Peeking behind the ordinal curtain: Improving distortion via cardinal queries

Fair and Efficient Resource Allocation with Partial Information

“…We characterize the exact value of k needed to achieve EF1. For MMS, we derive almost tight bounds the best possible approximation as a function of k. For the case of complete rankings (k = m), our results show that 1 /(2Hn)-MMS is achievable, almost matching the asymptotic upper bound of 1 /Hn due to Amanatidis et al [2016]. Thus, we establish, for the first time, that the best approximation to MMS given ordinal preference information scales logarithmically in the number of agents.…”

“…Here, each agent i places a non-negative value v i (g) on each good g and her value for a bundle of goods S ⊆ M is assumed to be the sum of her values for the individual goods in S, i.e., g∈S v i (g). Theoretically, this valuation class gives way to algorithms achieving strong fairness guarantees [Amanatidis et al, 2016;Caragiannis et al, 2019;Chaudhury et al, 2020;Ghodsi et al, 2018;Garg and Taki, 2020]. Practically, additive valuations are much simpler to elicit than fully combinatorial valuations, which has led to their adoption by popular fair division tools such as Spliddit and Adjusted Winner.…”

“…An interesting tradeoff can be achieved by eliciting ordinal preferences from the agents, but viewing them as partial information regarding underlying cardinal preferences. This idea originates from the related field of voting theory, where a growing body of work on the distortion framework uses ordinal preferences of voters over candidates as means to pick a candidate approximately maximizing social welfare according to the underlying cardinal preferences [Procaccia and Rosenschein, 2006;Boutilier et al, 2015;Mandal et al, 2019;Mandal et al, 2020;Kempe, 2020;Amanatidis et al, 2020].…”

“…With access to agents' full preference rankings over the goods, it is known that EF1 can be achieved via the round robin algorithm [Lipton et al, 2004;Caragiannis et al, 2019], under which agents take turns picking goods in a cyclic fashion. For MMS, Amanatidis et al [2016] show that, using just ordinal preferences over the goods, it is impossible to guarantee better than a 1 /Hn approximation of MMS, where H n = Θ(log n) is the n th harmonic number and n is the number of agents; in contrast, given additive cardinal preferences, even 3 /4-MMS can be achieved [Ghodsi et al, 2018;Garg and Taki, 2020]. What is the best MMS approximation that can be achieved given agents' preference rankings over all the goods?…”

Peeking Behind the Ordinal Curtain: Improving Distortion via Cardinal Queries

“…Due to space constraints, some details have been omitted and can be found in the full version (Amanatidis et al 2019).…”