[Paper Review] Modularity and Optimality in Social Choice

Gennaro Amendola, Simona Settepanella|arXiv (Cornell University)|Apr 20, 2010

Game Theory and Voting Systems22 references3 citations

TL;DR

This paper introduces u-local optima in social choice theory to analyze modularity and optimality when social outcomes are bundles of interdependent elements. By linking geometric models to tournament theory, it presents an O(M³ log M) algorithm to compute the universal basin of attraction, revealing that global optima are rare in complex choice settings.

ABSTRACT

Marengo and the second author have developed in the last years a geometric model of social choice when this takes place among bundles of interdependent elements, showing that by bundling and unbundling the same set of constituent elements an authority has the power of determining the social outcome. In this paper we will tie the model above to tournament theory, solving some of the mathematical problems arising in their work and opening new questions which are interesting not only from a mathematical and a social choice point of view, but also from an economic and a genetic one. In particular, we will introduce the notion of u-local optima and we will study it from both a theoretical and a numerical/probabilistic point of view; we will also describe an algorithm that computes the universal basin of attraction of a social outcome in O(M^3 logM) time (where M is the number of social outcomes).

Motivation & Objective

• To formalize and analyze the concept of u-local optima in social choice models with modular, interdependent outcomes.
• To resolve mathematical challenges in Marengo and Settepanella's geometric model of social choice among bundles of elements.
• To explore the implications of object construction power—where bundling and unbundling affect social outcomes—through topological and algorithmic tools.
• To compare the classical social choice model with the new modular framework, particularly regarding the existence and probability of optima.
• To develop a computationally efficient method for determining the universal basin of attraction of a social outcome.

Proposed method

• Introduces the notion of u-local optima as a refinement of local optima, defined via the universal basin of attraction across all possible object constructions.
• Applies tournament theory and hyperplane arrangements to model preference relations among social outcomes in modular settings.
• Uses algebraic topology and real central arrangements to analyze the structure of social choice functions and their optima.
• Develops an algorithm that computes the universal basin of attraction in O(M³ log M) time, where M is the number of social outcomes.
• Employs probabilistic and numerical analysis to estimate the likelihood of global optima in both classical and modular social choice models.
• Leverages geometric and combinatorial techniques to prove that certain outcomes are not global optima by contradiction, using object set constraints.

Experimental results

Research questions

• RQ1 How do u-local optima differ from local and global optima in modular social choice settings?
• RQ2 What is the computational complexity of determining the universal basin of attraction for a given social outcome?
• RQ3 How does the probability of a social rule having a global optimum change when preferences are structured as bundles of interdependent elements?
• RQ4 Can an outcome be an u-local optimum without being a global optimum, and what conditions allow this?
• RQ5 How does the object construction power—via bundling and unbundling—alter the social choice outcome in a way not captured by classical models?

Key findings

• The probability that a social rule with M outcomes has a global optimum in the classical model is M / 2^(M-1), which drops sharply with M, e.g., 0.001709 for M=14.
• The paper constructs a minimal example with 8 outcomes where a global optimum, an u-local optimum (000), and a local optimum (101) coexist, demonstrating the existence of distinct optima types.
• The social outcome 000 is an u-local optimum but not a global optimum, as shown by contradiction using object set constraints and order relations.
• The outcome 101 is a local optimum but not an u-local optimum, since its basin of attraction does not cover all alternatives under any object construction.
• The algorithm to compute the universal basin of attraction runs in O(M³ log M) time, significantly improving on brute-force enumeration.
• In the classical model, a social rule with fewer than eight outcomes can have at most two local optima, and only one global optimum if features are limited.