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Condorcet VS Borda, round n + 1 W. S. Zwicker

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Condorcet VS Borda, round n + 1

W. S. Zwicker

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Department of Mathematics, Union College, Schenectady NY, USA, zwickerw@union.edu

A voting rule is manipulable if it is sometimes possible for a voter to change the election’s outcome to one she prefers, by switching away from her sincere ballot – one that represents her actual preferences – to an insincere ballot. Loosely speaking, the Gibbard-Satterthwaite Theorem tells us that every voting rule suffers from this problem, when there are more than two candidates. More precisely, the theorem states that with three or more alternatives every resolute, non-imposing, and non-dictatorial social choice function (SCF) is manipulable. We prove a variant of this theorem that uses stronger hypotheses and provides additional information, revealing a fundamental dichotomy in the type of strategic vulnerability for an SCF.

Theorem 1 (Stated loosely) With four or more alternatives every resolute, neutral, and anonymous SCF f is either:

• Free from the reversal paradox, but manipulable on the subdomain D Condorcet , or

• Strategy-proof on D Condorcet but suffers from the reversal paradox for some profile P ∈ / D Condorcet , or

• Suffers from both problems.

Terminology A resolute SCF is a voting rule that returns a unique winning alternative (no ties allowed) for each profile of strict (linear) preferences over a finite set A of alternatives (aka “candidates").

In a reversal paradox a voter changes the winner to one she strictly prefers by completely reversing her sincere ranking. Such a reversed ballot is maximally insincere – for every pair of alternatives, it misstates which of the two the voter prefers – so a reversal paradox represents a particularly extreme form of manipulability.

A Condorcet alternative x is preferred to each other alternative y by a strict majority of voters, and D Condorcet is the subdomain consisting of all profiles for which a Condorcet alternative exists. Pairwise Majority Rule selects the Condorcet alternative for each profile in D Condorcet . A Condorcet extension is an SCF that agrees with Pairwise Majority Rule on D Condorcet .

Comment The precise statement of Theorem 1 relaxes the neutrality and anonymity hypotheses but requires a minimum number of voters (which is increasing in the number of alternatives). The proof combines two earlier results. Campbell and Kelly [1] show that Condorcet extensions are the only strategy-proof SCFs on D Condorcet (assuming weak forms of neutrality and anonymity), while Sanver and Zwicker [2] show that resolute Condorcet extensions for four or more alternatives suffer from the reversal paradox. The latter result is a strong form of Moulin’s theorem in [3], [4] that resolute Condorcet extensions suffer from the no-show paradox.

References

[1] D. E. Campbell and J. S. Kelly, A strategy-proofness characterization of majority rule, Econ Theor

22, pp. 557-568 (2003).

[2] M. R. Sanver and W. S. Zwicker, One-way monotonicity as a form of strategy-proofness, Int J Game Theory

39, pp. 553-574 (2009).

[3] H. Moulin, Axioms of cooperative decision making, Econometric Soc Monograph #15, Cambridge U. Press, London (1988).

[4] H. Moulin, Condorcet’s principle implies the no-show paradox, J Econ Theory

45, pp. 53-64 (1988).

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