Outset and Pairwise Voting Map
3. An Axiomatic Foundation of CI with a Threshold
The purpose of developing CI with or without a threshold is to partition ImF 3 into the following regions; , , : c i wins, , , , , mod 3: c i and c i+1 are tied at the top, : no winner is decided. CI without a threshold introduces a set of axioms on which these regions are identified as an acceptable modification of the Condorcet winner criterion. I introduce a threshold of size ε into CI for reducing the influence of “almost” confused voters in a profile. Confused voters are profiles making cyclic votes. In ImF 3 , the direct influence of confused voters appears on the line segment , , , , , , which is the indecision region for CI without a threshold. By introducing a threshold, the indecision region must be expanded to include the line segment in its interior. In Saari 1994 and Saari 1995, a coordinate system on the set of profiles based on pairwise voting vectors, called as the cycle coordinate system, is introduced. It helps for identifying effects of positive and negative confused voters in the pairwise voting. The cyclic coordinate of a profile p is a tuple , , , where , , , , , , , , , , , and . By letting ≡ ∈ ∑ , , and ≡ ∈ ∑ , , , ∈ and ∈ . The map defined by , , is one-to-one except for profiles for which the sum of components for the value by F 3 is either 1 or -1. For such profiles, either or can be neglected in an obvious way. It is clear that . 5 The positive confused voter p c + is the profile , . , , , , and the negative confused voter p c - is the profile , , . , , . Any convex combination of these profiles is called as a confused voter. The former is a unique profile that has the value of F 3 to be , , , while the latter is a unique profile that has the value of F 3 to be , , . The difference between and , , is a measure for the degree of freedom from the positive confused voter, and that between and , , is that from the negative confused voter. It is intended that a threshold is used to identify a profile p for which each of and are close to positive or negative confused voters, correspondingly. An axiom for CI without a threshold claims that, if a profile consists of voter types for which an alternative is ranked at the bottom and the other two alternatives are tied in the pairwise vote, then it must be decided that those two alternatives are tied at the top. A confused voter can be perturbed in any small degree to make two alternatives dominate another but are tied each other. A threshold is used to guarantee that such a perturbation does not rank any alternative bellow others socially if the size of perturbation is small. All confused voters are treated equally, so that the effect of a threshold must be uniform over confused voters. Based on this argument, Kaneko 2016 proposes the following set of axioms to refine CI with a threshold . Axiom 1. For each ∈ , , , D i is an open convex set. For each ∈ , , mod 3. D i,i+1 is a subset of the intersection of boundaries of D i and D i+1 and a convex set. E is a closed convex set. Axiom 2. The set is dense in ImF 3. Axiom 3. For each ∈ , , , let C i be the set of all pairwise voting vectors for which the alternative c i is the winner by CI without a threshold. Then ∖ for all ∈ , , . Axiom 4. For some ∈ , , the following statements hold. 4.1. Let p c + i, λ p c - i, λ be the profile obtained by modifying the positive negative confused voter in the way that the type which ranks c i at the bottom is increased by and the type which ranks c i at the top is reduced by . Then F 3 p c + i, λ F 3 p c - i, λ belongs to E if and only if .4.2. If a profile p satisfies αp=αp
c + i, λ and βp=βp c - i, λ, then F 3 p is in E if and only if .4.3. For a profile p, let p
c be the profile having the cycle coordinate αp c + , βp c - , dp . Also, for each ∈ , , , let p c i, λ be the profile having the cycle coordinate αp c + i, λ, βp c - i, λ, dp. Then, if ∈ , ∈ if and only if F 3 p is non-negatively spanned by F 3 p c , F 3 p c i+1, ε and F 3 p c i+2, ε mod 3. With this set of axioms, Kaneko 2016 derives the following refinement of CI. Theorem. A partition of ImF 3 into D i i=1,2,3, D i, i+1 ∈ , , , mod 3 and E satisfies the axioms 1- 4 if and only if these sets are determined as follows; ∈ , ∈ , ∈ , , ∖ i=1,2,3, 6 , ∖ ∈ , , , mod 3}. where bdC i is the boundary of C i in ImF 3 for i=1,2,3, and , , , , , , , , , , , , , , , , , . In Saari 1994 and Saari 1995, it is derived that ∈ | , mod 3, for each ∈ , , . On the section of ImF 3 by the plane ∑ , , ∈ , , E becomes the closed triangle with vertices , , , , , and , , . The rest of the partition on the section can be easily figured out with these. The proof of the theorem in Kaneko 2016 derives the partition on the section directly by an elementary calculation. Let H t be the intersection of ImF 3 and the plane ∑ , , . Consider any profile p such that ∈ . It is trivial to see that for such profiles, so that they have the same d-coordinate. An elementary calculation shows that F 3 p c + i-1, λ i+k = 13 + k λ for ∈ , , and ∈ , , mod 3. Similarly, F 3 p c - i-1, λ i+k = -13 + k λ for ∈ , , and ∈ , , mod 3. Therefore, we have , for ∈ , , and ∈ , , mod 3. Let ≡ ∈ | for i=1,2,3. Then F 3 p c i-1, λ i is in and not in C i- 1 . Since , , , , guarantees that F 3 p c i-1, λ is in . By the axiom 3 and 4, ∖ , , , , , , mod 3. By the axiom 1 and 2, mod 3. They imply that , , , , , , , , . The theorem follows easily by integrating these sets for ∈ , , or along the line segment , , , , , . 4. Remarks Kaneko 2016 includes more results related to the theorem. It proposes that, for all scoring voting methods, the image of the expanded indecision region for CI by the one-to-one map from a pairwise voting vector to a scoring voting vector with the normalized Borda scoring vector is adopted as the indecision region, though the maximal translation is proved to be larger than that for non-Borda scoring methods. But it is proved that, by doing so, the threshold value for a pairwise voting on a pair of alternatives which guarantees that a scoringParts
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