Abstract
The shortcomings of the widely used plurality procedure are discussed. Elimination procedures are introduced as attempts to avert the shortcomings of the plurality procedure. The Condorcet tournament (commonly known as the round robin tournament) with its graph representation and related concepts are introduced and special attention is given to the Condorcet paradox and how it can be used in a parliamentary procedure to manipulate the voting outcome by creating a killer amendment. McGarvey’s theorem - which establishes the existence of a scenario of voting that can be represented by any complete directed graph - is presented. The vulnerability of Borda count to the influence of irrelevant alternatives and its frequent failure to elect an existing Condorcet winner are demonstrated. The principe of elimination is brought to the Borda count, thus creating the Hare-Borda procedure which satisfies the Condorcet winner criterion. An introduction of the social welfare theory is presented including the theorems of Duncan Black, Kenneth May, Kenneth Arrow and Allen Gibbard, with complete detailed proofs. The standard social choice procedures and Condorcet’s concepts are adapted to allow ties and indifference in voter’s rankings of the alternatives. The manipulation of such procedure is studied and the theorem of Peter Gärdenfors on the manipulability of social choice functions that satisfy the Condorcet winner criterion in addition to anonymity and neutrality is proven.
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El-Helaly, S. (2019). Social Choice. In: The Mathematics of Voting and Apportionment. Compact Textbooks in Mathematics. Birkhäuser, Cham. https://doi.org/10.1007/978-3-030-14768-6_1
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DOI: https://doi.org/10.1007/978-3-030-14768-6_1
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