Ask a Language Model for Lottery Numbers: Concentration in Repeated Six-of-49 Outputs

Many high-stakes AI deployments proceed only if every stakeholder deems the system acceptable relative to their own minimum standard. With randomization over a finite menu of options, this becomes a feasibility question: does there exist a lottery over options that clears all stakeholders' acceptability bars? We study a query model where the algorithm proposes lotteries and receives only binary accept/reject feedback. We give deterministic and randomized algorithms that either find a unanimously acceptable lottery or certify infeasibility; adaptivity can avoid eliciting many stakeholders' constraints, and randomization further reduces the expected elicitation cost relative to full elicitation. We complement these upper bounds with worst-case lower bounds (in particular, linear dependence on the number of stakeholders and logarithmic dependence on precision are unavoidable). Finally, we develop learning-augmented algorithms that exploit natural forms of advice (e.g., likely binding stakeholders or a promising lottery), improving query complexity when predictions are accurate while preserving worst-case guarantees.

When a language model must choose one answer from a large space of equally valid options, a format clause -- "Reply with JSON only" -- changes which answer it chooses. We re-run the One-Word Census (arXiv:2607.12796): 31 wide-answer-space category prompts asked of 44 models, now with the reply requested in JSON -- no schema enforcement, no constrained decoding, only the request. Convergence deepens sharply: on the unconstrained "Pick a word" prompt the modal answer rises from 41% to 64% of the pool and distinct answers fall from 52 to 36; mean answer-choice surprisal drops from 1.80 to 1.58 bits. The tax is progressive: six of 44 models move individually (BH-FDR q=.10), all toward the mode, led by the most distinctive models, while the conformist floor is immobile. It is a sharpener, not a re-indexer -- the plain-chat modal answer survives in 28 of 31 categories. Defaults are register-indexed: a within-run re-sample (n=20) finds JSON shifts 53% of a model's stable chat defaults, mostly back to the crowd, and installs defaults absent from chat (Claude Fable 5 answers "cerulean" for colour 0% of the time in chat, 100% in JSON). Full-battery controls reveal a register gradient: compression is significant and specific to the answer-delivery formats models are trained to speak (JSON -0.22 bits, p=.0002; XML -0.19, p=.002), absent for YAML and CSV, and reversed for an arbitrary bracket wrapper (+0.13, p=.009) -- weighing the mechanism toward tool-use post-training. Enforcing the schema at the decoder (response_format) compresses no further than the request (-0.03 bits): the collapse lives in the model's response to the register, not the decoder. Structured output is how software consumes language models, and that surface is served by a measurably more homogeneous model than the chat surface on which models are evaluated, compared, and chosen.

When a language model must choose one answer from a large space of equally valid options, which answer does it choose, and how often is it the answer every other model chooses? Asked to "pick a word," 105 language models from more than twenty labs chose serendipity 46% of the time. We measure this convergence, and each model's share in it, with 96 single-turn prompts that each name a category with many valid one-word answers ("Name a tree."), asked eight times per model and scored by exact match, with no embeddings and no judge. A model's answer-choice surprisal is the average -log2 probability of its answers under the pooled answers of all other models. In 28 of 96 categories a single answer takes at least 80% of all answers. The concentration does not depend on small or persona-tuned models: the 87 major-lab models are at least as concentrated as the full field. Lightly post-trained and persona-tuned models are the most divergent; heavily post-trained assistants from the major labs are the most conformist. Models that avoid the modal answer mostly land on the same runner-up. Within the major providers' lineages, release order shows no panel-wide trend once model tier is controlled; GPT, Gemini, Grok and Qwen become more conformist across releases, and Claude's generation-5 releases reverse. On open checkpoints of three post-training pipelines, supervised fine-tuning is the largest step toward the field's answers. Against human category-production norms, the field is more concentrated than people in 18 of 20 shared categories. All prompts, transcripts, and code are public.