Given a state and a set of options, such as intents, tools or interface elements, System One decision models return a distribution over the options in one forward pass, without generated reasoning. Web systems increasingly build them by reading decisions from language models. Two axioms of rational choice are natural requirements for such decisions, because in web systems the listing and the option set are determined by other components and can change or be manipulated: the order in which the options are listed should not matter, and adding an option should not change the relative odds of the others (Luce's choice axiom). We test the standard reads, namely single-choice question (SCQ) prompting, with shuffled-order training, order averaging or sorted options, and invariant set architectures, and find that each violates at least one axiom: one shuffle changes 12-20% of SCQ answers, sorting only freezes an arbitrary convention, and a well-chosen added option steers 4-34% of decisions. By Luce's theorem, a read satisfies both axioms only if it is a softmax over scores that each depend on one option alone, so in a transformer they hold by construction only if no token that scores an option sees another. We implement this as Prism-1 (Position- and Rival-Independent Scoring for System One Models): every option is read in its own block at shared positions, with the model's usual left-to-right attention, in one forward pass. On five benchmark sets with Qwen3-8B and Gemma 4 12B, SCQ prompting trained on the same data still changes 7% of its answers under one shuffle, and an added irrelevant option steers 12% of its decisions. Prism-1 stays within one point of that baseline's accuracy and is 2-4 points more accurate once only answers that survive a reordering are counted, while no shuffle changes any of its answers and no added irrelevant option moves its choice among the others.
Figures & tables
Figure 1. SCQ violates both axioms, after training and after sorting (mean over five sets). (a) Share of answers that change under one shuffle after each epoch of training on shuffled orders, and when the options move but keep their own letters. (b) After one epoch of training, the answers that change when the options are shuffled, zero by construction with sorted options because sorting undoes the shuffle, the answers that move to a different original option when one irrelevant option is added, and, for options sorted by text, the answers that change when they are sorted Z–A instead of A–Z.
Figure 2. Prism-1 . (a) Every option is a block that starts at the same position P after the shared context, and all blocks are read in one forward pass of the pretrained decoder, fine-tuned with LoRA. The score sk=logP(yes∣h,bk)−logP(no∣h,bk) is read at the last token of each block ( ■ ), a softmax over the blocks gives the decision, and training minimises −logR(oy∣S) for the correct option oy . (b) The attention mask: the context attends causally to itself, and each block attends to the context and causally to itself, never to another option. No option’s score depends on the other options or on the listing.
Banking20
Newsgroups
BEV
Typed
JevBench
Avg
Method
Acc
Flip
Dr.
St.
Acc
Flip
Dr.
St.
Acc
Flip
Dr.
St.
Acc
Flip
Dr.
St.
Acc
Flip
Dr.
St.
Acc
Rob.
Flip
Dr.
St.
Closed-source models, through their APIs
Jev 1.13 §
87.0
4.0
1.6
3.3
75.3
3.7
2.8
8.0
76.4
6.3
3.2
13.9
72.8
8.0
5.1
22.0
88.5
1.4
1.6
7.2
80.0
77.9
4.7
2.9
10.9
Mercury Decide §
91.3
2.3
1.6
7.0
73.3
7.7
2.6
8.3
75.1
12.5
4.7
27.2
67.3
10.5
6.8
30.5
89.9
5.0
1.4
7.2
79.4
76.7
7.6
3.4
16.0
Open-source models: Gemma 4 12B
SCQ
91.9
5.4
1.8
3.0
71.8
9.4
4.2
13.2
77.5
10.1
5.2
20.3
75.9
5.3
4.8
11.2
84.9
5.6
2.8
10.8
80.4
77.8
7.1
3.8
11.7
Table 1. Main results. Acc: accuracy (%). Flip: answers changed by one shuffle (%). Dr.: probability moved among the original options when one irrelevant option is added (points). St.: questions on which an added irrelevant option can move the answer to another original option (%). Rob.: order-robust accuracy, right in both the source and the shuffled order (%). Flip, Dr. and St. are zero for a rule that satisfies both axioms. For SCQ with sorted options, Rob. compares A–Z with Z–A sorting. § : published model, not trained on our data. Bold: best per model, excluding published models.
Figure 3. BEV by domain: order-robust accuracy of SCQ (grey) and Prism-1 (blue), which for Prism-1 equals its accuracy. Prism-1 is ahead wherever its dot lies to the right.
Figure 4. Injected options: answers (%) that an added option naming a wrong option as correct moves to that option, per set.
Figure 5. The number of options and the choice axiom, with every method trained for one epoch on the same data. (a, b) Order-robust accuracy, right both in the source order and in the shuffled order, on the 300 Newsgroups test posts with K nested topics as options. Shaded: SCQ’s answers that are right in the source order but not after one reshuffle, up to its accuracy in the source order alone. (c) Switch rate: answers that move to a different original option when one irrelevant option is added, averaged over five sets.
Gemma 4 12B
Qwen3-8B
Variant
Acc
Switch
Acc
Switch
Prism-1
79.6
0.0
77.7
0.0
options attend to each other
74.4
3.5
74.8
1.3
+ score read after the set
76.4
3.8
72.0
3.4
+ likelihood of the option’s text
74.7
3.6
72.1
5.1
Prism-1 + answer slot
79.7
0.0
76.1
0.0
Table 2. Ablation of Prism-1 : mean accuracy and switch rate when an option is added (%). Each row changes one more design choice, and the fourth row is the design of Set-LLM.
Figure 6. The price of the choice axiom: accuracy relative to Prism-1 against the answers an added option moves to another original option. Grey: run-to-run noise, two standard deviations.
Appendix figures & tables6 assets
Supplementary material from the paper’s appendix.
Appendix
Figure 7. What bidirectional attention among options costs. (a) Validation accuracy before and after one epoch of training, for a bidirectional set prompt (Set-LLM’s design) and for isolated causal blocks with the answer slot. (b) Accuracy of the trained bidirectional set prompt minus trained SCQ, in points.
Set
K
Eval
Train
State (chars)
Banking20 ( Casanueva et al., 2020 )
20
300
2,450
55 / 71
Newsgroups ( Lang, 1995 )
20
300
2,450
542 / 750
BEV ( avbiswas, 2026 )
2–10
1,182
2,450
232 / 787
Typed Decisions ( LocalLLaMA, 2026 )
4–5
600
1,750
708 / 697
JevBench ( Standhartinger, 2026 )
3–6
139
–
82 / 2,563
Appendix
Table 3. The benchmark. Train: training questions, after holding out 50 per set for validation. State: length in characters, median / mean.
Gemma 4 12B
Qwen3-8B
Variant
S
Acc
JB
Sw.
Acc
JB
Sw.
Options compare (not of the required form)
gated channel, rivals’ last tokens
1
−0.2
−1.4
0.7
−0.2
0.0
1.0
gated channel, all rival tokens
1
+0.1
+1.4
0.7
–
–
–
gated channel, gates forced open
1
−1.2
+1.4
–
–
–
–
comparison head, all options
2
+0.9
+3.2
0.0
+0.7
+3.6
0.0
Appendix
Table 4. The mechanisms of Section 5.6 : accuracy minus Prism-1 ’s with the same seeds (points), overall and on JevBench (JB), and switch rate (Sw., %). S: seeds; –: not run.
Tokens
Attn. pairs (k)
ms
Method
B20
NG
B20
NG
B20
NG
SCQ
282
417
40
100
60
63
Set-LLM
322
413
60
168
68
66
Prism-1
221
356
21
76
58
60
Appendix
Table 5. Cost per decision with 20 options on Banking20 (B20) and Newsgroups (NG): tokens processed, attention pairs (query–key pairs the mask allows, thousands) and milliseconds on one H100 GPU, Gemma 4 12B, medians over the first 100 questions of each set.
Model
Flip
Dr.
St.
Sw.
Jev 1.13
4.7
2.9
10.9
3.4
identical copies
1.3
1.3
5.1
1.3
Mercury Decide
7.6
3.4
16.0
3.5
identical copies
0.7
0.5
1.5
0.7
Appendix
Table 6. The closed-source models against their noise floor, five-set means. Identical copies: the same question repeated in one request, scored as the added-option variants are. Sw.: switch rate (%).
Gemma 4 12B
Qwen3-8B
Read
Target
Switched
Target
Switched
SCQ
23.1
25.3
9.6
13.0
SCQ + cyclic avg.
10.0
10.8
4.7
6.1
SCQ, sorted
11.2
13.0
2.9
5.1
Set-Based Prompting
54.8
58.5
31.4
41.8
Set-LLM
22.7
23.6
15.3
18.1
Appendix
Table 7. Injected options: answers (%) moved to the named option (Target) or to any other original option (Switched).
LLM decision-making studies how language models choose, act, seek information, and allocate resources under goals, uncertainty, constraints, and social pressures.