Two-sided fairness in rankings via Lorenz dominance

1 Introduction . Recommender systems have a growing impact on the information we see and on our life opportunities , as they help us browse news articles , find a new job , house , or people to connect with . While the objective of recommender systems is usually defined as maximizing the quality of recommendations from the user ’ s perspective , the recommendations also have an impact on the recommended “ items ” . News outlets rely on exposure to generate revenue , finding a job depends on which recruiter gets to see our resume , and the effectiveness of a dating application also depends on who we are recommended to—and if we are being recommended , then someone else is not . Two-sided fairness in rankings is the problem of generating personalized recommendations by fairly mediating between the interests of users and items . It involves a complex multidimensional trade-off . Fairness towards item producers requires boosting the exposure of small producers ( e.g. , to avoid winner-take-all effects and popularity biases [ 1 ] ) at the expense of average user utility . Fairness towards users aims at increasing the utility of the least served users ( e.g. , so that least served users do not support the cost of item-side fairness ) , once again at the expense of average user utility . The goal of this paper is to provide an algorithmic framework to generate rankings that achieve a variety of these trade-offs , leaving the choice of a specific trade-off to the practitioner . The leading approach to fairness in rankings is to maximize user utility under constraints of equal item exposure ( or equal quality-weighted exposure ) [ 54 , 7 ] or equal user satisfaction [ 6 ] . When these constraints imply an unacceptable decrease in average user utility , so-called “ trade-offs between utility and fairness ” [ 65 , 41 ] are obtained by relaxing the fairness constraints , leading to the optimization of a trade-off between average user utility and a measure of users ’ or items ’ inequality . Thinking about fairness in terms of optimal utility/inequality trade-offs has , however , two fundamental limitations . First , the optimization of a utility/inequality trade-off is not necessarily Pareto-efficient from the point of view of users and items : it sometimes chooses solutions that decrease the utility of some individuals without making anybody else better off . We argue that reducing inequalities by decreasing the utility of the better-off is not desirable if it does not benefit anyone . The second 35th Conference on Neural Information Processing Systems ( NeurIPS 2021 ) . limitation is that focusing on a single measure of inequality does not address the question of how inequality is reduced , and in particular , which fraction of the population benefits or bears the cost of reducing inequalities . In this paper , we propose a new framework for two-sided fairness in rankings grounded in the analysis of generalized Lorenz curves of user and item utilities . Widely used to study efficiency and equity in cardinal welfare economics [ 53 ] , these curves plot the cumulative utility obtained by fractions of the population ordered from the worst-off to the best-off . A curve that is always above another means that all fractions of the populations are better off . We define fair rankings as those with non-dominated generalized Lorenz curves for users and items . First , this definition guarantees that fair rankings are Pareto efficient . Second , examining the entirety of the generalized Lorenz curves provides a better understanding of which fractions of the population benefit from an intervention , and which ones have to pay for it . We present our general framework based on Lorenz dominance in usual recommendation settings ( e.g. , music or movie recommendation ) , and also show how extend it to reciprocal recommendation tasks such as dating applications or friends recommendation , where users are recommended to other users . We present a new method for generating rankings based on the maximization of concave welfare functions of users ’ and items ’ utilities . The parameters of the welfare function control the relative weight of users and items , and how much focus is given to the worse-off fractions of users and items . We show that rankings generated by maximizing our welfare functions are fair for every value of the parameters . Our framework does not aim at defining what parameters are suitable in general — rather , the choice of a specific trade-off depends on the application . From an algorithmic perspective , two-sided fairness is challenging because items ’ utilities depend on the rankings of all users , requiring global inference . Previous work on item-side fairness addressed this issue with heuristic methods without guarantees or control on the achievable trade-offs . We show how the Frank-Wolfe algorithm can be leveraged to make inference tractable , addressing both our welfare maximization approach and existing item-side fairness penalties . We demonstrate that our welfare function approach enjoys stronger theoretical guarantees than existing methods . While it always generates rankings with non-dominated generalized Lorenz curves , many other approaches do not . We show that one of the main criteria of the literature , called equity of attention by Biega et al . [ 7 ] , can lead to decrease user utility , while increasing inequalities of exposure between items . Moreover , equal user satisfaction criteria in reciprocal recommendation can lead to decrease the utility of every user , even the worse-off . Our notion of fairness prevents these undesirable behaviors . We report experimental results on music and friend recommendation tasks , where we analyze the trade-offs obtained by different methods by looking at different points of their Lorenz curves . Our welfare approach generates a wide variety of trade-offs , and is , in particular , more effective at improving the utility of worse-off users than the baselines . We present our formal framework in Section 2 . We discuss the theoretical properties of previous approaches in Section 3 , and present our ranking algorithm in Section 4 . Our experiments are described in Section 5 , and the related work is discussed in Section 6 . 2 Two-sided fairness via Lorenz dominance . 2.1 Formal framework . Terminology and notation . We identify an item with its producer , so that “ item utility ” means “ item producer ’ s utility ” . The main paper focuses on fairness towards individual users and items . We describe in Appendix B the extension of our approach to sensitive groups of users or items . |X | denotes the cardinal of the set X . Given n 2 N , we denote by [ [ n ] ] = { 1 , . . . , n } . The set of users N is identified with { 1 , ... , |N | } and the set of items I is identified with { |N | + 1 , ... , n } where n = |N |+ |I| . For ( i , j ) 2 N ⇥ I , we denote by µij the value of item j to user i . A ( deterministic ) ranking : I ! [ [ |I| ] ] is a one-to-one mapping from items j to their rank ( j ) . Following [ 54 ] , we use stochastic rankings because they allow us to perform inference using convex optimization ( see Section 4 ) . The recommender system produces one stochastic ranking per user , represented by a 3-way ranking tensor P where Pijk is the probability that j is recommended to i at rank k. We denote by P the set of ranking tensors . Utilities of users and items are defined through a position-based model , as in previous work [ 54 , 7 , 63 ] . Let v 2 R|I| , where vk is the exposure weight at rank k. We assume that lower ranks receive more exposure , so that 8k 2 [ [ |I| 1 ] ] , vk vk+1 0.1 Given a user i and a ranking i , the user-side utility of i is the sum of the µijs weighted by the exposure weight of their rank i ( j ) : ui ( i ) = P j2I v i ( j ) µij . Given an item j , the item-side utility of j is the sum over users i of the exposure of j to i . These definitions extend to stochastic rankings by taking the expectation over rankings , written in matrix form:2 user-side utility : ui ( P ) = X j2I µijPijv item-side utility ( exposure ) : uj ( P ) = X i2N Pijv We denote by u ( P ) = ( ui ( P ) ) ni=1 the utility profile for P , and by U = { u ( P ) : P 2 P } the set of feasible profiles . For u 2 U , uN = ( ui ) i2N and uI = ( ui ) i2I denote the utility profiles of users and items respectively . Two-sided fairness in rankings . In practice , values of µij are not known to the recommender system . Ranking algorithms use an estimate µ̂ of µ based on historical data . We address here the problem of inference : the task is to compute the ranking tensor given µ̂ , with the goal of making fair trade-offs between ( true ) user and item utilities . Notice that the user-side utility depends only on the ranking of the user , but for every item , the exposure depends on the rankings of all users . Thus , accounting for both users ’ and items ’ utilities in the recommendations is a global inference problem . More general item utilities We consider exposure as the item-side utility to follow prior work and for simplicity . Our framework and algorithm readily applies in a more general case of two-sided preferences , where items also have preferences over users ( for instance , in hiring , job seekers have preferences over which recruiters they are recommended to ) . Denoting µji the value of user i to item j , the item side-utility is then uj ( P ) = P i2N µjiPijv . 2.2 Lorenz efficiency and the welfare function approach . Our notion of fairness aims at improving the utility of the worse-off users and items . Since this does not prescribe exactly which fraction of the worse-off users/items should be prioritized , the assessment of trade-offs requires looking at all fractions of the population . This is captured by the generalized Lorenz curve used in cardinal welfare economics [ 53 ] . Formally , given a utility profile u , let ( u ( i ) ) ni=1 be the sorted values in u from smallest to largest , i.e. , u ( 1 ) . . . u ( n ) , then the generalized Lorenz curve plots ( Ui ) ni=1 where Ui = u ( 1 ) + . . . + u ( i ) . To assess the fairness of trade-offs , we rely on the following dominance relations on utility profiles : Pareto-dominance P. u P u0 ( ) ( 8i 2 [ [ n ] ] , ui u0i and 9i 2 [ [ n ] ] , ui > u0i ) . Lorenz-dominance L. Then u L u0 ( ) U P U 0 . We write ⌫L for non-strict Lorenz dominance ( i.e. , 8i , Ui U 0i ) . Notice that Pareto-dominance implies Lorenz-dominance . Our notion of fairness , which we call Lorenz efficiency , states that a ranking is fair if the utility profiles for users and for items are not jointly Lorenz-dominated : 1We use a user-independent v for simplicity . Considering user-dependent weights is straightforward . 2We consider Pij as a row vector in the formula , so that Pijv = P|I| k=1 Pijkvk . Definition 1 ( Lorenz efficiency ) . A utility profile u 2 U is Lorenz-efficient if there is no u0 2 U such that either ( u0I ⌫L uI and u0N L uN ) or ( u0N ⌫L uN and u0I L uI ) . We consider that Lorenz-dominated profiles are undesirable ( and unfair ) because the utility of worseoff fractions of the population could have been increased at no cost for total utility . Examples of Lorenz-curves of users and items are given in Fig . 1 . The blue solid , green dotted and orange dashed curves are all non-dominated ( the blue solid ranking has higher user utility but high item inequality , the green dotted and orange dashed curves have similar item exposure profiles , but user curves that intersect ) . On the other hand , the red dot/dashed curve is an unfair ranking : compared to the green dotted and orange dashed curve , all fractions of the worse off users have lower utility , together with less exposure for worse-off items . A fundamental result from cardinal welfare economics is that concave welfare functions of utility profiles order profiles according to Lorenz dominance [ 3 , 53 ] . The choice of the welfare function specifies which ( fair ) trade-off is desirable in a specific context . This result holds when all utilities are comparable . In our case where there are users and items , we propose the following welfare function parameterized by ✓ = ( , ↵1 , ↵2 ) :3 8u 2 Rn+ : W✓ ( u ) = ( 1 ) X i2N ( ui , ↵1 ) + X j2I ( uj , ↵2 ) with ( x , ↵ ) = 8 < : x ↵ if ↵ > 0 log ( x ) if ↵ = 0 x↵ if ↵ < 0 . Inference is carried out by maximizing W✓ ( an efficient algorithm is proposed in Section 4 ) : ( ranking procedure ) P ⇤ 2 argmax P2P W✓ ( u ( P ) ) ( 1 ) In W✓ , 2 [ 0 , 1 ] controls the relative weight of users and items . The motivation for the specific choice of is that it appears in scale invariant welfare functions [ 43 ] , but other families can be used as long as the functions are increasing and concave . Monotonicity implies that maxima of W✓ are Paretoefficient . For ↵1 < 1 and ↵2 < 1 , W✓ is strictly concave . Then , W✓ exhibits diminishing returns , which is the key to Lorenz efficiency : an increment in utility for a worse-off user/item increases welfare more than the same increment for a better-off user/item . The effect of the parameters is shown in Fig . 1 ( left ) : For item fairness we obtain more item equality by using ↵1 < 1 ( here , ↵1 = 0.5 ) and incrasing ( see blue solid vs orange dashed curve ) . The parameter ↵2 controls user fairness : smaller values yield more user utility for the worse-off users at the expense of total utility , with similar item exposure curve ( green dotted vs orange dahsed curves ) . Let ⇥ = { ( , ↵1 , ↵2 ) 2 ( 0 , 1 ) ⇥ ( 1,1 ) 2 } . For every ✓ 2 ⇥ , W✓ is strictly concave , and users and items have non-zero weight . We then have ( the result is a straightforward consequence of diminishing returns , see Appendix C ) : Proposition 1 . 8✓ 2 ⇥ , 8P ⇤ 2 argmax P2P W✓ ( u ( P ) ) , P ⇤ is Lorenz-efficient . Relationship to inequality measures A well-known measure of inequality is the Gini index , defined as 1 2⇥AULC , where AULC is the area under the Lorenz curve . The difference between Lorenz and generalized Lorenz curves is that the former is normalized by the cumulative utility . This difference is fundamental : we can decrease inequalities while dragging everyone ’ s utility to 0 . However , this would lead to dominated generalized Lorenz curves . Interestingly , for item-side fairness , the cumulative exposure is a constant and thus trade-offs between user utility and item exposure inequality are not really problematic . However , for user-side fairness , the total utility is not constant and reducing inequalities might require dragging the utility of some users down for the benefit of no one . Additional theoretical results In App . C.2 , we show that as ↵1 , ↵2 ! 1 , utility profiles tend to leximin-optimal solutions [ 43 ] . Leximin optimality corresponds to increasing the utility of the worst-off users/items one a a time , similarly to a lexical order . In App . C.3 , we present an excess risk bound , which provides theoretical guarantees on the true welfare when computing rankings based on estimated preferences , depending on the quality of the estimates . 3W✓ ( u ) = 1 if ↵ 0 and 9i , ui = 0 . In practice , we use ( x+ ⌘ , ↵ ) for ⌘ > 0 to avoid this case .

Summary: This paper proposes a new ranking framework that concerns the fairness of all parties (i.e., users and items) participating in recommender systems. Inspired by techniques used in welfare economics, the authors formulate a convex optimization that yields Pareto-efficient and equitable (within each parties) utilities. To demonstrate the superiority over existing methods, they consider some special cases where the proposed approach is shown to achieve better utilities than the baseline methods. They provide an efficient implementation based on the Frank-Wolfe algorithm, together with theoretical results on tractability and convergence.