Simplex Game Theory Lab

A three-strategy simplex model for the advisory-control story. The simplex tracks the population mix of Loyal, Traitorous, and Skeptical/Resistant strategic tendencies. Dynamics are driven by a payoff matrix and trust-alignment parameters.

Quick presets

Interpretation

The simplex point is the current strategic composition. Flow arrows show replicator dynamics induced by the current observer-game payoff matrix. The trust score bends the payoff landscape by changing how plausible the ruler finds advice.

Population on the simplex

Values are renormalized to sum to 1.

Observer-game parameters

0.45
0.55
0.50
0.35
0.62
0.60
0.020
4
Loyal vertex Traitor vertex Skeptical vertex Current state Trajectory
Trust score
0.000
Influence weight
0.000
Dominant tendency
Loyal

Model notes

The top-right panel is a pseudo-3D lift of the simplex by an effective trust potential. Higher regions correspond to stronger advisory influence under the current observer parameters.

We use a replicator equation on the 2-simplex, \(\dot x_i = x_i[(Ax)_i - x^T A x]\), where the payoff matrix \(A\) is built from the observer-game parameters. Loyal strategy benefits from belief and discernment. Traitor strategy benefits from opacity, masking, and persuasion. Skeptical strategy benefits when trust becomes unreliable and resistance is high.

This is not a full solution of the underlying repeated Bayesian game. It is a geometric laboratory: a reduced mean-field picture showing how hidden alignment and observer distortion bend strategic flow on a simplex. It is a nice way to see capture basins, unstable mixtures, and skeptical stabilization.