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Decidability in complex social choices


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.

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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.

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Correspondence to Gennaro Amendola.



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 \(([x]_{\sim _{\{i\}}},\succ )\).

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Amendola, G., Marengo, L., Pirino, D. et al. Decidability in complex social choices. Evolut Inst Econ Rev 12, 141–168 (2015).

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

Mathematics Subject Classification

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

JEL Classification

  • D03
  • D71
  • D72