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arxiv: 2505.07014 · v3 · submitted 2025-05-11 · ⚛️ physics.soc-ph · cs.SI

When cardinals strategize: An agent-based model of influence and ideology for the papal conclave

Nuno Crokidakis This is my paper

Pith reviewed 2026-05-22 16:47 UTC · model grok-4.3

classification ⚛️ physics.soc-ph cs.SI
keywords agent-based modelpapal conclaveideological polarizationstrategic votingsocial influencevoting dynamicshistorical calibration
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The pith

An agent-based model shows strategic responsiveness among cardinals can offset election delays from ideological polarization.

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

The paper builds two agent-based models of papal conclave voting to track how social imitation, shifts toward current leaders, and strategic switches affect the number of rounds needed to reach a two-thirds majority. Cardinals update votes by copying a random peer or following the most-supported candidate from the last round, and they drop weak candidates for viable alternatives. Adding ideological groups where cardinals start by favoring their own side produces longer elections under polarization. Raising the rate of strategic response to the leading candidate shortens the process even when groups remain divided. The authors adjust model parameters to match the durations of real conclaves from 1939 to 2025.

Core claim

Numerical simulations show that ideological polarization tends to delay the election by increasing the number of voting rounds required. However, higher values of strategic responsiveness q can restore efficiency even under polarization. The model initializes the electorate with 20 percent conservative cardinals and allows cardinals to cross ideological lines for strategic reasons. Calibration to historical data from conclaves held between 1939 and 2025 reproduces observed convergence times with good agreement.

What carries the argument

The agent-based update rule in which each cardinal imitates a random peer with probability p or adopts the most-voted candidate from the previous round with probability q, combined with a useful-voting threshold that redirects support away from low-viability candidates and an ideological assignment that initially biases votes toward same-group candidates.

If this is right

  • Ideological polarization among cardinals increases the rounds required to reach a qualified majority.
  • Raising the strategic responsiveness parameter q shortens convergence time even when cardinals are split into opposing groups.
  • The model reproduces historical convergence times when parameters are calibrated to conclave data from 1939 to 2025.
  • Rapid elections such as the 2025 conclave point to informal consensus mechanisms operating before formal voting begins.

Where Pith is reading between the lines

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

  • The same update rules could be applied to other repeated voting settings where participants hold group identities and observe running tallies, such as legislative or corporate elections.
  • Small increases in attention paid to current leaders might measurably reduce gridlock in any multi-round voting system with visible tallies.
  • Controlled experiments with human participants repeating the voting protocol could test whether the model's predicted effects on round counts hold outside the specific conclave rules.

Load-bearing premise

Cardinals begin by supporting only candidates from their own ideological group but will switch to a more viable alternative when their preferred candidate falls below a support threshold.

What would settle it

If simulations with high polarization and low strategic responsiveness q produce average round counts that exceed the actual ballot numbers recorded in historically polarized conclaves, the predicted delay effect would be contradicted.

Figures

Figures reproduced from arXiv: 2505.07014 by Nuno Crokidakis.

Figure 1
Figure 1. Figure 1: Typical output of the numerical simulations, illustrating th [PITH_FULL_IMAGE:figures/full_fig_p007_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Average number of voting rounds until a candidate is elect [PITH_FULL_IMAGE:figures/full_fig_p008_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: Average number of voting rounds in the ideological model, a [PITH_FULL_IMAGE:figures/full_fig_p009_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: Victory probabilities of conservative and progressive can [PITH_FULL_IMAGE:figures/full_fig_p010_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: Comparison between historical data for papal conclaves [PITH_FULL_IMAGE:figures/full_fig_p013_5.png] view at source ↗
read the original abstract

We propose and analyze two agent-based models to investigate the dynamics of papal conclaves, focusing on how social influence, strategic voting, and ideological alignment affect the time required to elect a pope. In the first model, cardinals interact through two mechanisms: with probability $p$, they imitate the choice of a randomly selected peer, and with probability $q$, they shift support to the most voted candidate from the previous round. Additionally, strategic behavior is introduced via ``useful voting'', where agents abandon their preferred candidate if he receives less than a threshold fraction of the votes, switching instead to the most viable alternative. A candidate must secure a qualified majority of two-thirds to be elected. We then extend the framework by incorporating ideological blocs, assigning each cardinal and candidate to one of two groups (e.g., progressives and conservatives). Cardinals initially vote for candidates from their own group but may cross ideological lines for strategic reasons. We initialize the electorate with $20\%$ conservative cardinals, reflecting the current composition shaped by papal appointments. Numerical simulations show that ideological polarization tends to delay the election by increasing the number of voting rounds required. However, higher values of strategic responsiveness $q$ can restore efficiency even under polarization. We further validate the model by calibrating parameters to historical data from conclaves held between 1939 and 2025. The model reproduces observed convergence times with good agreement, supporting its explanatory power across institutional contexts. The rapid outcome of the 2025 conclave, despite ideological divisions, suggests the importance of informal consensus-building, possibly prior to voting, as a key mechanism for accelerating convergence.

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

2 major / 2 minor

Summary. The manuscript proposes two agent-based models of papal conclave dynamics. The base model includes imitation (probability p) and strategic responsiveness (probability q) plus a useful-voting rule in which agents abandon candidates below a threshold vote fraction. The extended model adds ideological blocs (progressive/conservative), with agents initialized to support their own bloc (20 % conservative fraction) but permitted to cross lines strategically. Simulations are reported to show that polarization lengthens convergence to the required two-thirds majority while higher q mitigates the delay. The authors calibrate the four free parameters (p, q, threshold, conservative fraction) to historical conclave data 1939–2025 and state that the model reproduces observed convergence times with good agreement, thereby supporting its explanatory power.

Significance. If the reported effects of polarization and strategic responsiveness survive scrutiny of the calibration procedure, the work supplies a concrete computational framework for studying how ideology and strategic voting interact under qualified-majority rules in small electorates. The explicit incorporation of both social-influence and ideological mechanisms is a strength, as is the attempt to anchor the model in real historical outcomes. However, the small number of calibration events and the absence of out-of-sample testing limit the strength of the explanatory claim.

major comments (2)
  1. [Abstract / Validation] Abstract and validation section: the central claim that the model possesses explanatory power rests on reproducing historical convergence times after calibrating p, q, threshold fraction and conservative fraction to the same 1939–2025 data set. With only ~10–12 scalar observations and a stochastic model, this procedure risks overfitting; the manuscript provides no description of the fitting algorithm, loss function, number of Monte Carlo realizations per parameter set, or any uncertainty quantification on the reproduced times.
  2. [Model extension] Ideological initialization paragraph: the 20 % conservative fraction is presented as reflecting current composition, yet the polarization-delay result is shown to depend on this choice. No sensitivity analysis with respect to the conservative fraction or to the initial distribution of candidates across blocs is reported, leaving open whether the reported q-mitigation effect is robust or an artifact of the chosen initialization.
minor comments (2)
  1. [Abstract] The abstract states that simulations 'reproduce observed convergence times with good agreement' but supplies no quantitative metric (e.g., mean absolute deviation, Kolmogorov–Smirnov statistic) or comparison against a null model without ideology or without the q mechanism.
  2. [Numerical results] Figure captions and methods text should explicitly state the number of independent runs used to generate each reported mean or distribution of voting rounds.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the constructive and detailed report. We address each major comment below and outline the revisions we will make to strengthen the manuscript.

read point-by-point responses

  1. Referee: [Abstract / Validation] Abstract and validation section: the central claim that the model possesses explanatory power rests on reproducing historical convergence times after calibrating p, q, threshold fraction and conservative fraction to the same 1939–2025 data set. With only ~10–12 scalar observations and a stochastic model, this procedure risks overfitting; the manuscript provides no description of the fitting algorithm, loss function, number of Monte Carlo realizations per parameter set, or any uncertainty quantification on the reproduced times.

    Authors: We acknowledge the referee's concern about the calibration procedure and the risk of overfitting given the limited number of historical observations. The calibration in the original manuscript was performed by exploring parameter combinations to achieve qualitative and quantitative agreement with observed convergence times, but we agree that the manuscript lacks sufficient methodological detail. In the revised version, we will add a dedicated subsection describing the calibration approach (a grid search over bounded ranges for each parameter), the loss function (mean squared error between simulated and historical mean convergence times), the number of Monte Carlo realizations per parameter set (100), and uncertainty quantification via standard errors across runs. These additions will allow readers to evaluate the robustness of the fit directly. revision: yes

  2. Referee: [Model extension] Ideological initialization paragraph: the 20 % conservative fraction is presented as reflecting current composition, yet the polarization-delay result is shown to depend on this choice. No sensitivity analysis with respect to the conservative fraction or to the initial distribution of candidates across blocs is reported, leaving open whether the reported q-mitigation effect is robust or an artifact of the chosen initialization.

    Authors: The 20% conservative fraction is chosen to approximate the current ideological composition of the College of Cardinals. We agree that the lack of sensitivity analysis limits the ability to assess robustness of the polarization-delay and q-mitigation results. In the revised manuscript, we will include a new set of simulations that systematically vary the conservative fraction (10%–40%) and the initial distribution of candidates across ideological blocs. We will report how these variations affect both the delay induced by polarization and the extent to which higher q reduces convergence time, thereby demonstrating that the main qualitative findings are not artifacts of the specific initialization. revision: yes

Circularity Check

1 steps flagged

Validation of convergence times achieved by calibrating parameters to the same historical data

specific steps
  1. fitted input called prediction [Abstract]
    "We further validate the model by calibrating parameters to historical data from conclaves held between 1939 and 2025. The model reproduces observed convergence times with good agreement, supporting its explanatory power across institutional contexts."

    Parameters p, q, the useful-voting threshold and imitation probability are adjusted to the 1939–2025 data set; the subsequent claim that the model reproduces the observed number of rounds is therefore the direct output of that calibration rather than a prediction generated from the ideological and strategic mechanisms alone.

full rationale

The paper's core simulation results on polarization and strategic responsiveness q are generated from the agent-based rules and initial conditions. The load-bearing validation step, however, consists of tuning parameters to match the 1939–2025 convergence times and then reporting agreement with those same times. This reduces the claimed explanatory power to a fitted reproduction rather than an independent test. No other circular patterns (self-definitional equations, self-citation load-bearing, or imported uniqueness theorems) appear in the provided text.

Axiom & Free-Parameter Ledger

4 free parameters · 2 axioms · 0 invented entities

The central claim rests on several free parameters (p, q, vote threshold) whose values are adjusted during calibration to historical data, plus the domain assumption of two-thirds majority and the ad-hoc initialization of 20% conservatives. No new physical entities are postulated.

free parameters (4)
  • p
    Probability that a cardinal imitates a randomly selected peer; central to imitation dynamics and likely tuned in calibration.
  • q
    Probability of shifting support to the most voted candidate from the previous round; directly affects strategic responsiveness and efficiency under polarization.
  • threshold fraction
    Vote share below which agents abandon their preferred candidate for a viable alternative; governs useful voting behavior.
  • conservative fraction
    Initial 20% assignment of conservative cardinals; chosen to reflect current composition and load-bearing for polarization results.
axioms (2)
  • domain assumption A candidate must secure a qualified majority of two-thirds to be elected.
    Standard institutional rule invoked to define election success.
  • domain assumption Cardinals interact through imitation with probability p and strategic shift with probability q, plus useful voting below threshold.
    Core behavioral rules of the agent-based model stated in the abstract.

pith-pipeline@v0.9.0 · 5826 in / 1713 out tokens · 143555 ms · 2026-05-22T16:47:34.804281+00:00 · methodology

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Reference graph

Works this paper leans on

17 extracted references · 17 canonical work pages

  1. [1]

    Pope Leo XIV conclave by the numbers: From number of cardinals to length of voting, https : //abcnews.go.com/International/leo− xiv − conclave − numbers − number − cardinals − length − voting/story ?id = 121606316

    work page

  2. [2]

    Edward Berger, screenplay by Peter Straughan, based on the novel by Robert Harris, Amazon MGM Studios / Prime Video, 2024

    Conclave, dir. Edward Berger, screenplay by Peter Straughan, based on the novel by Robert Harris, Amazon MGM Studios / Prime Video, 2024

    work page 2024

  3. [3]

    Ribeiro-Rodrigues, A

    E. Ribeiro-Rodrigues, A. P. Bortoleto, A systematic review of agent-based modeling and simulation applications for anal yzing pro- environmental behaviors, Sustainable Production and Consumption 47, 343-362 (2024)

    work page 2024

  4. [4]

    Castellano, S

    C. Castellano, S. Fortunato, V. Loreto, Statistical physics of social dy- namics, Rev. Mod. Phys. 81, 591-646 (2009)

    work page 2009

  5. [5]

    Vazquez, S

    F. Vazquez, S. Redner, Ultimate fate of constrained voters , Journal of Physics A: Mathematical and General 37, 8479 (2004)

    work page 2004

  6. [6]

    Jusup et

    M. Jusup et. al. , Social Physics , Physics Reports 948, 1-148 (2022)

    work page 2022

  7. [7]

    Perc, The social physics collective , Scientif Reports 9, 16549 (2019)

    M. Perc, The social physics collective , Scientif Reports 9, 16549 (2019). 15

    work page 2019

  8. [8]

    Bianconi et

    G. Bianconi et. al., Complex systems in the spotlight: next steps after the 2021 Nobel Prize in Physics , Journal of Physics: Complexity 4, 010201 (2024)

    work page 2021

  9. [9]

    Doghmi, D

    A. Doghmi, D. Lepelley, F. Valognes, An axiomatic analysis of the papal conclave, Economic Theory 70 755-781 (2020)

    work page 2020

  10. [10]

    M. Gao, Z. Wang, K. Wang, C. Liu, S. Tang, Forecasting elections with agent-based modeling: Two live experiments , PLOS ONE 17 (6), e0270194 (2022)

    work page 2022

  11. [11]

    Atkinson, S

    N. Atkinson, S. Ganz, J. Mantus, A simple agent-based model f or simulating single winner elections, Georgetown McDonough School of Business Research Paper No. 4911226 (2024), available at https : //ssrn.com/abstract= 4911226

    work page 2024

  12. [12]

    Zhu, ConclaveSim: Simulating the 2025 Papal Election with LLM Agents, GitHub repository (2025)

    M. Zhu, ConclaveSim: Simulating the 2025 Papal Election with LLM Agents, GitHub repository (2025). Available at: https : //github.com/michaelbzhu/conclave − sim

    work page 2025

  13. [13]

    Antonioni, M

    A. Antonioni, M. Re Fiorentin, E. Valdano, Complex totopapa: pre- dicting the successor to Pope Francis , arXiv preprint arXiv:2505.01553 (2025)

    work page arXiv 2025

  14. [14]

    https : //www.bbc.com/portuguese/articles/c3r8r2lld5eo

    work page

  15. [15]

    F. J. Baumgartner, Behind Locked Doors: A History of the Papal Elec- tions, Palgrave Macmillan (2003), ISBN 0 − 312 − 29463 − 8

    work page 2003

  16. [16]

    Pentin, The Next Pope: The Leading Cardinal Candidates (Sophia Institute Press, 2020)

    E. Pentin, The Next Pope: The Leading Cardinal Candidates (Sophia Institute Press, 2020)

    work page 2020

  17. [17]

    Galam, Sociophysics: A review of Galam models , International Jour- nal of Modern Physics C 19, 409-440 (2008)

    S. Galam, Sociophysics: A review of Galam models , International Jour- nal of Modern Physics C 19, 409-440 (2008). 16

    work page 2008