Efficient portfolio selection through preference aggregation with Quicksort and the Bradley–Terry model

Allocating limited resources to a set of alternatives with uncertain long-term benefits is a common challenge in innovation management, research funding, and participatory budgeting. Related problems arise in emerging applications such as ranking outputs of large language models and coordinating decisions in agentic systems. All settings include multiple agents tasked with estimating the true value of a potentially large number of alternatives. These estimates, or quantities derived from them, are then aggregated to select a final portfolio that maximizes overall benefit, ideally using efficient methods. Standard sorting algorithms are ill-suited as they do not account for uncertainties associated with each agent’s estimate. Furthermore, the cognitive load on agents can be demanding, especially if the number of alternatives to evaluate is large. Building on the Quicksort algorithm and the Bradley–Terry model, we develop four new, efficient aggregation protocols based on agent-assigned win probabilities of pairwise comparisons that are then globally aggregated. The pairwise comparisons we introduce not only reduce cognitive load on agents, but lead to aggregation protocols that outperform existing ones, which we confirm via numerical simulations. Our methods can be combined with sampling strategies to further reduce the number of pairwise comparisons.