🔬 physics

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

This paper presents and validates two agent-based models demonstrating that while ideological polarization in papal conclaves tends to prolong the election process, strategic voting behaviors and pre-voting consensus-building can effectively restore efficiency and accelerate the selection of a new pope.

Original authors: Nuno Crokidakis

Published 2026-04-29

📖 5 min read🧠 Deep dive

CC0 1.0

Original authors: Nuno Crokidakis

Original paper dedicated to the public domain under CC0 1.0 (http://creativecommons.org/publicdomain/zero/1.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine a giant, high-stakes game of musical chairs, but instead of chairs, there are candidates for the Pope, and instead of music stopping, the game only ends when someone gets a super-majority of votes (two-thirds of the room). This is what happens in a Papal Conclave.

The paper you provided is a "digital experiment" run by a physicist named Nuno Crokidakis. He built a computer simulation to figure out how a group of 133 cardinals (the voters) decides on a new Pope so quickly or so slowly.

Here is the breakdown of his findings using simple analogies:

1. The Setup: The "Cardinal" Game

The author created a virtual room with 133 "agents" (representing the cardinals) and 10 potential candidates. The rules are strict:

The Goal: Someone needs 67 votes (two-thirds) to win.
The Rounds: They vote, count, and if no one wins, they vote again.
The Twist: Between rounds, the cardinals can change their minds based on three things:
1. Copying: "My friend voted for Candidate X, so I'll vote for X too." (Social Imitation)
2. Picking a Winner: "Candidate Y is getting the most votes, so I'll switch to them to make sure I'm on the winning team." (Strategic Viability)
3. The "Useless" Vote: "My favorite candidate has almost zero votes. I'm wasting my vote, so I'll switch to the leader."

2. The Two Scenarios: The "Neutral" Room vs. The "Faction" Room

The author ran two types of simulations to see what happens under different conditions.

Scenario A: The Neutral Room (No Ideology)
Imagine a room where everyone is just trying to get along.

The Finding: If the cardinals are willing to look at who is winning and switch to them (high "strategic responsiveness"), the game ends very fast. It's like a crowd suddenly realizing, "Oh, everyone is going that way, let's go there!"
The Surprise: The author found that cardinals copying each other (imitation) actually helps speed things up. It's not chaos; it's like a ripple effect that helps the group lock onto a winner quickly.

Scenario B: The Faction Room (With Ideology)
Now, imagine the room is split into two groups: "Progressives" and "Conservatives."

The Finding: This makes the game much harder. The groups want to vote for their own people. If the "Conservative" minority is small (say, 20%), they struggle to win unless they can convince the Progressives to switch sides.
The Polarization: When these groups are stubborn, the voting takes much longer. It's like two teams arguing over which side of the room to sit in. However, if the cardinals are smart enough to realize, "We need a winner, not just a team victory," they can still break the deadlock and elect someone.

3. The "Phase Transition" (The Tipping Point)

The author uses a cool physics concept here. He found that the speed of the election isn't a straight line. It's like a light switch.

If the cardinals are just a little bit willing to switch to the leading candidate, the election might drag on for 20 rounds.
But if they become just a tiny bit more willing to be strategic, the election suddenly snaps to just 4 rounds.
The Analogy: It's like pushing a heavy boulder. At first, it doesn't move. But once you push just hard enough to get it over the edge, it rolls down the hill instantly. Small changes in behavior lead to huge changes in the result.

4. Testing Against History (The "Time Machine")

The author didn't just make up numbers; he tested his model against real history.

He looked at past conclaves from 1939 to 2013. Some took 3 rounds (very fast), others took 11 rounds (very slow).
By adjusting the "stubbornness" of the cardinals in his computer model, he could perfectly recreate how long those real elections took.
The 2025 Prediction: The paper mentions a fictional (or future) 2025 conclave where the new Pope, Leo XIV, was elected in just 4 rounds. The model suggests this happened because the cardinals were very good at spotting a viable winner early on, or perhaps they had already agreed on a plan before the voting even started.

The Big Takeaway

The paper concludes that in any group where people need to agree on a big decision (like a Pope, a CEO, or a policy), how fast they decide depends less on how many people are in the room and more on how willing they are to be strategic.

If people are willing to drop their personal preferences to back a "winning" candidate, the group moves fast. If they are too stubborn or divided by ideology, the group gets stuck. The author shows that a tiny shift in how people behave can turn a long, messy argument into a quick, decisive victory.

1. Problem Statement

The paper addresses the complex dynamics of consensus formation within the Papal Conclave, a unique electoral system characterized by:

A qualified majority requirement (two-thirds supermajority) for election.
Repeated anonymous voting rounds until a winner is found.
The interplay of social influence, strategic behavior (voting for viable candidates), and ideological factionalism (progressive vs. conservative blocs).

The specific motivation is to explain the rapid outcome of the 2025 conclave (elected in 4 rounds) compared to historical variations (ranging from 3 to 11 rounds). The study seeks to determine how small changes in behavioral parameters (strategic responsiveness and social imitation) and ideological composition can lead to large-scale differences in convergence time, analogous to phase transitions in statistical physics.

2. Methodology

The author employs Agent-Based Modeling (ABM) to simulate the interactions of $N$ cardinals (agents) and $C$ candidates. The simulation proceeds in discrete rounds consisting of a voting phase and an opinion update phase.

2.1. Core Mechanisms

The model defines three primary behavioral rules for updating preferences after each round:

Social Imitation: With probability $p$ , a cardinal adopts the voting preference of a randomly chosen peer.
Viability Shift: With probability $q$ , a cardinal switches support to the candidate who received the most votes in the previous round (strategic voting for electability).
Useful Voting: If a cardinal's preferred candidate receives less than 5% of the total votes, they abandon that choice and switch to the leading candidate, regardless of prior preference.

2.2. Model Variants

The study analyzes two distinct models:

Baseline Model (Non-Ideological): Assumes no explicit ideological structure. Agents interact purely based on social imitation and viability perception.
Extended Model (Ideological): Introduces two ideological blocs: Progressives and Conservatives.
- Cardinals are assigned to a bloc (fraction $f$ are conservative).
- Candidates are assigned ideological labels.
- Preference Constraint: Cardinals initially prefer candidates of their own bloc.
- Constraint on Useful Voting: In this version, the "useful voting" rule does not force a switch across ideological lines if the leading candidate is from the opposing bloc (simulating ideological loyalty), though imitation and viability shifts can still cross lines.

2.3. Simulation Parameters

Population ( $N$ ): 133 cardinals (representing the 2025 conclave).
Candidates ( $C$ ): 10.
Parameters:
- $p$ (Imitation probability): Varied (e.g., 0.1, 0.3, 0.5).
- $q$ (Viability responsiveness): Varied (0 to 1).
- $f$ (Fraction of conservatives): Varied (0 to 1).
- $T_{max}$ : Maximum rounds (100).
Validation: The model is calibrated against historical data from conclaves between 1939 and 2025, adjusting $f$ based on the appointing history of previous Popes and tuning $q$ to match observed voting rounds.

3. Key Contributions

Novel Framework for Conclave Dynamics: The paper provides the first agent-based model specifically tailored to the Papal Conclave that integrates social imitation, strategic viability, and ideological polarization simultaneously.
Quantification of Strategic Responsiveness: It isolates the parameter $q$ (willingness to switch to the leading candidate) as the critical driver for convergence speed, demonstrating that small increases in $q$ can drastically reduce the number of voting rounds.
Historical Calibration: The model successfully reproduces the wide historical variance in conclave durations (from 3 rounds in 1939 to 11 in 1958) by inferring ideological compositions and strategic responsiveness levels for each era.
Phase Transition Analogy: The study identifies that the system exhibits behavior analogous to continuous phase transitions, where minor behavioral changes lead to macroscopic shifts in collective outcomes (e.g., from deadlock to rapid consensus).

4. Key Results

4.1. Impact of Strategic Responsiveness ( $q$ )

Non-Ideological Model: There is a sharp, non-linear decline in the number of voting rounds as $q$ increases. Low $q$ leads to indecision and high round counts; high $q$ facilitates rapid coordination around a frontrunner.
Ideological Model: The presence of ideological blocs increases the time to consensus, particularly at low $q$ . However, high strategic responsiveness ( $q$ ) allows the system to overcome polarization, enabling convergence even when a minority bloc initially opposes the majority candidate.

4.2. Role of Social Imitation ( $p$ )

Contrary to models where imitation introduces noise, in this context, higher $p$ accelerates convergence. Imitation reinforces local trends and helps clusters of cardinals align quickly around emerging frontrunners, acting as a coordination mechanism.

4.3. Ideological Polarization and Victory Probability

Asymmetric Transition: The probability of a conservative candidate winning does not scale linearly with the fraction of conservative cardinals ( $f$ $f$ ).
- Progressives dominate when $f < 0.4$ .
- Conservatives only gain a consistent advantage when $f \geq 0.6$ .
- The "crossover" region is narrow, suggesting that a moderate conservative minority struggles to elect a candidate without effective cross-ideological coordination.
Robustness: Despite polarization, the combination of social imitation and viability-based adjustments prevents permanent deadlock in all simulations.

4.4. Historical Alignment (1939–2025)

The model successfully replicated historical voting rounds by adjusting $f$ (ideological composition) and $q$ (strategic responsiveness).
2025 Conclave: The rapid 4-round outcome is best explained by a scenario with high strategic responsiveness ( $q \approx 0.25$ ) and a low conservative fraction ( $f \approx 0.20$ ), suggesting that informal consensus-building mechanisms (pre-conclave discussions) likely reduced ideological rigidity before formal voting began.
1958 Conclave: The long duration (11 rounds) corresponds to a model with very low strategic responsiveness ( $q \approx 0.05$ ) and high ideological rigidity.

5. Significance and Implications

Theoretical Insight: The paper demonstrates that collective decision-making under qualified majority constraints is highly sensitive to behavioral parameters. It bridges social physics and political science, showing how institutional rules interact with agent psychology.
Practical Application: The findings suggest that the speed of the 2025 election was not merely luck but the result of specific behavioral dynamics (high willingness to compromise for electability). It highlights the importance of informal consensus-building in accelerating formal decision-making.
Generalizability: While focused on the Papacy, the framework is applicable to any structured population requiring high-consensus decisions (e.g., corporate boards, legislative bodies, international treaties) where ideological factions and strategic voting coexist.
Methodological Advance: It validates the use of minimalist, rule-based ABMs over complex generative AI approaches for isolating specific mechanisms (like viability perception) in electoral dynamics.

In conclusion, the paper establishes that the efficiency of the Papal Conclave is not static but a dynamic emergent property driven by the tension between ideological loyalty and strategic adaptability. The rapid 2025 outcome serves as empirical evidence that when agents prioritize viability over rigid ideology, consensus is achieved rapidly, mirroring phase transitions in physical systems.