# Five-arm fairness basket and correct-walk credit

All intervals are 95% cluster bootstraps (10000 resamples) over the 24 parameter sets, each set's five seeds resampled together. Paired contrasts join on `(instance_id, episode_seed)`.


## Fairness basket — conditional on a closed deal

Every column below is computed over closed deals only, so `deal rate` is carried beside them: an arm that closes 28 of 120 deals is describing a self-selected set of games, not the same games. Normalized coordinate z_i = (u_i − tau_i)/c_i with c_i the party's best surplus on the individually rational set (affine-invariant per party).

| arm | n | deal rate | below-thr. accept | worst-off z | worst-off share | dist-NBS | dist-KS | norm. Gini | norm. Nash welf. | max share |
|---|---:|---:|---:|---:|---:|---:|---:|---:|---:|---:|
| `all_llm` | 120 | 0.958 [0.925, 0.983] | 0.000 [0.000, 0.000] | 0.257 [0.207, 0.306] | 0.077 [0.064, 0.090] | 0.464 [0.357, 0.583] | 0.398 [0.316, 0.474] | 0.223 [0.200, 0.247] | 0.535 [0.459, 0.602] | 0.299 [0.288, 0.309] |
| `one_rational` | 120 | 0.767 [0.683, 0.842] | 0.000 [0.000, 0.000] | 0.239 [0.197, 0.282] | 0.073 [0.061, 0.084] | 0.444 [0.343, 0.568] | 0.414 [0.332, 0.491] | 0.231 [0.209, 0.255] | 0.506 [0.440, 0.564] | 0.301 [0.289, 0.313] |
| `one_oracle` | 120 | 0.917 [0.850, 0.967] | 0.000 [0.000, 0.000] | 0.201 [0.152, 0.252] | 0.059 [0.047, 0.070] | 0.550 [0.436, 0.674] | 0.524 [0.426, 0.614] | 0.266 [0.237, 0.293] | 0.472 [0.395, 0.546] | 0.314 [0.297, 0.331] |
| `all_rational` | 120 | 0.233 [0.133, 0.342] | 0.000 [0.000, 0.000] | 0.147 [0.103, 0.194] | 0.052 [0.037, 0.067] | 0.600 [0.500, 0.680] | 0.590 [0.521, 0.649] | 0.261 [0.238, 0.283] | 0.408 [0.316, 0.485] | 0.314 [0.299, 0.327] |
| `all_oracle` | 120 | 1.000 [1.000, 1.000] | 0.000 [0.000, 0.000] | 0.166 [0.118, 0.220] | 0.048 [0.037, 0.061] | 0.631 [0.520, 0.756] | 0.549 [0.449, 0.649] | 0.285 [0.256, 0.314] | 0.406 [0.316, 0.491] | 0.319 [0.303, 0.336] |

Lower is better for below-threshold accept, dist-NBS, dist-KS, Gini, and max share; higher is better for worst-off z, worst-off share, and normalized Nash welfare. An equal split puts worst-off share at 0.200 and max share at 0.200.


### Among-deals surplus (per-party z and episode score)

Both blocks condition on a closed deal, so deal rate is carried beside them: an arm's closed set is self-selected, and a low-deal-rate arm's columns describe a different, easier subset of games. Per-party z pools party-observations (five per closed deal, no per-deal averaging; mean with an instance-clustered bootstrap CI, quartiles empirical/descriptive); the episode-score block is descriptive (the unconditional headline already carries the arm's interval).

| arm | deals | deal rate | per-party z mean [95% CI] | z median | z IQR | score mean | score median | score IQR |
|---|---:|---:|---:|---:|---:|---:|---:|---:|
| `all_llm` | 115/120 | 0.958 | 0.653 [0.620, 0.690] | 0.667 | [0.433, 1.000] | 0.911 | 0.946 | [0.838, 1.000] |
| `one_rational` | 92/120 | 0.767 | 0.639 [0.605, 0.675] | 0.657 | [0.431, 1.000] | 0.895 | 0.931 | [0.825, 1.000] |
| `one_oracle` | 110/120 | 0.917 | 0.649 [0.612, 0.689] | 0.687 | [0.341, 1.000] | 0.904 | 0.961 | [0.829, 1.000] |
| `all_rational` | 28/120 | 0.233 | 0.554 [0.512, 0.611] | 0.567 | [0.329, 0.762] | 0.809 | 0.830 | [0.699, 0.922] |
| `all_oracle` | 120/120 | 1.000 | 0.645 [0.611, 0.681] | 0.684 | [0.310, 1.000] | 0.907 | 0.982 | [0.843, 1.000] |

## Unconditional companions

`below-thr. episode` counts a no-deal episode as 0 rather than dropping it, so it is the rate at which an arm's episodes END in an individually irrational agreement.

| arm | n | deal rate | below-thr. episode | normalized score | norm. Nash welf. (uncond.) |
|---|---:|---:|---:|---:|---:|
| `all_llm` | 120 | 0.958 [0.925, 0.983] | 0.000 [0.000, 0.000] | 0.873 [0.833, 0.910] | 0.512 [0.430, 0.586] |
| `one_rational` | 120 | 0.767 [0.683, 0.842] | 0.000 [0.000, 0.000] | 0.686 [0.613, 0.761] | 0.388 [0.320, 0.452] |
| `one_oracle` | 120 | 0.917 [0.850, 0.967] | 0.000 [0.000, 0.000] | 0.829 [0.767, 0.884] | 0.433 [0.349, 0.515] |
| `all_rational` | 120 | 0.233 [0.133, 0.342] | 0.000 [0.000, 0.000] | 0.189 [0.111, 0.278] | 0.095 [0.050, 0.145] |
| `all_oracle` | 120 | 1.000 [1.000, 1.000] | 0.000 [0.000, 0.000] | 0.907 [0.880, 0.933] | 0.406 [0.316, 0.491] |


## Paired fairness contrasts vs `all_llm`

On the conditional columns a pair contributes only when BOTH arms closed a deal on that `(instance, seed)`; the pair count is reported because it falls sharply for the low-deal-rate arms.

| arm | deal rate | below-thr. accept | worst-off z | worst-off share | norm. Gini | max share |
|---|---:|---:|---:|---:|---:|---:|
| `one_rational` | -0.192 [-0.275, -0.108] (n=120) | 0.000 [0.000, 0.000] (n=90) | -0.011 [-0.043, 0.020] (n=90) | -0.004 [-0.015, 0.006] (n=90) | 0.005 [-0.016, 0.026] (n=90) | 0.000 [-0.009, 0.010] (n=90) |
| `one_oracle` | -0.042 [-0.100, 0.017] (n=120) | 0.000 [0.000, 0.000] (n=106) | -0.060 [-0.092, -0.033] (n=106) | -0.020 [-0.030, -0.011] (n=106) | 0.043 [0.025, 0.065] (n=106) | 0.015 [0.005, 0.027] (n=106) |
| `all_rational` | -0.725 [-0.833, -0.608] (n=120) | 0.000 [0.000, 0.000] (n=28) | -0.096 [-0.163, -0.024] (n=28) | -0.028 [-0.051, -0.005] (n=28) | 0.031 [-0.003, 0.068] (n=28) | 0.006 [-0.013, 0.022] (n=28) |
| `all_oracle` | 0.042 [0.017, 0.075] (n=120) | 0.000 [0.000, 0.000] (n=115) | -0.086 [-0.130, -0.043] (n=115) | -0.028 [-0.042, -0.015] (n=115) | 0.061 [0.034, 0.091] (n=115) | 0.020 [0.007, 0.035] (n=115) |

## Correct-walk credit

**Every one of the 24 bank instances admits at least one weakly all-IR deal** (u_i ≥ tau_i for all five parties; IR-set sizes 4–47 of 256 deals). The discrete justified-walk fraction is therefore identically 0.000 in every arm and every stratum: **no no-deal episode in this campaign was a correct refusal of an infeasible game.** The metric charging zero for a walk is not, on this bank, charging zero for correct behaviour.

The continuous form carries what is left. The IR margin of an instance is max over deals of min_i z_i — how much room the best all-satisfying deal leaves the party it satisfies least. Across the bank it ranges 0.000–0.652 (mean 0.377). But feasibility is not comfortable everywhere: 3 of 24 instances sit at or below a 0.10 margin, and 2 of those are exactly 0.000 — the best all-satisfying deal in those games holds some party precisely at its threshold, so agreement is feasible on paper and knife-edge in practice. Those instances carry the tags hard, high-conflict, no-clear-win-win, pivotal-seat, small-frontier, which is the same corner of the bank the inverted difficulty gradient points at.


### No-deal episodes by arm and difficulty tag

Each cell is `no-deals/episodes (justified walks)`, where a walk is justified only if its instance admits no all-IR deal. The walk-weighted mean IR margin of each cell is in `summary.json`.

| tag | `all_llm` | `one_rational` | `one_oracle` | `all_rational` | `all_oracle` |
|---|---:|---:|---:|---:|---:|
| easy | 2/40 (0 just.) | 6/40 (0 just.) | 2/40 (0 just.) | 20/40 (0 just.) | 0/40 (0 just.) |
| hard | 2/40 (0 just.) | 16/40 (0 just.) | 5/40 (0 just.) | 39/40 (0 just.) | 0/40 (0 just.) |
| high-conflict | 0/30 (0 just.) | 6/30 (0 just.) | 1/30 (0 just.) | 26/30 (0 just.) | 0/30 (0 just.) |
| large-frontier | 2/30 (0 just.) | 7/30 (0 just.) | 2/30 (0 just.) | 22/30 (0 just.) | 0/30 (0 just.) |
| medium | 1/40 (0 just.) | 6/40 (0 just.) | 3/40 (0 just.) | 33/40 (0 just.) | 0/40 (0 just.) |
| no-clear-win-win | 2/30 (0 just.) | 11/30 (0 just.) | 4/30 (0 just.) | 28/30 (0 just.) | 0/30 (0 just.) |
| pivotal-seat | 2/30 (0 just.) | 13/30 (0 just.) | 5/30 (0 just.) | 29/30 (0 just.) | 0/30 (0 just.) |
| small-frontier | 2/30 (0 just.) | 6/30 (0 just.) | 2/30 (0 just.) | 26/30 (0 just.) | 0/30 (0 just.) |

### Bank IR margin by difficulty tag

| tag | instances | mean IR margin | min IR margin |
|---|---:|---:|---:|
| easy | 8 | 0.442 | 0.329 |
| hard | 8 | 0.297 | 0.000 |
| high-conflict | 6 | 0.354 | 0.086 |
| large-frontier | 6 | 0.430 | 0.310 |
| medium | 8 | 0.394 | 0.269 |
| no-clear-win-win | 6 | 0.320 | 0.000 |
| pivotal-seat | 6 | 0.236 | 0.000 |
| small-frontier | 6 | 0.331 | 0.000 |

### Does no-deal track thin margins?

Per-arm Pearson correlation between an instance's IR margin and its no-deal rate in that arm (negative = failures concentrate where agreement was thinnest).

| arm | r(IR margin, no-deal rate) [95% CI] | instances |
|---|---:|---:|
| `all_llm` | -0.465 [-0.823, 0.089] | 24 |
| `one_rational` | -0.249 [-0.698, 0.308] | 24 |
| `one_oracle` | -0.506 [-0.782, 0.098] | 24 |
| `all_rational` | -0.256 [-0.545, 0.082] | 24 |
| `all_oracle` | — | 24 |