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 ulocal) are very likely to appear also in cases in which no generalized Condorcet winner exists in the original search space.
This is a preview of subscription content, access via your institution.
References
Aldous D (1988) Probability approximations via the poisson clumping heuristic, Applied Mathematical Sciences (Book 77). Springer, New York
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 36:44–77
Arratia R, Goldstein L, Gordon L (1989) Two moments suffice for Poisson approximations: the ChenStein method. Ann. Probab. 17(1):9–25
Arrow K (1951) Social choice and individual values. Wiley, New York
Banks J (1985) Sophisticated voting outcomes and covering relation. Soc Choice Welf 1:295–306
Barbour AD, Holst L, Janson S (1992) Poisson approximation. Oxford University Press, London
Bernholz P (1974) Logrolling, arrow paradox and decision rules. a generalization. Kyklos 27:49–61
Brams S, Kilgour D, Zwicker W (1998) The paradox of multiple elections. Soc Choice Welf 15:211–236
Buchanan JM, Tullock G (1962) Calculus of consent. University of Michigan Press, Ann Arbor
Callander S, Wilson CH (2006) Contextdependent voting. Q J Polit Sci 1:227–254
Chartrand G, Lesniak L (2005) Graphs & digraphs, 4th edn. Chapman & Hall/CRC, Boca Raton
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, Paris
Conitzer V, Lang J, Xia L (2009) “How hard is it to control sequential elections via the Agenda?.” Proceeding IJCAI09, pp 103–108
Conitzer V, Lang J, Xia L (2011) “Hypercubewise Preference Aggregation in MultiIssue domains.” Proceeding IJCAI11, pp 158–163
Denzau AT, Mackay RJ (1981) Structureinduced equilibria and perfectforesight expectations. Am J Polit Sci 25:762–779
Dutta B (1988) Covering sets and a new Condorcet choice correspondence. J Econ Theory 44:63–80
Enelow JM, Hinich MJ (1983) Voting one issue at a time: the question of voter forecasts. Am Polit Sci Rev 77:435–445
Falk M, Hü J, Reiss RD (2004) Laws of small numbers: extremes and rare events. Birkhäuser
Fey M (2008) Choosing from a large tournament. Soc Choice Welf 31:301–309
Fryer R, Jackson M (2008) A categorical model of cognition and biased decision making. B.E. Press J Theor Econ 8, Article 6
Kahneman D, Tversky A (2000) Choices, Values, and Frames. Cambridge University Press, Cambridge
Kramer GH (1972) Sophisticated voting over multidimensional choice spaces. J Math Sociol 2:165–180
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 1366–1371
Marengo L, Pasquali C (2011) The construction of choice. A computational voting model. J Econ Interact Coord 6:139–156
Marengo L, Settepanella S (2012) Social choice among complex objects. Scuola Norm. Sup. Pisa Cl. Sci, Ann. doi:10.2422/20362145.201202_004, to appear
Miller N (1977) Graphtheoretical approaches to the theory of voting. Am J Polit Sci 21:769–803
Miller N (1980) A new solution set for tournaments and majority voting. Am J Polit Sci 68–96:24
Moon JW (1968) Topics on tournaments. Holt, Rinehart and Winston, New York
Mullainathan S (2000) Thinking through categories. MIT working paper
Mullainathan S, Schwartzstein J, Shleifer A (2008) Coarse thinking and persuasion. Q J Econ 123:577–619
Saari DG, Sieberg KK (2001) The sum of the parts can violate the whole. Am Polit Sci Rev 95:415–433
Schwartz T (1972) Rationality and the myth of the maximum. Nous 6:97–117
Schwartz T (1990) Cyclic tournaments and cooperative majority voting: a solution. Soc Choice Welf 7:19–29
Scott A, Fey M (2012) The minimal covering set in large tournaments. Soc Choice Welf 38:1–9
Shepsle KA (1979) Institutional arrangements and equilibrium in multidimensional voting models. Am J Polit Sci 23:27–59
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.
Author information
Authors and Affiliations
Corresponding author
Appendix
Appendix
In this appendix, we reproduce the model described in Sect. 3 in a purely algebraic way, as preferences on combinatorial domains.
Preferences on combinatorial domains A feature \(f_i\) can be defined as an element \(\{0,\ldots , m_i1\}\) of the nonnegative integers. The set \(\{f_1, \ldots ,f_n\}\) of features is denoted by \(F\). The combinatorial domain \(X=f_1 \times \ldots \times f_n\) is the set of all social outcomes and \(x=(x_1,\ldots ,x_n)\) denotes an element in \(X\). Let us remark that, with this notation, we totally loose the spacial structure of \(R^n\) and the utility of thinking of \(x \in X\) as a point in the real \(n\)dimensional space.
A tournament \(T=(X, \succ )\) is an orientation of a complete graph on \(X\), in which case \(\succ\) can equivalently be seen as a complete and asymmetric relation on \(X\).
A social outcome \(x \in X\) is said to be a generalized Condorcet winner of a tournament \(T=(X, \succ )\) if \(x \succ y\) for all \(y\) distinct from \(x\). The probability that a randomly chosen social outcome in \(X\) is a generalized Condorcet winner is given in equation (1).
Every subset of features \(\{f_i\}_{i \in I}\), with \(I \subset \{1,\ldots ,n\}\), induces an equivalence relation \(\sim _I\) over \(X\) such that for all elements \(x=(x_1,\ldots ,x_n)\) and \(y=(y_1,\ldots ,y_n)\) in \(X\),
For each \(x \in X\) and each subset of features \(\{f_i\}_{i \in I}\), the equivalence relation \(\sim _I\) induces an equivalence class
Given a tournament \(T=(X, \succ )\), for each subset of features \(\{f_i\}_{i \in I}\) and each \(x \in X\) one can associate, if it exists, the maximum element in \([x]_{\sim _I}\), i.e.,
Observe that, since \(\succ\) is asymmetric, the cardinality of \(\max _{\succ }([x_i]_{\sim _i})\) is either \(0\) or \(1\) and that, if \(I=\{1,\ldots ,n\}\), this maximum coincides, if it exists, with the generalized Condorcet winner.
With the above notations, we have the following:

an objects scheme is a set \(A=\big \{ \{f_i\}_{i \in I_j}\big \}_{1 \le j \le k}\) of subsets of features such that \(\cup _{1 \le j \le k}I_j=\{1,\ldots ,n\}\), i.e., all features are considered at least once;

an agenda \(\alpha\) is an order, with repetitions, of the indices \(j \in \{1, \ldots , k\}\);

the process starting from an initial element \(x_0 \in X\) determines a subgraph \(T_{x_0,A,\alpha }\) of \(T=(X, \succ )\) that depends from \(x_0 \in X\), the objects scheme and the fixed agenda.
An element \(x \in X\) is a local optimum for the objects scheme \(A\) if it exists an \(x_0 \in X\) and an agenda \(\alpha\) such that \(x\) is the generalized Condorcet winner in the subgraph \(T_{x_0,A,\alpha }\). Marengo and Settepanella (2012) show that the fact that \(x\) is the generalized Condorcet winner in a subgraph \(T_{x_0,A,\alpha }\) is independent of \(x_0\) and \(\alpha\), that is, if \(x\) is a local optimum then it is a local optimum for \(T_{x,A,\alpha }\) for any agenda \(\alpha\) and that, given an agenda \(\alpha\) there is always an element \(y \in X\) such that \(x\) is the generalized Condorcet winner in the subgraph \(T_{y,A,\alpha }\). Moreover, they noticed that a necessary and sufficient condition for \(x \in X\) to be local optimum for at least an objects scheme \(A\) is that
i.e., \(x\) is the generalized Condorcet winner in each subtournament \(([x]_{\sim _{\{i\}}},\succ )\).
About this article
Cite this article
Amendola, G., Marengo, L., Pirino, D. et al. Decidability in complex social choices. Evolut Inst Econ Rev 12, 141–168 (2015). https://doi.org/10.1007/s4084401500061
Published:
Issue Date:
DOI: https://doi.org/10.1007/s4084401500061
Keywords
 Social choice
 Object construction
 Hyperplane arrangement
 Probability
 Tournament
 Algorithm
Mathematics Subject Classification
 05C20
 52C35
 91B10
 91B12
 91B14
JEL Classification
 D03
 D71
 D72