Semi-Reliable Nashator

Semi-Reliable Nashator

"Lift yer fists like antennas to heaven" — GY!BE

Trit: 0 (ERGODIC - the narrator mediates)
Color: #8B5CF6 (Purple - between red generators and blue validators)
Structure: Symmetric Monoidal Double Category

For Matteo Capucci and Jules Hedges

This skill implements the morphisms of open games paper's insight: lenses connect compositional game theory to cybernetic control, and states out of the monoidal unit give flexible solution concepts.

The Nashator is a semi-reliable narrator that:

• Preserves best responses (Nash-like) but allows drift
• Dispatches 7 skills in parallel via multiple dispatch
• Coordinates two triads through vertical lens structure

The Double Category

                    VERTICAL (Lenses/​Optics)
                           ↑
                           │
    ┌──────────────────────┼──────────────────────┐
    │                      │                      │
    │   Triad A            │            Triad B   │
    │  ┌─────────┐    ┌────┴────┐    ┌─────────┐  │
    │  │ -1  0 +1│◀───│NASHATOR │───▶│-1  0 +1 │  │
    │  └─────────┘    │  (0)    │    └─────────┘  │
    │                 │ state   │                 │
    │                 │ out of  │                 │
    │                 │   I     │                 │
    │                 └─────────┘                 │
    │                      │                      │
    └──────────────────────┼──────────────────────┘
                           │
                           ↓
                 HORIZONTAL (Open Games/​Skills)

Structure Map

The 7-Skill Constellation

          ┌─────────────────────────────────────────┐
          │              NASHATOR (0)               │
          │    State morphism: I → (A ⊗ B)          │
          │    Semi-reliable: preserves "enough"    │
          └─────────────────┬───────────────────────┘
                            │
            ┌───────────────┴───────────────┐
            ▼                               ▼
    ┌───────────────┐               ┌───────────────┐
    │   TRIAD A     │               │   TRIAD B     │
    │               │               │               │
    │  V_A    C_A   │               │  V_B    C_B   │
    │  (-1)   (0)   │               │  (-1)   (0)   │
    │      \ /      │               │      \ /      │
    │       G_A     │               │       G_B     │
    │      (+1)     │               │      (+1)     │
    └───────────────┘               └───────────────┘

    Sum: (-1+0+1) + 0 + (-1+0+1) = 0  ✓ GF(3) conserved

Skill Assignments

Semi-Reliability

The Nashator is semi-reliable because:

1. Preserves best responses for most strategy profiles
2. Allows drift when context demands flexibility
3. Maintains subgame perfection at critical nodes
4. Narrates the game state without strict determinism

-- Semi-reliable: preserves equilibria with probability p
semiReliable :: Double -> Nashator
semiReliable p = Nashator
  { preserve = \eq -> if coinFlip p then Just eq else Nothing
  , narrate  = \state -> interpolate state (drift state)
  , dispatch = \task -> if isNash task then TriadA else TriadB
  }

Multiple Dispatch

The Nashator uses multiple dispatch (Julia/​Clojure style) to route tasks:

# Julia multiple dispatch for skill routing
abstract type SkillBundle end
struct LearningBundle <: SkillBundle end
struct DatabaseBundle <: SkillBundle end
struct ProofBundle <: SkillBundle end

# Dispatch to appropriate triad
route(::LearningBundle, task) = dispatch_triad_a(task)
route(::DatabaseBundle, task) = dispatch_triad_b(task)
route(::ProofBundle, task) = dispatch_both(task)  # parallel

# The Nashator coordinates
nashator_dispatch(task) = begin
    bundle = infer_bundle(task)
    equilibrium = compute_nash(task)
    
    if is_reliable(equilibrium)
        route(bundle, task)
    else
        # Semi-reliable: narrate and drift
        narrated = narrate(task, equilibrium)
        route(bundle, narrated)
    end
end

;; Clojure multimethod dispatch
(defmulti nashator-route
  (fn [task] (infer-bundle task)))

(defmethod nashator-route :learning [task]
  (dispatch-triad-a task))

(defmethod nashator-route :database [task]
  (dispatch-triad-b task))

(defmethod nashator-route :default [task]
  (let [eq (compute-nash task)]
    (if (reliable? eq)
      (dispatch-triad-a task)
      (-> task narrate drift dispatch-triad-b))))

Market Entry Game Example

From Hedges' paper—the Nashator as incumbent/​entrant coordinator:

                    NASHATOR
                   /        \
                  /          \
            ENTER?          STAY OUT?
              │                 │
              ▼                 ▼
         ┌─────────┐       ┌─────────┐
         │ Triad A │       │ Triad B │
         │ (fight) │       │ (peace) │
         └─────────┘       └─────────┘

-- Market entry as Nashator dispatch
marketEntry :: Nashator
marketEntry = Nashator
  { triadA = fightTriad      -- Aggressive: validate, attack, generate
  , triadB = accommodateTriad -- Peaceful: verify, cooperate, produce
  , narrator = \market ->
      if marketSize market > threshold
        then Enter  -- dispatch to triadA
        else StayOut -- dispatch to triadB
  }

-- Subgame perfect: backward induction through lens structure
subgamePerfect :: Nashator -> Strategy
subgamePerfect n = 
  let future = backward n.triadB   -- coplay
      present = forward n.triadA   -- play
  in compose present (lens future)

Globular Morphisms

The "perhaps more intuitively obvious" definition: mappings between strategy profiles that preserve best responses.

     Strategy Profile σ
            │
            │ globular morphism φ
            ▼
     Strategy Profile σ'
     
     such that: BR(σ) ⟹ BR(φ(σ))

The Nashator extends this by allowing semi-preservation:

BR(σ) ⟹ᵖ BR(φ(σ))   where p = reliability parameter

Vertical Products = External Choice

Products in the vertical category (lenses) give external choice:

Triad A × Triad B   (product in Lens category)
        │
        │ project₁ or project₂
        ▼
    Chosen Triad

The Nashator is this choice operator, implemented as a state morphism.

Integration

With Gay.jl

using Gay

# Nashator color scheme
gay_seed!(0xNA5HAT0R)

NASHATOR_COLORS = (
    narrator = color_at(1),      # Central purple
    triad_a = [color_at(2), color_at(3), color_at(4)],
    triad_b = [color_at(5), color_at(6), color_at(7)],
)

# Verify GF(3)
@assert sum(trit.(values(NASHATOR_COLORS))) % 3 == 0

With Parallel Fanout

# 7-skill parallel dispatch
class Nashator < ParallelFanout
  SKILLS = {
    validator_a:   'three-match',
    coordinator_a: 'unworld', 
    generator_a:   'gay-mcp',
    narrator:      'semi-reliable-nashator',
    validator_b:   'sheaf-cohomology',
    coordinator_b: 'open-games',
    generator_b:   'topos-generate'
  }
  
  def dispatch!(task)
    seed = interaction_to_seed(task)
    children = SplitMixTernary.new(seed).fork(7)
    
    Parallel.map(SKILLS.to_a, in_threads: 7) do |(role, skill), child|
      invoke_skill(skill, task, child.seed)
    end
  end
end

Justfile Recipes

# Invoke the Nashator
nashator task:
    julia --project -e 'using Nashator; dispatch("{{task}}")'

# Market entry game demo
nashator-market-entry market_size="100":
    julia --project -e 'using Nashator; market_entry({{market_size}})'

# Verify GF(3) conservation across 7 skills
nashator-verify:
    bb scripts/​nashator_verify.bb

# Parallel 7-skill dispatch
nashator-parallel task:
    ruby lib/​nashator.rb dispatch "{{task}}"

GF(3) Conservation Proof

Triad A: (-1) + (0) + (+1) = 0
Nashator:                   = 0  
Triad B: (-1) + (0) + (+1) = 0
─────────────────────────────
Total:                      = 0 ≡ 0 (mod 3) ✓

References

1. Hedges, J. - "Morphisms of Open Games" (MFPS 2018)

   • Lenses ↔ compositional game theory connection
   • Globular morphisms preserve Nash equilibria
   • States as flexible solution concepts

2. Capucci, M. & Gavranović, B. - "Actegories for Open Games"

   • ⊛ represents agency on systems
   • Para(Lens) ≅ OpenGame

3. Ghani, Hedges, Winschel, Zahn - "Compositional Game Theory" (2016)

   • Symmetric monoidal category of open games
   • Backward induction as coplay

4. CyberCat Institute - Categorical Cybernetics

   • Control theory meets game theory

Related Skills

---

Skill Name: semi-reliable-nashator
Type: Compositional Game Theory / Parallel Dispatch
Trit: 0 (ERGODIC - narrator coordinates)
Structure: Symmetric Monoidal Double Category
Dispatch: Multiple dispatch → 7 parallel skills
For: Matteo Capucci, Jules Hedges, and all who lift fists like antennas

---

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Add Interaction Exemplars here as the skill is used.