Computing with Voting Trees

A central challenge in social choice theory arises from the fact that there is no fair way to deterministically select a winner in an election among more than two candidates; the only definite collective preferences are between individual pairs of candidates. Combinatorially, one may summarize this information with a graph-theoretic tournament on $n$ vertices (one per candidate), placing an edge from $u$ to $v$ if $u$ would beat $v$ in an election between only those two candidates (no ties are permitted). One well-studied procedure for selecting a winner is to specify a binary tree whose leaves are labeled by the candidates and evaluate it by running pairwise elections between the pairs of leaves, sending the winners to successive rounds of pairwise elections which ultimately terminate with a single winner. This structure is called a voting tree. Much research has investigated which functions on tournaments are computable in this way. Fischer, Procaccia, and Samorodnitsky quantitatively studied the comput...