Stochastic Stability of Ordinal Aggregation under Majorization Dynamics: A Drift–Diffusion Analysis of Political Failure Modes
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2026
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| _version_ | 1866901206869213184 |
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| author | Fathi, Kevin |
| author_facet | Fathi, Kevin |
| contents | <p>Classical social choice theory is organized around “no-go” results: Arrow’s impossibility<br>theorem and Condorcet cycles show that no ordinal aggregation rule can universally convert<br>individual rankings into a coherent transitive social ordering while satisfying minimal fairness<br>axioms. This paper pivots from the static existence question to a dynamic stability question.<br>We treat paradoxes as regimes of a stochastic dynamical system.<br><br>Our first contribution is a measurement layer that replaces an often-undefined social ranking<br>with an observable state variable: the ordinal-profile distribution p over ballot types or joint<br>ordinal profiles. Majorization (equivalently Lorenz order) supplies a label-free concentration<br>order on p, and Shannon ordinal majorization entropy (OME) supplies a scalar potential that is<br>ordinally invariant and exactly decomposable across subgroups. We provide full proofs of Lorenz<br>consistency, decomposability, and a characterization theorem identifying Shannon entropy as the<br>unique continuous Lorenz-monotone refinement-decomposable index (up to positive scale).<br><br>Our second contribution is a dynamical layer based on Dynamic Inequality Equilibrium (DIE):<br>we map political dynamics into a two-dimensional phase space Φ = (M, D) where D is Lorenz<br>drift (an inequality-velocity statistic) and M is rank mobility (a micro-turbulence statistic). We<br>prove that a persistent Condorcet cycle is a churning equilibrium: macro-stationary (D = 0) but<br>micro-volatile (M is order one).<br><br>Our third contribution is a mechanism design result. We formalize a constitutional circuit<br>breaker as an affine reinjection (restart) update pt+1 = (1 − ε)Ppt + εs, which is exactly the<br>PageRank/random-walk-with-restart operator. A contraction theorem guarantees convergence<br>to a unique stationary distribution for any ε > 0, damping persistent cycling without pretending<br>to “solve” Arrow.</p> |
| format | Recurso digital |
| id | zenodo_https___doi_org_10_5281_zenodo_18156054 |
| institution | Zenodo |
| language | eng |
| publishDate | 2026 |
| publisher | Zenodo |
| record_format | zenodo |
| spellingShingle | Stochastic Stability of Ordinal Aggregation under Majorization Dynamics: A Drift–Diffusion Analysis of Political Failure Modes Fathi, Kevin social choice theory ordinal aggregation majorization dynamics Dynamic Inequality Equilibrium Lorenz order Shannon entropy Condorcet cycles stochastic stability political instability PageRank mechanism <p>Classical social choice theory is organized around “no-go” results: Arrow’s impossibility<br>theorem and Condorcet cycles show that no ordinal aggregation rule can universally convert<br>individual rankings into a coherent transitive social ordering while satisfying minimal fairness<br>axioms. This paper pivots from the static existence question to a dynamic stability question.<br>We treat paradoxes as regimes of a stochastic dynamical system.<br><br>Our first contribution is a measurement layer that replaces an often-undefined social ranking<br>with an observable state variable: the ordinal-profile distribution p over ballot types or joint<br>ordinal profiles. Majorization (equivalently Lorenz order) supplies a label-free concentration<br>order on p, and Shannon ordinal majorization entropy (OME) supplies a scalar potential that is<br>ordinally invariant and exactly decomposable across subgroups. We provide full proofs of Lorenz<br>consistency, decomposability, and a characterization theorem identifying Shannon entropy as the<br>unique continuous Lorenz-monotone refinement-decomposable index (up to positive scale).<br><br>Our second contribution is a dynamical layer based on Dynamic Inequality Equilibrium (DIE):<br>we map political dynamics into a two-dimensional phase space Φ = (M, D) where D is Lorenz<br>drift (an inequality-velocity statistic) and M is rank mobility (a micro-turbulence statistic). We<br>prove that a persistent Condorcet cycle is a churning equilibrium: macro-stationary (D = 0) but<br>micro-volatile (M is order one).<br><br>Our third contribution is a mechanism design result. We formalize a constitutional circuit<br>breaker as an affine reinjection (restart) update pt+1 = (1 − ε)Ppt + εs, which is exactly the<br>PageRank/random-walk-with-restart operator. A contraction theorem guarantees convergence<br>to a unique stationary distribution for any ε > 0, damping persistent cycling without pretending<br>to “solve” Arrow.</p> |
| title | Stochastic Stability of Ordinal Aggregation under Majorization Dynamics: A Drift–Diffusion Analysis of Political Failure Modes |
| topic | social choice theory ordinal aggregation majorization dynamics Dynamic Inequality Equilibrium Lorenz order Shannon entropy Condorcet cycles stochastic stability political instability PageRank mechanism |
| url | https://doi.org/10.5281/zenodo.18156054 |