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.

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