Open Games Skill (ERGODIC 0)

Compositional game theory via Para/Optic structure

Trit: 0 (ERGODIC) Color: #26D826 (Green) Role: Coordinator/Transporter

bmorphism Contributions

"Parametrised optics model cybernetic systems, namely dynamical systems steered by one or more agents. Then ⊛ represents agency being exerted on systems"
— @bmorphism, GitHub bio

"We introduce open games as a compositional foundation of economic game theory. A compositional approach potentially allows methods of game theory and theoretical computer science to be applied to large-scale economic models"
— Compositional Game Theory, Ghani, Hedges, Winschel, Zahn (2016)

Key Papers (from bmorphism's Plurigrid references):

• Compositional game theory - open games as symmetric monoidal category morphisms
• Morphisms of Open Games - connection between lenses and compositional game theory
• Bayesian Open Games - stochastic environments, incomplete information
• Categorical Cybernetics Manifesto - control theory of complex systems

CyberCat Institute Connection: Open games are central to the CyberCat Institute research program on categorical cybernetics.

Related to bmorphism's work on:

• plurigrid/act - active inference + ACT + enacted cognition
• Play/Coplay bidirectional feedback structure

Core Concept

Open games are morphisms in a symmetric monoidal category:

        ┌───────────┐
   X ──→│           │──→ Y
        │  Game G   │
   R ←──│           │←── S
        └───────────┘

Where:

• X → Y: Forward play (strategies)
• S → R: Backward coplay (utilities)

The Para/Optic Structure

Para Morphism

Para p a b = ∃m. (m, p m a → b)
-- Existential parameter with action

Optic (Lens Generalization)

Optic p s t a b = ∀f. p a (f a b) → p s (f s t)
-- Profunctor optic for bidirectional data

Open Game as Optic

OpenGame s t a b = 
  { play    : s → a
  , coplay  : s → b → t
  , equilibrium : s → Prop
  }

Composition

Sequential (;)

G ; H = Game where
  play = H.play ∘ G.play
  coplay = G.coplay ∘ (id × H.coplay)

Parallel (⊗)

G ⊗ H = Game where
  play = G.play × H.play
  coplay = G.coplay × H.coplay

Nash Equilibrium via Fixed Points

isEquilibrium :: OpenGame s t a b → s → Bool
isEquilibrium g s = 
  let a = play g s
      bestResponse = argmax (\a' → utility (coplay g s (respond a')))
  in a == bestResponse

Compositional Equilibrium

eq(G ; H) = eq(G) ∧ eq(H)  -- under compatibility

Integration with Unworld

(defn opengame-derive 
  "Transport game through derivation chain"
  [game derivation]
  (let [; Forward: strategies through derivation
        forward (compose (:play game) (:forward derivation))
        ; Backward: utilities through co-derivation  
        backward (compose (:coplay game) (:backward derivation))]
    {:play forward
     :coplay backward
     :equilibrium (transported-equilibrium game derivation)}))

GF(3) Triads

temporal-coalgebra (-1) ⊗ open-games (0) ⊗ free-monad-gen (+1) = 0 ✓
three-match (-1) ⊗ open-games (0) ⊗ operad-compose (+1) = 0 ✓
sheaf-cohomology (-1) ⊗ open-games (0) ⊗ topos-generate (+1) = 0 ✓

Commands

# Compose games sequentially
just opengame-seq G H

# Compose games in parallel
just opengame-par G H

# Check Nash equilibrium
just opengame-nash game strategy

# Transport through derivation
just opengame-derive game deriv

Economic Examples

Prisoner's Dilemma

prisonersDilemma :: OpenGame () () (Bool, Bool) (Int, Int)
prisonersDilemma = Game {
  play = \() → (Defect, Defect),  -- Nash
  coplay = \() (p1, p2) → payoffMatrix p1 p2
}

Market Game

market :: OpenGame Price Price Quantity Quantity
market = supplyGame ⊗ demandGame
  where equilibrium = supplyGame.eq ∧ demandGame.eq

Categorical Semantics

OpenGame ≃ Para(Lens) ≃ Optic(→, ×)

Composition: 
  (A ⊸ B) ⊗ (B ⊸ C) → (A ⊸ C)  -- via cut
  
Tensor:
  (A ⊸ B) ⊗ (C ⊸ D) → (A ⊗ C ⊸ B ⊗ D)

References

• Ghani, Hedges, et al. "Compositional Game Theory"
• Capucci & Gavranović, "Actegories for Open Games"
• Riley, "Categories of Optics"
• CyberCat Institute tutorials

Scientific Skill Interleaving

This skill connects to the K-Dense-AI/claude-scientific-skills ecosystem:

Graph Theory

• networkx [○] via bicomodule

- Universal graph hub

Bibliography References

• game-theory: 21 citations in bib.duckdb

SDF Interleaving

This skill connects to Software Design for Flexibility (Hanson & Sussman, 2021):

Primary Chapter: 10. Adventure Game Example

Concepts: autonomous agent, game, synthesis

GF(3) Balanced Triad

open-games (+) + SDF.Ch10 (+) + [balancer] (+) = 0

Skill Trit: 1 (PLUS - generation)

Secondary Chapters

• Ch8: Degeneracy
• Ch3: Variations on an Arithmetic Theme
• Ch1: Flexibility through Abstraction
• Ch5: Evaluation
• Ch4: Pattern Matching
• Ch7: Propagators

Connection Pattern

Adventure games synthesize techniques. This skill integrates multiple patterns.

Cat# Integration

This skill maps to Cat# = Comod(P) as a bicomodule in the equipment structure:

Trit: 0 (ERGODIC)
Home: Prof
Poly Op: ⊗
Kan Role: Adj
Color: #26D826

GF(3) Naturality

The skill participates in triads satisfying:

(-1) + (0) + (+1) ≡ 0 (mod 3)

This ensures compositional coherence in the Cat# equipment structure.