IndisputableMonolith.GameTheory.NashEquilibriumFromJCost
This module constructs a minimal two-player game in Recognition Science where strategies are positive reals and payoffs are built from the J-cost function. It defines profiles, joint costs, Nash conditions, and certifies that the canonical profile satisfies equilibrium. Researchers linking RS cost minimization to classical game theory cite it. The module consists of definitions and direct algebraic checks that draw on the imported Cost module.
claimLet $J$ denote the J-cost function. A two-player profile is a pair of positive reals $(x,y)$. The joint J-cost is defined as $J(x)+J(y)+J(xy)+J(x/y)$. The canonical profile is $(1,1)$. A profile is a Nash equilibrium when neither player can unilaterally lower the joint cost by changing its strategy ratio.
background
Recognition Science derives physics from the J-cost function satisfying the Recognition Composition Law. The module imports the fundamental time quantum from Constants and the cost definitions from Cost. It treats the simplest non-trivial case of two-player interaction by encoding strategies as positive real ratios whose joint cost is assembled from the four terms $J(x)$, $J(y)$, $J(xy)$, and $J(x/y)$.
proof idea
This is a definition module, no proofs.
why it matters in Recognition Science
The module supplies the game-theoretic reading of J-cost minimization and thereby connects to the J-uniqueness result (T5) in the forcing chain. It prepares the ground for later applications that embed classical equilibria inside the RS cost framework, even though no downstream theorems yet depend on it.
scope and limits
- Does not treat games with more than two players.
- Does not incorporate continuous time or quantum strategies.
- Does not address arbitrary payoff matrices outside the J-cost construction.
- Does not prove uniqueness of the Nash equilibrium.