Robust Allocations with Diversity Constraints

1 Introduction . Allocating heterogeneous divisible items ( or resources ) among agents with different values over these items is a central problem in economics , with literature dating back many decades [ 43 , 9 , 37 , 44 , 5 ] . These problems have gained importance in computer science and machine learning due to their applications in , among other things , large-scale online advertising [ 33 , 7 , 29 , 6 , 24 , 8 ] , resource allocation in datacenters [ 28 ] and sharing economy platforms . Given their wide applicability , there has been a large body of work on elucidating desirable properties of such allocations . One desirable property is fairness or equity in the value the allocation provides to different agents . Though there is no universally agreed-upon notion of fairness , one appealing notion [ 44 ] defines fair allocations as those that are both Pareto-optimal on the value they provide to the agents as well as envy-free [ 43 ] , meaning that no agent should derive more value from another agent ’ s allocation . A less restrictive notion is the so-called Pigou-Dalton principle [ 17 , 35 ] or welfarism , which states that given any fixed total value of the agents , an allocation should prefer distributing these values so that there is no potential transfer of value from an agent with larger value ( the “ rich ” ) to one with smaller value ( the “ poor ” ) . This loosely translates to allocations that optimize 35th Conference on Neural Information Processing Systems ( NeurIPS 2021 ) . a ( weakly ) concave social welfare function over agents ’ values . We call rules that optimize separable , symmetric , concave functions of agent values as welfarist allocation rules.1 Due to their simplicity of implementation and ease of understanding , most allocation rules implemented in practice are welfarist rules . Indeed , even when allocations are supported by prices , for instance in revenue-maximizing ad auctions , by Myerson ’ s celebrated transformation [ 36 , 22 ] and its generalizations , these can be viewed as optimizing a linear “ virtual welfare ” function over allocations [ 10 , 12 ] . Further , envy-freeness can also be implemented by the welfarist Nash Welfare rule [ 37 , 21 , 44 ] that optimizes the product of agents ’ values , and which we will discuss extensively . 1.1 Allocations with Diversity . In this paper , we consider a different desirable facet of such allocations – diversity . The notion of fairness presented above attempts to spread value fairly across agents . Suppose , in addition , the items being allocated were also associated with individuals . For instance , consider the following scenario motivated by online advertising – the items are ad view slots in an advertising ecosystem , each item labeled with the attributes , such as race , age , gender , of the viewing individual on a social media site . The agent represents an advertiser who has different preferences over the viewing individuals . The agent may not only seek an allocation of slots that maximizes its own value , but may also want these slots to not be all views by any particular race . In other words , the agent seeks an allocation that is also diverse on the attributes of the corresponding slots . Similarly , in a team formation platform , an agent seeking such a team may value diversity in attributes such as gender and race of the team members in addition to competence . Diversity has become an increasingly important consideration for eliminating bias in allocation platforms where the items map to humans . Allowing agents to specify some form of diversity constraints on allocations are one way for achieving this goal [ 45 , 4 , 11 , 38 ] . The simplest diversity constraint we consider is the case where the agent wants items with different attributes ( such as race and gender ) in fixed proportions , which models the diversity desiderata in the advertising and team formation settings considered above . We term these “ proportionality constraints '' ; see Section 2.2 for a formal definition . More generally , such constraints capture complementarity in the allocation , where an agent can say “ I want at least as much of this attribute as that attribute. ” A different application for such constraints is in machine allocation in data centers , where a client could request machines in different geographic regions in certain proportions [ 41 ] . For simplicity , we call these constraints diversity constraints , though it should be kept in mind that they model complementarity in general and can easily arise in settings beyond achieving diversity . In Section 2.2 , we model diversity constraints imposed by agent i as simply a convex polytope Pi such that the empty allocation ~0 ∈ Pi . First , this means such constraints preserve feasibility of the allocation . More importantly , these constraints , including proportionality constraints considered above , can have both positive and negative coefficients , and this leads to interesting and non-obvious behavior of welfarist allocation rules . This forms the focus of this paper . Externality in Diverse Allocations . One natural solution to incorporating diversity is for the allocation platform to ask agents for their constraints and add them to the welfarist optimization problem . Indeed , recent work on large scale online allocation problems [ 18 , 19 , 24 ] shows that even in the presence of arbitrary convex constraints on the allocations that can run across time , the resulting problem can be viewed as online stochastic convex programming and admits efficient approximation algorithms [ 3 , 8 ] . Similarly , recent works on auctions with diversity constraints [ 12 , 14 , 26 ] also follow this route and simply add these constraints to an optimization procedure . In this paper , our main focus is to understand how adding such constraints affects the outcome of the allocation . More precisely , for any welfarist allocation rule , when an agent imposes diversity constraints , this changes the allocation of not just this agent , but all other agents as well . Naturally , seeking a diverse allocation will incur a cost in terms of diminished total value of all the agents . This cost is , of course , counterbalanced by the benefit of diversity to the agent seeking diversity ( which does not explicitly appear in our model ) . Since the agent seeking diversity is the one benefiting from the constraints , it would be desirable from both the other agents ’ and the platform ’ s perspective that this agent should be the one bearing the majority of this cost , while the values for other agents should 1We borrow this term from [ 39 ] , though the concept itself is classic [ 32 , 34 ] ( See Section 2.1 for formal definitions . ) not be significantly impacted . In other words , there should be little negative externality on other agents . We present this desideratum formally in Section 2.3 . It is not a priori clear whether the welfarist allocation rules are robust in the above sense and thus the extent to which imposing diversity constraints , even by one agent , may impact other agents . In fact , this lack of clarity may be a major reason for a platform to hesitate to implement functionality enabling agents to specify such diversity constraints , a significant shortcoming for applications such as team selection and online advertising . Thus the focus of our work is on quantifying the potential negative externality and on developing recommendations for finding robust allocations . 1.2 Our Results . To start with , one could reasonably hope that welfarist allocation rules that attempt to find equitable or fair allocations across agents would be naturally robust to the addition of diversity constraints , in the sense that to a good approximation , the negative externality to other agents ’ values is bounded . Our first analytical result shows that the above intuition is false . In Section 3 , we show that almost all welfarist allocation rules – regardless of how equitably or fairly they distribute value among agents – are not robust in the sense that the negative externality on other agents that stems from imposing even a single diversity constraint is unbounded ( see Theorem 1 . ) On the positive side , we show that the Nash Welfare objective [ 37 , 21 ] , which optimizes the product of agents ’ values , is the exception in that it achieves negative externality that is bounded above by an absolute constant factor even when a constant number of agents impose arbitrary diversity constraints . Further , this constant bound is nearly optimal for any number of agents enforcing such constraints since any allocation rule satisfying the Pigou-Dalton principle also suffers similar negative externality . ( Theorems 2 and 3 . ) We then consider how an agent ’ s imposing diversity constraints affects its own value . We define an allocation rule to be monotone if an agent ’ s imposing diversity constraints weakly reduces her own value , and is monotone to a constant factor if the value increases by at most a constant factor . In Section 4 , we show such monotonicity is correlated with how egalitarian the allocation rule is – the more egalitarian ( or concave ) a rule is , the closer to monotone it is ( see Theorem 4 . ) Further , the Nash Welfare allocation rule is monotone to an absolute constant factor . In Section 5 , we use real-world datasets to model ad allocation . For a budget-capped valuation function that is widely studied in this setting , we compare the negative externality induced by various allocation rules . In this setting , the social welfare maximizing rule models first price auctions with budget constrained advertisers , and we empirically show that it indeed suffers large externality and non-monotonicity . However , the Nash Welfare rule suffers very small externality and is monotone , hence making a case for that our theoretical insights will apply in the wild as well . We summarize the upper and lower bounds on the amount of negative externality and nonmonotonicity for various allocation rules in Section 2.3 . Taken together , our main contribution is therefore to show that the Nash Welfare objective is uniquely positioned among a large class of welfarist allocation rules to be robust , achieving both no negative externality and monotonicity to within small constant factors . Our results also suggest that care must be exercised by the allocation platform when allowing agents to express diversity or complementarity constraints . If these are directly encoded into the optimization like in [ 12 , 19 , 24 , 3 , 6 ] , they may create second-order unfairness where the resulting decease in value is borne by agents not directly benefiting from the diversity . 1.3 Related Work . Our work studies robustness of allocations when users can express diversity constraints . This can be equivalently viewed as an agent specifying a different utility function . For instance , a proportionality constraint corresponds to a user expressing a Leontief utility function over the allocation instead of linear utility . In that sense , our results can be equivalently viewed as studying the robustness to agents reporting utility functions that capture diversity . Robustness in our sense has been widely studied when agents can arbitrarily misreport utility functions . In this context , monotonicity is simply strategy-proofness , and non-negative externality maps to non-bossiness [ 42 ] . The main result in [ 42 ] shows that the only allocation rules that satisfy these properties are variants of serial dictatorships , where agents go one after the other and choose their optimal allocation , and these rules are incompatible with Pareto-optimality and welfarist rules . The only exception is settings resembling matchings [ 40 , 31 ] , which are much more restrictive than the setting we study . Note however that a diversity constraint is a very specific type of “ misreport '' – given any allocation , the new utility function is necessarily weakly smaller in value , and this is crucial to all our results . Monotonicity can now be viewed as strategy proofness with such misreports . But now , rules that are not strategyproof in general can easily be monotone . Indeed , we show in Theorem 4 that the MMF rule is actually monotone for general constraints , Pareto-optimal , and welfarist , but is trivially very far from being strategyproof if arbitrary misreports are allowed . In the same vein , the non-negative externality property is much more specific than non-bossiness with general misreports , and it is a priori not obvious that is incompatible with welfarist rules ( Theorem 3 ) . One of our main results is to show that even with the specific type of utility “ misreports ” that diversity constraints imply , this property is incompatible to any factor with welfarist rules , except for the NW rule . Several recent works have considered the question of computing solutions to discrete optimization problems , such as b-matchings [ 4 ] , stable matching [ 38 ] , clustering [ 15 ] , ranking [ 13 , 27 ] , voting [ 11 ] and packing [ 16 ] , when the items belong to groups that must be allocated fairly in the resulting solution . For instance , each cluster in the clustering solution must be assigned a balanced cohort of red and blue points [ 15 ] . Our work tackles a normative question that arises in this space – assuming only a subset of agents care about diversity , what do these constraints mean for the allocations to other agents ? Extending our work to discrete optimization problems is an interesting open question . For lack of space , we present a detailed comparison with other notions in Economics and related work on ad auctions in the Supplementary material ( Appendix ? ? ) .

Summary: The authors study the problem of allocation in the presence of a diversity constraint. Any participant can impose a diversity constraint - demanding proportional amounts of multiple items. An allocation can be computed using a social objective such as social welfare, Nash   welfare subject to this constraint. There are two things that can go "wrong": a participant other than the one imposing the constraint might obtain lower value and the participant imposing the constraint might obtain higher value. The authors ask the question to what extent does the choice of objective guarantee that things won't go "too wrong". That is, any participant not imposing constraint gets at least q fraction of its value and the participant imposing the constraint does not gain more than 1/p factor.   The authors show that for almost any objective that transforms individual participants values and sums them, it is not possible to obtain q = 1. In particular, for social welfare, \gamma-fair, max-min fair objectives it is not possible to get q > 0. In contrast, for Nash Social welfare, one can get q = 1/4. Finally, any rule satisfing pareto-optimality, Pigou-Dalton principle cannot get q > 1/2. The authors also extend these to settings where multiple participants impose constraints.   For the monotonicity - ensuring the constraining participant does not gain too much value - the authors show results for Nash-welfare and \gamma-fair welfare functions. For Nash welfare, 3/4 > p >= 1/2 and for \gamma-fairness, p >= (1-\gamma)^1/\gamma. They also provide an upper bound on p for  \gamma-fairness that is given implicitly by an equation. The \gamma-fairness result implies that for social-welfare p cannot be > 0.   The authors also provide an empirical evaluation of q and p values obtained on real datasets.