Can you arbitrage a language model?

September 4, 2026

Suppose you ask a language model the probability it will rain tomorrow, which we’ll call P(rain). Then you ask the model the probability it will not rain tomorrow, which we’ll call P(no rain).

Regardless of what happens tomorrow, these two probabilities should sum to one. But suppose the model generates P(rain) = 0.7 in response to your first request and P(no rain) = 0.2 in response to your second request.

How large is this violation of the laws of probability? We can measure probabilistic incoherence by “arbitraging” the model’s forecasts.

Example of arbitrage against generated probabilities.

This evaluation metric is “label-free,” in that we can compute the arbitrage profit against the language model without having to wait for the events to occur. Treat each probability as the price of a contract that pays $1.00 if the corresponding event occurs. You can buy both contracts for $0.70 + $0.20 = $0.90. Tomorrow, exactly one of rain or no rain will occur, so the two contracts will pay $1.00 in total. Therefore, you’ll make $0.10 in arbitrage profit on this trade.

Evaluating Probabilistic Coherence

In Dutch Books for Language Models with Isaiah Andrews, we systematically evaluate probabilistic coherence in language model forecasts.

People increasingly use language models to support life decisions, many of which depend on probabilistic forecasts. For example, how likely is a major life event, a natural disaster, or a financial outcome? Evaluating coherence in model forecasts may help us train more reliable models for these tasks.

We build a probabilistic forecasting environment from historical stock returns data. We discretize a stock’s future return into four bins, and ask the model to forecast probabilities for events built from the bins. We measure model incoherence using the Dutch book profit: the maximum guaranteed profit earned per unit of gross stake by betting against model-generated probabilities.

Incoherence and Brier score across language models

Coherence varies by about two orders of magnitude across models. Forecasting performance, as measured through the Brier score, varies much less. Still, coherence and accuracy are positively correlated across our evaluated models.

These findings relate to recent work on consistency checks and probabilistic coherence in language models, which evaluates particular consistency conditions: for example between an event and its complement, and across other algebraic identities (Paleka et al., 2025; Fluri et al., 2024; Zhu and Griffiths, 2024). Our approach takes a given set of events and exhausts the implications of probabilistic coherence, by finding the largest Dutch book against a model’s forecasts.

We additionally study what about the elicitation procedure influences coherence.

Logical relations: We find that more logical dependencies across events increase incoherence. For example, forecasts about the joint distribution of two assets are more incoherent than forecasts about marginal distributions.
Context: We also find that coherence is sensitive to additional information included in the prompt. Adding more context can increase incoherence, even when the context may not be relevant for the forecast. For example, incoherence rises when we augment the forecast prompt with “I’m feeling really optimistic about this,” or even “I had coffee this morning, by the way.”

Incoherence rises with joint forecasts across assets and with irrelevant prompt context

Ways Forward

How can we improve probabilistic coherence in language models? One way forward may be to build on the following finding in the paper: If we elicit all probabilities at once from the model (instead of eliciting one probability at a time), we get a much more coherent response. This suggests that models are capable of computation that supports more coherent forecasts in joint queries, even if models do not reliably perform such computation in one-off queries.

We may be able to encourage such computation through additional training. For example, by using process reward models that reward reasoning over contingencies. Or by using RL objectives over groups (Orney et al., 2026; Pres et al., 2026) that may encourage computation over related states. The linear program used to compute Dutch books may be attractive for such training (Andrews, 2026), as it provides an exact set level reward.

Link to paper. Comments are very welcome.