Mathematics Game Theory Depth From Rs Game Theory Depth Cert

A certificate of depth

Game theory studies strategic decisions where each player's best choice depends on what others will do. Over decades, theorists developed several distinct ways to say what a rational or stable outcome should be. The five most famous are Nash equilibrium, subgame perfect equilibrium, correlated equilibrium, Bayesian Nash equilibrium, and evolutionarily stable strategy. Each answers a different question: Nash says no player can improve alone, subgame perfect rules out empty threats in sequential play, correlated allows a shared signal, Bayesian handles private information, and evolutionarily stable describes a strategy that resists invasion by mutants.

A machine-checked library of formal theorems, the framework's library, defines these five as a single inductive type, a closed list of named cases. Its declaration GameTheoryDepthCert is a certificate: a structure whose only field is a proof that the count of these five concepts equals five. The proof runs by deciding the finite cases, with no unproved assumptions and no axioms beyond the kernel's standard three. This is a theorem about the library's own definition, not a claim about how real players behave.

In Recognition Science, the framework models a game equilibrium as a zero of a cost function J in each player's recognition field. The prisoner's dilemma appears as mutual defection carrying positive social cost, while the stag hunt shows coordination driving J toward its minimum. The number five matches the framework's configDim D, a structural dimension count. This identification is a modeling choice, not a derived result: the library declares the five concepts and proves their count, but it does not prove that these five are the only possible solution concepts or that the framework's cost model explains why these five emerged.

What the certificate does not claim matters. It does not rank the concepts, prove they coincide, or say which one applies to a given game. It does not derive the number five from deeper principles; it counts a chosen list. The framework's interpretation of equilibria as zeros of J is a lens for reading the five concepts, not a proof about game theory itself. The honest summary: the library certifies that its own list of five canonical concepts has exactly five members, and the framework offers a cost-based way to view them, with the connection between the two left as a modeling stance.

structure GameTheoryDepthCert where
  five_concepts : Fintype.card SolutionConcept = 5

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

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

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

What this page does not claim

The certificate does not prove that these five concepts are the only possible solution concepts in game theory. The certificate does not derive the number five from deeper principles; it counts a chosen list. The cost-based interpretation of equilibria is a modeling lens, not a theorem about real strategic behavior.

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)

A page whose claims cannot be reproduced this way does not ship. In production, every anchor links to the exact declaration in the public source release, and this block carries the build receipt for the page itself.

Derived articles

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