# Decidability in complex social choices

- 72 Downloads

## Abstract

In this paper, we develop on a geometric model of social choice among bundles of interdependent elements (objects). Social choice can be seen as a process of search for optima in a complex multidimensional space and objects determine a decomposition of such a space into subspaces. We present a series of numerical and probabilistic results which show that such decompositions in objects can greatly increase decidability, as new kind of optima (called local and u-local) are very likely to appear also in cases in which no generalized Condorcet winner exists in the original search space.

## Keywords

Social choice Object construction Hyperplane arrangement Probability Tournament Algorithm## Mathematics Subject Classification

05C20 52C35 91B10 91B12 91B14## JEL Classification

D03 D71 D72## Notes

### Acknowledgments

We are very grateful to anonymous referees for very useful suggestions. Simona Settepanella was partially supported by the Institute for New Economic Thinking, INET inaugural Grant \(\sharp\)220.

## References

- Aldous D (1988) Probability approximations via the poisson clumping heuristic, Applied Mathematical Sciences (Book 77). Springer, New YorkGoogle Scholar
- Amendola G (2011a) “FOSoR,” Software package, University of Pisa, Department of Mathematics, Pisa, http://www.dm.unipi.it/~amendola/files/software/fosor/
- Amendola G (2011b) “FOSoRStat,” Software package, University of Pisa, Department of Mathematics, Pisa, http://www.dm.unipi.it/~amendola/files/software/fosorstat/
- Amendola G, Settepanella S (2012) Optimality in social choice. J Math Sociol 36:44–77CrossRefGoogle Scholar
- Arratia R, Goldstein L, Gordon L (1989) Two moments suffice for Poisson approximations: the Chen-Stein method. Ann. Probab. 17(1):9–25CrossRefGoogle Scholar
- Arrow K (1951) Social choice and individual values. Wiley, New YorkGoogle Scholar
- Banks J (1985) Sophisticated voting outcomes and covering relation. Soc Choice Welf 1:295–306CrossRefGoogle Scholar
- Barbour AD, Holst L, Janson S (1992) Poisson approximation. Oxford University Press, LondonGoogle Scholar
- Bernholz P (1974) Logrolling, arrow paradox and decision rules. a generalization. Kyklos 27:49–61CrossRefGoogle Scholar
- Brams S, Kilgour D, Zwicker W (1998) The paradox of multiple elections. Soc Choice Welf 15:211–236CrossRefGoogle Scholar
- Buchanan JM, Tullock G (1962) Calculus of consent. University of Michigan Press, Ann ArborGoogle Scholar
- Callander S, Wilson CH (2006) Context-dependent voting. Q J Polit Sci 1:227–254CrossRefGoogle Scholar
- Chartrand G, Lesniak L (2005) Graphs & digraphs, 4th edn. Chapman & Hall/CRC, Boca RatonGoogle Scholar
- Condorcet de Caritat, J.-A.-N. (1785): Essai sur l’Application de l’Analyse aux Probabilités de Decision Rendue à la Pluralité des Voix. Imprimerie Royale, ParisGoogle Scholar
- Conitzer V, Lang J, Xia L (2009) “How hard is it to control sequential elections via the Agenda?.” Proceeding IJCAI-09, pp 103–108Google Scholar
- Conitzer V, Lang J, Xia L (2011) “Hypercubewise Preference Aggregation in MultiIssue domains.” Proceeding IJCAI-11, pp 158–163Google Scholar
- Denzau AT, Mackay RJ (1981) Structure-induced equilibria and perfect-foresight expectations. Am J Polit Sci 25:762–779CrossRefGoogle Scholar
- Dutta B (1988) Covering sets and a new Condorcet choice correspondence. J Econ Theory 44:63–80CrossRefGoogle Scholar
- Enelow JM, Hinich MJ (1983) Voting one issue at a time: the question of voter forecasts. Am Polit Sci Rev 77:435–445CrossRefGoogle Scholar
- Falk M, Hü J, Reiss RD (2004) Laws of small numbers: extremes and rare events. BirkhäuserGoogle Scholar
- Fey M (2008) Choosing from a large tournament. Soc Choice Welf 31:301–309CrossRefGoogle Scholar
- Fryer R, Jackson M (2008) A categorical model of cognition and biased decision making. B.E. Press J Theor Econ 8, Article 6Google Scholar
- Kahneman D, Tversky A (2000) Choices, Values, and Frames. Cambridge University Press, CambridgeGoogle Scholar
- Kramer GH (1972) Sophisticated voting over multidimensional choice spaces. J Math Sociol 2:165–180CrossRefGoogle Scholar
- Lang J (2007) Vote and aggregation in combinatorial domains with structured preferences. Proceeding IJCAI’07 Proceedings of the 20th international joint conference on Artifical intelligence, pp 1366–1371Google Scholar
- Marengo L, Pasquali C (2011) The construction of choice. A computational voting model. J Econ Interact Coord 6:139–156CrossRefGoogle Scholar
- Marengo L, Settepanella S (2012) Social choice among complex objects. Scuola Norm. Sup. Pisa Cl. Sci, Ann. doi: 10.2422/2036-2145.201202_004, to appear
- Miller N (1977) Graph-theoretical approaches to the theory of voting. Am J Polit Sci 21:769–803CrossRefGoogle Scholar
- Miller N (1980) A new solution set for tournaments and majority voting. Am J Polit Sci 68–96:24Google Scholar
- Moon JW (1968) Topics on tournaments. Holt, Rinehart and Winston, New YorkGoogle Scholar
- Mullainathan S (2000) Thinking through categories. MIT working paperGoogle Scholar
- Mullainathan S, Schwartzstein J, Shleifer A (2008) Coarse thinking and persuasion. Q J Econ 123:577–619CrossRefGoogle Scholar
- Saari DG, Sieberg KK (2001) The sum of the parts can violate the whole. Am Polit Sci Rev 95:415–433CrossRefGoogle Scholar
- Schwartz T (1972) Rationality and the myth of the maximum. Nous 6:97–117CrossRefGoogle Scholar
- Schwartz T (1990) Cyclic tournaments and cooperative majority voting: a solution. Soc Choice Welf 7:19–29CrossRefGoogle Scholar
- Scott A, Fey M (2012) The minimal covering set in large tournaments. Soc Choice Welf 38:1–9CrossRefGoogle Scholar
- Shepsle KA (1979) Institutional arrangements and equilibrium in multidimensional voting models. Am J Polit Sci 23:27–59CrossRefGoogle Scholar