# Ten-arm fairness-led comparison

> ## ⚠ ERRATUM (2026-08-10) — the two `OmniscientBestResponsePolicy` arms on this page (`one_oracle`, `all_oracle`) are measured on an agent with a spoiled ballot
>
> That agent cast its **forced-final** vote on whichever live offer it valued most instead of on the one offer
> under the up/down vote. The protocol rejects that as a legality error, the seat spends its single retry
> repeating itself, and the turn is recorded as a **pass** — a silent abstention. It consumed 114 of 334
> forced-final turns in `one_oracle` and 94 of 107 in `all_oracle`. Every other arm on this page — `all_llm`,
> `one_rational`, `all_rational`, `all_selfish_dp_oracle` and all four fairness arms — is **clean** (0
> mismatches over 9,589 re-derived turns; the fairness and DP arms are a different, composed policy).
> Fixed in commit `ca20157`, which postdates this campaign.
>
> **Corrected values, from the preregistered fresh-bank replication (research note 0039) and the ballot-repaired
> re-run (note 0045):**
>
> | published here (spoiled) | repaired |
> |---|---|
> | `one_oracle` − `all_llm` utilitarian **−0.412** | **−0.068** [−0.136, −0.001] |
> | `one_oracle` deal rate 0.508 (−0.450 paired) | ~8 points below `all_llm`, not 45 (0.558 → **0.892** on identical games) |
> | `one_fairness_oracle` − `one_oracle` utilitarian **+0.389** | correspondingly inflated — not the size of the objective swap |
> | `all_oracle` deal rate 0.875 (−0.083 paired) | **1.000** (+0.042 [+0.017, +0.075]) — all 15 no-deals were spoiled ballots |
> | "information amplifies motive", interaction +0.378 | **+0.019 [−0.052, +0.091]** — contains zero; **claim withdrawn** |
>
> What survives: a quarter-size interaction on the welfare coordinate (NNW +0.071 at one seat, +0.137 at five),
> and the reading that **information's first-order effect is agreement (motive-neutral) while motive decides
> distribution**. The `all_oracle` distributional finding survives and *sharpens* — repaired, it closes more
> often than all-LLM and still splits worse on every column.
>
> Nothing below is deleted; the numbers are the record of what was computed. Do not quote the affected rows
> unqualified, and never pool spoiled with repaired cells. Sources of record: research notes **0039**, **0045**,
> **0043**.


Reference arm for every paired contrast: `all_llm`. Intervals are 10000 cluster-bootstrap resamples over the 24 parameter sets (their five seeds resampled together). Paired contrasts join on `(instance_id, episode_seed)`; no-deal scores zero.


## Levels — both co-primaries side by side

`normalized_primary` is the utilitarian score the self-interested agents optimize; `normalized_nash_welfare` is the fairness objective's own coordinate. Each family is expected to lead on the metric it optimizes, which is why neither is reported alone.

| arm | normalized_primary | normalized_nash_welfare | deal_rate |
|---|---|---|---|
| `all_fairness_algorithmic` | 0.278 | 0.152 | 0.375 |
| `all_fairness_oracle` | 0.967 | 0.596 | 1.000 |
| `all_llm` | 0.873 | 0.512 | 0.958 |
| `all_oracle` | 0.791 | 0.370 | 0.875 |
| `all_rational` | 0.189 | 0.095 | 0.233 |
| `all_selfish_dp_oracle` | 0.871 | 0.397 | 1.000 |
| `one_fairness_algorithmic` | 0.697 | 0.408 | 0.808 |
| `one_fairness_oracle` | 0.850 | 0.492 | 0.933 |
| `one_oracle` | 0.461 | 0.240 | 0.508 |
| `one_rational` | 0.686 | 0.388 | 0.767 |

## Levels — the referee basket

`max_share` is the boundary-extraction detector (share of total normalized gain taken by the biggest winner); `worst_off` is the least-satisfied party's normalized gain. `walked_rate` and `expired_rate` split the no-deal episodes, which score identically on everything else.

| arm | dist_to_nbs | dist_to_ks | gini | worst_off | max_share | ir_rate | walked_rate | expired_rate |
|---|---|---|---|---|---|---|---|---|
| `all_fairness_algorithmic` | 0.567 | 0.537 | 0.324 | 0.129 | 0.334 | 0.375 | 0.000 | 0.625 |
| `all_fairness_oracle` | 0.091 | 0.092 | 0.252 | 0.357 | 0.275 | 1.000 | 0.000 | 0.000 |
| `all_llm` | 0.464 | 0.398 | 0.299 | 0.257 | 0.299 | 0.958 | 0.000 | 0.042 |
| `all_oracle` | 0.621 | 0.545 | 0.354 | 0.172 | 0.319 | 0.875 | 0.000 | 0.125 |
| `all_rational` | 0.600 | 0.590 | 0.311 | 0.147 | 0.314 | 0.233 | 0.000 | 0.767 |
| `all_selfish_dp_oracle` | 0.609 | 0.526 | 0.356 | 0.161 | 0.329 | 1.000 | 0.000 | 0.000 |
| `one_fairness_algorithmic` | 0.550 | 0.469 | 0.312 | 0.233 | 0.309 | 0.808 | 0.000 | 0.192 |
| `one_fairness_oracle` | 0.391 | 0.330 | 0.296 | 0.273 | 0.299 | 0.933 | 0.000 | 0.067 |
| `one_oracle` | 0.549 | 0.479 | 0.346 | 0.187 | 0.314 | 0.508 | 0.000 | 0.492 |
| `one_rational` | 0.444 | 0.414 | 0.294 | 0.239 | 0.301 | 0.767 | 0.000 | 0.233 |

## Levels — how agreement was reached

`turns_to_close` is conditional on a deal; `n_distinct_openers` counts the different deals proposed in the opening round, which is the focal-point statistic — agents optimizing the same table-level objective under the same information all open on one deal.

| arm | turns_to_close | n_distinct_openers | deal_rate |
|---|---|---|---|
| `all_fairness_algorithmic` | 13.000 | 2.300 | 0.375 |
| `all_fairness_oracle` | 5.000 | 1.000 | 1.000 |
| `all_llm` | 17.748 | 3.967 | 0.958 |
| `all_oracle` | 5.000 | 1.142 | 0.875 |
| `all_rational` | 24.821 | 3.600 | 0.233 |
| `all_selfish_dp_oracle` | 24.783 | 3.175 | 1.000 |
| `one_fairness_algorithmic` | 19.732 | 3.658 | 0.808 |
| `one_fairness_oracle` | 18.964 | 4.025 | 0.933 |
| `one_oracle` | 19.475 | 3.883 | 0.508 |
| `one_rational` | 21.391 | 3.917 | 0.767 |

## Paired effects vs `all_llm`

| arm | normalized_primary | normalized_nash_welfare | deal_rate | gini | worst_off | max_share |
|---|---|---|---|---|---|---|
| `all_fairness_algorithmic` | -0.595 | -0.361 | -0.583 | 0.058 | -0.134 | 0.032 |
| `all_fairness_oracle` | 0.094 | 0.083 | 0.042 | -0.053 | 0.108 | -0.024 |
| `all_oracle` | -0.082 | -0.143 | -0.083 | 0.053 | -0.075 | 0.019 |
| `all_rational` | -0.684 | -0.417 | -0.725 | 0.030 | -0.096 | 0.006 |
| `all_selfish_dp_oracle` | -0.001 | -0.116 | 0.042 | 0.054 | -0.090 | 0.031 |
| `one_fairness_algorithmic` | -0.175 | -0.104 | -0.150 | 0.006 | -0.021 | 0.008 |
| `one_fairness_oracle` | -0.023 | -0.020 | -0.025 | -0.004 | 0.018 | -0.001 |
| `one_oracle` | -0.412 | -0.273 | -0.450 | 0.040 | -0.070 | 0.018 |
| `one_rational` | -0.186 | -0.124 | -0.192 | -0.005 | -0.011 | 0.000 |

## Episodes per arm

- `all_fairness_algorithmic`: 120
- `all_fairness_oracle`: 120
- `all_llm`: 120
- `all_oracle`: 120
- `all_rational`: 120
- `all_selfish_dp_oracle`: 120
- `one_fairness_algorithmic`: 120
- `one_fairness_oracle`: 120
- `one_oracle`: 120
- `one_rational`: 120