8 research outputs found
Excluding pairs of tournaments
The Erd\H{o}s-Hajnal conjecture states that for every given undirected graph
there exists a constant such that every graph that does not
contain as an induced subgraph contains a clique or a stable set of size at
least . The conjecture is still open. Its equivalent directed
version states that for every given tournament there exists a constant
such that every -free tournament contains a transitive
subtournament of order at least . We prove in this paper that
-free tournaments contain transitive subtournaments of
size at least for some and several
pairs of tournaments: , . In particular we prove that
-freeness implies existence of the polynomial-size transitive
subtournaments for several tournaments for which the conjecture is still
open ( stands for the \textit{complement of }). To the best of our
knowledge these are first nontrivial results of this type
Majority rule in the absence of a majority
Which is the best, impartially most plausible consensus view to serve as the basis of democratic group decision when voters disagree? Assuming that the judgment aggregation problem can be framed as a matter of judging a set of binary propositions (“issues”), we develop a multi-issue majoritarian approach based on the criterion of supermajority efficiency (SME). SME reflects the idea that smaller supermajorities must yield to larger supermajorities so as to obtain better supported, more plausible group judgments. As it is based on a partial ordering, SME delivers unique outcomes only in special cases. In general, one needs to make cardinal, not just ordinal, trade- offs between different supermajorities. Hence we axiomatically characterize the class of additive majority rules, whose (generically unique) outcome can be interpreted as the “on balance most plausible” consensus judgment