Improving Cooperation in Language Games with Bayesian Inference and the Cognitive Hierarchy

Christopher Archibald; Diego Blaylock; Joseph Bills

REVIEW 5 major objections 5 minor 37 references

Improving Cooperation in Language Games with Bayesian Inference and the Cognitive Hierarchy

T0 review · 5 major / 5 minor · reviewed 2026-08-11 · deepseek-v4-flash

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Pith's one-line read A Bayesian spymaster that tracks probabilities over possible teammate embeddings and hierarchy levels outperforms the best static spymaster against out-of-distribution guessers in Codenames.

desk verdict Bayesian spymaster in Codenames shows real out-of-distribution gains, but the paper doesn't isolate whether Bayesian updating or noise-robust clue selection drives them. read the letter →

arxiv 2412.12409 v1 pith:UGHUAEPN submitted 2024-12-16 cs.AI cs.GT

Joseph Bills , Christopher Archibald , Diego Blaylock This is my paper · ORCID

classification cs.AIcs.GT

keywords CodenamesBayesianinferencecognitivehierarchywordembeddingscooperativelanguagegamesteammatemodelingadhocteamworksemanticsandpragmatics

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

This paper argues that an agent in a cooperative language game can adapt to an unknown partner by keeping a probability distribution over possible partner models, updating it with Bayes' rule after each observed message, and choosing actions that maximize expected utility under that posterior. The authors build such Bayesian agents for Codenames, using a small set of word embeddings for semantic uncertainty and a deductive/cognitive hierarchy for pragmatic uncertainty, with Gaussian noise added to embeddings so every observation stays possible. The central result is that the Bayesian spymaster that assumes noise outperforms the best static spymaster against out-of-distribution guessers, raising average win rate from 0.718 to 0.872 in the deterministic environment and from 0.468 to 0.625 in the stochastic one. This matters because previous Codenames AI required knowing the partner's embedding in advance; a single adaptive agent can now play well with unseen partners.

What carries the argument

The engine is a posterior distribution over partner models, where each model is a pair of a word embedding and a cognitive-hierarchy level. The likelihood of an observed guess or clue is evaluated by Monte Carlo sampling: the agent perturbs the relevant word's embedding with isotropic Gaussian noise (sigma = 1.0 for the noisy agents), simulates the partner's deterministic strategy, and uses the frequency of the observed action as its probability. The agent then chooses the clue or guess that maximizes the expected value of a turn-level heuristic utility function. This turns a discrete set of language models into a smoothly adaptable policy.

What would settle it

Pair the noisy Bayesian spymaster with a guesser whose embedding is a deterministic rotation or translation of one of the internal embeddings, a clearly non-Gaussian semantic shift. If the Bayesian spymaster does not beat the best static spymaster in this setting, the claim that it adapts to any unseen semantics fails. A complementary check is to collect the actual distribution of inter-agent guess differences and test whether they are Gaussian with variance near 1.0.

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Extended reading notes

Core claim

The central claim is that a Bayesian spymaster can learn which of a set of candidate guessers it is playing with, and that this inference, combined with hedging under embedding noise, yields better cooperation than any fixed-embedding spymaster when the partner is out of distribution. Concretely, the noisy Bayesian spymaster (S-hat-MB) achieves average win rates of 0.872 against guessers built from embeddings not in its model set, versus 0.718 for the best static spymaster; in the stochastic environment the gap is 0.625 versus 0.468. The paper also claims this is the first agent with a probability distribution over a set of language models, updated from observations, and that the same framework extends to guessers, although the Bayesian guessers are robust rather than dominant.

Load-bearing premise

The whole approach hinges on the assumption that any real teammate's behavior is well approximated by one of a small finite set of word-embedding plus hierarchy-level models, with all leftover variation being isotropic Gaussian noise added to word embeddings with a fixed standard deviation of 1.0; real semantic differences that are not noise-like would send the Bayesian updates off course.

Editorial extensions

If this is right

A single Bayesian spymaster can outperform any fixed-embedding spymaster against out-of-distribution guessers, so Codenames agents no longer need to know their partner's embedding in advance.
The framework generalizes the deductive hierarchy: with one model and no noise, the Bayesian spymaster reduces exactly to a level-k spymaster, and with multiple models it hedges across them.
Even in a deterministic environment with an in-distribution partner (the d2v guesser), the Bayesian spymaster beats the matching static spymaster by distinguishing card types and avoiding the assassin when forced to give a bad clue.
Bayesian guessers are more robust than static ones but rarely beat the best static guesser, because clues carry less information about the spymaster's identity than guesses carry about the guesser; the paper identifies this as a limitation.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

A large part of the gain may come from the smoothing that Gaussian noise injects rather than from accurate model identification; comparing the noisy Bayesian spymaster to a non-Bayesian ensemble that averages over all candidate models without updating would isolate the value of the posterior itself.
If the Gaussian-noise assumption holds, the same recipe could be applied to human partners by treating human semantic and pragmatic variation as a mixture over embedding-based hierarchy models, a step the authors leave untested.
A productive stress test would be to use deliberately structured, non-Gaussian semantic shifts such as rotating or translating one internal embedding; the current design has no mechanism for such shifts, so the claim of adapting to any unseen semantics would be sharply tested.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

The paper is worth a serious look. It gives the first Bayesian agent that keeps a posterior over language models and updates it from teammate behavior in Codenames, which is a genuine step beyond the static spymasters and guessers that dominate prior work. The Table 1 spymaster results are the main event: with out-of-distribution guessers, the noise-perturbed Bayesian spymaster (S-hat-MB) beats the best static spymaster by a wide margin in both deterministic (0.872 vs. 0.718) and stochastic (0.625 vs. 0.468) settings. That is a practically meaningful improvement for cross-embedding play, a known failure mode in Codenames AI. Full credit for releasing code and using a broad set of internal and external embeddings.

The soft spots are real, but they are mostly addressable rather than fatal. The stress-test concern about confounding is the important one: S-hat-MB differs from the best-model baseline by retaining an updating posterior, but it also uses Gaussian-perturbed clue embeddings and expected-utility maximization over the internal model set. There is no ablation with a static uniform-posterior spymaster that uses the same noise. The authors attribute the gain to “rapidly learns which of its model guessers is the best fit,” but the data do not isolate that mechanism. They need either a no-update baseline or a belief-convergence analysis. Similarly, Algorithm 11 multiplies by unnormalized pseudo-counts from the Monte Carlo samples; this is not obviously a well-normalized Bayesian update, and the dependence on sample count should be clarified. The Table 2 guesser results are also weaker than the abstract implies—the Bayesian guessers usually lose to the best static model—and the Discussion says this, but the abstract overstates the case. Minor but real: there are unfinished manuscript artifacts (e.g., “dosomethings” in Algorithm 2, “What is the input to this function?” in Algorithm 14). No error bars or significance tests are reported, though 500 games per pairing should be enough to make them meaningful.

Who gets value from this? Researchers in ad hoc teamwork, language games, and Codenames AI. The central spymaster finding is likely to survive a rewrite; the hard part is stating what exactly the Bayesian inference contributes versus robust clue selection under noise. I would send this to peer review with a clear request for an ablation and cleanup.

For me: bring it to reading group, yes; cite it when discussing cross-embedding Codenames agents; and treat it as an interesting empirical result whose mechanism needs one more experiment.

Referee Report

5 major / 5 minor

Summary. The paper proposes Bayesian agents for the cooperative language game Codenames. The agents maintain a probability distribution over teammate models that combine semantic uncertainty (a finite set of word embeddings) with pragmatic uncertainty (levels of a deductive/cognitive hierarchy), and add Gaussian noise to embeddings to model fine-grained uncertainty. The Bayesian spymaster updates beliefs from observed guesses and selects clues by maximizing expected heuristic utility; the Bayesian guesser updates beliefs from observed clues and uses thresholds to decide guesses. Experiments compare these agents against static baselines across in-distribution and out-of-distribution word embeddings, in deterministic and stochastic environments. The main positive result is that the noise-assuming Bayesian spymaster (S-hat-MB) improves win rates over the best static spymaster, especially out-of-distribution (Table 1: 0.872 vs 0.718 deterministic; 0.625 vs 0.468 stochastic). The Bayesian guessers, however, mostly underperform the best static guesser (Table 2).

Significance. If the spymaster improvement is robust and can be attributed to Bayesian inference, the paper would be a useful contribution to ad hoc teamwork in language games, demonstrating a principled way to combine semantic and pragmatic uncertainty and adapt to an unknown teammate. The paper's strengths include the explicit construction of a model set over embeddings and hierarchy levels, evaluation with external out-of-distribution embeddings, 500 games per pairing, and released source code. The significance is currently conditional: the central claim lacks an ablation that isolates posterior updating from the other mechanisms that differ between the Bayesian and baseline spymasters, and the guesser results contradict any broad claim that the Bayesian agents generally perform better. With the missing attribution and statistical detail supplied, the framework would be a solid contribution to the Codenames AI literature.

major comments (5)

[Section 6, Table 1 and Section 7] The central claim that Bayesian inference improves cooperation is not supported by an ablation that isolates Bayesian posterior updating from the other differences between S-hat-MB and the Best Model baseline. S-hat-MB differs from the baseline in at least four ways: it maintains and updates a posterior, it uses Gaussian-perturbed clue embeddings, it aggregates expected utility across the internal model set, and it uses the heuristic utility function with assassin avoidance. The Discussion attributes the gain to the spymaster 'rapidly learning which of its model guessers is the best fit,' but a static spymaster using the same noise and a fixed uniform posterior (a non-adaptive ensemble) is not reported. Without this control, the observed improvement could come entirely from noise-robust clue selection or from the heuristic, with Bayesian learning contributing little or nothing. Please add an ablation with a uniform-posterior, same-noise spymaster, and ideally a version that updates beliefs but uses noiseless clue selection.
[Appendix B, Algorithm 11 and 'Estimating Likelihood'] The belief update in Algorithm 11 multiplies the previous posterior by h_t(g,l,n), an unnormalized pseudo-count that grows with the number of Monte Carlo samples s used during clue selection. As written, the posterior depends on the arbitrary sampling count: with s=10, the likelihood factor is roughly an integer between 1 and 10, and multiplying by this factor each turn can inflate the belief values without normalization. The appendix says the pseudo-count is used as the likelihood 'since it does not need to be normalized,' but a product of unnormalized counts is not a likelihood and its magnitude is not invariant to s. Please specify the normalized likelihood (or the Dirichlet posterior mean) actually used in the implementation, or demonstrate that the update is invariant to s in the reported experiments.
[Abstract, Section 1, and Section 6, Table 2] The broad claim that the Bayesian agents 'perform better' in experiments is contradicted by the guesser results. In Table 2, all Bayesian guessers have lower average win rates than the Best Model for out-of-distribution spymasters, and in the stochastic in-distribution environment only DGB-hat marginally exceeds the baseline (0.733 vs 0.708). The Discussion acknowledges this, but the abstract and conclusion still present a general improvement claim. The paper should either restrict the contribution to the Bayesian spymaster or provide a clear explanation and supporting evidence for why the guesser results do not undermine the claim that Bayesian inference improves cooperation.
[Section 6, 'Experimental Results'] No error bars, confidence intervals, or significance tests are reported despite 500 games per pairing. Some comparisons in Table 1 are close (e.g., SMB vs Best Model in the stochastic in-distribution column, 0.370 vs 0.370, and 0.477 vs 0.468 out-of-distribution), so it is impossible to tell which differences are reliable. Please report standard errors or confidence intervals for the win rates, and ideally paired or per-board variance information, especially for the headline out-of-distribution comparisons.
[Section 3 and Section 5] The likelihood model assumes isotropic Gaussian perturbation of word embeddings with a fixed variance (sigma=1.0 in the experiments), but no evidence is provided that inter-agent semantic variation is well described by this noise model. A misspecified likelihood can cause the posterior to concentrate on the wrong teammate model. Since the Bayesian mechanism is the paper's contribution, please add a sensitivity analysis over sigma and, if feasible, a comparison with a heavier-tailed noise distribution, to show that the reported gains are not an artifact of this specific Gaussian assumption.

minor comments (5)

[Table 2] The entry '604' in the stochastic in-distribution column for BGB appears to be missing a leading decimal point and should likely be 0.604.
[Appendix B, Algorithm 14] Algorithm 14 contains the placeholder line 'Require: What is the input to this function?' and does not specify its inputs; please complete the pseudocode.
[Appendix A, Algorithms 2 and 6] Algorithm 2 contains the placeholder line 'dosomethings', and Algorithm 6 has typos such as 'vecor' and an unclosed parenthesis in 'order.insert(c,i'; these should be fixed for reproducibility.
[Section 5] The likelihood integral in Section 5 uses the notation N(m_t(w), sigma, x) without defining whether sigma is a scalar, vector, or covariance matrix, and without specifying the normalization; please clarify the notation.
[Section 1 and Section 2] The claim that this is 'the first example of an agent having a probability distribution over a set of language models' is qualified as 'to our knowledge,' but the paper would be stronger if it briefly stated the scope of the literature search or cited related work on probabilistic mixtures of embedding spaces, so readers can evaluate the novelty claim.

Circularity Check

0 steps flagged · score 2.0 of 10

No significant circularity: the Bayesian pipeline is empirically self-contained; the lone self-citation (deductive hierarchy, [4]) is a non-load-bearing component.

full rationale

The paper's derivation chain is self-contained. The Bayesian spymaster and guesser maintain a posterior over a fixed, a priori chosen model set (w2v, g3, cnnb, d2v), with likelihoods estimated by Monte Carlo sampling under Gaussian embedding noise (sigma=1.0); no parameter is fitted to the reported win-rate data and then re-presented as a prediction. Table 1 and Table 2 report empirical win rates against static baselines, including out-of-distribution guessers and spymasters using external embeddings (g1, ftxt, wg, elmo) not in the agent's model set, so the central claim (Bayesian adaptation improves cooperation under semantic uncertainty) is benchmarked against data external to the agent's construction. The only self-citation is the deductive hierarchy [4], which supplies the level-k pragmatics models as a component; the paper explicitly frames the Bayesian framework as an extension of that prior work ('Our proposed Bayesian framework can be viewed as an extension of the hierarchy'), and the hierarchy's properties are not invoked to derive the win-rate outcomes. The equivalence statements ('it behaves identically to the level-k guesser' at skip=0, belief=1; 'equivalent to a level-k spymaster from [4]' with no noise and a singleton model set) are consistency sanity checks, not load-bearing reductions. Remaining concerns, such as the absence of an ablation isolating posterior updating from noise-robust clue selection, or the unnormalized pseudo-count likelihood in Algorithm 11, bear on causal attribution and numerical correctness rather than circularity: the empirical comparison remains meaningful regardless of which mechanism drives the gains. The score of 2 reflects only the minor, non-load-bearing self-citation, with no circular step exhibited.

Assumptions & free parameters 4 free parameters · 6 assumptions · 0 invented entities

The central result depends on the coverage of the finite model set, the Gaussian noise model, the utility heuristic, and the clue filter; none of these are derived from first principles, but they are standard ad hoc engineering choices for this game.

free parameters (4)

Embedding noise sigma = 1.0
Chosen by hand to model fine-grained semantic uncertainty; no sensitivity analysis is reported, and likelihood estimates and win rates depend on it.
Skip threshold s = 0 (DGB), 1 (BGB), 0.5 (MGB)
Hand-selected to define three Bayesian guesser variants; affects which cards are skipped as non-team cards and thus the guesser's behavior.
Belief threshold u = 1 (DGB), 0 (BGB), 0.5 (MGB)
Hand-selected confidence threshold for guessing a card; directly controls how many cards are guessed.
Monte Carlo samples = 10 for spymaster, 1000-10000 for guesser
Number of noisy embedding samples used to estimate likelihoods and expected utilities; higher counts reduce variance, but 10 for the spymaster is small and may add noise to the results.

assumptions (6)

standard math Bayes' rule and the laws of probability as used in the posterior updates.
Invoked in Sections 3-5 without proof; standard.
domain assumption Word embeddings of the chosen set (w2v, g3, cnnb, d2v) adequately represent the semantics of Codenames words, and similarity in the embedding space predicts which words a spymaster connects.
Section 2: 'All language models we utilize in this work are vector-space word embeddings.' This is the basis of both static and Bayesian agents.
domain assumption The deductive hierarchy from [4] correctly models pragmatic differences between Codenames agents, including best-response relationships between levels.
Section 3: 'The deductive hierarchy is an organization of agents for playing Codenames initially proposed in [4].' The Bayesian agents use hierarchy levels as the pragmatic dimension.
ad hoc to paper Gaussian noise added to word embeddings simulates fine-grained semantic uncertainty and guarantees non-zero likelihood for all actions.
Section 3: 'we use Gaussian noise to simulate uncertainty in the implied communication channel.' This is a modeling choice with no independent justification; the variance sigma=1.0 is fixed for all experiments.
ad hoc to paper The heuristic utility function v(gamma,A) with u(red)=1, u(blue)=-1, u(bystander)=0, u(assassin)=-|R| is an adequate proxy for the true objective of winning the game.
Section 3 and Appendix C define this heuristic and claim it computes marginal contribution to score; its use to select actions is an approximation, not a proven optimality.
ad hoc to paper The clue optimization in Algorithm 15, which discards clues where all model guessers guess an incorrect card under no noise, does not exclude the optimal clue.
Section 4 and Appendix C state this optimization 'performs well in practice' but offer no guarantee; it can in principle skip a clue that would be best under noise.

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Cite this review

Pith. "Pith review of Improving Cooperation in Language Games with Bayesian Inference and the Cognitive Hierarchy." pith.science (2026). https://pith.science/paper/UGHUAEPN

@misc{pith2026241212409,
  author       = {Pith},
  title        = {Pith review of: Improving Cooperation in Language Games with Bayesian Inference and the Cognitive Hierarchy},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/UGHUAEPN}},
  note         = {Machine review of arXiv:2412.12409}
}

read the original abstract

In two-player cooperative games, agents can play together effectively when they have accurate assumptions about how their teammate will behave, but may perform poorly when these assumptions are inaccurate. In language games, failure may be due to disagreement in the understanding of either the semantics or pragmatics of an utterance. We model coarse uncertainty in semantics using a prior distribution of language models and uncertainty in pragmatics using the cognitive hierarchy, combining the two aspects into a single prior distribution over possible partner types. Fine-grained uncertainty in semantics is modeled using noise that is added to the embeddings of words in the language. To handle all forms of uncertainty we construct agents that learn the behavior of their partner using Bayesian inference and use this information to maximize the expected value of a heuristic function. We test this approach by constructing Bayesian agents for the game of Codenames, and show that they perform better in experiments where semantics is uncertain

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Reference graph

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Algorithm 9 is called after each guessed card is revealed, incrementing i, in order to update the Bayesian guesser. Algorithm 2 Run Bayesian Guesser Require: Set of spymasters M , posterior probability distribution over spymasters p(M ), clue history l1:t W← up to n sampled co...

Pith tools

Reviewed August 11, 2026 · model on record in the stance chip above.

[1] [1]

Baarslag, M

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