Paper deep dive
Nash Equilibrium Text: A Game-Theoretic Decoding Framework for Text Generation
Alireza Jafari, Arman Adibi, Mohammad Ghavamzadeh, Hadi Daneshmand
Intelligence
Status: not_run | Model: - | Prompt: - | Confidence: 0%
Entities (0)
Relation Signals (0)
No relation signals yet.
Cypher Suggestions (0)
No Cypher suggestions yet.
Abstract
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.
Tags
Links
- Source: https://arxiv.org/abs/2610.05817v1
- Canonical: https://arxiv.org/abs/2610.05817v1
Trouble viewing inline? Open PDF directly →
Full Text
115,677 characters extracted from source content.
Expand or collapse full text
Nash Equilibrium Text: A Game-Theoretic Decoding Framework for Text Generation Alireza Jafari Affiliation: Department of Computer Science Affiliation: University of Virginia Affiliation: Charlottesville, VA, USA Email: jrp5td@virginia.edu Arman Adibi Affiliation: School of Computer and Cyber Sciences Affiliation: Augusta University Affiliation: Augusta, GA, USA Email: aadibi@augusta.edu Mohammad Ghavamzadeh Affiliation: Qualcomm AI Research Affiliation: San Diego, CA, USA Email: mohammad.ghavamzadeh51@gmail.com Hadi Daneshmand Affiliation: Department of Computer Science Affiliation: University of Virginia Affiliation: Charlottesville, VA, USA Email: dhadi@virginia.edu 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 ε -Nash equilibrium in O(1/ε)O(1/ ) 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 18×18× larger, without any fine-tuning or retraining, at the cost of additional test-time computation. Code is available at https://github.com/alireza-jafari/Nash-Decoding. 1 Introduction “The first draft of anything is shit,” Ernest Hemingway reportedly remarked (Samuelson, 1984). The first draft of this manuscript was no exception. Accomplished writers, at least before the advent of LLMs, have devoted considerable effort to revision. Hemingway, for instance, rewrote the final page of A Farewell to Arms 39 times (Plimpton, 1958). This raises a scientific question: When does the revision terminate, and what is the equilibrium text that editing converges to? Revision has become the core in text generation with language models where the first draft often is the standard de facto autoregressive text obeying xi+1(AR)=argmaxx∈p(x∣x≤i(AR);prompt), x_i+1^(AR)= _x \;p(x x_≤ i^(AR);prompt), (1) where p denotes the probability function (Radford et al., 2019) and xix_i are tokens selected from a vocabulary V. Autoregressive generation calls for revision because each token is produced without access to the tokens that follow. Existing approaches to revision fall into four broad families: (1) remasking tokens in diffusion and masked language models (Ghazvininejad et al., 2019; Sahoo et al., 2024; Shi et al., 2024); (2) reinforcement learning with external feedback (Ouyang et al., 2022; OpenAI, 2024; Guo et al., 2025); (3) Markov chain Monte Carlo (MCMC) sampling (Karan and Du, 2026; Zhao et al., 2026; Xu et al., 2025; Wang and Cho, 2019); and (4) hybrid decoding techniques (Xia et al., 2017; Lee et al., 2018; Gu et al., 2019; Welleck et al., 2023). We focus on revision driven solely by conditional probabilities without external feedback. Such revision need not reach an equilibrium; the text may continue to change indefinitely. We therefore formulate revision as a game that admits an equilibrium. 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. Revision game: Given a prompt, n players are the positions i∈[n]i∈[n], and each player’s action is a token from V and player i receives utility logp(xi∣x−i;prompt) p(x_i x_-i;prompt), where x−ix_-i denotes all tokens except xix_i. Since p(xi∣x−i;prompt)=p(x1,…,xn∣prompt)/p(x−i∣prompt)p(x_i x_-i;prompt)=p(x_1,…,x_n )/p(x_-i ) and the denominator does not depend on xix_i, this is an exact potential game with potential logp(x1,…,xn∣prompt) p(x_1,…,x_n ) (Monderer and Shapley, 1996). Being finite, the game admits at least one pure Nash equilibrium x(nash)=(x1(nash),…,xn(nash))x^(nash)=(x_1^(nash),…,x_n^(nash)), which satisfies ∀i∈[n]:xi(nash)=argmaxx∈p(x∣x−i(nash);prompt). ∀ i∈[n]: x_i^(nash)= _x \;p (x x_-i^(nash);prompt ). (2) In other words, a sequence is a Nash equilibrium when no single position can increase its conditional probability by changing its token. Comparing to Equation 1, the above conditioning is on x−ix_-i rather than x<ix_<i. As an example, we simulate this revision game between tokens using GPT-2 prompted by the question “What famous city is the capital of Spain?”. Figure 1 shows revisions of the first draft generated by autoregressive decoding that is “The city of Barcelona is the capital of Spain.” As the game proceeds, the token “Barcelona” is revised to “Madrid”, increasing the likelihood of the full generated text by a factor of 1.261.2611 1 In this example, the increase in probability happens to coincide with improved factual correctness; in general, higher probability does not ensure correctness.. We will show that Nash equilibria can attain substantially higher likelihood than autoregressive generations: the ratio p(x(nash)∣prompt)p(x(AR)∣prompt) p(x^(nash) )p(x^(AR) ) can grow exponentially in n. Computing an exact pure Nash equilibrium in general potential games is PLS-complete (Fabrikant et al., 2004). In contrast, we propose a revision mechanism called Nash decoding that obtains an ϵε-Nash equilibrium in O(1/ε)O\! (1/ ) time. Furthermore, the conditional probability of revised text monotonically increases through the revision. Empirically, we show that masked language models can be used to compute a Nash equilibrium. Thus, we evaluate the question-answering performance of Nash equilibria against autoregressive generation on question-answering benchmarks. At the cost of additional computation, iterative Nash updates allow a masked language model to surpass the F1 and ROUGE of autoregressive models with 18 times as many parameters on CoQA (Reddy et al., 2019), PubMedQA (Jin et al., 2019), and CLAPNQ (Rosenthal et al., 2025) datasets. We also report negative results showing that scores of equilibrium vary and depend on the initialization and the underlying model. 2 Text Generation as a Game Why a Nash equilibrium? We motivate finding a Nash equilibrium by proving that its likelihood can be significantly larger than that of autoregressive text. Lemma 1 (Separation between greedy AR decoding and Nash equilibrium) For every positive even integer n, there exists a strictly positive autoregressive distribution on 0,1n\0,1\^n such that p(x(nash))p(x(AR))=exp(γn),γ:=12log(539289)≈0.31164. p(x^(nash))p(x^(AR))= (γ n), γ:= 12 \! ( 539289 )≈ 0.31164. The proof is postponed to Appendix B. While the above result is proven from a specific joint probability function, we will later show that for every probability function there exists at least one Nash equilibrium with higher likelihood than the autoregressive output (if it is not an equilibrium). 2.1 Approximate Equilibrium Is Attainable We study when revision terminates. We prove that there exists a revision process that terminates at an approximate Nash equilibrium. To make this notion precise, we first define the token gap at position i as the largest increase in conditional probability that a single token replacement at that position can achieve: gi(x)=maxv∈p(v∣x−i;prompt)−p(xi∣x−i;prompt).g_i(x)= _v \;p(v x_-i;prompt)-p(x_i x_-i;prompt). (3) The Nash gap is the largest improvement available at any position: G(x)=maxi∈[n]gi(x)G(x)= _i∈[n]g_i(x). The equilibrium condition for x(nash)x^(nash) in Equation 2 is therefore equivalent to G(x(nash))=0G(x^(nash))=0: no position can improve its utility through a unilateral token replacement. More generally, for a tolerance ε≥0 ≥ 0, x is an additive ε -Nash equilibrium if G(x)≤εG(x)≤ : no single-token replacement improves its position’s conditional probability by more than ε . Recall the formulation of conditional density as p(xi∣x−i;prompt)=p(x∣prompt)p(x−i∣prompt).p(x_i x_-i;prompt)= p(x )p(x_-i ). (4) The denominator is positive and unchanged by a unilateral deviation. Thus, whenever a player increases its utility, the joint probability also increases. In game-theoretic terms, the joint probability is an ordinal potential (Monderer and Shapley, 1996). This property yields a bound on the number of improving token refinements. We define Nash decoding as the maximum-gap best-response procedure in Algorithm 1. Given a tolerance 0<ε≤10< ≤ 1, start from any completed sequence x(0)∈nx^(0) ^n. At iteration k≥0k≥ 0, compute gi(x(k))g_i(x^(k)) for every position i∈[n]i∈[n]. If G(x(k))≤εG(x^(k))≤ , return x(k)x^(k). Otherwise, select a position with the largest gap and replace its token with the best response, leaving all other tokens unchanged. It terminates when the maximum Nash gap is at most ε , yielding an ε -Nash equilibrium. Algorithm 1 Nash Decoding 1: Prompt, initial sequence x(0)∈nx^(0) ^n, tolerance 0<ε≤10< ≤ 1 2: Sequence x satisfying G(x)≤εG(x)≤ 3: k←0k← 0 4: while G(x(k))>εG(x^(k))> do 5: Compute gi(x(k))g_i(x^(k)) for all i∈[n]i∈[n] 6: G(x(k))←maxi∈[n]gi(x(k))G(x^(k))← _i∈[n]g_i(x^(k)) 7: Select ik∈argmaxi∈[n]gi(x(k))i_k∈ _i∈[n]g_i(x^(k)) 8: Set xik(k+1)∈argmaxv∈p(xik=v∣x−ik(k);prompt)x_i_k^(k+1)∈ _v p\! (x_i_k=v x_-i_k^(k);prompt ) 9: Set xi(k+1)←xi(k)x_i^(k+1)← x_i^(k) for all i≠iki≠ i_k 10: k←k+1k← k+1 11: Break if x(k)x^(k) is visited before 12: end while 13: return x(k)x^(k) The next theorem shows that Nash decoding monotonically increases the sequence probability in each iteration and reaches an ε -Nash equilibrium in a polynomial number of iterations. Theorem 1 (Convergence Rate) Let 0<ε≤10< ≤ 1 and let p0=p(x(0)∣prompt)>0p_0=p(x^(0) )>0 be the initial sequence probability. Every performed update of the above dynamics satisfies p(x(k+1)∣prompt)p(x(k)∣prompt)≥1+G(x(k))⏟>0. p(x^(k+1) )p(x^(k) )≥ 1+ G(x^(k))_>0. (5) The dynamics terminate at an additive ε -Nash equilibrium after KεK_ token replacements, with Kε≤log(1/p0)log(1+ε)≤2log(1/p0)ε.K_ ≤ (1/p_0) (1+ )≤ 2 (1/p_0) . (6) Proof. Let pk=p(x(k)∣prompt)p_k=p(x^(k) ). Equation 4 and maximum-gap selection imply pk+1−pk=p(x−ik(k)∣prompt)G(x(k))≥pkG(x(k)),p_k+1-p_k=p(x_-i_k^(k) )G(x^(k))≥ p_kG(x^(k)), since the marginal probability is at least the probability of the completed sequence. This proves Equation 5. Each performed update has G(x(k))>εG(x^(k))> , so any K performed updates satisfy 1≥pK≥p0(1+ε)K.1≥ p_K≥ p_0(1+ )^K. Taking logarithms bounds the number of updates; hence the dynamics must terminate with G(x)≤εG(x)≤ . Finally, log(1+ε)≥ε/2 (1+ )≥ /2 for 0<ε≤10< ≤ 1. □ The bound depends logarithmically on the inverse initial probability: a more probable starting sequence yields a tighter guarantee. To distinguish replacements from their computational cost, count one conditional query as obtaining p(⋅∣x−i;prompt)p(· x_-i;prompt) over the entire vocabulary at one position. Recomputing all gaps after each replacement requires O(nlog(1/p0)ε−1)O\! (n (1/p_0) ^-1 ) density evaluations, including the final scan that certifies termination. A consequence of this result is: for every x(AR)x^(AR) in Equation 1, there exists a Nash equilibrium whose likelihood is higher as long as it is not an equilibrium. This follows because Nash decoding, initialized at the autoregressive output, monotonically increases the likelihood until it reaches an equilibrium. 2.2 Nash Decoding with Pretrained Language Models Nash decoding relies on the conditional probability p(v∣x−i;prompt)p(v x_-i;prompt) of token v at position i given the rest of the sequence. This quantity can be estimated with either autoregressive or masked language models. Masked language models. A masked language model (MLM), such as BERT, provides the conditional estimates needed for Nash decoding. For each position i, we replace xix_i with [MASK] while retaining the prompt and all other tokens. The model produces a logit zθ,i(v)z_θ,i(v) for each candidate token v∈v at the masked position. Applying softmax gives qθ(v∣x−i;prompt)=exp(zθ,i(v))∑w∈exp(zθ,i(w)),q_θ(v x_-i;prompt)= (z_θ,i(v)) _w (z_θ,i(w)), where w ranges over the vocabulary and the logits depend on the masked input. Because the model is pretrained to reconstruct masked tokens from their surrounding context, we use this distribution as an estimate of p(v∣x−i;prompt)p(v x_-i;prompt) (Devlin et al., 2019). Autoregressive models. Nash decoding can also use an AR model, as illustrated by the motivating example in Section 1. An AR model defines a joint distribution pθAR(x∣prompt)p_θ^AR(x ) as the product of its left-to-right token probabilities, so its full-context conditional is qθAR(v∣x−i;prompt)=pθAR(x<i,v,x>i∣prompt)∑w∈pθAR(x<i,w,x>i∣prompt).q_θ^AR(v x_-i;prompt)= p_θ^AR(x_<i,v,x_>i ) _w p_θ^AR(x_<i,w,x_>i ). (7) But computing the above conditional probability is expensive because each replacement can change subsequent token probabilities, requiring up to |||V| candidate suffix evaluations per position. The challenge of joint density estimation. The conditional distributions supplied by a masked language model need not be compatible with a common joint distribution (Torroba Hennigen and Kim, 2023) with a joint probability p where p(x∣prompt) p(x ) =p(xi∣x−i;prompt)p(x−i∣prompt) =p(x_i x_-i;prompt)\,p(x_-i ) (8) =p(xj∣x−j;prompt)p(x−j∣prompt) =p(x_j x_-j;prompt)\,p(x_-j ) holds for every sequence x and every pair of positions i and j. Consequently, the convergence guarantee in Theorem 1 does not necessarily hold for the conditional probability estimates; hence, we empirically validate the convergence to Nash equilibrium. 2.3 Empirical Validation of Theorem 1 (a) Convergence Test (b) Monotonicity Test Figure 2: Convergence of Nash decoding on WikiText-103. (a) Running minimum of the Nash gap, compared with a 1/k1/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. We experimentally validate (1) the convergence rate and (2) likelihood monotonicity in Theorem 1. Using ModernBERT (Warner et al., 2025), we generate 64-token continuations for 500 prompts from WikiText-103 (Merity et al., 2017). Each continuation starts from 64 [MASK] tokens and repeatedly replaces the maximum-gap token with its best response (see details in Appendix C). (1) Convergence test: Figure 2(a) compares the empirical convergence rate of the Nash gap with the linear rate established in Theorem 1. We track the rate only when all tokens are unmasked (see Appendix C.4). The Nash Gap remains below the theoretical rate 1/k1/k. The plateaus near the end are due to the fact that most texts have already reached equilibrium at a faster rate compared to the theoretical rate O(1/k)O(1/k). Notably, this observation holds only when all tokens are unmasked. (2) Monotonicity test: To verify the likelihood increase, we report the perplexity of text obtained across iterations of Nash decoding. Figure 2(b) shows ModernBERT pseudo-perplexity and perplexity under an external autoregressive model, GPT-2 XL (Radford et al., 2019). Both scores decrease with iterations, indicating an increase in likelihood. Although the likelihood increases does not necessarily hold for conditional probability estimates, as discussed in the last section, the observed decrease in pseudo-perplexity is consistent with the likelihood increase in Theorem 1. 3 Related Work Nash decoding, introduced in this work, links to a family of decoding methods which we categorize into three main groups: (1) multi-pass decoders, (2) parallel decoders, and (3) speculative decoders. (1) Multi-pass decoders. This work is inspired by a rich body of prior work on refining autoregressive text generation to improve the likelihood of the generated text. Xia et al. (2017) address this limitation with a first-pass autoregressive decoder that produces a complete draft and a trained second-pass decoder that refines it. The first generator might itself prefer to refine any token once the draft is complete. Self-Refine instead uses the same model to generate and revise its output (Madaan et al., 2023), but its revisions target task-specific quality criteria rather than certifying token-level equilibrium. Zhou et al. (2026) define a competitive MCMC sampling (remasking) mechanism that bridges the gap between masked and autoregressive language models. (2) Parallel decoders. Like masked diffusion language models (Sahoo et al., 2024; Shi et al., 2024), Nash decoding iteratively revises tokens. However, diffusion language models focus on parallel text generation (Austin et al., 2021; Nie et al., 2025), whereas our procedure for finding a Nash equilibrium is sequential, updating one token at a time. This reflects a difference in objectives: rather than computational efficiency of decoding, we focus on characterizing the likelihood properties of the Nash equilibrium. (3) Speculative decoders. Speculative decoding accelerates autoregressive generation by using a draft model to propose tokens and a target model to verify them (Leviathan et al., 2023; Chen et al., 2023). In SpecDec, a drafted token is revised if it falls outside the verifier’s top tokens or if its log-likelihood is more than a threshold below that of the verifier’s top prediction. At the first such position, the verifier replaces the token with its own prediction and discards the remaining draft (Xia et al., 2023). Nash decoding differs from these methods in two ways: (1) it selects tokens for revision based on the Nash gap, and (2) it provably converges to an approximate Nash equilibrium. A growing body of work studies game-theoretic interactions among language-model agents and response policies (Zhang et al., 2025; Jacob et al., 2024; Munos et al., 2024; Hagen et al., 2026), including test-time methods for human-value alignment (Zhang et al., 2024; Tamine et al., 2025). Nash learning from human feedback (Munos et al., 2024), for example, seeks an equilibrium between competing response policies. In contrast, we formulate decoding itself as a game in which positions within a single generated sequence are players and tokens are their actions. 4 Empirical Results and Benchmarking We assess the question-answering performance of Nash decoding compared to existing autoregressive and masked diffusion decoding. For conditional probability estimates, we used four pretrained models: ModernBERT-Large (395M) (Warner et al., 2025), RoBERTa-Large (355M) (Liu et al., 2019), mmBERT-base (Marone et al., 2026), and Ettin-400M (396M) (Weller et al., 2025). Datasets. We used three standard question-answering benchmarks akin to prior works (Reddy et al., 2019; Dong et al., 2019; Rosenthal et al., 2025; Luo et al., 2022; Huang et al., 2023): (1) CLAPNQ (Rosenthal et al., 2025) pairs a Natural Questions passage with either a cohesive long-form answer or no answer at all; we use the 300300 answerable questions from its development split (mean answer length 63.863.8 tokens). (2) From PubMedQA (Jin et al., 2019), we use the 500500 PubMedQA-Labeled test questions, where the model must infer the abstract’s withheld conclusion rather than extract or repeat text from the input (mean answer length 48.348.3 tokens). (3) From CoQA (Reddy et al., 2019), we use the 1,8141,814 development questions whose reference answer contains at least 5 tokens (using ModernBERT tokenizer); this threshold excludes bare yes/no and other very short answers, leaving enough positions for meaningful token interactions (mean answer length 7.57.5 tokens) 22 2 To verify that the ≥5≥ 5-token filter does not drive the CoQA results, we also evaluate on the full development set, including all questions; see Appendix F.. Appendix D describes how the long-answer questions were selected and reports their statistics. Protocol. We use a fixed prompt format for all models within each dataset. For CLAPNQ and PubMedQA, the passage and question are followed by the soft instruction “This question is answered completely with evidence from the passage. Answer:” and an Answer: prefix. For CoQA, which consists of conversational question–answer turns, we instead use the standard Q: and A: dialogue format following the original evaluation setup (Radford et al., 2019). Across all datasets, every model receives the same oracle answer-length budget T, computed from the reference answer using that model’s tokenizer. End-of-sequence and structural tokens are banned, so every model emits exactly T tokens. For Nash decoding, the masked model is used without additional training: we append a canvas of T [MASK] tokens to the prompt and iteratively update tokens using the model’s full-context conditionals until the Nash gap satisfies G(x)≤10−4G(x)≤ 10^-4. We report SQuAD-style token F1 (Rajpurkar et al., 2016), scored against the most favorable reference, together with ROUGE-L and ROUGE-Lsum (Lin, 2004). Appendix E provides the complete protocol. 4.1 Question-Answering Performance Table 1 reports the question-answering performance of masked language models with and without Nash decoding, alongside strong autoregressive baselines. With Nash decoding, masked language models become effective zero-shot question answerers, outperforming autoregressive models 18×18× their size in F1, ROUGE-L, and ROUGE-Lsum. These gains come at the cost of additional test-time computation, which we analyze in Section 5. 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. 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 OPT-350M (Zhang et al., 2022) 331M 27.16 29.86 29.94 23.43 20.43 21.41 33.21 32.14 33.39 SmolLM2-360M (Allal et al., 2025) 362M 52.77 54.30 54.32 21.56 19.71 20.57 32.62 31.02 32.66 GPT-2 L (Radford et al., 2019) 774M 42.61 44.31 44.38 22.09 19.85 20.84 33.06 31.50 33.68 GPT-2 XL (Radford et al., 2019) 1.5B 46.83 48.26 48.35 23.78 20.77 22.00 30.58 29.51 30.73 BLOOM (5, 2022) 3B 56.69 56.25 56.62 22.60 19.53 21.46 34.00 32.18 34.31 OLMo-1.7 (Groeneveld et al., 2024) 7B 51.44 53.53 53.57 19.67 17.71 20.09 30.12 28.63 32.15 Nash decoding with same masked LMs RoBERTa-L 355M 41.35 44.02 44.04 24.99 21.17 22.69 25.25 25.59 25.17 ModernBERT-L 395M 58.12 58.71 58.71 24.28 20.33 21.36 38.04 34.79 37.42 Ettin-400m 396M 61.79 61.98 62.03 22.77 20.05 21.65 34.04 31.61 33.84 While Nash decoding turns frozen masked language models into zero-shot text generators without additional training or fine-tuning, the model performance varies across datasets, showing that Nash decoding remains dependent on the underlying model used for conditional density estimation. Better estimation leads to better performance. Remarkably, the results in Table 1 must not be compared directly with dataset leaderboards, which typically include models fine-tuned on task-specific training data. For example, fine-tuned models on CoQA reach 90.790.7 F1 (Ju et al., 2019). In contrast, every model in Table 1 is evaluated zero-shot. Our comparison therefore isolates decoding methods with pretrained models under a shared evaluation protocol rather than targeting state-of-the-art performance. For reference, Radford et al. (2019) report that GPT-2 XL achieves approximately 54.854.8 F1 on all CoQA questions. Our lower score reflects the subset used in Table 1, which contains only questions with answers of at least five tokens and thus excludes yes/no and other short answers. We exclude these because, for a single-token answer, the Nash equilibrium coincides with autoregressive decoding, leaving no room for revision. 4.2 Ablation Analysis We study how F1 performance evolves over Nash decoding iterations and how initialization affects performance at equilibrium. Additional ablations are provided in Appendix G. Effect of Token Replacements. Figure 3: F1 score and Nash gap during Nash decoding iterations on CLAPNQ. Figure 3 tracks F1 score and the Nash gap throughout Nash decoding with ModernBERT-Large on CLAPNQ. F1 measures movement toward the reference answer, whereas the Nash gap measures the remaining incentive for any player to change its token. Generation starts from an entirely masked sequence, whereas ModernBERT was pretrained with a 30%30\% masking rate (Warner et al., 2025). With short initial context, the model is uncertain, but as early tokens are filled, the emerging context sharpens the context length and the maximum gap can increase. Once sufficient context is placed, Nash updates increasingly resolve unstable positions and drive the Nash gap toward zero. F1 continues to increase throughout the trajectory, but more slowly in the second half as most examples reach equilibrium, leaving only a smaller subset with profitable token deviations (results for other datasets are available in Appendix G.1). Qualitative example of Nash equilibrium. Beyond aggregate scores, we inspect a representative Nash-equilibrium answer on CLAPNQ. For the question “where do they shoot guy’s grocery games”, Nash decoding using ModernBERT-Large fills a 4848-slot canvas and returns, “Guy’s Grocery Games Season 1 was shot inside of an actual grocery store, Field’s Market in West Hills, California. For Season 2, the market was built in a 15,500 square foot warehouse in Santa Rosa.” It contains two well-formed sentences with proper nouns and numbers, scoring 98.598.5 F1, showing that Nash equilibrium can be a coherent text even if the masked model is not fine-tuned (Appendix G.2). Table 2: Nash Equilibria Basins on PubMedQA. 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 mmBERT-base 308M 3.89 5.30 4.59 RoBERTa-Large 355M 24.99 21.17 22.69 ModernBERT-Large 395M 24.28 20.33 21.36 Ettin-400m 396M 22.77 20.05 21.65 Equilibrium Multiplicity. We now show empirically that multiple Nash equilibria exist. To reach different equilibria, we run Nash decoding from different initializations. Table 2 compares two of them. In left-to-right initialization, tokens are unmasked sequentially from left to right and are not revisited until the entire sequence has been unmasked. In contrast, all-mask initialization starts from a fully masked answer and lets Nash decoding update any position, including revising previously unmasked tokens before all positions are filled. The two schemes therefore follow different early decoding trajectories, after which both proceed to equilibrium. The resulting differences in final performance indicate that the two trajectories can converge to different equilibria (see Appendix G.4 for more details). Although the performance of Nash decoding depends on initialization, we find that all-mask initialization is often effective, as shown in Table 1. 4.3 From Diffusion Updates to Equilibrium Updates Masked diffusion models generate text through iterative full-context refinement, typically updating multiple masked positions in parallel. The closest existing method to Nash decoding is the particularly strong single-token masked diffusion model studied in dLLM (Zhou et al., 2026). dLLM provides a powerful instruction-tuned masked model, ModernBERT-chat, together with iterative decoding. We analyze the performance of Nash decoding using the same underlying model from dLLM. Both decoding methods start from an all-[MASK] sequence and update one position at a time; dLLM selects masked positions using uniform or confidence-based ordering, whereas Nash decoding selects the position with the largest Nash gap and can revisit previously generated tokens. Because both methods can use the same masked language model, this provides a controlled comparison of the position-selection and revision rules. 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. 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 compares Nash decoding with the two decoding rules used by dLLM while holding the model, initialization, prompt, and generation budget fixed. All methods use the released ModernBERT-chat checkpoint, which is based on ModernBERT-Large and supervised fine-tuned by dLLM for masked generation. dLLM’s uniform rule selects a masked position at random, whereas its confidence rule commits to the position whose predicted token has the highest conditional probability; committed tokens are not revisited. Nash decoding instead continues revising tokens until reaching equilibrium. Across all three datasets, equilibrium-based decoding with the dLLM fine-tuned model yields higher F1 and ROUGE scores, showing that diffusion masked language models achieve even stronger question-answering performance with Nash revision, at the cost of additional test-time computation (see Appendix H). 5 Discussion and Limitations Our paper establishes the potential of game-theoretic decoding by proving that it runs in polynomial time and by demonstrating its performance on question-answering benchmarks. These results call for future work that addresses the limitations of Nash decoding. The convergence of Nash decoding established in Theorem 1 requires conditional probabilities to be compatible with a common joint distribution. However, the conditional distributions estimated by masked language models do not necessarily satisfy this requirement, since training does not enforce compatibility conditions such as those in Equation 8. A promising direction for future research is to enforce these conditions, which would provide a provable convergence guarantee to an approximate Nash equilibrium. A second limitation is computational cost. Although Nash decoding may require only a modest number of token replacements to reach equilibrium, each replacement is expensive with current masked architectures because computing the maximum Nash gap requires evaluating leave-one-out conditionals across all n token positions. Consequently, the number of model calls per refinement step grows linearly with the generation length n, and the total query cost scales as O(nK)O(nK) for K token replacements. Figure 4 illustrates this overhead on CoQA, our shortest-answer benchmark. The statistical gains of Nash decoding currently come at substantially higher test-time computation, and the gap becomes more pronounced for longer text. This overhead is primarily architectural rather than inherent to the equilibrium objective: masked models that can evaluate conditional probabilities for multiple positions in parallel, without a need for masking, can substantially reduce the cost (see Appendix I). 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. References Allal et al. (2025) L. B. Allal, A. Lozhkov, E. Bakouch, G. M. Blázquez, G. Penedo, L. Tunstall, A. Marafioti, H. Kydlíček, A. P. Lajarín, V. Srivastav, J. Lochner, C. Fahlgren, X. Nguyen, C. Fourrier, B. Burtenshaw, H. Larcher, H. Zhao, C. Zakka, M. Morlon, C. Raffel, L. von Werra, and T. Wolf SmolLM2: when smol goes big – data-centric training of a small language model. External Links: 2502.02737, Link Cited by: Table 7, Table 7, Table 8, Table 8, Table 9, Table 9, Table 1. Almazrouei et al. (2023) E. Almazrouei, H. Alobeidli, A. Alshamsi, A. Cappelli, R. Cojocaru, M. Debbah, É. Goffinet, D. Hesslow, J. Launay, Q. Malartic, D. Mazzotta, B. Noune, B. Pannier, and G. Penedo The falcon series of open language models. External Links: 2311.16867, Link Cited by: Table 8, Table 9. Austin et al. (2021) J. Austin, D. D. Johnson, J. Ho, D. Tarlow, and R. van den Berg Structured denoising diffusion models in discrete state-spaces. In Advances in Neural Information Processing Systems, Vol. 34, p. 17981–17993. External Links: Link Cited by: §3. Biderman et al. (2023) S. Biderman, H. Schoelkopf, Q. G. Anthony, H. Bradley, K. O’Brien, E. Hallahan, M. A. Khan, S. Purohit, U. S. Prashanth, E. Raff, A. Skowron, L. Sutawika, and O. van der Wal Pythia: a suite for analyzing large language models across training and scaling. In Proceedings of the 40th International Conference on Machine Learning, p. 2397–2430. Cited by: Table 7, Table 7, Table 7, Table 8, Table 8, Table 9, Table 9. BigScience Workshop et al. (2022) BigScience Workshop, T. L. Scao, A. Fan, C. Akiki, E. Pavlick, S. Ilić, D. Hesslow, R. Castagné, A. S. Luccioni, F. Yvon, et al. BLOOM: a 176B-parameter open-access multilingual language model. External Links: 2211.05100, Link Cited by: Table 7, Table 8, Table 9, Table 1. Chen et al. (2023) C. Chen, S. Borgeaud, G. Irving, J. Lespiau, L. Sifre, and J. Jumper Accelerating large language model decoding with speculative sampling. External Links: 2302.01318, Document, Link Cited by: §3. Devlin et al. (2019) J. Devlin, M. Chang, K. Lee, and K. Toutanova BERT: pre-training of deep bidirectional transformers for language understanding. In Proceedings of the 2019 Conference of the North American Chapter of the Association for Computational Linguistics: Human Language Technologies, Volume 1 (Long and Short Papers), p. 4171–4186. External Links: Document, Link Cited by: §2.2. Dong et al. (2019) L. Dong, N. Yang, W. Wang, F. Wei, X. Liu, Y. Wang, J. Gao, M. Zhou, and H. Hon Unified language model pre-training for natural language understanding and generation. In Advances in Neural Information Processing Systems, Vol. 32, p. 13063–13075. External Links: Link Cited by: §D.1, §4. Fabrikant et al. (2004) A. Fabrikant, C. H. Papadimitriou, and K. Talwar The complexity of pure nash equilibria. In Proceedings of the Thirty-Sixth Annual ACM Symposium on Theory of Computing, p. 604–612. External Links: Document Cited by: §1. Geng and Liu (2023) X. Geng and H. Liu OpenLLaMA: an open reproduction of LLaMA. Note: GitHub repository External Links: Link Cited by: Table 8. Ghazvininejad et al. (2019) M. Ghazvininejad, O. Levy, Y. Liu, and L. Zettlemoyer Mask-Predict: parallel decoding of conditional masked language models. In Proceedings of the 2019 Conference on Empirical Methods in Natural Language Processing and the 9th International Joint Conference on Natural Language Processing (EMNLP-IJCNLP), p. 6112–6121. External Links: Document, Link Cited by: §1. Groeneveld et al. (2024) D. Groeneveld, I. Beltagy, E. Walsh, A. Bhagia, R. Kinney, O. Tafjord, A. Jha, H. Ivison, I. Magnusson, Y. Wang, S. Arora, D. Atkinson, R. Authur, K. Chandu, A. Cohan, J. Dumas, Y. Elazar, Y. Gu, J. Hessel, T. Khot, W. Merrill, J. Morrison, N. Muennighoff, A. Naik, C. Nam, M. Peters, V. Pyatkin, A. Ravichander, D. Schwenk, S. Shah, W. Smith, E. Strubell, N. Subramani, M. Wortsman, P. Dasigi, N. Lambert, K. Richardson, L. Zettlemoyer, J. Dodge, K. Lo, L. Soldaini, N. Smith, and H. Hajishirzi OLMo: accelerating the science of language models. In Proceedings of the 62nd Annual Meeting of the Association for Computational Linguistics (Volume 1: Long Papers), Bangkok, Thailand, p. 15789–15809. External Links: Document, Link Cited by: Table 7, Table 8, Table 9, Table 1. Gu et al. (2019) J. Gu, C. Wang, and J. Zhao Levenshtein Transformer. In Advances in Neural Information Processing Systems, Vol. 32, p. 11179–11189. External Links: Link Cited by: §1. Guo et al. (2025) D. Guo, D. Yang, H. Zhang, J. Song, P. Wang, Q. Zhu, R. Xu, R. Zhang, S. Ma, X. Bi, et al. Deepseek-r1: incentivizing reasoning capability in llms via reinforcement learning. arXiv preprint arXiv:2501.12948. Cited by: §1. Hagen et al. (2026) L. Hagen, J. P. Müller, W. Zhang, M. Qiao, and B. Kainz Wasserstein equilibrium decoding for reliable medical visual question answering. arXiv preprint arXiv:2605.18313. Cited by: §3. Huang et al. (2023) S. Huang, L. Qin, and Z. Cao Diffusion language model with query-document relevance for query-focused summarization. In Findings of the Association for Computational Linguistics: EMNLP 2023, Singapore, p. 11020–11030. External Links: Document, Link Cited by: §D.1, §4. IBM Research (2024) IBM Research Granite-7B-Base. Note: Hugging Face model card External Links: Link Cited by: Table 8. Jacob et al. (2024) A. P. Jacob, Y. Shen, G. Farina, and J. Andreas The consensus game: language model generation via equilibrium search. In The Twelfth International Conference on Learning Representations, External Links: Link Cited by: §3. Jiang et al. (2023) A. Q. Jiang, A. Sablayrolles, A. Mensch, C. Bamford, D. S. Chaplot, D. de las Casas, F. Bressand, G. Lengyel, G. Lample, L. Saulnier, L. R. Lavaud, M. Lachaux, P. Stock, T. L. Scao, T. Lavril, T. Wang, T. Lacroix, and W. E. Sayed Mistral 7b. External Links: 2310.06825, Link Cited by: Table 8, Table 8, Table 9. Jin et al. (2019) Q. Jin, B. Dhingra, Z. Liu, W. Cohen, and X. Lu PubMedQA: a dataset for biomedical research question answering. In Proceedings of the 2019 Conference on Empirical Methods in Natural Language Processing and the 9th International Joint Conference on Natural Language Processing (EMNLP-IJCNLP), p. 2567–2577. External Links: Document, Link Cited by: §D.1, §D.2, §1, item (2). Ju et al. (2019) Y. Ju, F. Zhao, S. Chen, B. Zheng, X. Yang, and Y. Liu Technical report on conversational question answering. External Links: 1909.10772, Link Cited by: §4.1. Karan and Du (2026) A. Karan and Y. Du Reasoning with sampling: your base model is smarter than you think. In International Conference on Learning Representations, External Links: Link Cited by: §1. Kim et al. (2025) W. Kim, G. Kim, and S. Kang Accelerating inference in retrieval-augmented generation models for long-form question answering via dynamic token pruning. Mathematics 13 (14), p. 2231. External Links: Document, Link Cited by: §D.1. Lee et al. (2018) J. Lee, E. Mansimov, and K. Cho Deterministic non-autoregressive neural sequence modeling by iterative refinement. In Proceedings of the 2018 Conference on Empirical Methods in Natural Language Processing, p. 1173–1182. External Links: Document, Link Cited by: §1. Leviathan et al. (2023) Y. Leviathan, M. Kalman, and Y. Matias Fast inference from transformers via speculative decoding. In Proceedings of the 40th International Conference on Machine Learning, Proceedings of Machine Learning Research, Vol. 202, p. 19274–19286. External Links: Link Cited by: §3. Lin (2004) C. Lin ROUGE: a package for automatic evaluation of summaries. In Text Summarization Branches Out, Barcelona, Spain, p. 74–81. External Links: Link Cited by: §4. Liu et al. (2019) Y. Liu, M. Ott, N. Goyal, J. Du, M. Joshi, D. Chen, O. Levy, M. Lewis, L. Zettlemoyer, and V. Stoyanov RoBERTa: a robustly optimized bert pretraining approach. arXiv preprint arXiv:1907.11692. Cited by: Table 7, Table 8, Table 9, Table 1, §4. Liu et al. (2023) Z. Liu, A. Qiao, W. Neiswanger, H. Wang, B. Tan, T. Tao, J. Li, Y. Wang, S. Sun, O. Pangarkar, R. Fan, Y. Gu, V. Miller, Y. Zhuang, G. He, H. Li, F. Koto, L. Tang, N. Ranjan, Z. Shen, X. Ren, R. Iriondo, C. Mu, Z. Hu, M. Schulze, P. Nakov, T. Baldwin, and E. P. Xing LLM360: towards fully transparent open-source LLMs. Cited by: Table 8, Table 9. Llama Team, AI @ Meta (2024) Llama Team, AI @ Meta The Llama 3 herd of models. External Links: 2407.21783, Link Cited by: Table 8. Luo et al. (2022) R. Luo, L. Sun, Y. Xia, T. Qin, S. Zhang, H. Poon, and T. Liu BioGPT: generative pre-trained transformer for biomedical text generation and mining. Briefings in Bioinformatics 23 (6), p. bbac409. External Links: Document, Link Cited by: §D.1, §4. Madaan et al. (2023) A. Madaan, N. Tandon, P. Gupta, S. Hallinan, L. Gao, S. Wiegreffe, U. Alon, N. Dziri, S. Prabhumoye, Y. Yang, S. Gupta, B. P. Majumder, K. Hermann, S. Welleck, A. Yazdanbakhsh, and P. Clark Self-refine: iterative refinement with self-feedback. In Advances in Neural Information Processing Systems, Vol. 36. External Links: Document, Link Cited by: §3. Marone et al. (2026) M. Marone, O. Weller, W. Fleshman, E. Yang, D. Lawrie, and B. Van Durme mmBERT: a modern multilingual encoder with annealed language learning. In International Conference on Machine Learning, External Links: Link Cited by: §4. Merity et al. (2017) S. Merity, C. Xiong, J. Bradbury, and R. Socher Pointer sentinel mixture models. In 5th International Conference on Learning Representations (ICLR 2017), Conference Track Proceedings, Toulon, France. External Links: Link Cited by: §C.1, §2.3. Monderer and Shapley (1996) D. Monderer and L. S. Shapley Potential games. Games and economic behavior 14 (1), p. 124–143. Cited by: §1, §2.1. Munos et al. (2024) R. Munos, M. Valko, D. Calandriello, M. Gheshlaghi Azar, M. Rowland, Z. D. Guo, Y. Tang, M. Geist, T. Mesnard, C. Fiegel, A. Michi, M. Selvi, S. Girgin, N. Momchev, O. Bachem, D. J. Mankowitz, D. Precup, and B. Piot Nash learning from human feedback. In Proceedings of the 41st International Conference on Machine Learning, Proceedings of Machine Learning Research, Vol. 235, p. 36743–36768. External Links: Link Cited by: §3. Nie et al. (2025) S. Nie, F. Zhu, Z. You, X. Zhang, J. Ou, J. Hu, J. Zhou, Y. Lin, J. Wen, and C. Li Large language diffusion models. arXiv preprint arXiv:2502.09992. External Links: Document, Link Cited by: §3. OpenAI (2024) OpenAI OpenAI o1 system card. External Links: 2412.16720, Link Cited by: §1. Ouyang et al. (2022) L. Ouyang, J. Wu, X. Jiang, D. Almeida, C. Wainwright, P. Mishkin, C. Zhang, S. Agarwal, K. Slama, A. Ray, et al. Training language models to follow instructions with human feedback. Advances in neural information processing systems 35, p. 27730–27744. Cited by: §1. Pfeiffer et al. (2024) P. Pfeiffer, P. Singer, Y. Babakhin, G. Fodor, N. Dhankhar, and S. S. Ambati H2O-Danube3 technical report. External Links: 2407.09276, Link Cited by: Table 7, Table 8. Plimpton (1958) G. Plimpton Ernest hemingway, the art of fiction no. 21. The Paris Review (18). Note: Interview with Ernest Hemingway External Links: Link Cited by: §1. Qwen Team (2024) Qwen Team Qwen2.5 technical report. External Links: 2412.15115, Document, Link Cited by: Table 7. Radford et al. (2019) A. Radford, J. Wu, R. Child, D. Luan, D. Amodei, and I. Sutskever Language models are unsupervised multitask learners. Cited by: §E.1, Table 7, Table 7, Table 7, Table 7, Table 8, Table 8, Table 9, Table 9, Table 9, Table 9, §1, §2.3, §4, §4.1, Table 1, Table 1. Rajpurkar et al. (2016) P. Rajpurkar, J. Zhang, K. Lopyrev, and P. Liang SQuAD: 100,000+ questions for machine comprehension of text. In Proceedings of the 2016 Conference on Empirical Methods in Natural Language Processing, p. 2383–2392. External Links: Link Cited by: §4. Reddy et al. (2019) S. Reddy, D. Chen, and C. D. Manning CoQA: a conversational question answering challenge. Transactions of the Association for Computational Linguistics 7, p. 249–266. External Links: Document, Link Cited by: §D.1, §D.2, §1, item (3), §4. Rosenthal et al. (2025) S. Rosenthal, A. Sil, R. Florian, and S. Roukos CLAPnq: cohesive long-form answers from passages in natural questions for RAG systems. Transactions of the Association for Computational Linguistics 13, p. 53–72. External Links: Document, Link Cited by: §D.1, §D.2, §1, item (1), §4. Sahoo et al. (2024) S. S. Sahoo, M. Arriola, Y. Schiff, A. Gokaslan, E. Marroquin, J. T. Chiu, A. Rush, and V. Kuleshov Simple and effective masked diffusion language models. In Advances in Neural Information Processing Systems, Vol. 37, p. 130136–130184. External Links: Document, Link Cited by: §1, §3. Samuelson (1984) A. Samuelson With hemingway: a year in key west and cuba. Random House, New York. External Links: ISBN 9780394539836 Cited by: §1. Shi et al. (2024) J. Shi, K. Han, Z. Wang, A. Doucet, and M. K. Titsias Simplified and generalized masked diffusion for discrete data. In Advances in Neural Information Processing Systems, Vol. 37. External Links: Document, Link Cited by: §1, §3. Tamine et al. (2025) M. Tamine, O. Sakhi, and B. Heymann Data valuation for llm fine-tuning: efficient shapley value approximation via language model arithmetic. arXiv preprint arXiv:2512.15765. Cited by: §3. Team OLMo et al. (2024) Team OLMo, P. Walsh, L. Soldaini, D. Groeneveld, K. Lo, S. Arora, A. Bhagia, Y. Gu, S. Huang, M. Jordan, N. Lambert, D. Schwenk, O. Tafjord, T. Anderson, D. Atkinson, F. Brahman, C. Clark, P. Dasigi, N. Dziri, M. Guerquin, H. Ivison, P. W. Koh, J. Liu, S. Malik, W. Merrill, L. J. V. Miranda, J. Morrison, T. Murray, C. Nam, V. Pyatkin, A. Rangapur, M. Schmitz, S. Skjonsberg, D. Wadden, C. Wilhelm, M. Wilson, L. Zettlemoyer, A. Farhadi, N. A. Smith, and H. Hajishirzi 2 OLMo 2 Furious. External Links: 2501.00656, Link Cited by: Table 9. Torroba Hennigen and Kim (2023) L. Torroba Hennigen and Y. Kim Deriving language models from masked language models. In Proceedings of the 61st Annual Meeting of the Association for Computational Linguistics (Volume 2: Short Papers), A. Rogers, J. Boyd-Graber, and N. Okazaki (Eds.), Toronto, Canada, p. 1149–1159. External Links: Link, Document Cited by: §2.2. Touvron et al. (2023) H. Touvron, T. Lavril, G. Izacard, X. Martinet, M. Lachaux, T. Lacroix, B. Rozière, N. Goyal, E. Hambro, F. Azhar, A. Rodriguez, A. Joulin, E. Grave, and G. Lample LLaMA: open and efficient foundation language models. External Links: 2302.13971, Document, Link Cited by: Table 8, Table 9. Wang and Cho (2019) A. Wang and K. Cho BERT has a mouth, and it must speak: BERT as a Markov random field language model. In Proceedings of the Workshop on Methods for Optimizing and Evaluating Neural Language Generation, p. 30–36. External Links: Document, Link Cited by: §1. Warner et al. (2025) B. Warner, A. Chaffin, B. Clavié, O. Weller, O. Hallström, S. Taghadouini, A. Gallagher, R. Biswas, F. Ladhak, T. Aarsen, G. T. Adams, J. Howard, and I. Poli Smarter, better, faster, longer: a modern bidirectional encoder for fast, memory efficient, and long context finetuning and inference. In Proceedings of the 63rd Annual Meeting of the Association for Computational Linguistics (Volume 1: Long Papers), Vienna, Austria, p. 2526–2547. External Links: Document, Link Cited by: §D.4, Table 7, Table 8, Table 9, §2.3, §4.2, Table 1, §4. Welleck et al. (2023) S. Welleck, X. Lu, P. West, F. Brahman, T. Shen, D. Khashabi, and Y. Choi Generating sequences by learning to self-correct. In The Eleventh International Conference on Learning Representations, External Links: Link Cited by: §1. Weller et al. (2025) O. Weller, K. Ricci, M. Marone, A. Chaffin, D. Lawrie, and B. Van Durme Seq vs Seq: an open suite of paired encoders and decoders. External Links: 2507.11412, Link Cited by: Table 7, Table 8, Table 9, Table 1, §4. Xia et al. (2023) H. Xia, T. Ge, P. Wang, S. Chen, F. Wei, and Z. Sui Speculative decoding: exploiting speculative execution for accelerating seq2seq generation. In Findings of the Association for Computational Linguistics: EMNLP 2023, Singapore, p. 3909–3925. External Links: Document, Link Cited by: §3. Xia et al. (2017) Y. Xia, F. Tian, L. Wu, J. Lin, T. Qin, N. Yu, and T. Liu Deliberation networks: sequence generation beyond one-pass decoding. In Advances in Neural Information Processing Systems, Vol. 30. External Links: Link Cited by: §1, §3. Xiao et al. (2020) D. Xiao, H. Zhang, Y. Li, Y. Sun, H. Tian, H. Wu, and H. Wang ERNIE-GEN: an enhanced multi-flow pre-training and fine-tuning framework for natural language generation. In Proceedings of the Twenty-Ninth International Joint Conference on Artificial Intelligence, IJCAI-20, p. 3997–4003. External Links: Document, Link Cited by: §D.1. Xu et al. (2025) M. Xu, T. Geffner, K. Kreis, W. Nie, Y. Xu, J. Leskovec, S. Ermon, and A. Vahdat Energy-based diffusion language models for text generation. In The Thirteenth International Conference on Learning Representations, External Links: Link Cited by: §1. Zhang et al. (2022) S. Zhang, S. Roller, N. Goyal, M. Artetxe, M. Chen, S. Chen, C. Dewan, M. Diab, X. Li, X. V. Lin, T. Mihaylov, M. Ott, S. Shleifer, K. Shuster, D. Simig, P. S. Koura, A. Sridhar, T. Wang, and L. Zettlemoyer OPT: open pre-trained transformer language models. External Links: 2205.01068, Link Cited by: Table 7, Table 8, Table 9, Table 1. Zhang et al. (2024) W. Zhang, C. Zang, and B. Kainz Truth or deceit? a bayesian decoding game enhances consistency and reliability. arXiv preprint arXiv:2410.01064. Cited by: §3. Zhang et al. (2025) W. Zhang, C. Zang, and B. Kainz Strategic llm decoding through bayesian games. In ICLR 2025 Workshop on Reasoning and Planning for Large Language Models, Cited by: §3. Zhao et al. (2026) X. Zhao, Z. Kang, A. Feng, S. Levine, and D. Song Learning to reason without external rewards. In International Conference on Learning Representations, Cited by: §1. Zhou et al. (2026) Z. Zhou, L. Chen, H. Tong, and D. Song dLLM: simple diffusion language modeling. In Proceedings of the 64th Annual Meeting of the Association for Computational Linguistics (Volume 3: System Demonstrations), p. 78–88. External Links: Document, Link Cited by: §3, §4.3. Technical appendices and supplementary material Appendix A The Motivating Experiment and The Illustrated Token Game Setup. All numbers in this section are produced with GPT-2 Small (124M) in float32 with TF32 matrix shortcuts disabled, using greedy decoding. Generation and refinement use the same prompt. Conditionals are computed exactly over the full 50,25750,257-token vocabulary, and refinement replaces, at each step, the single position with the largest gap, halting when the Nash gap falls below 10−910^-9. First example: The example illustrated in the introduction concerns the prompt Prompt What famous city is the capital of Spain? Greedy continuation The city of Barcelona is the capital of Spain. Its ten tokens are the ten players; positions are numbered from 1. The city of Barcelona is the capital of Spain . 1 2 3 4 5 6 7 8 9 10 Greedy decoding commits to the factually incorrect city at player 4. At the moment that token was emitted, the decoder had seen only “The city of,” and under that left context alone it preferred “Barcelona” to “Madrid” by a factor of 1.151.15. Because an autoregressive decoder conditions only on x<ix_<i, it cannot revisit the choice once the rest of the sentence is written. Conditioning instead on all other tokens, including the six to the right, reverses the preference: p(x4∣x<4)p(x_4 x_<4) p(x4∣x−4)p(x_4 x_-4) token autoregressive full context Madrid 0.09040.0904 0.33430.3343 Barcelona 0.10360.1036 0.26550.2655 Valencia 0.05190.0519 0.15070.1507 Spain 0.01600.0160 0.06660.0666 Catalonia 0.01160.0116 0.03050.0305 The best response is “Madrid,” giving g4=0.0688g_4=0.0688. Every one of the other nine players already plays its best response, so G(x)=g4G(x)=g_4 and player 4 is the unique improvable position. Replacing it yields Nash equilibrium The city of Madrid is the capital of Spain. and the next scan reports zero gap at all ten positions. The sequence log-probabilities under the prompt are −20.2943-20.2943 before and −20.0639-20.0639 after, a ratio of 1.2591×1.2591×. Another example. The effect is not confined to that one prompt. A longer continuation, from a different phrasing of the question, produces a game that takes two steps and in which the players visibly influence one another. Prompt What city is the political capital of Spain? Greedy continuation The answer is Barcelona. The city is the capital of Spain. It is the capital of Spain. Here all twenty tokens participate, with the erroneous city at player 4 and the pronoun “It” opening the third sentence at player 14. The first scan reports player token p(current)p(current) best response gap 4 Barcelona 0.30390.3039 Madrid (0.3978)(0.3978) 0.09380.0938 14 It 0.27090.2709 Madrid (0.3144)(0.3144) 0.04350.0435 with zero gap at the remaining eighteen positions, so G(x)=0.0938G(x)=0.0938 and player 4 moves first: After the first replacement The answer is Madrid. The city is the capital of Spain. It is the capital of Spain. This is where the coupling between players shows itself. Correcting player 4 does not leave player 14 untouched: its gap more than doubles, from 0.04350.0435 to 0.09720.0972 (p( It)=0.3160p( It)=0.3160 against p( Madrid)=0.4132p( Madrid)=0.4132), because the repaired first sentence makes the proper noun a better subject for the third. Player 14 therefore moves second: Nash equilibrium The answer is Madrid. The city is the capital of Spain. Madrid is the capital of Spain. A final scan reports a gap of exactly zero at all twenty positions, so the text is a Nash equilibrium, and the algorithm halts. The sequence log-probabilities are −25.8889-25.8889 for the greedy text, −25.6198-25.6198 after the first replacement and −25.3516-25.3516 at equilibrium: each step raises the likelihood, by 1.309×1.309× and 1.308×1.308× respectively, for a total of 1.711×1.711×. Appendix B Proof of the Autoregressive–Nash Separation Lemma description. Lemma 1 shows that greedy autoregressive decoding and Nash equilibrium can differ substantially even when both are defined by the same joint distribution. Specifically, for every even sequence length n, there exists a strictly positive autoregressive distribution for which the greedy output is not an equilibrium, with Nash gap greater than 0.300.30. The unique Nash equilibrium differs from the greedy output at exactly half of the token positions and has exponentially larger joint probability: p(xNash)p(xAR)=exp(γn),γ=12log(539289)≈0.31164. p(x^Nash)p(x^AR)= (γ n), γ= 12 \! ( 539289 )≈ 0.31164. Thus, the lemma establishes that locally optimal left-to-right decisions can accumulate into a sequence that is both far from equilibrium and exponentially less probable than a full-context equilibrium. Proof. Consider a block (a,b)∈0,12(a,b)∈\0,1\^2 with p(a,b)=p1(a)p2(b∣a)p(a,b)=p_1(a)p_2(b a), where p1(0)=0.51p_1(0)=0.51, p1(1)=0.49p_1(1)=0.49, p2(1∣0)=0.51p_2(1 0)=0.51, and p2(1∣1)=0.99p_2(1 1)=0.99. Greedy decoding selects (a,b)=(0,1)(a,b)=(0,1). However, p(0,1)=0.2601p(0,1)=0.2601 and p(1,1)=0.4851p(1,1)=0.4851, so given b=1b=1, the full-conditional best response is a=1a=1. Hence, G(0,1)≥(0.4851−0.2601)(0.4851+0.2601)=125/414>0.30.G(0,1)≥ (0.4851-0.2601)(0.4851+0.2601)=125/414>0.30. Moreover, b=1b=1 is the strict best response for either value of a, while a=1a=1 is the strict best response given b=1b=1. Thus, (1,1)(1,1) is the unique Nash equilibrium of the block. Now take m=n/2m=n/2 independent copies of this block. Greedy decoding produces xAR=(0,1,…,0,1)x^AR=(0,1,…,0,1), whereas the unique Nash equilibrium is xNash=(1,1,…,1,1)x^Nash=(1,1,…,1,1). Therefore, dH(xAR,xNash)=n/2d_H(x^AR,x^Nash)=n/2, and p(xNash)p(xAR)=(0.49⋅0.990.51⋅0.51)n/2=(539289)n/2=exp(γn)≈exp(0.31164n). p(x^Nash)p(x^AR)= ( 0.49· 0.990.51· 0.51 )^n/2= ( 539289 )^n/2= (γ n)≈ (0.31164\,n). Appendix C Equilibrium Dynamics on WikiText-103 Section 2.3 replaces the exact conditionals assumed by Theorem 1 with estimates from a pretrained masked model and asks whether the dynamics still behave as the analysis predicts. This appendix specifies that experiment and reports the measurements behind Figure 2. C.1 Setup What is tested is the convergence rate and likelihood increase in Theorem 1, not the question-answering performance. We take 500500 prompts of exactly 512512 tokens from the WikiText-103 test split (Merity et al., 2017) and generate a 6464-token continuation of each with ModernBERT-Large. To estimate the likelihood, we used an external autoregressive referee rather than the model likelihood estimate. Using an external model lets us evaluate the likelihood without overfitting to our own model’s estimates. Numerics follow Appendix E: greedy selection throughout, float32 with TF32 disabled and verified against a float64 reference, the SDPA attention implementation, and a 10241024-token context. The action set is the full vocabulary minus the four non-mask special IDs, leaving 50,36450,364 actions. [MASK] is one of them. Keeping it legal is what makes the token gap of Equation 3 well defined at a position that has not yet been written, and it allows a player to return its position to the masked state if that is its best response; no player ever does in practice. C.2 Procedure Nash decoding runs Algorithm 1 with tolerance ε=0 =0. At each iteration, the algorithm computes the token gap at all 6464 positions, writes or revises the largest-gap position with its best response, and stops when G(x)≤10−4G(x)≤ 10^-4, which certifies an exact equilibrium. Figure 2 measures the trajectory only after all positions unmasked; we index this initial fully instantiated sequence as k=1k=1, and subsequent iterations correspond to revisions of that sequence. We begin measurement at this point because sequence-level quantities such as likelihood and perplexity are not meaningfully comparable while the sequence still contains [MASK] positions. C.3 Convergence and Cycle Termination Table 4 summarizes refinement on the 500500 WikiText-103 continuations. Refinement performs 1,3001,300 token replacements in total, with a median of two per continuation. Of the 500500 completed initial sequences, 118118 already satisfy the equilibrium criterion before any refinement. The final sequence differs from its initialization at an average of 2.292.29 of the 6464 token positions. Equilibrium and cycle termination. We distinguish reaching an equilibrium from stopping at a repeated state. An equilibrium is certified when a full scan of all positions finds G(x)=0G(x)=0: no unilateral token replacement improves the position’s conditional probability under the masked model. This certificate concerns the estimated conditionals and the numerical precision used in the experiment; it does not establish global optimality, factual correctness, or compatibility with a joint distribution. For cycle detection, we retain the complete token sequence of every visited state, including the initial sequence. After each token replacement, we check whether the resulting sequence has appeared previously. If so, refinement stops and returns that repeated sequence, with its termination status recorded as a cycle. The returned sequence remains in the reported statistics and likelihood diagnostics; it is not counted as an equilibrium. Here, a cycle refers to recurrence of the complete sequence across updates, not merely to repeated words or phrases within a continuation. While Theorem 1 rules out cycles, they may still arise when the true conditional probabilities are replaced by a transformer’s estimates, as these estimates may not be compatible with a common joint distribution (Section 2.2). A best-response update improves the conditional probability at the selected position but can change the incentives at other positions. Subsequent updates can therefore return the sequence to an earlier state. Thus, we terminate the revision process once a cycle emerges. Table 4: Refinement on 500500 WikiText-103 continuations with ModernBERT-Large, counted from the completed sequence x(1)x^(1). Construction is a separate max-gap order pass of 6464 model calls. Quantity Value Prompts, tokens per prompt, continuation slots T 500500, 512512, 6464 Action set (vocabulary minus four special IDs) 50,36450,364 Trajectories reaching an exact equilibrium 494494 (98.8%98.8\%) Trajectories stopped at a repeated state 66 (1.2%1.2\%) Already at equilibrium when construction ends 118118 (23.6%23.6\%) Refinement updates per text (mean / median / max) 2.602.60 / 22 / 3636 Tokens changed by refinement (mean) 2.292.29 Updates that write [MASK] 00 of 1,3001,300 Model calls per text (construction ++ refinement) 64+23064+230 Seconds per text 3.33.3 C.4 Equilibrium Test Figure 2(a) plots lnk k against the natural logarithm of the mean over the 500500 texts of the running minimum minj≤kG(x(j)) _j≤ kG(x^(j)). On these axes, a 1/k1/k rate is a straight line of slope −1-1, so the reference curve fixes the rate implied by Theorem 1 and not its constant. The empirical curve falls from 0.3420.342 at k=1k=1 to 6.9×10−56.9× 10^-5 at k=36k=36, and stays below the reference throughout. We also examine a numerical analogue of the bound using ModernBERT’s pseudo-likelihood. Define p~0=∏iqθ(xi(1)∣x−i(1);prompt), p_0= _iq_θ(x_i^(1) x_-i^(1);prompt), where p~0 p_0 is a product of full-context conditionals rather than a joint sequence probability. We compare the observed running minimum with the proxy 2log(1/p~0)/k2 (1/ p_0)/k, which has the same form as the bound in Equation 6. The empirical running minimum remains below this proxy for all 500500 texts and all k, with a median ratio of 71×71×; the mean value of 2log(1/p~0)2 (1/ p_0) is 21.421.4. Because ModernBERT’s conditionals need not be induced by a common joint distribution, this comparison is only an empirical test. C.5 Where the Guarantee Does Not Transfer Equation 5 requires the sequence probability to rise by a factor of at least 1+G(x(k))1+G(x^(k)) at every update. Substituting the masked model’s pseudo-likelihood for p, that inequality holds on 71.5%71.5\% of updates, and the pseudo-likelihood increases on 75.2%75.2\%. These failures are the expected consequence of Section 2.2. An update raises the estimated conditional at the position it rewrites, but it also perturbs the conditionals at every other position, and without a common joint distribution nothing relates the two. Monotone improvement is a property of the idealized game rather than of the only relying transformers estimates. However, the Nash gap does not suffer from this issue, since it depends only on conditional probabilities rather than the joint probability. Appendix D The Three Question-Answering Benchmarks Section 4 evaluates Nash decoding on CLAPNQ, PubMedQA, and CoQA. This appendix explains the choice of benchmarks, specifies the evaluated subsets and decoding budgets, and describes their differences in sequence length and lexical overlap with the supplied context. D.1 Why These Three Benchmarks We choose these benchmarks to compare Nash decoding across three datasets for conversational text analysis, long-answer generation, and scientific analysis. CoQA requires short answers used in prior studies (Reddy et al., 2019; Dong et al., 2019; Xiao et al., 2020). CLAPNQ evaluates the quality of long-answers: answers often combine information from non-contiguous parts of a passage (Rosenthal et al., 2025; Kim et al., 2025). For example, asked where the Narmada river is located, the reference answer joins the passage’s first sentence (“a river in central India”) with its fourth (“rises from Amarkantak Plateau near Anuppur district”), skipping the two sentences in between. PubMedQA extends the comparison to a technical domain, where our target is a biomedical conclusion withheld from the input abstract (Jin et al., 2019). Prior evaluations on PubMedQA include autoregressive models and diffusion models, under their respective task formulations (Luo et al., 2022; Huang et al., 2023). The benchmarks differ in both the amount of text to generate and how that text relates to the context. In our evaluated subsets, CoQA questions average 6.66.6 ModernBERT tokens and often depend on conversational history, whereas CLAPNQ answers average 63.863.8 tokens and require cohesive composition from passage material. PubMedQA adds scientific evidence and specialized terminology, with substantially less answer wording available directly in the context. Together, these settings let us examine whether the gains from Nash decoding persist across diverse question-answering scenarios. D.2 Preprocessing CLAPNQ. CLAPNQ (Rosenthal et al., 2025) pairs a question from Natural Questions with a single gold passage and a cohesive, long-form answer. Because its test split is not publicly available, we use the 300300 answerable development questions. We exclude the 300300 unanswerable questions because the evaluated decoding protocol does not include abstention; the dataset paper also reports answerable-only evaluations. The retained questions cover 297297 distinct passages and have 485485 reference answers: 160160 questions have one reference, 131131 have two, and 99 have seven, giving a mean of 1.621.62 references per question. Multi-reference evaluation takes the best-reference score. PubMedQA. PubMedQA (Jin et al., 2019) pairs title-style biomedical research questions with scientific abstracts. We use the official 500500-question test half of PQA-L, reconstructed from the released seed-0 two-fold split and checked against the expected split size. Each question has a distinct abstract and one reference answer: the abstract’s conclusion, which is excluded from the input context. The corpus additionally labels each question yes / no / maybe (276276 / 169169 / 5555 here), but we never use the label as a target: the task in our setting is to generate the long conclusion, not to classify. CoQA. CoQA (Reddy et al., 2019) contains conversations about passages, with questions about the conversations. The development split contains 7,9837,983 questions over 500500 passages. We retain questions whose gold answer contains at least five ModernBERT tokens, yielding 1,8141,814 questions from 460460 conversations and 459459 distinct passages. The retained questions span five domains: Wikipedia (447447), CNN (400400), RACE (360360), Project Gutenberg (351351), and MCTest (256256). Dialogue history is included in the prompt. The rationale for the length filter is given in the next subsection; results on the unfiltered development set, all 7,9837,983 questions, are reported in Table 7. D.3 CoQA Filtering by Answer Length Our evaluation focuses on interactions among token players during Nash decoding. Nearly half of the gold answers in the full CoQA development split are a single word, and 18.4%18.4\% are bare yes or no responses. Such answers often occupy only one or two tokens. For single-token answers, the game has one player: its best response depends on the fixed prompt, with no other generated tokens to influence or respond to. These cases therefore provide limited scope for studying the interactions that motivate Nash decoding. Requiring at least five ModernBERT tokens focuses the evaluation on answers with multiple token players, whose conditional preferences can change as other players replace their tokens. This lets us study coordination and refinement within an answer. The threshold is an experimental design choice, not an algorithmic requirement, and answer quality still depends on the underlying model’s capabilities. All models are evaluated on the same retained questions; the reported CoQA results apply to this filtered subset. D.4 Tokenization and Reference Processing Tokenization. Descriptive token counts use answerdotai/ModernBERT-Large (Warner et al., 2025), except where a different tokenizer is explicitly identified. Answers are tokenized with a leading space, as " " + target, matching the dataset builder. Word counts are whitespace-delimited. The decoding budget is computed separately in each evaluated model’s tokenizer, as described in Appendix D.5. Normalization and reference handling. Lexical statistics use the evaluation normalizer: lowercasing, removing ASCII punctuation and the articles a, an, and the, and collapsing whitespace. Copying and overlap therefore refer to normalized text, consistent with the F1 metric. When an example has multiple reference answers, we compute each lexical statistic against every reference and report the highest value, matching the best-reference convention used for F1 evaluation. D.5 Scale, Length, and the Decoding Budget Table 5 summarizes the exact evaluated questions. The benchmarks differ by almost an order of magnitude in mean answer length. CoQA has the shortest questions and answers but the longest prompts because its input includes dialogue history. For each example, the reference answer determines an oracle continuation-length budget in the evaluated model’s own tokenizer. Masked generators fill that many continuation positions, and autoregressive baselines generate that many tokens with end-of-sequence generation disabled. ModernBERT counts provide a common basis for describing dataset lengths; the GPT-2 BPE row illustrates how the same answers yield different budgets under another tokenizer. CLAPNQ PubMedQA CoQA† Split dev, answerable PQA-L test (seed 0) dev, gold ≥5≥ 5 tok. Questions scored 300300 500500 1,8141,814 Distinct contexts 297297 500500 459459 (460460 conv.) References per question 1.621.62 11 44 (2.362.36 distinct) Mean length in ModernBERT tokens (words in parentheses) Context 195.3195.3 (170.8170.8) 287.0287.0 (202.3202.3) 355.2355.2 (267.0267.0) Question 10.410.4 (9.19.1) 17.817.8 (13.113.1) 6.66.6 (5.35.3) Dialogue history ++ question — — 140.7140.7 Full prompt 222.7222.7 319.7319.7 495.9495.9 Reference answer length Mean ModernBERT tokens (words) 63.863.8 (51.751.7) 48.348.3 (39.139.1) 7.57.5 (5.65.6) Median / 90th percentile / maximum 56/107/23156/107/231 45/77/15445/77/154 7/11/407/11/40 Mean GPT-2 BPE tokens 63.163.1 50.250.2 7.47.4 Mean sentences per answer 2.262.26 1.911.91 1.011.01 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). D.6 Lexical Overlap and Answer Composition Table 6 characterizes how reference answers reuse the supplied context. Token and content-word overlap measure which answer words occur in the context; novel bigrams measure word pairs absent from it. The contiguous-span and longest-run statistics describe how this shared wording is arranged. All comparisons use the normalization specified in Appendix D.4. The sentence-selection diagnostic greedily selects context sentences to cover distinct answer content words, with a target coverage of 90%90\%. It records the number of selected sentences and the distance, in sentence positions, between the first and last selected sentences. We also score two copying baselines: the entire context and the single context sentence with the highest reference F1. The latter uses the reference to select the sentence. These scores describe the potential of these particular copying strategies at their natural output lengths; they do not impose the oracle length budgets used for model comparisons or bound all possible copying strategies. CLAPNQ PubMedQA CoQA† Lexical overlap with the context Gold tokens occurring in the context 97.6%97.6\% 60.0%60.0\% 99.1%99.1\% Gold content words occurring in the context 97.4%97.4\% 49.3%49.3\% 98.9%98.9\% Novel bigrams (absent from the context) 10.9%10.9\% 76.8%76.8\% 6.5%6.5\% Answers with <50%<50\% content-word overlap 0.3%0.3\% 52.0%52.0\% 0.6%0.6\% Arrangement of shared wording Gold answer is a contiguous span after normalization 4.0%4.0\% 0.0%0.0\% 79.8%79.8\% Longest shared contiguous run / answer length 53.7%53.7\% 12.8%12.8\% 91.9%91.9\% Sentences selected (90%90\% coverage target) 2.502.50 3.393.39 1.011.01 Answers with ≥2≥ 2 selected context sentences 89.3%89.3\% 95.0%95.0\% 2.4%2.4\% Answers with ≥3≥ 3 selected context sentences 37.3%37.3\% 72.6%72.6\% 0.3%0.3\% Distance between first and last selected sentence 3.473.47 6.196.19 0.060.06 Copying baselines (F1) Entire context as the answer 50.750.7 18.418.4 4.94.9 Best single context sentence 67.667.6 34.734.7 45.045.0 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%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. CoQA: predominantly span-based answers. In the retained subset, approximately four in five answers are contiguous context spans after normalization. The longest shared run accounts for 91.9%91.9\% of answer length on average, the sentence-selection diagnostic uses at most one sentence in 97.6%97.6\% of cases, and only 0.9%0.9\% of answer tokens are absent from the context. The task therefore largely involves selecting a short answer span, with its length supplied by the oracle budget. The full-context copying score of 4.94.9 F1 reflects the substantial length mismatch between a roughly 355355-token passage and a 7.57.5-token answer; this diagnostic does not measure the difficulty of locating the appropriate span. CLAPNQ: composition from passage wording. Although 97.6%97.6\% of reference tokens occur in the passage, only 4%4\% of complete answers form a contiguous span. The longest shared run covers just over half of an answer. The sentence-selection diagnostic typically combines material from two or three sentences, with the first and last selected sentences approximately three and a half positions apart on average. CLAPNQ thus combines high lexical overlap with the need to assemble a cohesive answer. Copying the entire passage scores 50.750.7 F1, while selecting the best single sentence scores 67.667.6 F1. These reference-informed diagnostics show that passage reuse can receive substantial credit under the overlap metric. PubMedQA: greater lexical novelty. Two fifths of reference tokens and more than three quarters of reference bigrams are absent from the abstract. About half of the answers have fewer than half their content words in the context, and none is a complete contiguous context span. The reported content-word overlap is only 49.3%49.3\%, so selecting additional sentences cannot recover words that are absent from the entire abstract. In contrast, 70.9%70.9\% of question tokens occur in the abstract, the highest proportion among the three benchmarks. The question therefore has substantial lexical overlap with its context even when the reference conclusion requires considerable new wording. This setting places greater emphasis on generating a technical answer from scientific evidence than on recovering a passage span. Appendix E Experimental configuration This appendix specifies the prompts, reference-derived output lengths, token restrictions, decoding procedures, stopping rules, and metrics used for the comparisons in Table 1. Rows differ in two respects: the model and the decoding procedure (autoregressive, one-shot masked model, or Nash gap decoding). Everything else, such as answer-length budget, token restrictions, precision, context window, and metric are the same across decoding methods. E.1 Prompt Two prompt formats are used in the experiments. CLAPNQ and PubMedQA use a soft-instruction prompt: the evidence text, the question, a one-line instruction, and the token Answer: on which generation begins. passage Question: question This question is answered completely with evidence from the passage. Answer: The instruction line is byte-identical across every item of a dataset and across every system. PubMedQA differs in one word, abstract instead of passage, because its evidence is an abstract. For CoQA, we follow the prompting format of Radford et al. (2019): the passage is followed by the dialogue history and the current question, using Q: and A: delimiters without an additional instruction. passage Q: q1 A: a1 ⋮ Q: qt A: at Q: question A: For every preceding turn, the dialogue history contains the corresponding gold answer rather than a model-generated response. Consequently, predictions from earlier turns cannot propagate errors to later turns, and all models receive the same conversational context. The masked language models receive a whitespace-normalized version of the Q: and A: prompt, where newline delimiters are replaced by spaces and a leading space is added. We therefore evaluate each question independently. No system receives a chat template, a system message, or few-shot examples. The initialization ablation uses the same dataset-specific prompt defined above and changes only the initialization procedure. E.2 Text length Length is not a free parameter. Each system is given an oracle budget B=|tok(‘ ’+a⋆)|B= (`\,'+a ) : the number of tokens the gold answer a⋆a , prefixed with a single space, occupies in that system’s own tokenizer. An autoregressive model emits exactly B tokens; a masked model is given a canvas of exactly T=BT=B slots. Budgets are therefore matched in content rather than in count, and systems that share a tokenizer receive byte-identical budgets. The budget is computed per example, never averaged or rounded. Mean budgets run from 7.37.3 to 8.58.5 tokens on CoQA, 47.647.6 to 59.259.2 on PubMedQA, and 63.163.1 to 73.973.9 on CLAPNQ, the spread being entirely a matter of tokenizer. If a reference tokenizes to zero tokens in some vocabulary, the budget is zero and that system emits the empty string; this does not occur for any system in Table 1. E.3 Banned tokens The action set excludes structural tokens only: each tokenizer’s all_special_ids, together with the untrained embedding slots [|tok|,vocab_size)[ ,\ vocab\_size) that exist when a checkpoint pads its embedding matrix past the end of its vocabulary. No content word, scaffold label, or answer-like string is ever banned. The size of this set is a property of the tokenizer, not a choice. Banning end-of-sequence is what makes the length comparison exact: with EOS unavailable, an autoregressive model cannot stop early, so it emits exactly B tokens, matching the masked decoders, which fill exactly T slots. For the masked encoders, [MASK] is deliberately not banned. It is a legal action at every position, which is what makes the Nash gap well-defined from the first step, when every position still holds a mask. E.4 Decoding and numerics Decoding is greedy everywhere: argmax over the unbanned logits, for the autoregressive models, the diffusion samplers, and the Nash update alike. No temperature, top-k, top-p, or sampling is used, and no random seed influences any reported number. All models run in float32 with TF32 disabled on both matmul and cuDNN, and with the SDPA attention implementation. TF32 being off is asserted rather than assumed: each run multiplies two 2048×20482048× 2048 matrices and requires the float32 result to agree with a float64 reference to a relative error below 10−510^-5, which fails by three orders of magnitude if TF32 is silently active. E.5 Context window The context limit is 10241024 tokens with one margin token reserved, so a prompt may use at most 1024−B−11024-B-1 tokens. Longer prompts are truncated from the left, preserving the question and the instruction, which sit at the end. CLAPNQ and PubMedQA never require truncation; on CoQA it affects between 0.4%0.4\% and 1.1%1.1\% of items, depending on the tokenizer. One exception is inherent to the checkpoint: RoBERTa-Large has 514514 position embeddings with a padding offset of two, giving it an effective limit of 512512 tokens. E.6 Nash gap decoding The masked encoder starts from an all-[MASK] canvas of T slots and repeatedly applies the update i⋆=argmaxi[maxvp(v∣x−i)−p(xi∣x−i)],i = _i [ _vp(v x_-i)-p(x_i x_-i) ], committing the best response at i⋆i and stopping when that gap is ≤0≤ 0, which is an exact equilibrium. The gap is recomputed at all T positions after every committed token, so a position that is already filled can be revised before the canvas is complete; construction and refinement are not separate phases. The gap tolerance is exactly 00, and the number of updates is capped at max(64,10T) (64,10T). The all-mask rows use that rule from the all-mask state. The L2R rows are a baseline that differs only in the order in which positions are first filled; they are otherwise the same decoder, the same budget, and the same bans. E.7 Metrics We report three numbers per dataset. Token F1 is the SQuAD-style score against the best reference, after normalization. ROUGE-L and ROUGE-Lsum are F-measures computed with rouge_score through score_multi against all references, with stemming disabled. Both are computed on sentence-split text; the split changes only ROUGE-Lsum, since ROUGE-L tokenization treats a newline like any other separator. CoQA has four references per turn, CLAPNQ one, two, or seven, and PubMedQA one. For CoQA, we also computed the official leave-one-annotator-out accumulation; it is 0.40.4–1.61.6 points below the best-reference number, the smaller gaps occurring for the weakest systems, and it preserves the ordering of systems. E.8 Reproducibility and Verification All decoding procedures are deterministic under the reported experimental configuration. Autoregressive baselines use greedy next-token decoding, while Nash decoding selects the position with the largest Nash gap and replaces it with its highest-probability token. We also fix the exact model checkpoint used in each experiment to ensure that the underlying model remains unchanged across runs, and selected experiments repeated on a different compute node produced identical outputs. Finally, we verify all reported results directly from the saved per-example predictions by recomputing the evaluation metrics and checking them against the values reported in all result tables, ensuring consistency between the reported numbers and the underlying experimental outputs. Appendix F Extended Question-Answering Results Tables 7–9 provide the full question-answering comparison underlying the results in Section 4. Nash decoding turns frozen masked models of only 308308–396396M parameters into zero-shot generators without additional training or fine-tuning. Despite their small size, these models outperform many substantially larger autoregressive models. For example, the 396396M Ettin model achieves the strongest Nash result on CoQA while outperforming several 66–88B autoregressive models, corresponding to nearly an 18×18× difference in parameter count. Similar behavior appears on PubMedQA and CLAPNQ, where the strongest Nash-decoded model also exceeds multiple multi-billion-parameter autoregressive baselines. The strongest model nevertheless varies across datasets: Ettin-400M performs best on CoQA, RoBERTa-Large on PubMedQA, and ModernBERT-Large on CLAPNQ. Since the decoding procedure is unchanged, the variation in performance reflects differences in the full-context conditional probability estimates provided by the underlying masked model. Nash decoding therefore does not imply that these small masked models dominate all larger autoregressive models; stronger autoregressive models can still achieve higher task performance, and models not included in our comparison may also do so. Rather, the results show that enforcing token-level equilibrium can yield surprisingly high F1 scores from comparatively small masked models, motivating the use of Nash decoding with larger and more accurate masked models. The Calls column also makes the computational tradeoff explicit. Autoregressive decoding requires approximately one forward pass per generated token, whereas Nash decoding repeatedly evaluates full-context token replacements until equilibrium is reached and therefore uses substantially more test-time computation. In contrast, masked language models provide a lower-cost non-autoregressive baseline: one-shot generation requires only a single forward pass. However, under the same evaluation protocol, these lower-compute baselines remain below the best Nash-decoding answers across all three benchmarks. To check that the length filter does not drive these conclusions, we repeated the CoQA comparison on the entire development set—all 7,9837,983 turns, including the 1,5381,538 yes/no/unknown answers and every short answer the ≥5≥ 5-token cohort removes—under an otherwise identical protocol. The ranking is unchanged: the same model with Nash decoding maintains a better score. Table 7: Results on CoQA under the GPT-2 paper prompt. The left block scores the 1,814 questions whose gold answer has ≥5≥ 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 ≥5≥ 5 cohort. Gold ≥5≥ 5 tokens (n=1,814n=1,814) All (n=7,983n=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 GPT-2 Small (Radford et al., 2019) 124M 21.20 24.50 24.55 7.4 24 26.03 28.64 28.65 SmolLM2-135M (Allal et al., 2025) 135M 32.59 34.92 34.95 7.6 25 40.88 42.66 42.67 Pythia-160M (Biderman et al., 2023) 160M 15.15 18.87 18.89 7.5 24 19.33 22.18 22.18 OPT-350M (Zhang et al., 2022) 331M 27.16 29.86 29.94 7.4 25 31.52 33.69 33.71 GPT-2 Medium (Radford et al., 2019) 355M 36.84 38.46 38.56 7.4 25 42.62 44.24 44.26 SmolLM2-360M (Allal et al., 2025) 362M 52.77 54.30 54.32 7.6 27 59.77 61.06 61.06 Pythia-410M (Biderman et al., 2023) 410M 34.34 36.41 36.47 7.5 25 42.01 43.63 43.65 Qwen2.5 (Qwen Team, 2024) 0.5B 58.35 59.47 59.55 7.5 29 62.63 63.76 63.78 GPT-2 Large (Radford et al., 2019) 774M 42.61 44.31 44.38 7.4 26 48.79 50.26 50.27 GPT-2 XL (Radford et al., 2019) 1.5B 46.83 48.26 48.35 7.4 26 53.77 55.12 55.15 BLOOM-3B (5, 2022) 3B 56.69 56.25 56.62 7.3 29 60.07 60.68 60.81 Danube3-4B (Pfeiffer et al., 2024) 4B 59.40 60.29 60.35 8.2 28 62.45 63.59 63.61 Pythia (Biderman et al., 2023) 6.9B 59.57 60.56 60.64 7.5 27 65.97 66.94 66.96 OLMo-1.7 (Groeneveld et al., 2024) 7B 51.44 53.53 53.57 7.5 24 42.79 45.02 45.03 Nash decoding with same masked LMs RoBERTa-Large 355M 41.35 44.02 44.04 47.9 28 53.74 54.86 54.87 ModernBERT-Large 395M 58.12 58.71 58.71 47.7 29 67.04 67.52 67.52 Ettin-400m 396M 61.79 61.98 62.03 47.5 28 72.76 72.91 72.92 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 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 Pythia-160M (Biderman et al., 2023) 160M 17.09 16.77 17.14 48.3 216 OPT-350M (Zhang et al., 2022) 331M 23.43 20.43 21.41 50.2 240 SmolLM2-360M (Allal et al., 2025) 362M 21.56 19.71 20.57 48.8 222 GPT-2 Large (Radford et al., 2019) 774M 22.09 19.85 20.84 50.2 228 GPT-2 XL (Radford et al., 2019) 1.5B 23.78 20.77 22.00 50.2 231 BLOOM-3B (5, 2022) 3B 22.60 19.53 21.46 50.1 257 Danube3-4B (Pfeiffer et al., 2024) 4B 12.71 12.57 14.49 56.2 222 Pythia (Biderman et al., 2023) 6.9B 22.06 18.80 20.96 48.3 224 OLMo-1.7 (Groeneveld et al., 2024) 7B 19.67 17.71 20.09 48.3 228 Llama 1 (Touvron et al., 2023) 7B 23.67 20.21 22.53 59.2 255 OpenLLaMA (Geng and Liu, 2023) 7B 24.03 20.74 22.54 52.1 234 Amber (Liu et al., 2023) 7B 23.43 20.40 21.95 59.2 239 Granite (IBM Research, 2024) 7B 19.13 17.53 18.67 59.2 244 Mistral v0.1 (Jiang et al., 2023) 7.2B 17.13 14.31 16.40 56.2 177 Mistral v0.3 (Jiang et al., 2023) 7.2B 17.32 14.42 16.30 56.2 167 Falcon (Almazrouei et al., 2023) 7.2B 22.31 19.18 21.35 49.3 234 Llama 3.1 (Llama Team, AI @ Meta, 2024) 8B 23.30 19.49 21.97 49.4 239 Nash decoding with same masked LMs RoBERTa-Large 355M 24.99 21.17 22.69 1716 247 Ettin-400m 396M 22.77 20.05 21.65 1548 221 ModernBERT-Large 395M 24.28 20.33 21.36 1520 206 Table 9: Full results on CLAPNQ. 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 SmolLM2-135M (Allal et al., 2025) 135M 30.26 28.92 30.46 66.0 284 Pythia-160M (Biderman et al., 2023) 160M 26.46 27.53 27.28 63.8 279 OPT-350M (Zhang et al., 2022) 331M 33.21 32.14 33.39 63.1 281 GPT-2 Medium (Radford et al., 2019) 355M 32.56 31.23 32.74 63.1 292 SmolLM2-360M (Allal et al., 2025) 362M 32.62 31.02 32.66 66.0 290 Pythia-410M (Biderman et al., 2023) 410M 31.29 30.55 31.52 63.8 276 GPT-2 Large (Radford et al., 2019) 774M 33.06 31.50 33.68 63.1 288 GPT-2 XL (Radford et al., 2019) 1.5B 30.58 29.51 30.73 63.1 289 BLOOM-3B (5, 2022) 3B 34.00 32.18 34.31 63.3 292 OLMo-1.7 (Groeneveld et al., 2024) 7B 30.12 28.63 32.15 63.8 279 Llama 1 (Touvron et al., 2023) 7B 37.04 35.01 37.12 73.8 300 Amber (Liu et al., 2023) 7B 35.34 33.15 35.61 73.8 311 Mistral v0.1 (Jiang et al., 2023) 7.2B 34.40 32.75 35.20 71.8 293 Falcon (Almazrouei et al., 2023) 7.2B 34.39 32.91 34.50 67.4 294 OLMo-2 (Team OLMo et al., 2024) 7.3B 35.44 33.26 37.37 64.3 302 Nash decoding with same masked LMs RoBERTa-Large 355M 25.25 25.59 25.17 2913 273 Ettin-400m 396M 34.04 31.61 33.84 2935 281 ModernBERT-Large 395M 38.04 34.79 37.42 2967 263 Appendix G Ablation analysis The main results establish the performance of Nash decoding under our default configuration. We next isolate the design choices that determine this behavior. Specifically, we study the decoding trajectory toward equilibrium, prompt formulation, sequence initialization and the resulting equilibrium basin, token-update rules on a shared model from prior masked-diffusion work, and the computational cost of reaching equilibrium. Together, these analyses clarify which gains arise from the equilibrium-based decoding rule itself, which depend on implementation choices, and how these choices affect both answer quality and convergence. G.1 Decoding dynamics on all three datasets Figure 5 extends the analysis in Figure 3 to all three benchmarks using the same ModernBERT-Large model and all-[MASK] initialization. The same qualitative pattern appears throughout: the Nash gap first increases and then decreases toward zero, while F1 improves most rapidly during the early part of the trajectory. Thus, the dynamics observed on CLAPNQ are not dataset-specific. (a) CoQA (b) PubMedQA (c) CLAPNQ Figure 5: Answer F1 (green, right scale) and the Nash gap G(x(k))G(x^(k)) (violet, left scale) against the update index k from the all-[MASK] canvas, for ModernBERT-Large with all-mask construction. The initial rise in Nash gap becomes larger and occurs later for longer-answer datasets. The mean gap starts at 0.580.58, 0.590.59, and 0.450.45 on CoQA, PubMedQA, and CLAPNQ, respectively, and peaks at 0.640.64 at k=3k=3, 0.810.81 at k=10k=10, and 0.760.76 at k=21k=21, corresponding to increases of 11%11\%, 36%36\%, and 71%71\%. At these peaks, the output is on average 45%45\%, 25%25\%, and 42%42\% filled. The largest incentive to revise therefore emerges while a substantial fraction of the answer is still masked, as newly generated tokens provide context that sharpens the remaining conditional distributions. Convergence is nearly universal. All 500500 PubMedQA examples and all 300300 CLAPNQ examples reach exact equilibrium, with final mean Nash gap below 10−410^-4. On CoQA, 1,8081,808 of 1,8141,814 examples converge; the remaining six enter a limit cycle and are terminated by the update cap. These six examples are the only non-converged trajectories in our experiments. Answer quality converges substantially earlier than the equilibrium criterion. F1 comes within 1%1\% of its final value by k=11k=11, 5555, and 7676 on CoQA, PubMedQA, and CLAPNQ, respectively, whereas the corresponding trajectories extend to 5050, 175175, and 250250 updates. Over the second half of each trajectory, F1 changes by at most 0.10.1 points. Only 16%16\%, 35%35\%, and 32%32\% of examples are still active at those points, and the remaining updates increasingly revise already-generated tokens. Revisions account for only 4.3%4.3\%, 2.5%2.5\%, and 4.4%4.4\% of all updates, although on CLAPNQ the final 16%16\% of updates contain 37%37\% of all revisions. Hence, most answer quality is obtained well before exact convergence; the additional computation primarily removes residual token-level instability and certifies equilibrium. G.2 Example Generations on CLAPNQ The example discussed in Section 4.2 is marked † ; the remaining nine are taken one at each evenly spaced percentile of the run’s F1 distribution, so the list spans the full range rather than a selected part of it. All text is verbatim output from ModernBERT-Large under Nash decoding. † Example 1 – F1 98.598.5 Question where do they shoot guy’s grocery games Nash decoding Guy’s Grocery Games Season 1 was shot inside of an actual grocery store , Field ’s Market in West Hills , California . For Season 2 , the market was built in a 15,500 square foot warehouse in Santa Rosa . Reference Guy’s Grocery Games Season 1 was shot inside of an actual grocery store , Field ’s Market in West Hills , California. For Season 2 , the market was built in a 15,500 square foot warehouse in Santa Rosa , CA. Example 2 – F1 83.383.3 Question where do you get an at hop card Nash decoding at ticket machines , at ticketing offices , at online retailers , and at selected retail outlets such as bookshops Reference AT HOP card can be purchased online, at ticket machines, at ticketing offices, and at selected retail outlets. Example 3 – F1 70.170.1 Question where does water go after it enters a storm drain Nash decoding Small storm drains may discharge into individual dry wells . Small storm drains may be interconnected using slotted pipe , to make a larger dry well system . Storm drains may discharge into man made reservoirs known as recharge basins . Reference Most drains have a single large exit at their point of discharge into a canal , river , lake , reservoir , sea or ocean, as well as discharge into man - made excavations known as recharge basins or retention ponds . Example 4 – F1 66.766.7 Question who wrote capitalism and underdevelopment in latin america Nash decoding andre gunder frank wrote capitalism and underdevelopment Reference Frank wrote Capitalism and Underdevelopment in Latin America. Example 5 – F1 60.160.1 Question what is the function of insulin and where is it produced Nash decoding Insulin is a peptide hormone produced by beta cells of the pancreatic islets . It is the main anabolic hormone of the body . It regulates the metabolism of carbohydrates , fats and protein by promoting the absorption of , especially glucose from the blood into liver , fat and skeletal muscle cells . In these tissues the absorbed glucose is converted into either glycogen or fats ( triglycerides ) via lipogenesis , or into glycogen ( triglycerides ) via glycogenesis , or , in the case of the liver , into both . Reference Insulin is a peptide hormone produced by beta cells of the pancreatic islets. It regulates the metabolism of carbohydrates, fats and protein by promoting the absorption of, especially glucose from the blood into liver, fat and skeletal muscle cells. Glucose production and secretion by the liver is strongly inhibited by high concentrations of insulin in the blood. Circulating insulin also affects the synthesis of proteins in a wide variety of tissues promoting the conversion of small molecules in the blood into large molecules inside the cells. Example 6 – F1 51.251.2 Question what happens to toby at the end of the office Nash decoding Toby was fired immediately after Dwight fired Kevin Malone ( it ’s likely that the Corporate office fired Toby and Dwight simply had the burden of telling him ) and Toby then moved to New York where he ’s trying to become a novelist and has six roommates . Toby surprises his former co-workers at the reunion panel when he says that his life feels meaningless now that he ’s not Reference In the series finale, it’s revealed that Toby was fired immediately after Dwight fired Kevin Malone and Toby then moved to New York where he’s trying to become a novelist and has six roommates. In a photo gallery NBC released of ” where are they now? ” info, it’s revealed that Toby has returned to his destiny by taking on an HR position at a TJ Maxx store. Example 7 – F1 48.048.0 Question what was agenda 21 of earth summit of rio de janeiro Nash decoding Agenda 21 is a non-binding , action plan of the United Nations with regard to sustainable development . It is a product of the Earth Summit ( UN Conference on Environment and Development ) held in Rio de Janeiro , Brazil , in 1992 . Answer: Earth Summit Reference Agenda 21 is a non-binding action plan of the United Nations with regard to sustainable development. It is an action agenda for the UN, other multilateral organizations, and individual governments around the world that can be executed at local, national, and global levels. Example 8 – F1 42.942.9 Question which sport awards the oldest trophy in international sports Nash decoding The America ’s Cup is the oldest international sporting trophy . It will next be raced for in the southern summer , in the early part of Reference The America ’s Cup is the oldest international sporting trophy awarded to the winner of the America ’s Cup match races between two sailing yachts. Example 9 – F1 39.039.0 Question where do you expect to find a composite cone volcano Nash decoding Answer: Shield volcano Shield volcanoes are sometimes called ” composite volcanoes ” because of their composite layered structure built up from sequential outpourings of eruptive materials . They are among the most common types of volcanoes , in contrast to the less common shield volcanoes . Two famous shield volcanoes are: . Question: where do you expect to find a shield volcano? This question is answered completely with evidence from the passage. Answer: Answer: Reference Stratovolcanoes are sometimes called ”composite volcanoes” because of their composite layered structure built up from sequential outpourings of eruptive materials. Two famous stratovolcanoes are Krakatoa, best known for its catastrophic eruption in 1883 and Vesuvius, famous for its destruction of the towns Pompeii and Herculaneum in 79 CE. In modern times, Mount Saint Helens and Mount Pinatubo have erupted catastrophically. Example 10 – F1 23.123.1 Question who takes photos of planes in the air Nash decoding Who takes photos of planes in the air? This question is answered completely with evidence from the passage. Answer: Who takes photos of planes in the air? Reference Air - to - air photography is the art of photographing aircraft in the air , while using another aircraft as a photo platform . It can be used for commercial use and advertising . G.3 Prompt Ablation. On CLAPNQ, Table 10 compares the soft-instruction prompt with a Question:/Answer: prompt that omits the instruction sentence. The passage, question, and decoding procedure are otherwise unchanged. The soft instruction improves all three metrics for both decoding paradigms, including Falcon-7B with autoregressive decoding and ModernBERT-Large with Nash decoding. Table 10: Prompt-format ablation on CLAPNQ. 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 G.4 Initialization Section 4.2 reports that the equilibrium a trajectory reaches depends on where it starts. This appendix defines the two constructions compared there, extends the comparison to every model across three datasets. G.5 What the Two Constructions Are Both constructions answer the same question — how does a completed sequence first come into existence, before any refinement — and both hand their output to the same max-gap refinement of Algorithm 1, run to G(x)≤10−4G(x)≤ 10^-4. Left-to-right construction. The answer canvas is T slots, all [MASK]. Left-to-right construction fills them in index order. At step i the model is given the prompt, the tokens already committed at positions 1,…,i−11,…,i-1, and [MASK] at positions i,…,Ti,…,T; it predicts position i, and the highest-probability token there is committed. That is one model call per position, T calls in total, and nothing is revised while construction is in progress: a token written at step i is not reconsidered until all T slots are full. [MASK] is removed from the action set during construction, so every step commits a real token rather than leaving the slot empty. This imitates the order of autoregressive decoding without being autoregressive. The model is bidirectional and does attend to the positions to the right of i; those positions simply carry no content yet. Remarkably, left-to-right construction is a masked model writing in left-to-right order, not a simulation of an autoregressive model. All-mask construction. All-mask construction does not separate the two phases. The max-gap rule is applied directly to the all-[MASK] canvas: the token gap of Equation 3 is evaluated at all T positions, and at a position that has not been written the current token is [MASK] itself, which is a legal action. Writing an empty slot and rewriting a filled one are therefore scored on the same scale and compete directly, so the two can interleave. They do: with ModernBERT-Large, 200200 of the 1,8141,814 CoQA trajectories and 276276 of the 500500 PubMedQA trajectories revise an already-written position before the canvas is complete. Under left-to-right construction that number is zero by definition. Appendix H Nash vs Diffusion Decoding Section 4.3 compares Nash decoding with single-token masked-diffusion decoding on one checkpoint. This appendix states that configuration and reports the comparison on all three benchmarks. H.1 Checkpoint and Coordinate Rules All three methods use the released ModernBERT-Large-chat checkpoint, which keeps the ModernBERT-Large architecture and is supervised fine-tuned on instruction data for masked-diffusion generation. Its identifier is dllm-hub/ModernBERT-Large-chat-v0.1. No weights are modified, and a single checkpoint serves every method. Each method begins from the same all-[MASK] canvas and writes one token per step. They differ only in which position is written. Uniform order selects a masked position at random; confidence order selects the masked position whose predicted token has the highest conditional probability; Nash gap selects the position with the largest token gap of Equation 3, evaluated at all T positions rather than only the masked ones. The two diffusion rules freeze a position once it is written, so each performs exactly T updates and halts. We run them in their single-token setting, with the number of steps and the block length both set to T, so one position is committed per step, and no block structure remains. Sampling is replaced by argmax in both methods, so no method draws a random token: the uniform rule is random only in which position it fills next. Nash decoding instead recomputes the gap at all T positions after every commitment, may rewrite a position it has already written, and halts only when G(x)≤10−4G(x)≤ 10^-4. H.2 Prompt and Canvas Every method receives byte-identical input. The prompt is the instruction of Appendix E with one clause appended: evidence Question: question This question is answered completely with evidence from the passage. Answer using the passage’s own words. Answer: It is presented as raw text, with no chat template, no system message and no added special tokens. The canvas is [CLS] prompt [MASK]×T× T [SEP], where T is the oracle answer-length budget of Appendix E computed in this checkpoint’s own tokenizer; its mean is 7.57.5 tokens on CoQA, 48.348.3 on PubMedQA and 63.863.8 on CLAPNQ. We checked that the prompt token ids and the canvas length agree exactly across the three methods on all 2,6142,614 examples, so a method can differ from another only through its coordinate rule. The longest prompt-plus-canvas in the three datasets is 1,2881,288 tokens, well inside the checkpoint’s 8,1928,192-token context, so nothing is truncated. Decoding is greedy in float32 with TF32 disabled and the SDPA attention implementation. Nash decoding uses tolerance ε=0 =0, halts at the first repeated state, and is capped at max(64,10T) (64,10T) updates. H.3 Banned Tokens One action set is used for every dataset and every method. From the checkpoint’s vocabulary we remove the four non-mask special ids and eleven further tokens: three pieces of its answer terminator ([/, Answer, ]), four chat-template markers ([, SYS, Response, Question), and the four newline tokens. That leaves 50,35250,352 actions common to all three methods, plus [MASK], which only Nash decoding may use. H.4 Results Table 3 reports the comparison. Nash decoding is ahead of both diffusion rules on all three benchmarks and on all three metrics. Against the stronger of the two rules, which is confidence order everywhere, the paired margin in F1 is +1.23+1.23 on CoQA (bootstrap CI [+0.61,+1.82][+0.61,+1.82], permutation p=0.000p=0.000), +0.37+0.37 on PubMedQA ([+0.09,+0.68][+0.09,+0.68], p=0.008p=0.008) and +0.32+0.32 on CLAPNQ ([−0.39,+1.03][-0.39,+1.03], p=0.386p=0.386). Cost and Convergence. Table 11 gives what the improvement costs and how often the equilibrium criterion is actually met. A diffusion method uses one forward pass per generated token, so its cost is T. Nash decoding rescans all T positions after each commitment, so its cost grows with the product of budget and updates: it is 6.6×6.6× the diffusion cost on CoQA, 33×33× on PubMedQA and 48×48× on CLAPNQ. 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. Calls Dataset Mean T Diffusion Nash gap CoQA (1,8141,814) 7.57.5 88 5353 PubMedQA (500500) 48.348.3 4848 1,6041,604 CLAPNQ (300300) 63.863.8 6464 3,0763,076 Appendix I Empirical cost of Nash decoding Table 1 reports answer quality under reference-derived token budgets. Figure 6 reports the associated computational cost for ModernBERT-Large with all-mask and left-to-right (L2R) construction, alongside three autoregressive baselines. All five systems were measured on a single B200 under the same numerical settings, with one evaluation process running at a time. CoQA and PubMedQA timings cover their full evaluation sets; CLAPNQ timings cover the shared 6060-item subsample specified in the caption. Reported costs include runs that stop at equilibrium or through the cycle and update-cap stopping rules. (a) CoQA (b) PubMedQA (c) CLAPNQ 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 6060-item subsample, consisting of every fifth item and shared by all five systems. Its mean reference answer length is 49.049.0 words, compared with 51.751.7 for the full set; mean L2R cost is 249249 calls, compared with 248248 on all 300300 items. All-mask decoding with ModernBERT-Large takes approximately 0.490.49, 10.510.5, and 17.417.4 s per question on the timed CoQA, PubMedQA, and CLAPNQ sets, respectively, requiring approximately 4848, 1,5201,520, and 2,7102,710 model calls. Falcon, the 7.27.2B-parameter autoregressive baseline, takes approximately 0.260.26, 0.80.8, and 1.11.1 s on the same sets. These values follow the rounding used in Figure 6. The larger overhead on the longer-answer tasks is consistent with the work required to recompute token gaps. For a canvas of T positions, a full gap scan can require up to T conditional evaluations. Repeating this scan across K token replacements therefore requires O(T(K+1))O(T(K+1)) conditional evaluations, including the final scan. Actual wall-clock cost also depends on prompt length, the number of replacements, the model, and how the evaluations are batched. L2R construction reduces this cost. It requires approximately 2020, 150150, and 249249 calls on the three timed sets, compared with 4848, 1,5201,520, and 2,7102,710 for all-mask decoding: reductions of approximately 2.4×2.4×, 10.1×10.1×, and 10.9×10.9×, respectively. On the full evaluation sets, the corresponding ModernBERT-Large F1 scores are 56.9356.93 versus 58.1258.12 on CoQA, 21.9121.91 versus 24.2824.28 on PubMedQA, and 36.1236.12 versus 38.0438.04 on CLAPNQ. Thus, L2R construction trades approximately 1.21.2, 2.42.4, and 1.91.9 F1 points for lower computational cost. The CLAPNQ quality comparison uses all 300300 questions, whereas its displayed timing comparison uses the 6060-item subsample. The two constructions also distribute computation differently. L2R fills the canvas in T sequential predictions before refinement begins, so subsequent refinement accounts for most of its model calls in these measurements. All-mask decoding repeatedly evaluates competing positions while the canvas is incomplete; most of its calls occur before the last mask is filled, and this interval can already include revisions. Model-call counts should be interpreted alongside wall-clock times because a call represents different amounts of work across the two model families. After its initial prompt evaluation, an autoregressive decoder processes each new token using a cached prefix, whereas the masked implementation re-encodes the prompt-and-canvas sequence for its conditional evaluations. The timings therefore characterize these models and this implementation under the stated hardware and numerical settings. They do not establish an implementation-independent cost of finding an equilibrium. Appendix J Use of Generative AI Generative AI tools were used to assist with implementing the experiments, developing and refining code for data visualization and figure generation, and editing the manuscript for grammar, clarity, and readability. All AI-assisted outputs were reviewed, tested where applicable, and revised by the authors, who take full responsibility for the final manuscript, experimental results, and conclusions.