Modularity and Optimality in Social Choice

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).

💡 Research Summary

The paper “Modularity and Optimality in Social Choice” builds on the geometric framework introduced by Marengo and his co‑author, which models social choice not as a selection among independent alternatives but as a process of bundling and unbundling interdependent elements. In this setting each possible social outcome corresponds to a point in a high‑dimensional polytope, and a single bundling operation moves the system from one vertex to another. The authors observe that an authority that controls the bundling sequence can steer the final outcome, a fact that reveals a hidden degree of power in collective decision‑making.

To give this observation a rigorous mathematical backbone, the authors map the bundling dynamics onto tournament theory. A tournament is a complete directed graph where every pair of vertices is connected by a single directed edge. In the authors’ translation, vertices are social outcomes, and a directed edge from outcome A to outcome B indicates that a single bundling operation can transform A into B and that B is preferred (or at least not worse) under the underlying preference structure. This representation allows the use of well‑studied concepts such as strength, centrality, and strongly connected components (SCCs) to analyse stability and manipulability.

The central novel concept introduced is the u‑local optimum (u‑local optimum). An outcome is a u‑local optimum if no single bundling move can improve the social welfare measured by a given utility function u. The authors prove that u‑local optima are in one‑to‑one correspondence with the SCCs of the tournament: each SCC collapses under repeated bundling into a single u‑local optimum, and the set of all states that eventually converge to that optimum constitutes its universal basin of attraction. This result gives a powerful convergence guarantee: regardless of the initial bundle configuration, the process will end in an SCC, i.e., a u‑local optimum.

From an algorithmic standpoint, the paper contributes an O(M³ log M) procedure for computing all universal basins of attraction, where M is the number of social outcomes (vertices). The algorithm proceeds in three phases: (1) it builds the adjacency matrix of the tournament; (2) it computes the transitive closure using a modified Floyd‑Warshall method, which yields reachability information for every pair of vertices; (3) it merges reachable sets using a heap‑based priority queue to keep the merging cost logarithmic. The final step extracts SCCs from the reachability matrix, thereby identifying each basin. Compared with a naïve O(M³) approach, the added log factor is modest but essential for handling realistic sizes (thousands of outcomes).

Empirical evaluation is carried out on two fronts. First, synthetic random tournaments are generated for M ranging from 500 to 5 000. The measured runtimes confirm the predicted O(M³ log M) scaling and show that memory consumption remains manageable. Second, a real‑world survey dataset is encoded as a bundling tournament. The experiments reveal that when the underlying elements are highly interdependent, the tournament collapses into a few large basins, indicating that an authority can exert substantial control. Conversely, low interdependence yields many small basins, suggesting a more dispersed decision landscape. These findings have direct policy implications: the structure of interdependencies determines how easily a planner can manipulate outcomes through bundling.

Beyond pure social choice theory, the authors discuss interdisciplinary applications. In economics, product bundling, portfolio construction, and policy package design can be modeled using the same framework; the u‑local optimum then represents a cost‑effective, stable bundle that cannot be improved by adding or removing a single component. In genetics, gene sets that jointly affect a phenotype can be treated as bundles; evolutionary dynamics that add or delete a gene correspond to the tournament moves, and the u‑local optima correspond to evolutionarily stable genotypic configurations.

The paper concludes with several open research directions: extending the model to incomplete tournaments where some bundling moves are infeasible; incorporating probabilistic transitions to capture stochastic environments; analysing multiple authorities within a game‑theoretic setting; and developing distributed algorithms for massive datasets.

In summary, the work unifies modularity (the decomposition of a complex choice problem into bundles) and optimality (the identification of u‑local optima) through tournament theory and provides both theoretical guarantees and a practical O(M³ log M) algorithm. This synthesis opens new avenues for rigorous analysis of collective decision processes across social sciences, economics, and biology.