Nash Equilibrium Reference
Module: Rock_Paper_Scissors
ID: nash_equilibrium
1. Game Setup#
- Players: {Agent A, Agent B}
- Action set: {Rock, Paper, Scissors}
- Payoff: zero-sum; u_A + u_B = 0
2. No Pure Strategy Equilibrium#
For any pure profile (s_A, s_B), the opponent can always deviate to win. Therefore no pure-strategy Nash equilibrium exists.
3. Mixed Strategy Derivation#
Let σ_A = (p_R, p_P, p_S). For B to be indifferent:
E_B[Rock] = p_P − p_S
E_B[Paper] = p_S − p_R
E_B[Scissors] = p_R − p_P
Setting all equal with p_R + p_P + p_S = 1 yields:
σ = (1/3, 1/3, 1/3)* — the unique Nash equilibrium.
4. Equilibrium Properties#
| Property | Value |
|---|---|
| Type | Mixed strategy (unique) |
| Expected payoff (each) | 0 |
| Pure strategy equilibria | None |
| Strategy support | Full (all 3 actions) |
| Evolutionary stability | ESS (neutrally stable orbit) |
5. Triadic Interpretation#
σ* = (1/3, 1/3, 1/3) is triadic equipoise: Firstness, Secondness, and Thirdness receive equal weight. No mode dominates — the equilibrium is the triadic null state: pure semiotic openness.
| Mode | Entity | Nash Weight | Triadic Meaning |
|---|---|---|---|
| Firstness | Rock | 1/3 | Equal weight on pure immediacy |
| Secondness | Paper | 1/3 | Equal weight on relational mediation |
| Thirdness | Scissors | 1/3 | Equal weight on differentiation |
6. Exploitability by Deviation#
| Deviation | Optimal Exploit | E[payoff] for Exploiter |
|---|---|---|
| Rock-heavy (p_R > 1/3) | Play Paper always | > 0 |
| Paper-heavy (p_P > 1/3) | Play Scissors | > 0 |
| Scissors-heavy (p_S > 1/3) | Play Rock | > 0 |
| Any pure strategy | Dominant counter | +1 |
7. Evolutionary Stability#
σ* is an ESS: a population playing σ* cannot be invaded by any mutant pure strategy. The replicator dynamic converges to a neutrally stable orbit around σ*, making RPS a canonical example of non-convergent cyclic dynamics in evolutionary game theory.