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arXiv · 2512.07901

The Theory of Strategic Evolution: Games with Endogenous Players and the Seven Laws of Strategic Replicators

Kevin Vallier
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Abstract

Von Neumann founded both game theory and the theory of self-reproducing automata, but the two programs never merged. Rational players do not control their replication, and replicators do not choose strategically. Contemporary AI systems expose this gap: they optimize objectives, yet the population of AI systems is not fixed but expands and contracts based on performance. When capital can spawn capital, we need a theory that captures both rationality and replication. The Theory of Strategic Evolution analyzes strategic replicators: entities that optimize under resource constraints and spawn copies of themselves. The framework is organized around Seven Laws: 1. Strategic Selection: Mean fitness serves as a Lyapunov function; dominated types are eliminated. 2. ESDI Characterization: Equilibria exist, are generically finite, and satisfy Nash-KKT-LP equivalence. 3. H-$\gamma$ Stability: Multi-level systems are stable iff the spectral radius $\rho(\Gamma) < 1$. 4. G$\infty$ Closure: A unique maximal class of safe modifications exists and is closed under composition. 5. Constitutional Duality: Shadow prices implement any frontier allocation; welfare theorems hold. 6. Alignment Impossibility: Full reachability destroys Lyapunov structure; alignment requires bounded modification. 7. Hopf Transition: At critical coupling, systems undergo supercritical bifurcation producing limit cycles. Applications range from AI deployment dynamics to institutional design. The framework shows why "personality engineering" fails under selection pressure and identifies constitutional constraints necessary for stable alignment. The Lean 4 formalization verifies every result with zero custom axioms. The framework generates empirical predictions that remain untested. The paper and its repository are therefore a prototype kernel: an instrument for studying strategic replicators, not a set of results about the world.

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Kevin Vallier. 2025-12-05. The Theory of Strategic Evolution: Games with Endogenous Players and the Seven Laws of Strategic Replicators. https://arxiv.org/abs/2512.07901

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