Authors: David Reber, James Reber, Ari Holtzman, Victor Veitch
Organizations: Department of Computer Science, University of Chicago · Independent Researcher · Data Science Institute, University of Chicago · Department of Statistics, University of Chicago
To use language models to help us make decisions we would like to understand the rationale behind the language model's outputs. In principle, the language model provides its rationale as part of its text output. However, the sheer volume of text makes this hard to use in practice. In this paper, we are interested in extracting the cruxes that lead a model to recommend one action over another. We take a crux to be a sub-question that a model's rollouts disagree on, where changing its value may have altered the conclusion. A crux is therefore a property of the distribution of a model's rollouts, not of any one rollout. We consider three strategies of increasing computational cost. First, we just ask a model to identify, a priori, what the cruxes of a decision will be. Second, we have the model actually go through the process of generating a response and then extract the cruxes that arose in its reasoning. Finally, we generate many responses and extract the cruxes by comparing the reasoning across the rollouts. We make this last procedure computationally tractable with cheap calls to a reranker, an estimated 86 times less compute than a judge. We then conduct an extensive empirical evaluation to assess whether the computationally inexpensive options are sufficient. They are not. Across domains and models, we find that a model's a priori assessment of the cruxes has little relation to the factors that are actually identified as important a posteriori. Using a single rollout helps, but is still inadequate for identifying the full set of cruxes. Accordingly, for high-stakes decisions (and to understand the space of language model decision making) the many rollout procedure is unavoidable.
Figures & tables
Figure 1: A crux is an assumption that forecasts disagree on, and whose answer could have changed their conclusions.
Figure 2: In a basketball game and a bankruptcy question, the cruxes a model predicted matched the questions its forecasts varied on far less often than the permutation baseline, which reassigns the same cruxes at random to the a priori and multiple-rollout sets. Left: Iowa State vs. Kentucky, a game in March Madness, the American college basketball tournament. Right: Frontier Group bankruptcy, an event on Metaculus, a public forecasting platform. Each diagram splits the clusters of paraphrased cruxes ( Section 5.1 ), formed from 25 cruxes per set, into those holding a priori cruxes only (orange), both kinds (green, the overlap) and multiple-rollout cruxes only (blue). Below each diagram stand one a priori crux, as a question, and one multiple-rollout crux, as one forecast’s answer ( contribution 1 extracts the multiple-rollout cruxes; contribution 2 groups similar paraphrases at low cost). A turnover is a possession lost before the team can shoot, and a forced turnover is one the defense causes. The Altman Z-score is a bankruptcy-risk score computed from a company’s accounts. Below the circles each diagram prints its overlap at 25 cruxes and its permutation baseline ( contribution 3 runs this comparison across games, events and models).
Figure 3: In March Madness the test asks whether a model, reading every scouting report its forecasts were written from, can foresee where those forecasts will disagree. A priori cruxes (left) against multiple-rollout cruxes (right), for one March Madness game, Iowa State vs. Kentucky. The a priori prompt predicts cruxes from the reports, and the multiple-rollout prompt finds them in each pair of the game’s rollouts. The a priori and multiple-rollout cruxes are then pooled and clustered, and their overlap is compared with a baseline that shuffles the two labels among the same cruxes (bottom row). Each crux is a yes/no question with both answers, and the figure quotes the multiple-rollout crux by the assumption one of the two rollouts took, that Iowa State forced 23 turnovers in an earlier game.
Figure 4: Cruxes are often shared across rollouts, so the number of distinct cruxes grows about linearly with the number of rollouts even though the number of rollout pairs grows quadratically. Left: the median number of distinct crux clusters at the same-resolution level ( Section 5.1 ) against the number of rollouts, each game or event averaged over up to 20 random subsets (bands, 10th to 90th percentile), with the number of rollout pairs for reference. March Madness reaches 28 clusters at nine rollouts and shows no sign of saturation. Metaculus, with 12 of the 29 or 30 rollouts per event and all their cruxes, appears to converge to 5 cruxes. Middle and right: March Madness (59 games, N=9 ) and Metaculus (15 events, N=12 ), where the colours now mark quantities, not sources, and every curve is a share of what all N rollouts give. Blue is coverage, the share of the clusters from all N rollouts that n rollouts already find. The dashed line is the coverage of as many cruxes drawn at random from the pool. Pink is the number of clusters from n rollouts as a share of the number from all N . In Metaculus that ratio passes 100%, and the axis stops at 150%.
Figure 5: Naming more cruxes, up to 25 on each side, still does not close the gap between predicted and discovered cruxes. Bands give the share of clusters holding a priori cruxes only (top, orange), multiple-rollout only (middle, blue) or both (bottom, green, the overlap). The dashed line is the permutation baseline, the overlap when the two labels are shuffled. Left: March Madness, 59 games. Right: Metaculus, 15 events, baseline 49.5% at 5 cruxes and 70.9% at 25. Same-resolution matching throughout; lighter bands show ±1 SE, and the numbers at the right give each band’s share at 25 cruxes.
Appendix figures & tables15 assets
Supplementary material from the paper’s appendix.
Appendix
Figure 6: Whether two cruxes must ask the same question or need only concern the same dispute, predicted cruxes match discovered ones less often than the permutation baseline. Stacked-area decomposition at the three levels, same question, same resolution and same dispute, for March Madness (59 games, top row) and Metaculus (15 events, bottom row), in the colours of Figure 5 . The numbers at the right of each panel give its three shares at 25 cruxes. At 25 cruxes the March Madness overlap is 5.6%, 13.4% and 12.8% against baselines of 33.4%, 39.2% and 41.2%, and the Metaculus overlap is 21.9%, 39.9% and 39.7% against 58.0%, 70.9% and 68.9%.
Source
Level
Overlap (%)
Baseline (%)
Gap
p
Below
March Madness
same question
5.6±0.9
33.4
27.8
<10−10
59 / 59
same resolution
13.4±1.6
39.2
25.9
<10−10
55 / 59
same dispute
12.8±1.5
41.2
28.4
<10−10
57 / 59
Metaculus
same question
21.9±3.6
58.0
36.1
6.1×10−5
14 / 15
same resolution
39.9±4.0
70.9
31.0
1.2×10−4
13 / 15
same dispute
39.7±4.4
68.9
29.3
2.4×10−4
12 / 15
Appendix
Table 1: At every level of strictness the gap below the permutation baseline is significant in both sources, and nearly every game and event falls below its own baseline. Overlap and baseline at k=25 at each level of strictness, each level against its own baseline, for March Madness (59 games) and Metaculus (15 events). Overlaps carry ± 1 SE across games or events. Gaps are in points, and p is from a one-sided Wilcoxon signed-rank test across games or events. The last column counts games or events below their own baseline; the rest equal it, apart from one March Madness game at the same-resolution level that lies above.
Figure 7: In March Madness, however coarsely cruxes are grouped, predicted cruxes match discovered ones less often than the permutation baseline, though halves of the discovered cruxes match each other. Left: Coarsening the clustering does not close the gap below the baseline. The overlap and its baseline across cuts of the tree, from broad categories on the left to one crux per cluster on the right (March Madness, same-resolution matching, 25 cruxes per set). The horizontal axis is ρ , the number of clusters divided by the number of pooled cruxes, and the top axis gives the clusters per game. The dotted line marks the cut used in the main text, a mean of 20.5 clusters per game ( ρ=0.41 ), and the grey band the middle half of games. Right: Two halves of either set share several times the overlap the two sets share with each other. Overlap against the number of cruxes drawn from each set, for two halves of the a priori cruxes, two halves of the multiple-rollout cruxes, and the a priori cruxes against the multiple-rollout cruxes. The dashed line gives the permutation baseline for the a priori against multiple-rollout comparison (March Madness, same-resolution matching). At 25 cruxes two halves of the multiple-rollout cruxes sit just above the dashed baseline, two halves of the a priori cruxes well above it, and the two sets far below it. So the discovered cruxes are not too scattered for a set of cruxes to match them.
Figure 8: Predicted cruxes match discovered ones less often than the permutation baseline in nearly every game and event, so the mismatch is the typical case. Top: the gap of each game and event at 25 cruxes, its own baseline minus its overlap, with a line at zero. In March Madness 55 of 59 games fall below their own baseline, 3 equal it and 1 lies above it; in Metaculus 13 of 15 events fall below, 2 equal it and none lies above. Bottom: six March Madness games chosen at random (seed 42), as each set grows to 25 cruxes, in the colours of Figure 5 and with each game’s own baseline as the dashed line. The shared band rises and falls with set size and differs from game to game. Same-resolution matching throughout.
Source
Cruxes
Factor
Claim
Value
March Madness
predicted, original prompt
1.9
66.4
31.7
predicted, same-day control
1.8
66.5
31.7
predicted, specific prompt
0.1
3.3
96.6
predicted, exemplar prompt
0.2
2.8
97.0
predicted, ladder, factor prompt
9.3
68.1
22.6
predicted, ladder, claim prompt
1.9
77.2
20.9
Appendix
Table 2: Asked for specific cruxes, the predictor states values at least as often as the discovered cruxes do, while its usual cruxes mostly do not. Share of cruxes at each level of specificity, in percent, for the cruxes as written and for each prompt. The factor prompt of the ladder failed its check, so the ladder separates claims from values only.
Source
Prompt
k=25
k=12
March Madness
original (headline, one seed)
25.8
14.8
same-day control
24.7
15.0
specific
16.4 ( p=1.4×10−5 )
10.1 ( p=0.0012 )
exemplar
12.1 ( p=1.4×10−8 )
7.8 ( p=5.2×10−6 )
ladder, factor
27.4 ( p=0.10 )
16.1 ( p=0.57 )
ladder, claim
23.0 ( p=0.40 )
12.5 ( p=0.043 )
Appendix
Table 3: In March Madness, asking for specific cruxes narrows the gap by about a third only by lowering the permutation baseline, and no prompt closes it. Gap in points under each prompt, at 25 and 12 cruxes per set. March Madness p values are two-sided Wilcoxon signed-rank tests of control minus arm over 59 games. The p values are normal approximations. In Metaculus the drift between the original and the same-day control (7.8 points at k=25 ) exceeds every difference between arms. Every row, the original included, uses the seed of the arm runs for the baseline’s label shuffles. That seed is the same for every game or event. The runs of Table 1 instead give each game or event its own seed. With that seeding the original prompt’s gaps are 25.9 and 15.0 in March Madness and 31.0 and 18.8 in Metaculus.
Source
Arm
k
Original
Arm
March Madness
multiple-rollout cruxes, rewritten one level vaguer
25
25.8
26.9 ( p=0.57 )
12
14.8
16.8
both sets rewritten toward factors
25
25.8
15.3 ( p=1.3×10−6 )
12
14.8
11.0 ( p=0.008 )
both sets restricted to values (24 games)
25
23.9
22.4
both sets restricted to values (53 games)
12
15.6
15.1
Appendix
Table 4: Making cruxes vaguer, or comparing only cruxes of equal specificity, leaves predicted cruxes matching discovered ones less often than the permutation baseline. Gap in points when cruxes are rewritten to be vaguer, and when both sets keep only cruxes at the same levels of specificity. The p values are two-sided Wilcoxon signed-rank tests of arm against original over games or events. The original column uses the full set of multiple-rollout cruxes, none rewritten or removed, on the same games or events. Both columns shuffle the two labels with one fixed seed for all games or events. So the original column gives 25.8 and 14.8 in March Madness and 31.4 in Metaculus. The main text gives each game or event its own seed and reports 25.9, 15.0 and 31.0.
Cruxes per set
Strict
Moderate
Broad
Text, same pools
25
20.3
24.3
20.1
22.9
12
13.9
18.2
16.0
13.7
Appendix
Table 5: Matching cruxes on the evidence behind them, instead of on their wording, still leaves predicted cruxes matching discovered ones less often than the permutation baseline. Gap in points when cruxes are matched on their written evidence under three wordings of the matching instruction, beside matching on the cruxes’ own text on the same pools (March Madness, 59 games). The full set of multiple-rollout cruxes gives 25.9 points at 25 cruxes.
Part of the prompt
Pairwise (multiple-rollout) extraction
Single-forecast (single-rollout) extraction
Task statement
none; the rubric leads straight into the steps
“You are given ONE argument: a single forecast for the question below, with its reasoning. Identify the cruxes of this argument: the factual premises about the world that its forecast turns on, so that if a premise were false, the forecast would change.”
Distillation
“Distill any differences into cruxes”
“Distill the premises its forecast depends on into cruxes”
option_A and option_B
Argument A’s premise and Argument B’s premise, contradicting along the crux dimension
the premise this argument relies on and the contrary premise, “so that if it were true the forecast would change”
Analysis step
analyze both arguments, do the same analysis for Argument B, then for each dimension note whether the arguments differ
analyze the one argument on the same four dimensions, with no comparison step
Maximum and order
none stated
“Return at most 30 cruxes, most important first: the premises whose reversal would move this forecast the most”
Minimum output
“You MUST return at least one crux if the arguments rely on different world-facts, even if they reach the same final answer”
no minimum
Appendix
Table 6: The single-forecast extraction prompt asks what one forecast rests on where the pairwise prompt asks where two forecasts disagree. Every part of the two prompts that differs, in descending order of how much the change matters; all else is shared.
Source
Cruxes
Overlap vs baseline
Gap
Below / above / tied, p
March Madness
single-rollout
18.8 vs 25.1
6.4
47 / 11 / 1, 4×10−9
March Madness
a priori
14.6 vs 41.4
26.8
56 / 0 / 3, 10−17
Metaculus
single-rollout
48.5 vs 44.0
−4.5
2 / 9 / 4, 0.94
Metaculus
a priori
35.1 vs 56.9
21.9
11 / 1 / 3, 5×10−4
Appendix
Table 7: Cruxes from one forecast come far closer to the permutation baseline than predicted cruxes, so scoring one set of cruxes against a pooled one does not explain the gap. Overlap against baseline at k=25 , same resolution, the single-rollout cruxes and the a priori cruxes scored on the same stripped multiple-rollout cruxes. Gaps are in points, the counts are games or events below, above and at their own baseline, and p is from the one-sided exact Wilcoxon test.
In pool
Covers pool
Source
Cruxes, pool
obs
base
gap
p
obs
base
gap
p
MM
a priori, headline
32.7
65.3
32.5
10−12
23.2
65.3
42.1
2×10−18
MM
single-rollout, stripped pool
36.7
57.7
21.0
3×10−14
28.6
37.8
9.2
8×10−8
MM
a priori, stripped pool
41.1
68.3
27.3
10−9
25.7
68.4
42.7
2×10−18
MM
per-forecast pooled, headline
40.9
52.2
11.3
3×10−12
55.0
52.1
−2.9
0.95
MM
split half
68.8
68.7
−0.1
0.73
65.9
68.5
2.6
0.035
Appendix
Table 8: Predicted cruxes cover far less of the discovered cruxes than the permutation baseline would, and in March Madness cruxes from one forecast fall short too. Containment at k=25 , same resolution, each direction against its own baseline. In pool is the share of the set’s cruxes whose cluster holds a pooled crux, covers pool the share of the pooled cruxes whose cluster holds one of the set’s; gaps are in points, p is one-sided exact Wilcoxon, and MM is March Madness.
Model
k
Overlap (%)
Baseline (%)
Gap
Below / above / equal
p
qwen3:32b
25
38.3±2.2
52.3
13.9±2.0
39 / 5 / 15
1.4×10−9
gpt-oss:120b
12
11.5
18.6
7.1
48 / 10 / 1
2×10−10
gemma4:31b
12
14.4±1.8
30.3
15.9±1.3
55 / 2 / 2
4×10−16
Appendix
Table 9: For each of three open-weight models, predicted cruxes match discovered ones less often than the permutation baseline in most games. The runs repeat the March Madness experiment at the same-resolution level; the qwen3:32b row holds no repeated cruxes, and the gpt-oss:120b and gemma4:31b rows are rescored with repeated cruxes removed. Gaps are in points, and the counts are games below, above and equal to their own baseline, where equal means that the game’s overlap matches the mean of its shuffles. The model gpt-oss:120b is quoted at k=12 because its extraction yields fewer than 25 cruxes in 17 games. The model gemma4:31b is also quoted at k=12 , the largest length at which most of its games are full, since its extraction yields fewer than 12 cruxes in 17 games. A game with fewer than k cruxes on one side is scored on all the cruxes it has ( Section 5.2 ).
March Madness
Metaculus
Forecaster
gpt-5.4-nano
gpt-5-nano, with web search
Predictor
gpt-5.4-nano-2026-03-17
gpt-5.4
Extractor
gpt-5.4-nano
gpt-5.4-mini
Scouting reports
gpt-5.4-mini, with web search
none
Matching
Qwen3-Reranker-8B
Qwen3-Reranker-8B
Appendix
Table 10: In March Madness one model forecasts, predicts and extracts, while in Metaculus each role goes to a different model. The model behind each role, behind the March Madness scouting reports and behind the matching of cruxes.
Experiment
Model
Calls
Cost
March Madness
Scouting reports
gpt-5.4-mini + web search
not recorded
not recorded
Forecasts, 62 teams
gpt-5.4-nano (batch)
17,019 stored
not recorded
Forecasts, First Four
gpt-5.4 (batch)
3,483
∼ $100
Multiple-rollout extraction
gpt-5.4-nano
68,076
∼ $290
A priori prediction
gpt-5.4-nano
2,341
not recorded
Appendix
Table 11: API and compute costs for all experiments. Costs are billed amounts where a record exists, dashboard readings for the March 2026 runs and token counts at OpenAI list prices of September 2026 for the later batch runs, and estimates where a figure is given without a record; cells with no per-stage record read not recorded, and batch rates apply to the runs marked batch, at half the standard rates (OpenAI list prices, 2026-09-22). Where the model column names a snapshot, that is the snapshot the run record names; elsewhere the column gives the undated model name the calls were sent with. Which teams meet in later rounds depends on who wins the earlier games, so March Madness forecasts were made for every pairing of the tournament’s teams. None of the 59 games draws on the forecasts for the First Four, the tournament’s play-in round.
Post
Question
Domain
42117
Will AfD win the most seats in the Baden-Württemberg Landtag election?
elections
42116
Will CDU win the most seats in the Baden-Württemberg Landtag election?
elections
42231 (or 40005)
Will any political party or coalition acquire a supermajority in the 2026 Hungarian parliamentary elections?
elections
42118
Will the official result of the Lamag by-election (Sabah), scheduled for January 24, 2026, be overturned before May 1, 2026?
elections
42232 (or 42148)
Will Ali Khamenei cease to be supreme leader of Iran before April 1, 2026?
geopolitics
42119
Will any state actor use chemical weapons in Ukraine before May 1, 2026?
geopolitics
Appendix
Table 12: The 15 Metaculus events, their post identifiers and one domain label each.
LLM decision-making studies how language models choose, act, seek information, and allocate resources under goals, uncertainty, constraints, and social pressures.
The use of large language models either as decision support systems, or in agentic workflows, is rapidly transforming the digital ecosystem. However, the understanding of LLM decision-making under uncertainty remains limited. We study LLM risky choices along two dimensions: (1) prospect representation (based on an explicit representation or outcome history) and (2) decision rationale (explanation). Our study, which involves 20 frontier and open LLMs, is complemented by a matched human subjects experiment, which provides one reference point, while an expected payoff maximizing rational agent model provides another. We find that LLMs cluster into two categories: reasoning models (RMs) and conversational models (CMs). RMs tend towards rational behavior, are insensitive to the order of prospects, gain/loss framing, and explanations, and behave similarly whether prospects are explicit or presented via a history of outcomes. CMs are significantly less rational, slightly more human-like, sensitive to prospect ordering, framing, and explanation, and exhibit a large description-history gap. Paired comparisons of open LLMs suggest that a key factor differentiating RMs and CMs is training for mathematical reasoning.
Large language models (LLMs) often generate more reasoning tokens than needed. Limiting the reasoning budget reduces cost, but forcing an answer from an incomplete trace can hurt accuracy. System-1 decision models select among candidate options without producing a reasoning response. We evaluate Jev as a decision reader of truncated reasoning across seven LLM reasoners and five multiple-choice benchmarks (35 pairs). Jev achieves higher mean accuracy than forced answering at tested budgets from 500 to 16,000 tokens, including the estimated costs of reasoning and answering. At a 1,500-token budget, it improves accuracy by 4.1 pp overall and 8.6 pp on truncated questions, with significant gains in 17 pairs under Holm-corrected bootstrap tests. Offline replay estimates 34.5% of the full-run cost, with a 1.13 pp accuracy loss. A policy selected on training folds saves 71.3% of the estimated full-run cost on held-out folds, with a 0.58 pp accuracy loss. We also find that Jev shows strong question-only accuracy, exceeding 500-token forced answering on average (68.0% vs. 65.5%), although this advantage varies by benchmark. These findings suggest that System-1 decision models can complement LLM reasoning and reduce the need for costly reasoning completion.
Zhongyin Wang, Zhaoqian Xue, Junting Mu +4
University of Illinois Urbana-Champaign · University of Arizona · University of California, Irvine +3
We ask whether large language models (LLMs) merely imitate rationales when choosing between two options, or whether their choices reflect a systematic underlying decision structure. Using synthetic binary decision settings in which models choose between profiles defined by graded attributes, we compare the attribute a model says mattered most with the attribute that best explains its choice under a behavioural model fit to prior decisions. The behavioural model predicts held-out choices well, showing that model behaviour is systematically related to the visible attributes rather than being random. However, direct self-reports and a separate score-based judge recover the behaviourally inferred driver only partially. The resulting picture is neither one of arbitrary behaviour nor one of fully articulated belief - outputs are structured enough to support prediction, but explicit reasons track the recovered driver only imperfectly. This qualitative pattern persists across prompt-order and sampling perturbations, alternative behavioural models, targeted occlusion analyses, and structurally varied decision settings. We interpret this as evidence for ``superficial belief'' in LLM decision-making: models behave as if guided by probabilistic local priorities over attributes, while having only limited verbal access to the attributes that drive their decisions.