Relation: Outcomes
Module: Rock_Paper_Scissors
Relation Type: Full Outcome Matrix
ID: outcomes
1. Outcome Matrix#
Rows = Player A's choice. Cell = outcome for Player A.
+1 = win · 0 = draw · -1 = loss
| vs. Rock | vs. Paper | vs. Scissors | |
|---|---|---|---|
| Rock | 0 | −1 | +1 |
| Paper | +1 | 0 | −1 |
| Scissors | −1 | +1 | 0 |
Matrix is antisymmetric and skew-symmetric (M = −Mᵀ) — hallmark of zero-sum symmetric games.
2. Formal Properties#
| Property | Value |
|---|---|
| Zero-sum | ✅ |
| Antisymmetric | ✅ |
| Diagonal | All 0 |
| Matrix rank | 2 |
3. JSON Encoding#
{
"relation_id": "outcomes",
"encoding": {"win": 1, "draw": 0, "loss": -1},
"matrix": {
"rock": {"vs_rock": 0, "vs_paper": -1, "vs_scissors": 1},
"paper": {"vs_rock": 1, "vs_paper": 0, "vs_scissors": -1},
"scissors": {"vs_rock": -1, "vs_paper": 1, "vs_scissors": 0}
},
"nash_equilibrium": "uniform_mixed_1/3"
}4. Expected Value Table (Nash Play)#
| Matchup Scenario | E[Payoff for A] |
|---|---|
| Nash vs. Nash | 0.000 |
| Nash vs. Rock-only | +0.333 |
| Nash vs. Paper-only | +0.333 |
| Nash vs. Scissors-only | +0.333 |
| Pure strategy vs. Nash | −0.333 |