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 paper considers the problem of allocating unit divisible items under several welfare objectives (i.e., social welfare, gamma-fairness, nash welfare, and max-min fairness) subject to general diversity constraints (modeling as a polytope for each agent). The paper presents two notions of measures, non-negative externality (NNE) and monotonicity (MON), that quantify the impact of the constraints and define their approximated notions of q-NNE and p-MON, respectively.   In particular, they provide a lower bound of q for a class of continuous functions and upper bounds of q for specific welfare functions that depend on the \delta-scaled. For the MON condition, they provide lower and upper bounds for Nash welfare and \gamma-fairness objectives.   Finally, they conduct experiments on two real-world ad allocation datasets (i.e., UCI Adult and Yahoo A3) and using the budget-capped valuation function (along with other parameter settings) to demonstrate the q and p values of various social welfare functions.