isSigmaConservative
The definition classifies a two-player joint outcome as sigma-conservative precisely when its sigma-charge is non-negative. Game theorists applying Recognition Science to the prisoner's dilemma cite this to identify which outcomes conserve the Noether charge sigma. It is implemented as a direct one-line abbreviation of the non-negativity condition on the sigma-charge function.
claimA joint outcome $o$ is $sigma$-conservative if $sigma(o) geq 0$, where $sigma(o)$ denotes the charge taking value $+1$ on mutual cooperation, $0$ on mixed strategies, and $-1$ on mutual defection.
background
In the module on Cooperation From sigma-Conservation, game-theoretic equilibria are characterized as J-cost minima on the multi-agent ledger. A JointOutcome pairs two player choices, each cooperate or defect. The sigma-charge function assigns +1 to mutual cooperation (creating coordinated value), 0 to mixed outcomes, and -1 to mutual defection (destroying coordinated value); this charge is the sigma-Noether charge of the multi-agent move.
proof idea
The declaration is a one-line definition that applies the sigma-charge function to the given outcome and asserts its non-negativity.
why it matters in Recognition Science
This definition is used by the GameTheoryCert structure to certify the properties of sigma-conservation, including that mutual defection violates it while all other outcomes satisfy it. It realizes the second prediction of the module: cooperation from sigma-conservation, which explains the dominance of reciprocal-altruism strategies that preserve sigma. The result sits within the Recognition Science framework as part of element 84 in the game theory domain.
scope and limits
- Does not prove that sigma-conservative outcomes are Nash equilibria.
- Does not address n-player generalizations.
- Does not quantify the coordination dividend 1/phi.
- Does not derive the charge values from more primitive axioms.
formal statement (Lean)
106def isSigmaConservative (o : JointOutcome) : Prop :=
proof body
Definition body.
107 jointSigma o ≥ 0
108
109/-- Mutual defection is not σ-conservative. -/