PIATRA . INSTITUTE

polity coalition attractors

basins of inclusion versus exclusion under stress, norms, and contact

Attractor basins (x₀ vs t₀)

Color shows which attractor dominates from each starting point.

x₀=0.35, t₀=0.55

Inclusive

ExclusionaryClick to set initial conditions.

Trajectory (time series)

x = exclusionary share, t = trust, threat = perceived threat

Phase portrait (x vs t)

Trajectory through state space; attractors appear as endpoints.

Snapshot (final step)

x (exclusionary)1.00

t (trust)0.00

perceived threat0.62

π(E)1.35

π(I)0.12

Δ = π(E)−π(I)1.23

AttractorExclusionary

When Δ is positive, exclusionary support tends to grow (scaled by x(1−x)). Trust can dampen or amplify that growth depending on parameter settings.

Coalition Dynamics Model

Two state variables evolve over time: $x$ (the share of the population supporting an exclusionary coalition) and $t$ (institutional trust / civic confidence). Both are bounded to [0, 1].

\frac{dx}{dt} = \Delta t \cdot x(1-x)(\pi_E - \pi_I) + \Delta t \cdot \text{noise}

\frac{dt}{dt} = \Delta t \cdot (0.55R + 0.45C + 0.25N - 0.55P - 0.65x - 0.45S - 0.2\theta)

The $x(1-x)$ factor ensures replicator dynamics: change is fastest at intermediate shares and stalls near the boundaries. The payoff differential $\pi_E - \pi_I$ determines whether exclusionary or inclusive support grows.

Attractor Classes

Inclusive ( $x < 0.2$ ): exclusionary support is marginal; trust-building feedbacks dominate.
Mixed ( $0.2 \leq x \leq 0.8$ ): neither coalition dominates; system is in a contested or transitional zone.
Exclusionary ( $x > 0.8$ ): exclusionary politics dominate; trust erodes in a self-reinforcing cycle.

Parameters

S (Stress): economic/security shocks that raise threat salience.
D (Diversity): salience of group boundaries in this toy model.
P (Polarization): fragmented information space; amplifies perceived threat.
N (Norms): rule-of-law / rights constraints that raise the cost of exclusion.
C (Contact): bridging social capital; reduces perceived threat.
R (Redistribution): material inclusion; increases trust and inclusive payoff.
O (Opportunism): elite identity entrepreneurship; strengthens exclusionary narrative feedback.

Notes

This is a toy model. Use it to reason about feedback loops and basins, not to estimate real-world quantities.
Presets are illustrative, not empirically calibrated. For data-grounded presets, map real indicators into S/D/P/N/C/R/O.
The basin map can be computationally expensive at high grid resolutions. Lower the grid setting if interaction feels slow.

Model Version

claude opus 4.8February 2026·first cut. a two-field dynamical model of coalition formation: an exclusionary support share x and institutional trust t evolve under replicator dynamics with a perceived-threat feedback driven by seven parameters (stress, diversity, polarization, norms, contact, redistribution, elite opportunism). includes an interactive basin map, trajectory and phase-portrait views, a library of historical parameter presets, calibration of the deterministic core (fixed points, basin membership, threat saturation), and six assumptions that keep the mechanism apart from the illustrative presets.

Calibration

Assumptions

Model Changelog

v1.0February 2026

state: two bounded fields, exclusionary share x and institutional trust t, both on [0, 1].
dynamics: replicator update dx = dt * x(1-x)(piE - piI) plus a slow trust update; x = 0 and x = 1 are fixed points by construction.
feedback: perceived threat as a clamped algebraic function of state and the seven parameters, with elite opportunism amplifying the existing exclusionary share.
attractor classes: inclusive (x < 0.2), mixed (0.2 to 0.8), and exclusionary (x > 0.8), with a clickable basin map over initial conditions.
presets: a library of historical regimes spanning ancient, nineteenth-century, and modern cases, set by hand to position each in parameter space.
calibration: the deterministic, noise-free core is checked against boundary fixed points, inclusive and exclusionary basin membership, and threat saturation.
framing kept honest: presets are illustrative parameter guesses, not data fits, and the calibration verifies only structural claims.

read the research companion →