Skip to main content

Decidability in complex social choices

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.

This is a preview of subscription content, access via your institution.

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8
Fig. 9
Fig. 10
Fig. 11
Fig. 12
Fig. 13
Fig. 14
Fig. 15
Fig. 16
Fig. 17
Fig. 18
Fig. 19
Fig. 20
Fig. 21
Fig. 22

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

    Article  Google Scholar 

  • Arratia R, Goldstein L, Gordon L (1989) Two moments suffice for Poisson approximations: the Chen-Stein method. Ann. Probab. 17(1):9–25

    Article  Google Scholar 

  • Arrow K (1951) Social choice and individual values. Wiley, New York

    Google Scholar 

  • Banks J (1985) Sophisticated voting outcomes and covering relation. Soc Choice Welf 1:295–306

    Article  Google Scholar 

  • Barbour AD, Holst L, Janson S (1992) Poisson approximation. Oxford University Press, London

    Google Scholar 

  • Bernholz P (1974) Logrolling, arrow paradox and decision rules. a generalization. Kyklos 27:49–61

    Article  Google Scholar 

  • Brams S, Kilgour D, Zwicker W (1998) The paradox of multiple elections. Soc Choice Welf 15:211–236

    Article  Google Scholar 

  • Buchanan JM, Tullock G (1962) Calculus of consent. University of Michigan Press, Ann Arbor

    Google Scholar 

  • Callander S, Wilson CH (2006) Context-dependent voting. Q J Polit Sci 1:227–254

    Article  Google Scholar 

  • Chartrand G, Lesniak L (2005) Graphs & digraphs, 4th edn. Chapman & Hall/CRC, Boca Raton

    Google 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, Paris

  • Conitzer V, Lang J, Xia L (2009) “How hard is it to control sequential elections via the Agenda?.” Proceeding IJCAI-09, pp 103–108

  • Conitzer V, Lang J, Xia L (2011) “Hypercubewise Preference Aggregation in MultiIssue domains.” Proceeding IJCAI-11, pp 158–163

  • Denzau AT, Mackay RJ (1981) Structure-induced equilibria and perfect-foresight expectations. Am J Polit Sci 25:762–779

    Article  Google Scholar 

  • Dutta B (1988) Covering sets and a new Condorcet choice correspondence. J Econ Theory 44:63–80

    Article  Google Scholar 

  • Enelow JM, Hinich MJ (1983) Voting one issue at a time: the question of voter forecasts. Am Polit Sci Rev 77:435–445

    Article  Google Scholar 

  • 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

    Article  Google Scholar 

  • 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

    Google Scholar 

  • Kramer GH (1972) Sophisticated voting over multidimensional choice spaces. J Math Sociol 2:165–180

    Article  Google 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 1366–1371

  • Marengo L, Pasquali C (2011) The construction of choice. A computational voting model. J Econ Interact Coord 6:139–156

    Article  Google 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 21:769–803

    Article  Google Scholar 

  • Miller N (1980) A new solution set for tournaments and majority voting. Am J Polit Sci 68–96:24

    Google Scholar 

  • Moon JW (1968) Topics on tournaments. Holt, Rinehart and Winston, New York

    Google Scholar 

  • 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

    Article  Google Scholar 

  • Saari DG, Sieberg KK (2001) The sum of the parts can violate the whole. Am Polit Sci Rev 95:415–433

    Article  Google Scholar 

  • Schwartz T (1972) Rationality and the myth of the maximum. Nous 6:97–117

    Article  Google Scholar 

  • Schwartz T (1990) Cyclic tournaments and cooperative majority voting: a solution. Soc Choice Welf 7:19–29

    Article  Google Scholar 

  • Scott A, Fey M (2012) The minimal covering set in large tournaments. Soc Choice Welf 38:1–9

    Article  Google Scholar 

  • Shepsle KA (1979) Institutional arrangements and equilibrium in multidimensional voting models. Am J Polit Sci 23:27–59

    Article  Google Scholar 

Download references

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

Authors

Corresponding author

Correspondence to Gennaro Amendola.

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 \(\{0,\ldots , m_i-1\}\) of the non-negative 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 \(X\),

$$(x_1,\ldots ,x_n)\sim _I (y_1,\ldots ,y_n) \qquad {\text{ if }} {\text{ and }} {\text{ only }} {\text{ if}} \qquad x_j=y_j\ {\text{ for }} {\text{ all }}\ j \notin I.$$

For each \(x \in X\) and each subset of features \(\{f_i\}_{i \in I}\), the equivalence relation \(\sim _I\) induces an equivalence class

$${}[x]_{\sim _I}=\{y \in X : x \sim _I y\}.$$

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.,

$$\max _{\succ }([x]_{\sim _I})=\big \{ y \in [x_I]_{\sim _I} : y \succ z {\text{ for }} {\text{ all }} z \in [x]_{\sim _I}\setminus \{y\}\big \}.$$

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

$$x=\max _{\succ }([x]_{\sim _{\{i\}}}) \quad {\text{ for }} {\text{ all }} i \in \{1,\ldots , n\},$$

i.e., \(x\) is the generalized Condorcet winner in each subtournament \(([x]_{\sim _{\{i\}}},\succ )\).

About this article

Verify currency and authenticity via CrossMark

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/s40844-015-0006-1

Download citation

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s40844-015-0006-1

Keywords

  • Social choice
  • Object construction
  • Hyperplane arrangement
  • Probability
  • Tournament
  • Algorithm

Mathematics Subject Classification

  • 05C20
  • 52C35
  • 91B10
  • 91B12
  • 91B14

JEL Classification

  • D03
  • D71
  • D72