Theorems

The results below concern rational finite games and classical continuous control. For correctness of executable checkers and floating-point error bounds, see verification and tests.

Static equilibrium

For general finite static games, checking every pure deviation is equivalent to checking every rational mixed deviation. The RPS adapter transfers equilibrium statements between the generic model and the original three-action model.

In rock-paper-scissors theory, an exact equilibrium is a population mix m for which no rational personal mix p has higher reward against fixed m. An ε-equilibrium adds ε ≥ 0 and permits improvement by at most ε. The proof shows that the uniform mix is the only exact equilibrium for the chosen win/loss signs.

Classical HJB equation

The classical solution structure records a smooth value candidate, its time and space derivatives, the HJB equation, and terminal condition. For dynamics x' = a and cost a²/2 + F(t,x), the equation is

An explicit quadratic candidate satisfies these conditions. The verification theorem gives a cost lower bound for any admissible path whose value derivative is interval integrable. The quadratic example proves that integrability condition for all of its admissible paths.

The current finite-state layer defines its equilibrium notion in the model module because the one-step game and its admissible policies are introduced together.