that all advertisers bid on the first user . Suppose that the mechanism assigns the minimum probability (or tied for minimum probability) on to some advertiser . Now, suppose that bids on every subsequent user, and every other advertiser bids on every subsequent user. Then the firstprice auction revenue is . This mechanism’s revenue is . The ratio is
Since there are advertisers, we know that , so this is:
Let . This is equal to
As , this approaches . ∎
a.2. Proofs for Section 4.3
First, we consider fairness of proportional allocation mechanisms.
Lemma A.4 ().
Suppose that total variation fairness is satisfied and we have a continuous, increasing function defining the allocation mechanism. Let be the bid ratio condition. Let
Then any bid ratio condition must satisfy
Moreover, the bid ratio condition
is sufficient.
Proof.
First, we show a sufficient bid ratio condition. Let’s consider the difference
where and . Let’s let , and , . WLOG, assume that . Let and let . We know that by the bid ratio condition. This means that . We have that
Observe that this expression can be upper bounded by the case where is maximized (i.e. where ) and is maximized (i.e. where ). Our expression becomes:
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