Replicator Dynamics - Information Geometry View

Fisher metric, geodesics, and gradient flow on the probability simplex

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Strategy Mixture

⚡ Information Geometry

Shannon Entropy H(p): 0.000
KL Divergence from uniform: 0.000
Fisher-Rao distance to center: 0.000
Coordinate View:

Current State (Barycentric)

Strategy 1 (x): 0.250
Strategy 2 (y): 0.250
Strategy 3 (z): 0.250
Strategy 4 (w): 0.250

Avg Fitness: 0.0000
Active Dimension: 3

Information Geometry Features:

The simplex is equipped with the Fisher information metric g_ij = δ_ij/p_i - 1. Replicator dynamics can be viewed as gradient flow.

Metric Visualizations:

Purple Grid Ellipses: Static grid showing overall metric structure across the simplex.
Green Current Ellipse: DYNAMIC! Moves with the red point and grows/shrinks to show how the local metric changes. Watch it expand dramatically as the point approaches boundaries - this is the Fisher metric stretching space! At center it's small and uniform; near edges it grows large showing negative curvature.

Geodesics: The "straight lines" in this curved geometry, computed by interpolation in log-space.

Coordinate Systems:

Barycentric: Standard probabilities (x,y,z,w) summing to 1.
Log-Ratio: Centered log coordinates log(p_i/g) where g is geometric mean. These reveal the natural linear structure of compositional data and are centered at 0 for uniform distribution. Note: CLR coordinates sum to 0!