We consider the problem of allocating divisible items among multiple agents, and consider the setting where any agent is allowed to introduce {\emph diversity constraints} on the items they are allocated. We motivate this via settings where the items themselves correspond to user ad slots or task workers with attributes such as race and gender on which the principal seeks to achieve demographic parity. We consider the following question: When an agent expresses diversity constraints into an allocation rule, is the allocation of other agents hurt significantly? If this happens, the cost of introducing such constraints is disproportionately borne by agents who do not benefit from diversity. We codify this via two desiderata capturing {\em robustness}. These are {\emph no negative externality} -- other agents are not hurt -- and {\emph monotonicity} -- the agent enforcing the constraint does not see a large increase in value. We show in a formal sense that the Nash Welfare rule that maximizes product of agent values is {\emph uniquely} positioned to be robust when diversity constraints are introduced, while almost all other natural allocation rules fail this criterion. We also show that the guarantees achieved by Nash Welfare are nearly optimal within a widely studied class of allocation rules. We finally perform an empirical simulation on real-world data that models ad allocations to show that this gap between Nash Welfare and other rules persists in the wild.

We consider the algorithmic question of choosing a subset of candidates of a given size k from a set of m candidates, with knowledge of voters' ordinal rankings over all candidates. We consider the well-known and classic scoring rule for achieving diverse representation: the Chamberlin-Courant (CC) or 1-Borda rule, where the score of a committee is the average over the voters, of the rank of the best candidate in the committee for that voter; and its generalization to the average of the top s best candidates, called the s-Borda rule. Our first result is an improved analysis of the natural and well-studied greedy heuristic. We show that greedy achieves a (1 - 2/k+1)-approximation to the maximization (or satisfaction) version of CC rule, and a (1 - 2s/k+1)-approximation to the s-Borda score. This significantly improves the existing submodularity-based analysis of the greedy algorithm that only shows a (1-1/e)-approximation. Our result also improves on the best known approximation algorithm for this problem. We achieve this result by showing that the average dissatisfaction score for the greedy algorithm is at most 2 m+1/k+1 for the CC rule, and at most 2s2 m+1/k+1 for s-Borda. We show these dissatisfaction score bounds are tight up to constants, and even the constant factor of 2 in the case of the CC rule is almost tight. For the dissatisfaction (or minimization) version of the problem, it is known that the average dissatisfaction score of the best committee cannot be approximated in polynomial time to within any constant factor when s is a constant (under standard computational complexity assumptions). As our next result, we strengthen this to show that the score of m+1/k+1 can be viewed as an optimal benchmark for the CC rule, in the sense that it is essentially the best achievable score of any polynomial-time algorithm even when the optimal score is a polynomial factor smaller. We show that another well-studied algorithm for this problem, called the Banzhaf rule, attains this benchmark. We finally show that for the s-Borda rule, when the optimal value is small, these algorithms can be improved by a factor of ~Ømega(√s) via LP rounding. Our upper and lower bounds are a significant improvement over previous results, and taken together, not only enable us to perform a finer comparison of greedy algorithms for these problems, but also provide analytic justification for using such algorithms in practice.