Narrowing the Collaboration Gap, Probably
Mirah Shi, Marcel Hussing, Ira Globus-Harris, Natalie Collina, Aaron Roth, Surbhi Goel
cs.LG, cs.AI, cs.MA
2026-10-14
On partial-information mazes, raw verbalized probabilities from four LLMs are no better than actions for most models; post-hoc calibration at least doubles four-round success.
Two agents share one task, and each holds a slice the other cannot see. Robots with different sensors, and coding agents handed different files, are cut the same way. Bandwidth, context length, and views the partner cannot read all block shipping the raw observation. Davidson et al. call the failure to pool those views the collaboration gap.
The cheapest message is the favorite action. One choice of right can mean the cell to the right is the only open path, or only that the maze starts at the top left and right is a strong prior. The receiver gets a single symbol, with no read on how much to trust it, and no way to fold it into a private map.
Agents send a distribution over actions, and it has to be calibrated. A report of 0.7 should be correct about 70% of the time. Across rounds the transcript also has to be conversation-calibrated: given the previous probability, the current one still equals the true conditional frequency.
The theory model is chained parity. There are K independent bits in {−1, +1}. Alice sees the odd indices, Bob the even ones. The target probability is 1/2 plus a weighted sum of prefix parities, with positive weights summing to less than 1/2. A prefix parity is an intermediate conclusion on a reasoning chain.
When distinct prefixes produce distinct partial sums, for example weights that halve at each step, the previous scalar identifies the prefix already revealed. The speaker folds in the new bit it sees. The message equals the Bayes posterior given everything revealed so far, and after K rounds it equals the full-information posterior. Excess squared loss drops by the square of that round's weight.
An action keeps only the sign. When the first weight exceeds the second, every round repeats the sign of the first bit. For K=3 and weights 0.20, 0.15, 0.10, that decision disagrees with the full-information Bayes action with probability 1/4.
Real reports get warped. A coefficient c inflates or shrinks confidence, and λ pulls the current report toward the previous one. That pull is sycophantic anchoring. Post-hoc calibration does not train the base model. Each round a map sees the raw forecast and the previous calibrated forecast, and is fit to the label with a strictly proper loss, whose population minimum is the true conditional probability. Once the previous message is fixed, an invertible warp is unwound back to the ideal posterior, and the result stays calibrated given that previous message.
The maze is the collaborative path-finding task from Davidson et al., restricted to a unique solution. Grids are 6 by 6 with 30% walls, from the top left to the bottom right, and each agent has half the cells hidden. A maze is solved only when every step of the solution path is correct. Small transformers send a softmax. Pretrained models compare a chosen action, a probability written in text, and the frequency of M samples. A one-hidden-layer MLP is fit round by round on 100 mazes and tested on 30 mazes with 10 rollouts each.
On chained parity, trained probability messages reach the theoretical optimum by round 5. Action messages stall.
On controlled mazes, unbiased probability communication improves success monotonically over four rounds. Action communication stalls. Warping the transformer probabilities with the affine map makes success fall across rounds, for global miscalibration and for miscalibration plus anchoring. Calibrating the current round alone recovers part of the loss, and the residual accumulates. Fitting the current round together with the previous one returns nearly to the unbiased baseline when both biases are present. Per-round point estimates are not in the text.
The pretrained models are GPT-4.1-mini, GPT-5, GPT-5.4-nano, and GPT-5.4-mini, with at most 2048 reasoning tokens. The eval uses 30 fixed mazes and 10 rollouts, with 90% bootstrap intervals. Verbalized probabilities do not beat actions on nano, GPT-5.4-mini, or GPT-5, and they help only slightly on GPT-4.1-mini.
| Method | Metric | Result |
| Probability exchange (chained parity) | Aggregation | Reaches the full-information posterior after K rounds; actions stay on the first bit and, for K=3, disagree with the Bayes action with probability 1/4 |
| Verbalized probabilities plus post-hoc calibration | Four-round maze success | At least 2× uncalibrated verbalized probabilities on all four models, with lower calibration loss |
| Post-hoc calibration (1 generation) | Versus M=10 samples | Within variance on GPT-4.1-mini; higher mean success on GPT-5.4-nano |
| Calibrated probabilities | Versus the full response | Full responses stay near 0 success every round; with the full maze, GPT-5.4-nano solves more than 85% |
The correct move is down or right 97% of the time and up or left 3%. A naive baseline that only knows this margin replaces a mode on up or left with 0.5 down and 0.5 right. On nano it is close to the calibrator. On GPT-5, after several rounds, the calibrator is slightly ahead, so the fit goes past the marginal frequencies.
With M=10, sampled probabilities beat verbalized ones on GPT-4.1-mini and land in the same variance band as calibration. On nano they stay close to verbalized probabilities, and calibration has the higher mean. In round 2, mini more often puts probability 1 on its top action and those calls are reasonably accurate, while nano is more overconfident below 1. One generation plus the calibrator can match or beat ten samples.
The small calibrator does not retrain the base model. It rewrites the probabilities that enter the next context, and all four models improve. At communication time, sampling spends 10 generations and calibration spends 1. A verbalized confidence is not a posterior. Forwarding a long chain of thought verbatim does not solve this maze either.
The mazes have one solution, side length 6, 30% walls, and fixed corners. With 97% of correct moves down or right, nano's calibrator is nearly the rule that replaces an up or left mode with half down and half right, and GPT-5 is only slightly better. These comparisons do not show that fine-grained uncertainty is in use.
At least double comes without per-round point estimates in the text. The intervals sit in the figures. Thirty mazes do not carry a claim about open-ended dialogue. The calibrator never reads the map, so a low-information report cannot be repaired, and the fit needs labels. LLM success stays far below a next-move model trained on maze data.
The action channel locks only when partial sums are distinguishable and the first weight is larger. Invertible warps can be undone. Real model bias need not be invertible. Anchor strength, and whether RLHF is what sharpens the samples, is not measured in these conversations. The full-response control for nano used an edited prompt, because its reasoning is hidden behind the API, and that difference is mixed into the near-zero scores.