  Aggregating the preferences of individuals into a collective decision is the
core subject of study of social choice theory. In 2006, Procaccia and
Rosenschein considered a utilitarian social choice setting, where the agents
have explicit numerical values for the alternatives, yet they only report their
linear orderings over them. To compare different aggregation mechanisms,
Procaccia and Rosenschein introduced the notion of distortion, which quantifies
the inefficiency of using only ordinal information when trying to maximize the
social welfare, i.e., the sum of the underlying values of the agents for the
chosen outcome. Since then, this research area has flourished and bounds on the
distortion have been obtained for a wide variety of fundamental scenarios.
However, the vast majority of the existing literature is focused on the case
where nothing is known beyond the ordinal preferences of the agents over the
alternatives. In this paper, we take a more expressive approach, and consider
mechanisms that are allowed to further ask a few cardinal queries in order to
gain partial access to the underlying values that the agents have for the
alternatives. With this extra power, we design new deterministic mechanisms
that achieve significantly improved distortion bounds and, in many cases,
outperform the best-known randomized ordinal mechanisms. We paint an almost
complete picture of the number of queries required by deterministic mechanisms
to achieve specific distortion bounds.
