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

TL;DR

Problem: understanding how social influence, strategic voting, and ideological polarization shape the speed and outcome of papal conclaves under the two-thirds rule ($2/3$). Approach: two agent-based models—one without explicit ideology and one with two ideological blocs—featuring imitation ($p$), perceived viability updates ($q$), and useful voting, calibrated to historical conclave data from 1939 to 2025. Key findings: ideological polarization tends to lengthen convergence, but higher $q$ can restore efficiency; higher $p$ can also speed convergence; the rapid outcome in 2025 supports the idea that informal pre-conclave consensus-building accelerates convergence. Significance: the framework captures cross-decade variation in conclave durations and is adaptable to other bounded, ideologically factional elections.

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.

Figures (5)

• Figure 1: Typical output of the numerical code.
• Figure 2: Average number of voting rounds until a candidate is elected, as a function of the probability $q$ of following the most voted candidate from the previous round. Results are shown for different values of the imitation probability $p$, with fixed $N=133$ cardinals and $C=10$ candidates. Each point represents the average over $1000$ independent simulations. The sharp decrease in the number of rounds for increasing $q$ indicates a coordination transition, where strategic responsiveness improves convergence efficiency.
• Figure 3: Average number of voting rounds in the ideological model, as a function of the probability $q$ of following the most voted candidate. The simulation assumes $N=133$ cardinals and $C=10$ candidates, with $20\%$ of the cardinals initialized as conservatives and $80\%$ as progressives. Results are shown for three values of the imitation probability $p$. Each point represents the average over $1000$ independent simulations. Compared to the non-ideological model, the presence of ideological blocs increases the number of rounds and shifts the coordination threshold to higher values of $q$. Despite the increased number of rounds in polarized scenarios, all simulations converged to a successful election within $T_{max}$.
• Figure 4: Victory probabilities of conservative and progressive candidates as a function of the fraction $f$ of conservative cardinals. Results are based on $1000$ independent simulations per data point, using the ideological model with $N=133$ cardinals and $C = 10$ candidates. Each candidate is randomly assigned an ideological label. The plot reveals a smooth transition around $f \approx 0.5$, with progressives dominating for low values of $f$, and conservatives gaining advantage as their representation increases.
• Figure 5: Comparison with historical data of papal conclaves, obtained from baumgartnerpentin, and results from model simulations. We fixed $p=0.10$ and used inferred values of the conservative fraction $f$, as discussed in the text (see Tab. \ref{['tab:fractions']}). The values of the strategic responsiveness parameter $q$ that best fit the historical data are: $q=0.40 (1939), q=0.05 (1958), q=0.15\, (1963), q=0.25\, (1978a), q=0.10\, (1978b), q=0.25\, (2005), q=0.20\, (2013)$ and $q=0.25\, (2025)$. Each data point corresponds to the average over $1000$ independent simulation runs.