Arrow's single peaked domains, richness, and domains for plurality and the Borda count (2401.12547v1)
Abstract: In this paper we extend the study of Arrow's generalisation of Black's single-peaked domain and connect this to domains where voting rules satisfy different versions of independence of irrelevant alternatives. First we report on a computational generation of all non-isomorphic Arrow's single-peaked domains on $n\leq 9$ alternatives. Next, we introduce a quantitative measure of richness for domains, as the largest number $r$ such that every alternative is given every rank between 1 and $r$ by the orders in the domain. We investigate the richness of Arrow's single-peaked domains and prove that Black's single-peaked domain has the highest possible richness, but it is not the only domain which attains the maximum. After this we connect Arrow's single-peaked domains to the discussion by Dasgupta, Maskin and others of domains on which plurality and the Borda count satisfy different versions of Independence of Irrelevant alternatives (IIA). For Nash's version of IIA and plurality, it turns out the domains are exactly the duals of Arrow's single-peaked domains. As a consequence there can be at most two alternatives which are ranked first in any such domain. For the Borda count both Arrow's and Nash's versions of IIA lead to a maximum domain size which is exponentially smaller than $2{n-1}$, the size of Black's single-peaked domain.
- ””, “Maximal Condorcet Domains,” http://abel.math.umu.se/~ klasm/Data/CONDORCET/.
- Akello-Egwell, Dolica, Charles Leedham-Green, Alastair Litterick, Klas Markström, and Søren Riis (2023) “Condorcet Domains of Degree at most Seven,” arXiv preprint arXiv:2306.15993.
- Aswal, Navin, Shurojit Chatterji, and Arunava Sen (2003) “Dictatorial Domains,” Economic Theory, 22 (1), 45–62, http://www.jstor.org/stable/25055669.
- Barbie, Martin, Clemens Puppe, and Attila Tasnádi (2006) “Non-manipulable domains for the Borda count,” Economic Theory, 27, 411–430.
- Black, Duncan (1948) “On the Rationale of Group Decision-making,” Journal of Political Economy, 56 (1), 23–34, http://www.jstor.org/stable/1825026.
- Dasgupta, Partha and Eric Maskin (2008) “On the robustness of majority rule,” Journal of the European Economic Association, 6 (5), 949–973.
- Egan, Patrick J (2014) “Do something” politics and double-peaked policy preferences,” The Journal of Politics, 76 (2), 333–349.
- Elkind, Edith, Martin Lackner, and Dominik Peters (2022) “Preference Restrictions in Computational Social Choice: A Survey.”
- Inada, Ken-Ichi (1964) “A Note on the Simple Majority Decision Rule,” Econometrica, 32 (4), 525–531, http://www.jstor.org/stable/1910176.
- List, Christian, Robert C Luskin, James S Fishkin, and Iain McLean (2013) “Deliberation, single-peakedness, and the possibility of meaningful democracy: evidence from deliberative polls,” The journal of politics, 75 (1), 80–95.
- Liversidge, Georgina (2020) “Counting Condorcet Domains,” Arxiv:2004.00751.
- Maskin, E (2023) “Borda’s Rule and Arrow’s Independence of Irrelevant Alternatives.”
- Moulin, H. (1980) “On Strategy-Proofness and Single Peakedness,” Public Choice, 35 (4), 437–455, http://www.jstor.org/stable/30023824.
- Nash, John F. (1950) “The Bargaining Problem,” Econometrica, 18 (2), 155–162, http://www.jstor.org/stable/1907266.
- Patty, John W. and Elizabeth Maggie Penn (2019) “A defense of Arrow’s independence of irrelevant alternatives,” Public Choice, 179 (1/2), pp. 145–164, https://www.jstor.org/stable/48704298.
- Puppe, Clemens (2018) “The single-peaked domain revisited: A simple global characterization,” Journal of Economic Theory, 176, 55–80, https://doi.org/10.1016/j.jet.2018.03.003.
- Ray, Paramesh (1973) “Independence of Irrelevant Alternatives,” Econometrica, 41 (5), 987–991, http://www.jstor.org/stable/1913820.
- Sen, A. K. (1966) “A Possibility theorem on majority decisions,” Econometrica, 34 (2), 491–499.
- Slinko, Arkadii (2019) “Condorcet domains satisfying Arrow’s single-peakedness,” Journal of Mathematical Economics, 84, 166–175, https://doi.org/10.1016/j.jmateco.2019.08.001.
- Zhou, Bei, Klas Markstrōm, and Søren Riis (2023) “CDL: A fast and flexible library for the study of permutation sets with structural restrictions,” arXiv preprint arXiv:2309.06306.
- (2024) “Local Diversity of Condorcet Domains,” arXiv preprint arXiv:2401.11912.
- Zhou, Bei and Søren Riis (2023) “New Record-Breaking Condorcet Domains on 10 and 11 Alternatives,” arXiv preprint arXiv:2303.06524.