IndisputableMonolith.GameTheory.ESSFromSigma
The module delivers the cooperator fraction threshold for an evolutionarily stable strategy as 1/φ in a kin-selected population. Game theorists studying cooperation would cite this when linking Recognition Science to prisoner's dilemma thresholds. The module supplies the core definitions for the threshold and the ESS predicate together with lemmas on their properties.
claimThe threshold for the cooperator fraction to be an evolutionarily stable strategy in a kin-selected population is $\frac{1}{\phi} \approx 0.618$.
background
This module sits in the GameTheory domain and imports the Constants module. The upstream result defines the fundamental RS time quantum as τ₀ = 1 tick. The module introduces cooperatorThreshold as the value 1/φ and isESS as the predicate on strategies, along with lemmas such as all_cooperator_isESS and no_cooperator_not_isESS. The setting derives stability from the J-cost function and the Recognition Composition Law in kin-selected populations.
proof idea
This is a definition module, no proofs.
why it matters in Recognition Science
The module feeds the Cooperation Cascade Theorem. The downstream result states that if the cooperator fraction in a kin-cluster crosses 1/φ then the J-cost gradient drives the entire cluster to full cooperation. This matches observed thresholds in n-person prisoner's dilemma experiments and supplies the game theory from first principles component of the Recognition Science framework.
scope and limits
- Does not derive φ from the T0-T8 forcing chain.
- Does not model non-kin selection or mutation.
- Does not simulate finite-population stochastic effects.
- Does not extend the threshold to multi-strategy games.