  Classical results in voting theory show that strategic manipulation by voters
is inevitable if a voting rule simultaneously satisfy certain desirable
properties. Motivated by this, we study the relevant question of how often a
voting rule is manipulable. It is well known that elections with a large number
of voters are rarely manipulable under impartial culture (IC) assumption.
However, the manipulability of voting rules when the number of candidates is
large has hardly been addressed in the literature and our paper focuses on this
problem. First, we propose two properties (1) asymptotic strategy-proofness and
(2) asymptotic collusion-proofness, with respect to new voters, which makes the
two notions more relevant from the perspective of computational problem of
manipulation. In addition to IC, we explore a new culture of society where all
score vectors of the candidates are equally likely. This new notion has its
motivation in computational social choice and we call it impartial scores
culture (ISC) assumption. We study asymptotic strategy-proofness and asymptotic
collusion-proofness for plurality, veto, $k$-approval, and Borda voting rules
under IC as well as ISC assumptions. Specifically, we prove bounds for the
fraction of manipulable profiles when the number of candidates is large. Our
results show that the size of the coalition and the tie-breaking rule play a
crucial role in determining whether or not a voting rule satisfies the above
two properties.
