Fairness in ad auctions through inverse proportionality

We study the tradeoff between social welfare maximization and fairness in the context of ad auctions. We study an ad auction setting where users arrive one at a time, $k$ advertisers submit values for each user, and the auction assigns a distribution over ads to each user. Following the works of Dwork and Ilvento (2019) and Chawla et al. (2020), our goal is to design a truthful auction that satisfies "individual fairness" in its outcomes: informally speaking, users that are similar to each other should obtain similar allocations of ads. We express the fairness constraint as a kind of stability condition: any two users that are assigned multiplicatively similar values by all the advertisers must receive additively similar allocations for each advertiser. This value stability constraint is expressed as a function that maps the multiplicative distance between value vectors to the maximum allowable $\ell_\infty$ distance between the corresponding allocations. Standard auctions do not satisfy this kind of value stability. Our main contribution is a new class of allocation algorithms called Inverse Proportional Allocation that achieve value stability with respect to an expressive class of stability conditions. These allocation algorithms are truthful and prior-free, and achieve a constant factor approximation to the optimal (unconstrained) social welfare. In particular, the approximation ratio is independent of the number of advertisers in the system. In this respect, these allocation algorithms greatly surpass the guarantees achieved in previous work. In fact, our algorithms achieve a near optimal tradeoff between fairness and social welfare under a mild assumption on the value stability constraint. We also extend our results to broader notions of fairness that we call subset fairness.