---
license: cc-by-4.0
pretty_name: Nash Equilibria of 2x2 Normal-Form Games
tags:
  - game-theory
  - nash-equilibrium
  - normal-form-games
  - combinatorics
  - synthetic
size_categories:
  - 100M<n<1B
configs:
  - config_name: default
    data_files:
      - split: payoffs_0_10
        path: matrices_2x2_0_to_10.parquet
      - split: payoffs_0_5
        path: matrices_2x2_0_to_5.parquet
      - split: payoffs_0_3
        path: matrices_2x2_0_to_3.parquet
      - split: payoffs_0_2
        path: matrices_2x2_0_to_2.parquet
---

# Nash Equilibria of 2×2 Normal-Form Games

An **exhaustive enumeration** of every two-player 2×2 normal-form (bimatrix) game with
non-negative integer payoffs in a fixed range, each annotated with its pure- and
mixed-strategy Nash equilibria and a set of game-theoretic classifications.

Because the payoff space is enumerated completely, the largest split contains **all
`11^8 = 214,358,881`** games with payoffs in `0..10` — not a sample.

## Splits

Each split is the complete enumeration for a payoff range. Payoffs `0..k` is a strict
subset of payoffs `0..10` (identical rows), so the smaller splits are convenience
subsets of `payoffs_0_10`.

| split | payoff range | games (rows) |
|---|---|--:|
| `payoffs_0_10` | 0–10 | 214,358,881 |
| `payoffs_0_5`  | 0–5  | 1,679,616 |
| `payoffs_0_3`  | 0–3  | 65,536 |
| `payoffs_0_2`  | 0–2  | 6,561 |

```python
from datasets import load_dataset
ds = load_dataset("AlexDunstan/nash-equilibria-matrices", split="payoffs_0_10")
```

## Game encoding

A 2×2 game has two players (p1, p2) choosing a row / column. Each of the 4 cells holds a
payoff pair. The 8 payoff columns are named `r{row}c{col}_p{player}` (row-major, 0-indexed):

|  | col 0 | col 1 |
|---|---|---|
| **row 0** | `r0c0_p1`, `r0c0_p2` | `r0c1_p1`, `r0c1_p2` |
| **row 1** | `r1c0_p1`, `r1c0_p2` | `r1c1_p1`, `r1c1_p2` |

## Columns (32)

**Payoffs (8)** — `r0c0_p1 … r1c1_p2`: `int8`, the integer payoff to each player in each cell.

**Equilibria**
- `num_equilibria` (`int8`): number of pure-strategy Nash equilibria.
- `equilibrium_positions` (`string`): list of `(row, col)` pure-NE cells, e.g. `"[(0, 0), (1, 1)]"`.
- `category` (`string`): `Solved` if ≥1 pure NE, else `Unsolved`.

**Structural flags** (`bool`)
- `p1_has_dominant`, `p2_has_dominant`, `both_dominant`: dominant-strategy existence.
- `is_zero_sum`, `is_symmetric`: structural properties of the payoff matrix.
- `has_pareto_dom_ne`, `all_ne_pareto_eff`: Pareto properties of the equilibria.

**Welfare**
- `max_welfare` (`int16`): maximum total payoff (`p1+p2`) over all cells.
- `ne_welfare` (`string`): total welfare at each NE, e.g. `"[0, 2]"`.
- `welfare_loss` (`int16`): efficiency gap between `max_welfare` and the equilibria.

**Mixed-strategy equilibrium** (populated for 2×2 games with no pure NE; `null`/empty otherwise)
- `mixed_exists` (`bool`).
- `mixed_p`, `mixed_q` (`double`): equilibrium mixing probabilities.
- `mixed_payoff_p1`, `mixed_payoff_p2` (`double`): expected payoffs under the mixed NE.

**Payoff asymmetry at equilibria**
- `ne_p1_payoffs`, `ne_p2_payoffs`, `ne_payoff_diffs` (`string`): per-NE payoff lists and differences.
- `ne_has_equal_payoffs` (`bool`).
- `ne_mean_abs_diff` (`double`): mean absolute payoff difference across the equilibria.

**Classification**
- `game_type` (`string`): a categorical label. Distribution over `payoffs_0_10`:

| game_type | rows |
|---|--:|
| Harmony | 56,150,160 |
| Dominant (P1 only) | 51,967,234 |
| Dominant (P2 only) | 51,967,234 |
| Other | 18,295,200 |
| No Equilibrium | 18,283,848 |
| Deadlock | 11,809,512 |
| Prisoner's Dilemma | 5,814,336 |
| Zero-Sum | 65,307 |
| Coordination | 6,050 |

> **List columns are stored as strings** (Python `repr` of a list), for exact
> round-trip fidelity. Parse them with `ast.literal_eval`:
> ```python
> import ast
> positions = ast.literal_eval(row["equilibrium_positions"])
> ```

## Provenance

Games were generated by exhaustive Cartesian enumeration of the payoff space, then
enriched with equilibrium detection and classification computed deterministically from
the payoff matrix (no sampling, no randomness). Payoffs are compact integer types and
repetitive categorical/list columns are dictionary-encoded, so the full 214M-row split is
~172 MB of Parquet.

## License

CC-BY-4.0 — free to use with attribution.