Mathematics Game Theory Depth From Rs Solution Concept Count

The five solution concepts

Game theory studies strategic situations where each player's best choice depends on what others will do. Over decades, theorists developed several distinct ways to say what a rational outcome is. The five most canonical are Nash equilibrium, subgame perfect equilibrium, correlated equilibrium, Bayesian Nash equilibrium, and evolutionarily stable strategy. Each answers a different question: Nash asks which strategy profiles survive when no player can improve unilaterally; subgame perfection adds credibility to threats; correlated equilibrium allows a public signal; Bayesian Nash handles private information; evolutionary stability asks which strategies resist invasion by mutants.

In the Recognition Science framework, these five are collected into a single inductive type named SolutionConcept, with one constructor for each concept. The framework then proves a theorem, solutionConceptCount (a machine-checked statement in its library of formal theorems), that the number of these concepts is exactly five. The proof is by computation: the type has exactly five elements, and the theorem verifies that count. The framework also models game equilibrium as a zero of its recognition cost function, where mutual defection in the Prisoner's dilemma carries positive social cost and coordination in the stag hunt drives cost toward its minimum.

The count itself is a definitional fact, not an empirical discovery. The framework chooses these five concepts as the canonical set, and the theorem certifies that the choice yields five. It does not claim that these are the only possible solution concepts, nor that game theory has no others. It does not assert that the five are equally important, mutually exclusive, or exhaustive of all strategic reasoning. The theorem's content is the count, and the count is five.

theorem solutionConceptCount : Fintype.card SolutionConcept = 5 := by decide

inductive SolutionConcept where
  | nash | subgamePerfect | correlated | bayesianNash | evolutionarilyStable
  deriving DecidableEq, Repr, BEq, Fintype

structure GameTheoryDepthCert where
  five_concepts : Fintype.card SolutionConcept = 5

What this page does not claim

The theorem does not claim these are the only possible solution concepts in all of game theory. The theorem does not rank the five concepts by importance or applicability. The framework does not claim that every game has exactly one equilibrium in its cost model.

Verify this page

Every tagged claim above names its theorem. To check one yourself rather than trust this page, elaborate the source module with Lean 4 and audit its axiom basis:

$ lake env lean IndisputableMonolith/Mathematics/GameTheoryDepthFromRS.lean
expected axiom basis: [propext, Classical.choice, Quot.sound] (the Lean kernel's standard three; no RS-specific axioms)

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Derived articles

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