The Complete Model

This model draws on three distinct mathematical frameworks that operate at different levels of description. (a) The continuous SDE/Langevin formulation (dL = f(L)dt + σdW) treats the liberty score as a continuous stochastic process driven by a deterministic drift toward basin equilibria plus Wiener noise. This provides analytical tractability: closed-form expressions for mean first-passage times, basin escape probabilities, and the stationary density via the Fokker-Planck equation. (b) The discrete Markov chain transition matrix (the 7×7 matrix above) bins countries into stages and estimates transition probabilities from historical panel data, capturing how often countries actually move between stages over discrete time intervals. (c) The Gaussian Mixture Model (GMM), implicit in the trimodal distribution of liberty scores, identifies the cross-sectional equilibrium structure — where the basins are at any given snapshot in time. In short: the GMM identifies WHERE the basins are; the Markov chain quantifies HOW often countries move between them; the SDE provides an analytical framework for deriving properties like escape times and basin stability.

These three frameworks are complementary but rest on different assumptions, and that tension should be stated explicitly. Strictly, the SDE and Markov models make different assumptions about the data-generating process. The SDE assumes continuous paths; the Markov model assumes discrete jumps. With irregular time spacing (5–20 year gaps pre-1972), neither is exact. We treat both as useful approximations and note where they diverge. The continuous SDE is most reliable for qualitative insights (basin depth comparisons, volatility ridgeline location) and for generating Monte Carlo trajectories; the discrete Markov chain is most reliable for empirical transition probabilities estimated directly from the data. Where the two frameworks agree — for instance, both predict that the tyranny well is the deepest attractor and that recovery from the hybrid trap is rare — we have greater confidence in the result. Where they diverge (e.g., the SDE predicts smooth traversal of the hybrid zone while the Markov chain shows significant dwell time there), the divergence itself is informative about the granularity of the political process.

Recommended future work: A continuous-time Markov chain (CTMC) would bridge both frameworks, modeling transitions as a rate matrix Q where off-diagonal entries q_ij give the instantaneous rate of transition from stage i to stage j. The CTMC handles irregularly-spaced observations naturally (the likelihood depends on matrix exponentials e^Qt evaluated at observed time gaps) and converges to the SDE in the limit of fine state discretization. Estimating Q via maximum likelihood from the irregularly-spaced panel would provide a single, unified framework that subsumes the discrete transition matrix and approximates the continuous dynamics, while respecting the actual temporal structure of the data.

The complete model—integrating state equations, transition matrices, attractor dynamics, and Monte Carlo projections—draws on and extends several established quantitative frameworks for studying regime transitions. This section traces those connections explicitly.

Przeworski et al. (2000), Democracy and Development. Przeworski and colleagues established the canonical finding that regime survival is a function of income: democracies above a per-capita GDP threshold (~$6,000 in 1985 PPP dollars) virtually never collapse, while poorer democracies are fragile. The complete model developed here takes a fundamentally different approach, treating the liberty score itself as the state variable driving survival probabilities rather than income. The 8-stage transition matrix makes no reference to GDP; instead, transition probabilities are estimated purely from the liberty trajectory. This is both a strength and a limitation. It is a strength because it reveals the internal dynamics of institutional erosion that income-based models obscure—the US, well above Przeworski's income threshold, is nonetheless eroding at −18/yr (2yr window), demonstrating that wealth alone does not immunize democracies. It is a limitation because the model cannot explain why countries enter particular stages—the economic, social, and geopolitical determinants that Przeworski's framework captures. A synthesis incorporating income as a covariate affecting transition probabilities (e.g., stage-dependent k values conditioned on GDP) would strengthen both frameworks.

Geddes, Wright & Frantz (2014), “Autocratic Breakdown and Regime Transitions.” GWF's influential typology distinguishes among personalist, military, party-based, and monarchic autocracies, demonstrating that regime subtype powerfully predicts breakdown mode and post-breakdown trajectory. The 8-stage ladder in the complete model complements GWF by providing an orthogonal dimension of classification: GWF classifies how autocracies are organized; the stage model classifies where they sit on the liberty continuum. A country at Stage 7 (Consolidated Autocracy, L=25–39) could be personalist (Russia), party-based (China pre-2012), or military (Myanmar)—and the GWF typology would predict different breakdown probabilities for each. The complete model's 2% escape rate from Stage 8 represents an average across all autocratic subtypes; disaggregating by GWF type would likely reveal that party-based autocracies have higher Stage 8 retention (they are harder to dislodge) while personalist regimes have slightly lower retention (they are vulnerable to the mortality of their leader). This integration represents a natural extension of both frameworks.

Polity V / Marshall & Gurr (2020). The Polity dataset has been the workhorse of quantitative regime analysis for decades, scoring countries on a −10 to +10 scale based on executive recruitment, constraints on the executive, and political competition. The complete model uses Freedom House rather than Polity as its primary data source—a choice with methodological implications. Freedom House provides finer granularity (0–100 vs. −10 to +10), enabling the detection of subtle within-regime variation that Polity's coarser scale obscures. Specifically, the hybrid trap (L≈20–70) spans a range that Polity often codes as a single "anocracy" category (−5 to +5), missing the internal structure that the GMM and transition matrix reveal: a shallow but genuine attractor at L≈47 with distinct entry and exit dynamics. However, Freedom House's shorter temporal coverage (1972–present vs. Polity's 1800–present) limits the historical depth of the analysis. The V-Dem dataset (1789–2024) is used to extend coverage, but measurement compatibility across datasets introduces additional uncertainty not fully captured in the confidence intervals. Future work should replicate the tristable basin analysis using Polity V scores to test whether the three-basin structure is robust to measurement instrument choice—if the GMM also identifies three clusters in the Polity data, confidence in the tristable finding would increase substantially.

Acemoglu & Robinson (2006), Economic Origins of Dictatorship and Democracy. The complete model's state equation (L_t+1 = L_t + v_t + ε_t) represents a methodological departure from Acemoglu and Robinson's game-theoretic approach. Their model derives regime transitions from strategic interactions between elites and citizens, yielding sharp predictions about when democratization or coups occur based on inequality levels and repression costs. The stochastic dynamics approach here is agnostic about micro-level mechanisms: the drift term v_t captures the net effect of all institutional pressures, and the shock term ε_t absorbs strategic interactions, economic fluctuations, and exogenous events alike. This loss of micro-foundation is offset by a gain in empirical tractability—the model can be estimated directly from time-series data without requiring difficult-to-measure variables like inequality perception or revolutionary threat. The attractor dynamics equation (dv/dt = −k(L − L*)) also introduces a feature absent from game-theoretic models: mean reversion within regimes. Acemoglu and Robinson's framework predicts discrete jumps between equilibria; the complete model shows that most political change is gradual drift within a basin, with discrete transitions being rare events driven by unusually large shocks.

Schedler (2006), The Politics of Uncertainty. Schedler's analysis of electoral authoritarianism—regimes that hold elections but manipulate them to ensure incumbent victory—maps precisely onto the complete model's Stages 5–6 (Electoral Autocracy and Soft Dictatorship, L=40–59). Schedler argued that the defining feature of these regimes is structured uncertainty: elections are neither free enough to produce genuine alternation nor rigged enough to eliminate all opposition hope. The transition matrix provides a quantitative signature of this uncertainty: Stage 5 has the lowest stay probability among non-extreme stages (64%), reflecting the inherent instability that Schedler identified. Countries at Stage 5 face a 24% annual probability of sliding to Stage 6 but only a 12% combined probability of recovery (9% to Stage 4 + 3% to Stage 2–3)—a 2:1 asymmetry between erosion and recovery that formalizes Schedler's observation that electoral authoritarian regimes are "menus of manipulation" that tend to expand their repressive toolkit over time rather than contract it. The complete model extends Schedler by showing that this instability is not random but directionally biased: the potential energy landscape slopes downward from the hybrid trap toward the tyranny well, giving electoral authoritarian regimes a gravitational pull toward deeper autocracy that Schedler's framework described qualitatively but could not quantify.