1. Collective choice with known preferences

A group of individuals has to choose an alternative. The individuals disagree about the desirability of the options; their preferences are known. Does a mechanism exist to select an alternative that depends on the individuals' preferences in a reasonable fashion? If only the individuals' ordinal preferences are known, then when the number of alternatives is two, majority voting is the only mechanism that satisfies a short list of appealing properties. When there are three or more alternatives, the existence of an alternative that beats head-to-head every other alternative, known as a Condorcet winner, is significant. If every collective choice problem that the individuals may face has a Condorcet winner, then the mechanism that selects that alternative is the only one that satisfies a short list of attractive properties. Otherwise, no such mechanism generally exists.

A related question is whether a mapping exists from profiles of the individuals' preference relations to a preference relation for the group that satisfies some reasonable properties. If for every profile of preference relations that is possible for the group a Condorcet winner exists, then such a mapping associates with each profile of preference relations the preference relation in which one alternative is preferred to another if and only if it is preferred by a majority of individuals. If all profiles of logically possible preference relations are possible, no such mapping exists.

If the individuals' preference intensities are known and can be compared then the only ordering of welfare profiles that satisfies a specific equity property ranks these profiles according to the welfare of the worst-off individual. If welfare differences, but not levels, can be compared across individuals, then the utilitarian ordering results.