On the reversal bias of the Minimax social choice correspondence

Extensions of the Simpson voting rule to the committee selection setting

“…Recall that a csr suffers the reversal bias when for a given preference profile a unique committee is selected and reversing the preferences of all the voters the same committee is still selected as unique outcome. Saari and Barney (2003) and Bubboloni and Gori (2016) prove several results about the reversal bias when k = 1. On the basis of our results and those in Bubboloni and Gori (2016) we can deduce interesting facts concerning the computations we are going to perform.…”

“…Saari and Barney (2003) and Bubboloni and Gori (2016) prove several results about the reversal bias when k = 1. On the basis of our results and those in Bubboloni and Gori (2016) we can deduce interesting facts concerning the computations we are going to perform. In particular, from Theorem A in Bubboloni and Gori (2016), we get that S and M are immune to the reversal bias for m = 3 and k = 1; from Proposition 33, S is immune to the reversal bias for m = 3 and k = 2 as well as for m = 4 and k = 3; when m = 4, from Theorem A in Bubboloni and Gori (2016), we have that S and M are immune to the reversal bias if and only if n ∈ {2, 3, 4, 5, 7}; from Theorem A in Bubboloni and Gori (2016) and Corollary 37, M is immune to the reversal bias for m = 3 and k = 2; from Corollary 36, M is immune to the reversal bias for m = 4 and k = 2; from Corollary 37, the probability for M to be immune to the reversal bias for m = 4 and k = 1 is the same as for m = 4 and k = 3.…”

“…On the basis of our results and those in Bubboloni and Gori (2016) we can deduce interesting facts concerning the computations we are going to perform. In particular, from Theorem A in Bubboloni and Gori (2016), we get that S and M are immune to the reversal bias for m = 3 and k = 1; from Proposition 33, S is immune to the reversal bias for m = 3 and k = 2 as well as for m = 4 and k = 3; when m = 4, from Theorem A in Bubboloni and Gori (2016), we have that S and M are immune to the reversal bias if and only if n ∈ {2, 3, 4, 5, 7}; from Theorem A in Bubboloni and Gori (2016) and Corollary 37, M is immune to the reversal bias for m = 3 and k = 2; from Corollary 36, M is immune to the reversal bias for m = 4 and k = 2; from Corollary 37, the probability for M to be immune to the reversal bias for m = 4 and k = 1 is the same as for m = 4 and k = 3. It is also immediate to show that the Borda csr is always immune to the reversal bias; the probability of suffering the reversal bias of the Plurality csr (Bloc csr) with k = 1 is the same as the one of the Negative Plurality csr (Bloc csr) with k = m − 1; the probability of suffering the reversal bias of the Negative Plurality csr with k = 1 is the same as the one of the Plurality csr with k = m − 1; when m is even and k = m 2 , the probability of suffering the reversal bias of the Plurality csr and the one of the Negative Plurality csr are the same; the Bloc csr is immune to the reversal bias.…”

“…7 Recall that a csr R is resolute if, for every (C, V, p) ∈ E and k ∈ [|C|−1], |R(C, V, p, k)| = 1; R suffers the reversal bias if there exist (C, V, p) ∈ E and k ∈ [|C| − 1] such that |R(C, V, p, k)| = 1 and R(C, V, p, k) = R(C, V, p r , k); R suffers the Condorcet loser paradox if there exist (C, V, p) ∈ E, k ∈ [|C| − 1] and W * ∈ 2 C k such that W * is a k-Condorcet loser set for (C, V, p), that is, for every x ∈ W * and y ∈ C \ W * , c p (x, y) < |V | 2 , and R(C, V, p, k) = {W * }; R suffers the leaving member paradox if there exist (C, V, p) ∈ E, k ∈ [|C| − 1] with k = 1, and 7 Those properties are largely studied in the literature. See for instance, Bubboloni and Gori (2016), Diss and Doghmi (2016), Diss and Gehrlein (2012), Diss and Tlidi (2018), Duggan and Schwartz (2000), Fishburn and Gehrlein (1976), Gehrlein and Lepelley (2010b), Jeong and Ju (2017), Kamwa and Merlin (2015), Saari and Barney (2003), and Staring (1986).…”

Breaking ties in collective decision-making

“…Along with the described analysis of spcs, we also study the problem to determine whether a k-scc admits V -anonymous and W -neutral resolute refinements 8 and whether some of them are immune to the reversal bias too. Recall that a k-scc is immune to the reversal bias if it never associates the same set of k alternatives with a preference profile and its reversal (see Saari and Barney 2003;Bubboloni and Gori 2016a). We prove that if (1) holds true, then any V -anonymous and W -neutral k-scc admits a V -anonymous and W -neutral resolute refinement (Theorem 20).…”