Abstract
This paper presents the conditions required for a profile in order to never exhibit either the strong or the strict Borda paradoxes under all weighted scoring rules in three-candidate elections. The main particularity of our paper is that all the conclusions are deduced from the differences of votes between candidates in pairwise majority elections. This way allows us to answer new questions and provide an organized knowledge of the conditions under which a given profile never shows one or the other of the two paradoxes.
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Notes
A rigorous definition of this notion is given in Sect. 2.1.
Saari and McIntee (2013) use the notation P(X, Y). Instead, we use the notation XY since it allows us to reduce the length of our results.
Notice that these results are found under the standard assumption that \(AB > 0\) and \(BC > 0\). This is established without loss of generality.
Remember that if the strongly non-cyclic condition is not satisfied this does not necessarily define a cycle. Although we use the name “cyclic” for this essential profile to be consistent with Saari and McIntee (2013), we need to consider \(AC>0\), in our framework of Borda’s paradox, in order for C to be a Condorcet loser.
For instance, the preference \(A\succ B\succ C\) type has a reverse given by \(C\succ B\succ A\). It is clear that removing the same number of voters with reversal pairs does not change XY values.
This is not true with more than three candidates.
It is equivalent to remove 5 voters of each of both \(A\succ B\succ C\) and \(C\succ B\succ A\) preference types.
Notice that n and all XY values have the same parity. This is true because, for each pair of candidates X and Y we have \(n = \{X\text {'s vote} + Y\text {'s vote}\}\), which leads to \(\{X\text {'s vote}\} = \frac{1}{2}(n + XY)\) since \(XY=\{X\text {'s vote} - Y\text {'}\text{ s } \text{ vote }\}\). In order for \(\{X\text {'s vote}\}\) to be an integer value, it follows that n and all XY values either are odd integers, or are all even integers. In other words, \(\alpha +\beta +\gamma \) is always an integer value and, even more, it has the same parity as n and all XY values.
Notice that for all U in \(\mathbb {Z}\), we have \(\lfloor U \rfloor = U - F(U)\) such that \(\lfloor U \rfloor \) stands for the greatest integer less than or equal to U. Notice that F(U) is defined in the same way for positive and negative numbers. However, it is important to precise that for instance \(\lfloor -1.3 \rfloor = -2 \) leading to \(F(-1.3)=0.7\) and \(\lfloor 1.3 \rfloor = 1 \) leading to \(F(1.3)=0.3\).
Notice that in this case \(U=\dfrac{AC-AB}{2}\) and \(F(U)=0\) since AB and AC have the same parity (see the discussion in Footnote 9). The same remark can be used for some other parts of our corollaries.
Let us explain more precisely the latter case. First, the condition \((1-s)(AC-AB)+2sBC\ge 0\) in 1-c of Theorem 1 leads to \(s\ge \dfrac{1}{3}\). In addition, the value of F(U) in the condition \((1-s)AC+sBC \ge \dfrac{1-2s}{3}(n-2+2 F(U))\) does not matter. Indeed, we know that \(0 \le F(U)\lesssim 1\). In addition, if \(F(U)=0\), the condition \((1-s)AC+sBC \ge \dfrac{1-2s}{3}(n-2+2 F(U))\) becomes \(s\ge \dfrac{1}{8}\). Moreover, if \(F(U)\simeq 1\), the condition \((1-s)AC+sBC \ge \dfrac{1-2s}{3}(n-2+2 F(U))\) becomes \(s\gtrsim \dfrac{1}{5}\). Both \(s\ge \dfrac{1}{8}\) and \(s\gtrsim \dfrac{1}{5}\) are relaxed in comparison with the first requirement given by \(s\ge \dfrac{1}{3}\).
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The authors thank two anonymous reviewers of Theory and Decision for their helpful feedback on an earlier version of this paper. We further thank Nicolas Barbaroux, Richard Baron, Rachid Chemsi, Rachid Dali, and Florent Pirot for their help in proofreading the paper. The first author gratefully acknowledges financial support by the National Agency for Research (ANR)—research program “Dynamic Matching and Interactions: Theory and Experiments” (DynaMITE) ANR-BLANC.
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Diss, M., Tlidi, A. Another perspective on Borda’s paradox. Theory Decis 84, 99–121 (2018). https://doi.org/10.1007/s11238-017-9649-1
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DOI: https://doi.org/10.1007/s11238-017-9649-1