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We initiate the study of distortion in stable matching. Concretely, we aim to design algorithms that have limited access to the agents' cardinal preferences and compute stable matchings of high quality with respect to some aggregate objective, e.g., the social welfare. Following the recent agenda of Amanatidis et al. (2021), we seek to find the best possible tradeoffs between the distortion of an algorithm and the amount of information elicited via means of cardinal value queries. As a warm-up, we show that purely ordinal algorithms (i.e., algorithms that only use the preference rankings of the agents but no value queries), including the classic Deferred Acceptance (DA) algorithm of Gale and Shapley (1962), have unlimited distortion. Despite this strong impossibility, we observe that a simple randomized version of the DA algorithm achieves a distortion of $2$, and show that this is in fact optimal among all randomized stable matching algorithms. We then consider algorithms that also use value queries on top of the ordinal ranking information. Again, it is easy to see that the same bound of $2$ can be achieved with only $1$ value query per agent, and thus we focus on the question of improving upon this bound. To this end, we provide two strong lower bounds, namely that any stable matching algorithm requires $Ω(n)$ non-adaptive value queries or $Ω(\log n)$ adaptive value queries per agent to achieve a distortion smaller than $2$, where $n$ is the number of agents. Finally, we show an asymptotically tight bound of $Θ(\log (1+n\varepsilon)/\varepsilon)$ on the number of queries that are necessary and sufficient to achieve near-optimal distortion, i.e., a distortion of $1+\varepsilon$, for any $\varepsilon >0$.

The distortion framework in social choice theory allows quantifying the efficiency of (randomized) selection of an alternative based on the preferences of a set of agents. We make two fundamental contributions to this framework. First, we develop a linear-programming-based algorithm for computing the optimal randomized decision on a given instance, which is simpler and faster than the state-of-the-art solutions. For practitioners who may prefer to deploy a classical decision-making rule over the aforementioned optimal rule, we develop an algorithm based on non-convex quadratic programming for computing the exact distortion of any (and the best) randomized positional scoring rule. For a small number of alternatives, we find that the exact distortion bounds are significantly better than the asymptotic upper bounds established in prior literature and lead to different recommendations on which rules to use. These results rely on a novel characterization of the instances yielding the worst distortion, which may be of independent interest.