Necessary and possible winners for top-cycles from preferences allowing ties and tentative incompleteness

In autonomous multiagent systems, collective choices among several proposed alternatives are made with the use of voting procedures such as the majority rule. When the elicitation of the agents’ preferences is incomplete but tentative, collective choice procedures commonly aggregate available individual preferences into an incomplete binary relation. From this relation, we can assess the Condorcet winner (the best alternative) if it exists, or, more generally, a choice set (the best compromise solution), by identifying alternatives that are able to be eligible (possible winners) and those that are definitely among the future chosen alternatives (necessary winners). In this context, here we characterise the possible and necessary winner sets for three choice sets: The weak top-cycle (or Smith-set), strong top-cycle (or Schwartz-set) and rational topcycle. We also introduce linear and quadratic time algorithms for computing the necessary winners for the three top-cycles and the possible winners for the weak and strong top-cycles. Finally, we propose efficient heuristics for finding a subset and a superset of possible winners for the rational top-cycle, with its computational tractability staying an open question.