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

Asymptotic Collusion-Proofness of Voting Rules: The Case of Large Number of Candidates