Matching, social welfare and ordinal approximation

Abstract

Many important problems involve agents with preferences for different outcomes. Such settings include, for example, social choice and matching problems. Although the quality of an outcome to an agent may be measured by a numerical utility, it is often not possible to obtain these exact utilities when forming a solution. This can occur because eliciting numerical information from the agents may be too difficult, the agents may not want to reveal this information, or even because the agents themselves do not know the exact numerical values. On the other hand, eliciting ordinal information (i.e., the preference ordering of each agent over the outcomes) is often much more reasonable. Because of this, there has been a lot of recent work on ordinal approximation algorithms: these are algorithms which only use ordinal preference information as their input, and yet return a solution provably close to the optimal one. In other words, these are algorithms which only use limited ordinal information, and yet can compete in the quality of solution produced with omniscient algorithms which know the true (possibly latent) numerical utility information.

Finally, we develop new voting mechanisms for social choice problems given voters' ordinal preferences as well as a small amount of information about the voters' preference strengths. We provide mechanisms with much better distortion when this extra information is known as compared to mechanisms which use only ordinal information. We quantify tradeoffs between the amount of information known about preference strengths and the achievable distortion. We further provide advice about which type of information about preference strengths seems to be the most useful.

Then we consider general facility location and social choice problems in the setting that besides ordinal preferences of the agents, the exact locations of the facilities/candidates are also given. Due to this extra information about the facilities, we are able to form powerful algorithms which have small distortion, i.e., perform almost as well as omniscient algorithms (which know the true numerical distances between agents and facilities) but use only ordinal information about agent preferences. We analyze many general problems including matching, k-center, and k-median, and present black-box reductions from omniscient approximation algorithms with approximation factor beta to ordinal algorithms with approximation factor 1+2beta; doing this gives new ordinal algorithms for many important problems, and establishes a toolkit for analyzing such problems in the future.

First, we study ordinal approximation algorithms for maximum-weight bipartite matchings. We designed and analyzed mechanisms for three different levels of ordinal preferences: one-sided, two-sided and total ordering. We also consider settings where only the top preferences of the agents are known to us, instead of their full preference orderings. The results show that the approximation improves as more ordinal information is revealed.

We study approximation algorithms for matching, social choice, facility assignment and other problems using agents' ordinal preferences and some other information in various settings. The basic assumption in this work is that the agents, facilities and candidates lie in a metric space, and the social welfare or cost depends on the distances among them in the metric.