Efficient Portfolio Selection through Preference Aggregation with Quicksort and the Bradley--Terry Model

TL;DR

The paper tackles portfolio selection under uncertainty when many projects must be evaluated by multiple agents. It proposes a framework that uses the Bradley–Terry model to derive pairwise win probabilities and integrates this with Quicksort and Newman's iteration to obtain robust project rankings. To reduce cognitive load, it introduces sampling schemes such as cyclic graph sampling and two-phase variants, achieving near-linear comparison complexity in practice. Across Monte Carlo experiments, methods based on win probabilities—particularly Quicksort and full Bradley–Terry—consistently outperform value- or score-based baselines, with larger gains as agent knowledge breadth increases, offering scalable tools for participatory budgeting and research-portfolio decisions.

Abstract

How to allocate limited resources to projects that will yield the greatest long-term benefits is a problem that often arises in decision-making under uncertainty. For example, organizations may need to evaluate and select innovation projects with risky returns. Similarly, when allocating resources to research projects, funding agencies are tasked with identifying the most promising proposals based on idiosyncratic criteria. Finally, in participatory budgeting, a local community may need to select a subset of public projects to fund. Regardless of context, agents must estimate the uncertain values of a potentially large number of projects. Developing parsimonious methods to compare these projects, and aggregating agent evaluations so that the overall benefit is maximized, are critical in assembling the best project portfolio. Unlike in standard sorting algorithms, evaluating projects on the basis of uncertain long-term benefits introduces additional complexities. We propose comparison rules based on Quicksort and the Bradley--Terry model, which connects rankings to pairwise "win" probabilities. In our model, each agent determines win probabilities of a pair of projects based on his or her specific evaluation of the projects' long-term benefit. The win probabilities are then appropriately aggregated and used to rank projects. Several of the methods we propose perform better than the two most effective aggregation methods currently available. Additionally, our methods can be combined with sampling techniques to significantly reduce the number of pairwise comparisons. We also discuss how the Bradley--Terry portfolio selection approach can be implemented in practice.

Figures (4)

• Figure 1: Comparison of three projects $p_1$, $p_2$, and $p_3$ by a single agent. Each node represents a project and each directed edge represents a pairwise comparison between projects.
• Figure 2: Portfolio selection with $n=30$ projects, $N=3$ agents, and a target of selecting $n^*=15$ projects. We show the performance $E(\beta; N, n, n^*)$ as a function of the knowledge breadth $\beta$ for the six aggregation methods (a--f).
• Figure 3: Portfolio selection with $n=30$ projects, $N=3$ agents, and a target of selecting $n^*=15$ projects. We show the performance $E(\beta; N, n, n^*)$ as a function of the knowledge breadth $\beta$ for the six aggregation methods (a--f). In all approaches that are based on win probabilities, agents select one of the following values $\{0.01, 0.1, 0.2, \ldots, 0.9, 0.99\}$.
• Figure 4: Number of pairwise project comparisons across aggregation methods (c--f) for knowledge breadths $\beta \in \{0, 5, 10\}$ used to determine the performance $E(\beta)$ in Figure \ref{['fig:bradley_terry_discrete probability']}.