Rock Paper Scissors — TriadicFrameworks Module
module.json— Agentic module schema role assignments
Path:
docs/Rock_Paper_Scissors/
Version: 1.0.0
Updated: 2026-09-09
Domain: Agentic Science · Game-Theoretic Triadics
Status:stable
What This Module Is#
Rock Paper Scissors (RPS) is encoded here not as a game implementation, but as a formal triadic object — a minimal, closed system of three entities whose structure instantiates the full logic of Peircean Firstness, Secondness, and Thirdness, cyclic zero-sum dominance, and multi-tier agentic decision-making.
This module is the canonical entry-level benchmark for the TriadicFrameworks project. It is small enough to be understood completely and rich enough to demonstrate every structural feature of the framework:
- Three entities, each fully defined as a triadic node
- A strict cyclic dominance relation with no transitive closure
- A complete signed payoff matrix
- A multi-tier agentic protocol grounded in Peircean sign types
- A formally derived Nash equilibrium with triadic interpretation
Module Map#
Rock_Paper_Scissors/
│
├── README.md ← You are here
├── index.md ← Module overview, entity registry, tag index
├── module.json ← Machine-readable manifest (full schema)
├── module.yaml ← Lightweight machine-readable manifest
│
├── entities/
│ ├── Rock.md ← Firstness · Qualisign · crushes Scissors
│ ├── Paper.md ← Secondness · Sinsign · covers Rock
│ └── Scissors.md ← Thirdness · Legisign · cuts Paper
│
├── relations/
│ ├── dominance.md ← Cyclic dominance: formal definition, Z₃ symmetry
│ └── outcomes.md ← Full 3×3 signed outcome matrix + EV analysis
│
└── agentic/
├── agent_protocol.md ← Tier 0 / 1 / 2 agent decision protocol (Python)
└── Nash_equilibrium.md ← Formal equilibrium derivation + triadic reading
Core Concept: The Triadic Structure#
Each of the three entities maps to one Peircean phenomenological category:
| Entity | Position | Sign Type | Defining Quality |
|---|---|---|---|
| Rock | Firstness | Qualisign | Pure immediacy; brute presence; solidity |
| Paper | Secondness | Sinsign | Relational coverage; dyadic mediation |
| Scissors | Thirdness | Legisign | Differentiation; law; habit-cutting |
The dominance cycle follows directly from this ordering:
Rock (Firstness)
↗ ↘
Paper Scissors
(Secondness) ← (Thirdness)
Firstness crushes Thirdness · Thirdness cuts Secondness · Secondness covers Firstness
No mode universally dominates — the cycle encodes the irreducibility of all three categories.
Entity Quick Reference#
| Entity | Beats | Beaten By | Draw | Nash Weight | Novice Freq. |
|---|---|---|---|---|---|
| Rock | Scissors | Paper | Rock | 1/3 | ~35.4% |
| Paper | Rock | Scissors | Paper | 1/3 | ~31.7% |
| Scissors | Paper | Rock | Scissors | 1/3 | ~32.9% |
Outcome Matrix#
Player A's payoff. +1 = win · 0 = draw · −1 = loss.
| vs. Rock | vs. Paper | vs. Scissors | |
|---|---|---|---|
| Rock | 0 | −1 | +1 |
| Paper | +1 | 0 | −1 |
| Scissors | −1 | +1 | 0 |
The matrix is skew-symmetric (M = −Mᵀ), consistent with zero-sum structure. The unique Nash equilibrium is the *uniform mixed strategy σ = (1/3, 1/3, 1/3)**.
Agentic Protocol Summary#
Three tiers of agent are defined, each corresponding to a Peircean sign mode:
| Tier | Name | Sign Mode | Strategy |
|---|---|---|---|
| 0 | Naive | Firstness | Uniform random — implements Nash |
| 1 | Adaptive | Secondness | Counters opponent's most frequent observed throw |
| 2 | Triadic | Thirdness | Models opponent's tier; counters their anticipated counter |
Full Python implementations and the agentic decision loop are in agentic/agent_protocol.md.
Nash Equilibrium#
The unique Nash equilibrium σ* = (1/3, 1/3, 1/3) is the triadic null state — perfect equipoise across all three Peircean modes. An agent at Nash is maximally unpredictable and therefore maximally robust against exploitation.
Any deviation from σ* is exploitable:
| Deviation | Optimal Counter | Exploiter EV |
|---|---|---|
| Rock-heavy (>1/3) | Paper always | > 0 |
| Paper-heavy (>1/3) | Scissors always | > 0 |
| Scissors-heavy (>1/3) | Rock always | > 0 |
Full derivation in agentic/Nash_equilibrium.md.
How to Use This Module#
As a Framework Reference#
Read index.md for the entity registry, then each entities/*.md file for full triadic definitions. The relations/ files specify the structural backbone; agentic/ files specify how agents operate within the module.
As an Agentic Benchmark#
Import module.json for a machine-readable graph of all entities, relations, and outcome weights. Use the agentic tier protocol as a baseline for evaluating an inference engine's strategic reasoning depth.
As a Teaching Example#
RPS is the minimum viable triadic system: 3 entities, 1 relation type, complete closure. Use it to explain triadic frameworks before introducing more complex modules.
Key Properties of This Module#
| Property | Value |
|---|---|
| Entity count | 3 |
| Relation type | Cyclic dominance (Z₃) |
| Payoff structure | Zero-sum, skew-symmetric |
| Nash equilibrium | Unique mixed strategy (1/3, 1/3, 1/3) |
| Triadic completeness | Full — all three positions occupied |
| Pure NE | None |
| Evolutionary stability | ESS (neutrally stable orbit) |
| Peirce sign coverage | Qualisign · Sinsign · Legisign |
Related Modules#
| Module | Relation |
|---|---|
Core_Schema |
Base triadic ontology this module instantiates |
Agentic_Protocols |
General agent decision framework; RPS is a benchmark case |
Tags#
triadic · game-theory · zero-sum · cyclic-dominance · Peircean · agentic · benchmark · Firstness · Secondness · Thirdness · Nash · Z3