Nash Equilibrium Text: A Game-Theoretic Decoding Framework for Text Generation

Alireza Jafari; Arman Adibi; Mohammad Ghavamzadeh; Hadi Daneshmand

cs.AIOct 5, 2026

Nash Equilibrium Text: A Game-Theoretic Decoding Framework for Text Generation

Authors: Alireza Jafari, Arman Adibi, Mohammad Ghavamzadeh, Hadi Daneshmand

Organizations: Department of Computer Science University of Virginia Charlottesville, VA, USA · School of Computer and Cyber Sciences Augusta University Augusta, GA, USA · Qualcomm AI Research San Diego, CA, USA

Abstract

Text revision has become an integral component of large language models. This paper formulates revision such that it admits a Nash equilibrium: Token positions are players, vocabulary items are actions, and each player's utility is the language model's log conditional probability. We motivate the revision by showing that Nash equilibria can have exponentially higher likelihood than autoregressive outputs as the sequence length grows. We further propose Nash decoding, an algorithm that reaches an $\varepsilon$ -Nash equilibrium in $O(1/\varepsilon)$ time given access to the joint probability of tokens conditioned on a prompt. In practice, we run Nash decoding using conditional probability estimates from large language models and evaluate the resulting equilibria on question-answering benchmarks. On CLAPNQ, PubMedQA, and CoQA, Nash equilibria obtained from masked language models achieve higher F1 and ROUGE scores than autoregressive models up to $18\times$ larger, without any fine-tuning or retraining, at the cost of additional test-time computation.

Figures & tables

Figure 1: Revision Game. Each token position is a player whose choice is evaluated given all context tokens. Autoregressive decoding leaves earlier tokens fixed, whereas the tokens are revised in the revision game. We used GPT-2 small for conditional probability estimates. At a Nash equilibrium, no player can increase its conditional probability by changing its token. Details are reported in Appendix A .

Panel of Figure 2 — Figure 2: Convergence of Nash decoding on WikiText-103. (a) Running minimum of the Nash gap, compared with a $1/k$ reference curve. The reference indicates a rate of decay, aligned with Theorem 1 . Both axes are log-transformed. (b) GPT-2 XL continuation perplexity and ModernBERT pseudo-perplexity. The x-axis is log-transformed.

		CoQA			PubMedQA			CLAPNQ
Model	Size	F1	R-L	R-Ls	F1	R-L	R-Ls	F1	R-L	R-Ls
One-shot masked LMs (not fine-tuned)
RoBERTa-L ( Liu et al., 2019 )	355M	28.00	31.45	31.46	7.99	8.21	8.74	5.94	10.38	10.59
ModernBERT-L ( Warner et al., 2025 )	395M	39.54	42.92	42.93	5.93	7.95	8.34	7.03	12.05	12.66
Ettin-400m ( Weller et al., 2025 )	396M	43.87	46.22	46.24	4.03	4.55	4.76	6.97	11.78	12.41
Autoregressive models

Table 1: Results across three question-answering benchmarks under a shared evaluation protocol. Size denotes the number of model parameters; for Nash decoding, it denotes the masked model size. Best results are bolded and second-best results are underlined . The complete benchmark results are reported in Appendix F .

Figure 3: F1 score and Nash gap during Nash decoding iterations on CLAPNQ.

Model	Size	F1	R-L	R-Ls
Left-to-right initialization
mmBERT-base	308M	4.22	6.11	5.38
RoBERTa-Large	355M	22.23	19.79	20.77
ModernBERT-Large	395M	21.91	19.88	19.99
Ettin-400m	396M	17.41	17.29	17.36
All-mask initialization

Table 2: Nash Equilibria Basins on PubMedQA.

	CoQA			PubMedQA			CLAPNQ
Decoding rule	F1	R-L	R-Ls	F1	R-L	R-Ls	F1	R-L	R-Ls
Uniform-ordered	38.65	40.05	40.08	30.50	23.51	27.14	56.15	50.08	55.00
Confidence-ordered	41.85	42.21	42.28	31.74	24.64	28.50	58.84	54.30	58.11
Max Nash gap	43.07	43.22	43.31	32.11	24.80	28.75	59.16	54.67	58.37

Table 3: Nash decoding versus single-token masked-diffusion decoding using the same masked model. All rows use the instruction-tuned ModernBERT-chat with its corresponding prompt; details in Appendix H .

Figure 4: Runtime comparison of Nash and autoregressive decoding on CoQA. The left panel reports wall-clock time per question, and the right panel reports model calls per question. For Nash decoding, stacked bars separate computation before the sequence is fully unmasked from subsequent refinement. All models are evaluated on a single NVIDIA B200 GPU.

Appendix figures & tables10 assets

Supplementary material from the paper’s appendix.

Appendix

Quantity	Value
Prompts, tokens per prompt, continuation slots $T$	$500$ , $512$ , $64$
Action set (vocabulary minus four special IDs)	$50{,}364$
Trajectories reaching an exact equilibrium	$494$ ( $98.8\%$ )
Trajectories stopped at a repeated state	$6$ ( $1.2\%$ )
Already at equilibrium when construction ends	$118$ ( $23.6\%$ )
Refinement updates per text (mean / median / max)	$2.60$ / $2$ / $36$

Appendix

Table 4: Refinement on

500

WikiText-103 continuations with ModernBERT-Large , counted from the completed sequence

x^{(1)}

. Construction is a separate max-gap order pass of

64

model calls.

	CLAPNQ	PubMedQA	CoQA †
Split	dev, answerable	PQA-L test (seed 0)	dev, gold $\geq 5$ tok.
Questions scored	$300$	$500$	$1{,}814$
Distinct contexts	$297$	$500$	$459$ ( $460$ conv.)
References per question	$1.62$	$1$	$4$ ( $2.36$ distinct)
Mean length in ModernBERT tokens (words in parentheses)
Context	$195.3$ ( $170.8$ )	$287.0$ ( $202.3$ )	$355.2$ ( $267.0$ )

Appendix

Table 5: Scale and length of the evaluated subsets. Token lengths use ModernBERT-Large unless otherwise stated; parenthesized word counts are whitespace-delimited. Reference counts are averages per question, with distinct references counted after normalization. Oracle decoding budgets use the reference answer length in each evaluated model’s tokenizer. † CoQA includes only development questions whose gold answer has at least five ModernBERT tokens (Appendix D.3 ).

	CLAPNQ	PubMedQA	CoQA †
Lexical overlap with the context
Gold tokens occurring in the context	$97.6\%$	$\mathbf{60.0\%}$	$99.1\%$
Gold content words occurring in the context	$97.4\%$	$\mathbf{49.3\%}$	$98.9\%$
Novel bigrams (absent from the context)	$10.9\%$	$\mathbf{76.8\%}$	$6.5\%$
Answers with $<50\%$ content-word overlap	$0.3\%$	$\mathbf{52.0\%}$	$0.6\%$
Arrangement of shared wording

Appendix

Table 6: Lexical overlap and arrangement of passage material. Statistics use normalized text and the most favorable reference for each measure. The sentence-selection diagnostic greedily covers distinct answer content words with a

90\%

target. Copying baselines score either the full context or its best single sentence against the references. † CoQA uses the filtered subset described in Appendix D.3 .

		Gold $\geq 5$ tokens ( $n=1{,}814$ )					All ( $n=7{,}983$ )
Model	Size	F1	R-L	R-Ls	Calls	Char	F1	R-L	R-Ls
One-shot masked LMs (not fine-tuned)
RoBERTa-Large ( Liu et al., 2019 )	355M	28.00	31.45	31.46	1.0	21	47.46	49.29	49.29
ModernBERT-Large ( Warner et al., 2025 )	395M	39.54	42.92	42.93	1.0	21	58.42	60.17	60.17
Ettin-400m ( Weller et al., 2025 )	396M	43.87	46.22	46.24	1.0	21	66.00	66.83	66.83
Autoregressive decoders

Appendix

Table 7: Results on CoQA under the GPT-2 paper prompt . The left block scores the 1,814 questions whose gold answer has

\geq 5

ModernBERT tokens; the right block scores all 7,983 CoQA development turns , including the yes/no/unknown answers. The protocol is identical throughout and identical to Table 1 : oracle answer length in each model’s own tokenizer, structural-token bans only, greedy decoding, fp32. Calls is the mean number of forward passes per example; for autoregressive models it equals the oracle token budget

B

. Char is the mean character length of the scored answer. Calls and Chars are reported for the

\geq 5

cohort.

Model	Size	F1	R-L	R-Ls	Calls	Char
One-shot masked LMs (not fine-tuned)
RoBERTa-Large ( Liu et al., 2019 )	355M	7.99	8.21	8.74	1.0	176
ModernBERT-Large ( Warner et al., 2025 )	395M	5.93	7.95	8.34	1.0	93
Ettin-400m ( Weller et al., 2025 )	396M	4.03	4.55	4.76	1.0	63
Autoregressive decoders
SmolLM2-135M ( Allal et al., 2025 )	135M	22.45	20.23	20.93	48.8	236

Appendix

Table 8: Full results on PubMedQA.

Model	Size	F1	R-L	R-Ls	Calls	Char
One-shot masked LMs (not fine-tuned)
RoBERTa-Large ( Liu et al., 2019 )	355M	5.94	10.38	10.59	1.0	195
ModernBERT-Large ( Warner et al., 2025 )	395M	7.03	12.05	12.66	1.0	142
Ettin-400m ( Weller et al., 2025 )	396M	6.97	11.78	12.41	1.0	117
Autoregressive decoders
GPT-2 Small ( Radford et al., 2019 )	124M	29.31	29.34	29.30	63.1	282

Appendix

Table 9: Full results on CLAPNQ.

Panel of Figure 5 — Figure 5: Answer F1 (green, right scale) and the Nash gap $G(x^{(k)})$ (violet, left scale) against the update index $k$ from the all- [MASK] canvas, for ModernBERT-Large with all-mask construction.

Model	Prompt	F1	R-L	R-Ls
Autoregressive decoding
Falcon-7B	Question: / Answer:	32.87	29.80	32.61
Falcon-7B	Soft-instruction	34.39	32.91	34.50
Nash decoding
ModernBERT-Large	Question: / Answer:	37.00	33.25	36.39
ModernBERT-Large	Soft-instruction	38.04	34.79	37.42

Appendix

Table 10: Prompt-format ablation on CLAPNQ.

		Calls
Dataset	Mean $T$	Diffusion	Nash gap
CoQA ( $1{,}814$ )	$7.5$	$8$	$53$
PubMedQA ( $500$ )	$48.3$	$48$	$1{,}604$
CLAPNQ ( $300$ )	$63.8$	$64$	$3{,}076$

Appendix

Table 11: Cost and convergence of the three coordinate rules. Calls is the mean number of forward passes per example. The diffusion variants always terminate after

T

commitments.

Panel of Figure 6 — Figure 6: Wall-clock seconds and model calls per question for the same five systems on each dataset, measured on a single B200 in float32 with TF32 disabled and one evaluation process running at a time. The Nash bars are split at the update that first fills the last [MASK] : the light segment includes all computation through that update, and the dark segment includes subsequent refinement. For all-mask decoding, the light segment can include revisions of previously filled positions. Horizontal scales differ between panels. CoQA and PubMedQA use their full evaluation sets. CLAPNQ uses a fixed $60$ -item subsample, consisting of every fifth item and shared by all five systems. Its mean reference answer length is $49.0$ words, compared with $51.7$ for the full set; mean L2R cost is $249$ calls, compared with $248$ on all $300$ items.

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What Suppresses Nash Equilibrium Play in Large Language Models? Mechanistic Evidence and Causal Control

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Nash Equilibrium Game Theory LLM Interpretability Causal Interventions in Language Models Multi-Agent LLMs Nash Equilibrium Computation

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LLM agents are known to deviate from Nash equilibria in strategic interactions, but nobody has looked inside the model to understand why, or asked whether the deviation can be reversed. We do both. Working with four open-source models (Llama-3 and Qwen2.5, 8B to 72B parameters) playing four canonical two-player games, we establish the behavioral picture through self-play and cross-play experiments, then open up the 32-layer Llama-3-8B model and examine what actually happens during a strategic decision. The mechanistic findings are clear. Opponent history is encoded with near-perfect fidelity at the first layer (96% probe accuracy) and consumed progressively, while Nash action encoding is weak throughout, never exceeding 56%. There is no dedicated Nash module. Instead, the model privately favors the Nash action through most of its forward pass, but a prosocial override rooted in pretraining on human text concentrated in the final layers reverses this, reaching 84% probability of cooperation at layer 30. Injecting a learned Nash direction into the residual stream shifts behavior bidirectionally and causally, confirmed through concept clamping. The behavioral experiments surface six scale- and architecture-dependent findings, the most notable being that chain-of-thought reasoning worsens Nash play in small models but achieves near-perfect Nash play above 70B parameters. The cross-play experiments reveal three phenomena invisible in self-play: a small model can unravel any partner's cooperation by defecting early; two large models reinforce each other's cooperative instincts indefinitely; and who moves first determines which Nash equilibrium the system reaches. LLMs do not lack Nash-playing competence. They compute it, then suppress it.

Abstract

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