Curvature Shadow paper resolves Kuhn poker equilibrium selection puzzle
A 0.021 coordinate gap in Kuhn poker explained by entropy curvature, not bias.
A new paper by Luis Leal (arXiv:2607.17543) tackles a subtle puzzle in game-theoretic AI: why does Regularized Nash Dynamics (R-NaD) select a slightly off equilibrium in Kuhn poker? Even though R-NaD attains 99.7% of the maximum entropy, its bluff coordinate lands at 0.180 instead of the expected 0.201—a gap of 0.021. The paper rigorously shows this is not a true bias but an artifact caused by the curvature of the entropy landscape. The author derives a factorization: gap ≈ √(2δ/κ), where δ is the entropy shortfall and κ is the curvature at the peak. Across five test games, the formula holds to within 2×10⁻⁴ (under 1% relative error).
Crucially, causal experiments with varying magnet strength drive δ→0, and the gap shrinks along the predicted √δ curve (fitted exponent 0.50, R² > 0.999999). The remaining tiny gap is purely a removable finite-precision effect hitting a stability floor—no fixed bias remains. The paper also flags a moving-target pitfall in Tsallis-entropy experiments. For professionals working with multi-agent systems and equilibrium selection, this means the maximum-entropy projection principle is upheld even in sequential games—any apparent failure is just a curvature shadow, not a fundamental limitation.
- R-NaD in Kuhn poker shows a 0.021 coordinate gap from the maximum-entropy equilibrium, but this is fully explained by entropy curvature (gap ≈ √(2δ/κ)).
- Across five games the formula matches with under 1% relative error; in matrix games δ≈0 yields exact selection, while Kuhn's sequential structure requires higher resolution.
- Causal experiments with increasing magnetic strength reduce the gap to near zero along a √δ curve (R² > 0.999999), confirming it's a removable artifact, not a fixed bias.
Why It Matters
Validates maximum-entropy equilibrium selection as robust, with practical implications for training stable multi-agent AI systems.