The paper is devoted to study of manipulability of scoring voting rules. We present a conjectural list of inequalities , which are satisfied by a  profile if and only if it is manipulable by a coalition Coal_{\mathcal{A_k} (of members of electorate who prefer A_k over the winner). The conjecture is verified for all positional rules with m=4 alternatives and for general m with mild restrictions on the weights. In particular, it is established for the plurality and anti-plurality rule (any m) and the Borda count (m\leq 6 candidates). The shares } of  manipulable outcomes for plurality, anti-plurality and Borda count in case of m=4 alternatives are presented and the Python code for carrying out the computation in case of Borda count is provided. We also study a specific type of manipulation (SM), which includes not only the fact of manipulability but the manipulability which leads to the winning of the desired alternative. The shares of SM results for plurality, anti-plurality, Borda and Coombs' procedures in case of m=3 alternatives are computed .
Coalitional manipulation in voting is considered to be any scenario in which a group of voters decide to vote in a way that does not represent their true preferences in order to improve the outcome of the election. The present paper is devoted to study coalitional manipulability within the class of scoring voting rules. For any such rule and any number of alternatives, we introduce a new approach allowing us to characterize the list of inequalities which are satisfied by a preference profile if and only if it is manipulable by a coalition _k} (of members of electorate who prefer A_k over the winner). The conjecture is verified for all positional rules with m=4 alternatives and for general m with mild restrictions on the weights. In particular, it is established for the plurality and anti-plurality rule (any m) and the Borda count (m\leq 6 candidates). The shares } of voters. Furthermore, the probability of manipulable outcomes for some well-studied scoring voting rules in case of small number of alternatives and large electorates are presented under a well-known assumption on individual preference profiles .
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Coalitional manipulation in voting is considered to be any scenario in which a group of voters decide to vote in a way that does not represent their true preferences in order to improve the outcome of the election. The present
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coalitional manipulability within the class
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We present a conjectural
For any such rule and any number of alternatives, we introduce a new approach allowing us to characterize the
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some well-studied scoring voting rules
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m=4 alternatives are presented and the Python code for carrying out the computation in case of Borda count is provided. We also study a specific type of manipulation (SM), which includes not only the fact of manipulability but the manipulability which leads to the winning of the desired alternative. The shares of SM results for plurality, anti-plurality, Borda and Coombs' procedures in case of m=3 alternatives are computed
small number of alternatives and large electorates are presented under a well-known assumption on individual preference profiles