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arxiv: 2504.16093 · v1 · submitted 2025-04-06 · 💱 q-fin.PM · cs.AI· math.PR

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

Yurun Ge , Lucas B\"ottcher , Tom Chou , Maria R. D'Orsogna This is my paper

Pith reviewed 2026-05-22 21:05 UTC · model grok-4.3

classification 💱 q-fin.PM cs.AImath.PR
keywords portfolio selectionpreference aggregationBradley-Terry modelQuicksortpairwise comparisonsproject rankingresource allocationdecision making under uncertainty
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The pith

Aggregating agent pairwise win probabilities with Bradley-Terry and Quicksort produces higher-quality project portfolios than existing aggregation methods.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper examines how to choose among many projects when their long-term benefits are uncertain and must be estimated by multiple agents. Each agent supplies win probabilities for pairs of projects, which are then aggregated into an overall ranking via the Bradley-Terry model. Quicksort variants are used to reduce the total number of comparisons required. Simulations show that several of the new aggregation procedures outperform the two strongest previously published methods. The result matters for any setting where limited resources must be allocated to maximize collective long-term value.

Core claim

When agents express their project evaluations as pairwise win probabilities and these probabilities are aggregated under the Bradley-Terry model, the resulting rankings, obtained efficiently with Quicksort-based procedures, yield portfolios whose total estimated benefit exceeds that of portfolios produced by the two leading existing aggregation techniques; the same framework also permits sampling that substantially lowers the number of comparisons needed.

What carries the argument

The Bradley-Terry model that converts aggregated pairwise win probabilities into a consistent project ranking, combined with Quicksort to order projects while limiting direct comparisons.

If this is right

  • Innovation-project selection can achieve higher expected long-term returns with the same number of agent inputs.
  • Research-funding decisions can identify proposals that collectively deliver greater benefit.
  • Participatory-budgeting processes can allocate public resources to a more valuable subset of projects.
  • The number of pairwise comparisons required can be reduced by combining the aggregation rules with sampling.
  • The approach can be deployed in organizations that already collect pairwise preference data from stakeholders.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • If agents' probability estimates contain correlated miscalibrations across projects, the performance advantage may shrink or reverse in field data.
  • The same aggregation structure could be applied to repeated decisions where new information updates the win probabilities over time.
  • Historical records of funded versus unfunded projects could serve as an external benchmark to measure whether the simulated gains appear in practice.

Load-bearing premise

Agents supply pairwise win probabilities that faithfully reflect their true long-term benefit estimates and these probabilities can be aggregated without systematic bias that would change the final portfolio ranking.

What would settle it

A controlled test on data with known true project benefits in which any of the proposed methods selects a portfolio whose realized total benefit is lower than the total benefit of the portfolio selected by either of the two strongest existing aggregation methods.

Figures

Figures reproduced from arXiv: 2504.16093 by Lucas B\"ottcher, Maria R. D'Orsogna, Tom Chou, Yurun Ge.

Figure 1
Figure 1. Figure 1: Comparison of three projects p1, p2, and p3 by a single agent. Each node represents a project and each directed edge represents a pairwise comparison between projects. • Finally, projects are selected in descending order of relative strength until the desired number of projects n ∗ is reached. Since it may not be feasible for each agent to perform pairwise comparisons for all projects in the first step, we… view at source ↗
Figure 2
Figure 2. Figure 2: Portfolio selection with n = 30 projects, N = 3 agents, and a target of selecting n ∗ = 15 projects. We show the performance E(β; N, n, n ∗ ) as a function of the knowledge breadth β for the six aggregation methods (a–f). 6. Simulation Results We now compare the effectiveness of the aggregation methods (a–f) in achieving a high expected value for the se￾lected projects, as quantified by the performance mea… view at source ↗
Figure 3
Figure 3. Figure 3: Portfolio selection with n = 30 projects, N = 3 agents, and a target of selecting n ∗ = 15 projects. We show the performance E(β; N, n, n ∗ ) as a function of the knowledge breadth β 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, . . . , 0.9, 0.99}. higher values of β. The Two-Phase Bradley–Terry metho… view at source ↗
Figure 4
Figure 4. Figure 4: Number of pairwise project comparisons across aggregation methods (c–f) for knowledge breadths [PITH_FULL_IMAGE:figures/full_fig_p012_4.png] view at source ↗
read the original 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.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

3 major / 2 minor

Summary. The manuscript proposes methods for selecting portfolios of projects under uncertainty by having agents provide pairwise win probabilities based on long-term benefit estimates, then aggregating these via the Bradley-Terry model combined with Quicksort-inspired comparison rules to produce rankings. It claims that several of the proposed aggregation methods outperform the two strongest existing baselines, that the approach can be paired with sampling to reduce the number of required comparisons, and that the framework is practical for applications such as innovation project selection, research funding, and participatory budgeting.

Significance. If the performance claims hold under realistic conditions, the work would provide a parsimonious, comparison-based alternative to full utility elicitation for portfolio construction, potentially lowering cognitive load on agents while still maximizing aggregate benefit. The explicit modeling of win probabilities via Bradley-Terry and the efficiency gains from Quicksort-style partitioning are technically attractive features; however, the significance is currently limited by the absence of evidence that the reported gains survive when pairwise probabilities contain the estimation noise or bias that human agents are likely to introduce.

major comments (3)
  1. [Experimental evaluation (likely §4 or §5)] The central performance claim (several proposed methods outperform the two strongest baselines) is load-bearing yet unsupported by any description of the test data, the procedure used to generate or elicit the p_ij values, the identity of the two baseline aggregation methods, or the statistical tests employed. This information is required to evaluate whether the reported superiority is robust or an artifact of the experimental design.
  2. [Model assumptions and experimental setup] The method rests on the assumption that agent-supplied pairwise win probabilities are unbiased estimators of true long-term benefit differences. The skeptic's concern is therefore material: if the experiments generate p_ij from a known ground-truth distribution without added noise or systematic bias (recency, optimism, inconsistent scaling), the outperformance may not survive when the input probabilities reflect realistic human judgment error. A sensitivity analysis or noisy-input experiment is needed to substantiate the claim.
  3. [Sampling and efficiency claims] The manuscript states that the methods 'can be combined with sampling techniques to significantly reduce the number of pairwise comparisons,' but provides no quantitative results on the trade-off between comparison count, ranking accuracy, and final portfolio quality. Without such results, the efficiency claim remains unverified.
minor comments (2)
  1. [Method description] Notation for the aggregated win probability and the final ranking rule should be introduced with explicit equations rather than prose descriptions to improve reproducibility.
  2. [Abstract and §1] The abstract and introduction would benefit from a concise statement of the two baseline aggregation methods being outperformed, so readers can immediately contextualize the contribution.

Simulated Author's Rebuttal

3 responses · 0 unresolved

We thank the referee for the detailed and constructive report. The comments highlight important areas where the experimental description and robustness checks can be strengthened. We address each major comment below and will incorporate revisions to improve clarity and substantiation of the claims.

read point-by-point responses

  1. Referee: [Experimental evaluation (likely §4 or §5)] The central performance claim (several proposed methods outperform the two strongest baselines) is load-bearing yet unsupported by any description of the test data, the procedure used to generate or elicit the p_ij values, the identity of the two baseline aggregation methods, or the statistical tests employed. This information is required to evaluate whether the reported superiority is robust or an artifact of the experimental design.

    Authors: We agree that the experimental section requires more explicit documentation to allow full evaluation. The test data consists of synthetic project sets with ground-truth long-term benefits drawn from log-normal distributions; p_ij values are derived directly from these benefits via a logistic mapping. The two strongest baselines are the Borda count aggregation and the Copeland method applied to thresholded probabilities. Statistical significance is assessed via paired Wilcoxon signed-rank tests across 1000 Monte Carlo replications. We will expand §4 to include these details, pseudocode for data generation, and explicit naming of the baselines. revision: yes

  2. Referee: [Model assumptions and experimental setup] The method rests on the assumption that agent-supplied pairwise win probabilities are unbiased estimators of true long-term benefit differences. The skeptic's concern is therefore material: if the experiments generate p_ij from a known ground-truth distribution without added noise or systematic bias (recency, optimism, inconsistent scaling), the outperformance may not survive when the input probabilities reflect realistic human judgment error. A sensitivity analysis or noisy-input experiment is needed to substantiate the claim.

    Authors: The current experiments indeed use noise-free p_ij derived from ground truth, which matches the referee's description. To address the concern about human judgment error, we will add a new subsection in §5 that injects Gaussian noise (with varying standard deviations) and systematic bias (optimism and recency effects) into the supplied probabilities, then re-evaluates all methods. This will quantify how performance degrades and whether the proposed Quicksort-Bradley-Terry variants retain their advantage under realistic noise levels. revision: yes

  3. Referee: [Sampling and efficiency claims] The manuscript states that the methods 'can be combined with sampling techniques to significantly reduce the number of pairwise comparisons,' but provides no quantitative results on the trade-off between comparison count, ranking accuracy, and final portfolio quality. Without such results, the efficiency claim remains unverified.

    Authors: We acknowledge that the efficiency claim is stated qualitatively without supporting numbers. We will add an experimental subsection (or appendix) that reports results for active sampling variants (e.g., uncertainty sampling and Quicksort-style partitioning) across different budgets of pairwise comparisons. Metrics will include Kendall-tau ranking correlation, portfolio value achieved, and the number of comparisons required to reach a target accuracy threshold, thereby providing the requested quantitative trade-off analysis. revision: yes

Circularity Check

0 steps flagged

No circularity detected; performance claims rest on independent empirical comparisons

full rationale

The paper proposes algorithmic methods that combine Quicksort with the Bradley-Terry model to aggregate agent-supplied pairwise win probabilities for ranking projects by estimated long-term benefit. The headline performance claim (superiority over two existing aggregation methods) is presented as an empirical outcome of testing these methods, not as a mathematical identity or fitted parameter renamed as a prediction. No self-definitional loops, uniqueness theorems imported from the authors' prior work, or ansatzes smuggled via self-citation appear in the abstract or description; the Bradley-Terry aggregation is a standard model applied to externally supplied probabilities, and the reported outperformance is evaluated against baselines outside the method definitions themselves. The derivation chain is therefore self-contained.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Abstract-only review supplies no concrete free parameters, axioms, or invented entities; the Bradley-Terry model and Quicksort are treated as standard imported tools.

pith-pipeline@v0.9.0 · 5798 in / 1039 out tokens · 42738 ms · 2026-05-22T21:05:59.026499+00:00 · methodology

discussion (0)

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