Soft Tournament Equilibrium: Differentiable Set-Valued Inference for Non-Transitive Pairwise Comparisons

Saad Alqithami

cs.AIApr 6, 2026

Soft Tournament Equilibrium: Differentiable Set-Valued Inference for Non-Transitive Pairwise Comparisons

Authors: Saad Alqithami

Abstract

Soft Tournament Equilibrium (STE) is a differentiable layer for Top-Cycle (TC) and Uncovered-Set (UC) inference from reciprocal pair probabilities. Normalized log-sum-exp reachability and covering give smooth scores with approximation, perturbation, and margin-recovery bounds. We distinguish structural supervision, posterior uncertainty, and the final set decision through controlled synthetic studies and reconstruction of recorded ordinal profiles. Matched-epoch training improves F1 under a common structural readout, but a separate equal-budget soft-UC comparison does not demonstrate an advantage. A prospective equal-budget hard-UC study improves selective F1 at 24 alternatives from 0.6327/0.6292 for ordinary/relational native heads to 0.6676, while exact recovery remains only 1.16%. On 36 human profiles held out by source, a separate exploratory comparison favors Jeffreys posterior inference over learned independent-edge and mixture distributions in native expected-F1 decoding (0.8742 versus 0.8704/0.8366). Only three references are non-singleton selective cores. Completion certificates and full-depth TC diagnostics clarify additional identification, smoothing, and computational limits. The evidence supports conditional synthetic overlap gains, while preserving the soft-set null and the absence of a demonstrated learning advantage over strong count-based human-profile controls.

Figures & tables

Figure 1: STE inference workflow. Comparison records define reciprocal pair probabilities. STE-LSE computes smooth TC/UC scores for supervision; STE-Post averages hard memberships under a specified posterior. Reporting a set requires a separate decision rule. Training differentiates the smooth scores, not the selected set.

Panel of Figure 2 — Figure 2: One relation, distinct TC and UC targets. Solid arrows show all six majority edges; blue edges certify that $b$ covers $d$ . Circles mark UC members and the square marks covered $d$ . The spanning cycle certifies full TC. The dashed leader is an annotation, not an edge.

1	Set $D_{aa}\leftarrow 0$ ; for $a\neq b$ , set $D_{ab}\leftarrow\sigma((P_{ab}-1/2)/\tau)$ .
2	$Q\leftarrow D$ , $L\leftarrow D/\gamma$ .
3	for $k=2,\ldots,K$ :
4	Allocate a new $n\times n$ matrix $Q^{\rm new}$ .
5	for $b\in\mathcal{A}$ , write column $b$ of $Q^{\rm new}$ :
6	$H_{ac}\leftarrow\operatorname{smin}_{\gamma}(Q_{ac},D_{cb})$ for all $a,c$ .
7	$Q^{\rm new}_{ab}\leftarrow\operatorname{smax}_{\gamma}\{H_{ac}:c\in\mathcal{A}\}$ for all $a$ .

Algorithm 1 Canonical STE-LSE forward pass

Core	$n$	STE	Aux.	$\Delta$ F1	95% CI	$p_{\rm Holm}$	Pos.
UC	24	.6078	.5761	$+.0317$	$[.0182,.0453]$	.0015	12/12
UC	48	.4931	.4638	$+.0293$	$[.0102,.0484]$	.0127	11/12
TC	24	.5405	.5111	$+.0294$	$[.0109,.0480]$	.0127	10/12

Table 1: Frozen primary contrasts under the same structural readout and development-selected cutoff.

\Delta

is STE minus auxiliary F1; CI uses

t_{11}

over twelve collection means, each averaging graphs and three initializations. Two-sided exhaustive sign-flip tests receive Holm adjustment over these three endpoints. Pos. counts positive collection effects.

Objective / readout	$n=24$	$n=48$
Pair / structural	$0.6245\pm 0.0043$	$0.4018\pm 0.0061$
Aux. / head	$0.6414\pm 0.0085$	$0.4280\pm 0.0068$
Rel. aux. / head	$0.6416\pm 0.0061$	$0.4263\pm 0.0051$
STE / structural	$0.6281\pm 0.0074$	$0.4076\pm 0.0046$

Table 2: Equal-budget native systems: selective non-singleton UC F1, mean

\pm

SEM across twelve collections, each averaging three initializations. Only

n=24

is primary;

n=48

is descriptive. SEM is not a confidence interval.

System / readout	F1, $n=24$	F1, $n=48$	Exact (%) $n=24$
Pair / hard	0.5865 (0.0078)	0.4012 (0.0056)	1.45
Ordinary / native	0.6327 (0.0045)	0.4309 (0.0042)	2.02
Relational / native	0.6292 (0.0051)	0.4318 (0.0055)	2.53
STE / hard	0.6676 (0.0043)	0.5159 (0.0030)	1.16
Ordinary / hard	0.5972 (0.0082)	0.4121 (0.0060)	1.41
Relational / hard	0.6237 (0.0031)	0.4324 (0.0053)	1.44

Table 3: Prospective equal-budget UC systems on selective cores. F1 is mean (SEM) over twelve collections, each averaging three initializations; exact recovery is a percentage. Only STE-hard versus native heads at

n=24

belongs to the separate two-test primary family; other entries are descriptive.

Method	UC F1	Exact (%)
STE-LSE	0.6491	9.64
Posterior core	0.7824	32.32
Posterior GFM	0.6920	11.43
Smooth Copeland	0.6820	22.50
BTL	0.6561	6.43
Copeland	0.6548	27.50

Table 4: Strict selective recorded UC reconstruction at the frozen 25%-record setting: three election and nine sports profiles, four equally weighted sources. Score methods use source-excluded absolute calibration; GFM and hard/all/none controls use native decisions. Exact is complete-set recovery.

System	F1	Exact	UC Brier
Jeffreys–GFM	0.8742	0.7656	0.0437
Empirical Bayes–GFM	0.8705	0.7522	0.0447
Learned edges–GFM	0.8704	0.7614	0.0400
Learned mixture–GFM	0.8366	0.7301	0.0486
Raw-count UC	0.8481	0.7950	—
Relational head	0.7944	0.6161	0.0681

Table 5: Exploratory native decisions at 10% records: mean within profile, then source, then six equally weighted sources. Exact is complete-set recovery; lower UC Brier measures agreement of membership scores with the finite reference. The cohort has 33 singleton and three selective targets. No significance or population-calibration claim is made.

Figure 3: Full-depth TC score error (

K=n-1

). (a) Families at

\tau=\gamma=0.035

. (b) Long-path sensitivity with

\tau=\gamma

; shading spans means at the three tested temperatures, not a confidence band. Each point averages vertex errors over three saved graphs. These are operator diagnostics, not trained-model transfer tests.

Claim	Evidence	Verdict
Structural training helps	Matched-readout contrasts (Table 1 )	Conditional support
Soft sets beat heads at equal budget	Selective primary contrasts (Table 2 )	Not demonstrated
Hard UC beats native heads	Frozen selective comparison (Table 3 )	Protocol-specific support
STE beats recorded-data controls	Count reconstruction and human-profile learning	Not supported
Learned dependence improves native F1	Posterior/control comparison (Table 5 )	Not supported

Table 6: Claims, evidence, and scope. Descriptive diagnostics do not replace the primary comparisons.

Appendix figures & tables29 assets

Supplementary material from the paper’s appendix.

Appendix

$n$	$K$	Original TC	Sharpened TC	Sharpened UC
12	11	1.280	0.954	0.084
24	23	3.200	2.557	0.110
48	47	7.618	6.367	0.135
64	63	10.819	9.170	0.145
96	95	17.616	15.176	0.159

Appendix

Table 7: Smoothing-only allowances at

\gamma=\gamma_{c}=0.035

K=n-1

. Values greater than one are uninformative for score recovery. Statistical/edge-smoothing error must additionally fit within the threshold allowance.

Target	$n$	Pair/struct.	Aux./struct.	STE/struct.	Aux./head	Rel./struct.	Rel./head
UC	12	0.6446	0.6527	0.6710	0.6873	0.6636	0.6894
UC	24	0.5572	0.5761	0.6078	0.5772	0.5858	0.5868
UC	48	0.4383	0.4638	0.4931	0.4588	0.4663	0.4669
UC	64	0.4018	0.4227	0.4499	0.4288	0.4238	0.4366
TC	12	0.6416	0.6524	0.6722	0.7106	0.6501	0.7104
TC	24	0.4816	0.5111	0.5405	0.5729	0.5128	0.5761

Appendix

Table 8: Complete-system descriptive mean F1 with absolute calibration. “Struct.” uses the relevant TC/UC operator; “head” is the model’s own supervised membership output. Each readout receives its own development selection. Relational control has 128 extra parameters.

Target	$n$	Objective	Readout	F1	Exact (%)	AP	Size	Truth
TC	12	all	native	0.5055	19.73	–	12.00	5.01
TC	12	aux	direct	0.7106	25.16	0.8638	6.15	5.01
TC	12	aux	structural	0.6524	32.34	0.8338	7.69	5.01
TC	12	none	native	0.0000	0.00	–	0.00	5.01
TC	12	pair	structural	0.6416	29.57	0.8277	8.03	5.01
TC	12	relational_aux	direct	0.7104	25.30	0.8650	6.14	5.01

Appendix

Table 9: All matched-epoch learned systems under absolute or native selection. Exact is complete-set recovery. AP omits full/empty reference targets.

Target	$n$	Reference class	STE	Aux.	$\Delta$ F1
UC	24	singleton	0.3881	0.3564	$+0.0317$
UC	24	selective non singleton	0.6835	0.6459	$+0.0376$
UC	24	full	0.9616	0.9530	$+0.0087$
UC	48	singleton	0.2361	0.2007	$+0.0355$
UC	48	selective non singleton	0.4842	0.4472	$+0.0370$
UC	48	full	0.9986	0.9979	$+0.0007$

Appendix

Table 10: Descriptive common-structural-readout effects by reference-core class. Within-class means average graphs/initializations and then twelve collections. These are inspected strata, not extra primary endpoints.

Core	$n$	F1	Exact (%)	Selected	Reference
UC	24	0.6835	5.76	13.16	9.66
UC	48	0.4842	1.79	29.34	13.13
TC	24	0.5014	1.89	21.08	8.48

Appendix

Table 11: STE structural readout on selective non-singleton targets only (

1<|C|<n

), with the original development-selected absolute cutoff. Metrics average the saved per-collection stratum means over twelve collections. Exact is complete-set recovery; selected and reference are mean cardinalities. These are inspected descriptive strata, not additional primary tests.

Target	$n$	STE-LSE	Post.	RC	BTL	Copeland	Smooth C.	All
UC	12	0.6303	0.6717	0.6109	0.5527	0.5439	0.5437	0.4605
UC	24	0.5459	0.4583	0.5386	0.4538	0.4383	0.4382	0.4069
UC	48	0.4269	0.3692	0.4656	0.3738	0.3678	0.3692	0.3663
UC	64	0.3930	0.3569	0.4424	0.3481	0.3454	0.3485	0.3562
TC	12	0.6266	0.6033	0.6244	0.5667	0.5687	0.5703	0.5055
TC	24	0.4669	0.4290	0.5494	0.4699	0.4727	0.4763	0.4282

Appendix

Table 12: Count-only descriptive F1 using absolute development calibration; all-selected is native. RC is Rank Centrality. These observations are reused from the learning collections and do not form an independent confirmation.

Source	Profile file	Record type	$n$	Records	UC size	TC size
00007	00007-00000068.soi	Ballots	11	902	3	5
00007	00007-00000086.soi	Ballots	4	279	3	3
00019	00019-00000004.toi	Ballots	8	28,660	3	6
00049	00049-00000288.soc	Laps	15	77	3	4
00049	00049-00000479.soc	Laps	15	51	12	15
00053	00053-00000094.soi	Laps	22	53	3	4

Appendix

Table 13: Every primary strict selective UC profile. “Records” is the raw PrefLib multiplicity total, called “voters” by its generic format; its meaning is ballots or laps according to source. Complete pair coverage concerns the aggregate decisive matrix, not whether every individual order is complete.

Source	STE-LSE	Posterior	$\Delta$ F1
00007	0.7967	0.8258	$-0.0292$
00019	0.6433	0.9500	$-0.3067$
00049	0.5290	0.6984	$-0.1693$
00053	0.6273	0.6552	$-0.0279$

Appendix

Table 14: Primary UC F1 by held-out source; STE and posterior both use absolute source-excluded calibration. Sources 00007/00019 are elections; 00049/00053 are sports lap rankings. Every source favors posterior. These four purposively assembled clusters limit population inference.

$n$	Ordered	Planted	Long path
8	0.0486	0.0963	0.2180
16	0.0733	0.1268	0.5181
32	0.0809	0.1313	0.7389
64	0.0932	0.1417	0.8213
96	0.0874	0.1382	0.8475

Appendix

Table 15: Full-depth TC mean absolute score error at the default coupled temperature 0.035. Each value averages three diagnostic graphs and their vertices. Full long-path cores do not make this a selective-set classification test.

Target	$n$	Aux. (ms)	STE (ms)	Ratio
TC	12	4.37	15.98	3.66 $\times$
UC	12	4.35	17.86	4.10 $\times$
TC	24	4.37	28.60	6.54 $\times$
UC	24	4.38	31.41	7.17 $\times$
TC	48	4.41	53.79	12.19 $\times$
UC	48	4.40	58.82	13.38 $\times$

Appendix

Table 16: Recorded L40S complete batch-four training-step medians. Ratios compare STE structural loss with the auxiliary own-head loss on the common base model. They exclude data preparation and posterior inference and are not fit-to-convergence timings.

Objective / readout	$n$	F1 (all)	F1 (selective)	Exact (%)	Size
Pair / structural	24	0.5252	0.6245	3.76	16.64
Pair / structural	48	0.4104	0.4018	0.91	39.14
Aux. / head	24	0.5865	0.6414	3.31	16.57
Aux. / head	48	0.4633	0.4280	0.31	35.04
Rel. aux. / head	24	0.5879	0.6416	2.87	16.56
Rel. aux. / head	48	0.4651	0.4263	0.23	35.17

Appendix

Table 17: Complete-system UC follow-up. Each method uses its own development-selected native readout. All-class F1 uses the natural mixture; selective columns restrict to

1<|\mathrm{UC}|<n

. Exact is complete-set recovery and selected size is predicted cardinality on that stratum. Means average graphs, then three initializations, then twelve collections. Only

n=24

selective F1 is primary.

Baseline	$\Delta$	95% CI	$p$	$p_{\rm Holm}$	Pos.
Ordinary head	$-0.01329$	$[-0.03286,+0.00628]$	0.1621	0.3193	5/12
Relational head	$-0.01345$	$[-0.03300,+0.00611]$	0.1597	0.3193	3/12

Appendix

Table 18: The two prespecified primary contrasts on selective non-singleton UC at

n=24

. Differences are STE minus the indicated direct-head system; intervals are individual paired

t_{11}

intervals. Holm adjustment applies to the two exact sign-flip tests.

Figure 4: All twelve independent collection differences for each primary contrast. Each small point averages the three paired initializations within one collection. The accent point and interval show the mean difference and individual, unadjusted 95%

t_{11}

interval. Zero marks equal mean F1; positive differences favor STE. No collection is excluded.

Objective / readout	$n$	Pair MSE	Brier	Log loss	Hard UC F1	Ties
Pair / structural	24	0.0417	0.2394	0.9518	0.5231	0.0252
Pair / structural	48	0.0416	0.2401	0.9541	0.4143	0.1407
Aux. / head	24	0.0427	0.2404	0.8550	0.4911	0.0002
Aux. / head	48	0.0421	0.2407	0.8558	0.4084	0.0017
Rel. aux. / head	24	0.0380	0.2357	0.8180	0.5703	0.0005
Rel. aux. / head	48	0.0366	0.2352	0.8150	0.4668	0.0016

Appendix

Table 19: Post hoc, descriptive all-class relation-quality and hard-decoding diagnostics for the selected systems. Hard UC uses each stored float32 direction

\widehat{P}_{ab}>1/2

without the learned membership cutoff. Pair errors use latent simulated truth unavailable to the predictor. Ties count upper-triangle probabilities equal to

1/2

per graph; numerical reciprocity can leave the reverse direction above

1/2

. These are not additional primary endpoints.

System	$n$	Native F1	Hard UC F1	Native exact (%)	Hard exact (%)	Pair MSE
Pair / structural	24	0.6245	0.6035	3.76	1.73	0.0412
Pair / structural	48	0.4018	0.3960	0.91	0.38	0.0410
Aux. / head	24	0.6414	0.5816	3.31	1.09	0.0420
Aux. / head	48	0.4280	0.3967	0.31	0.54	0.0414
Rel. aux. / head	24	0.6416	0.6420	2.87	1.61	0.0402
Rel. aux. / head	48	0.4263	0.4445	0.23	0.84	0.0377

Appendix

Table 20: Post hoc diagnostics restricted to selective non-singleton cores. Native and hard readouts use the same selected weights; checkpoint selection was not repeated for hard decoding. Exact is complete-set recovery. Pair MSE is measured against latent simulated truth. Means average twelve collections, each averaging graphs and three initializations. No additional inferential tests are claimed.

Native comparator	Difference	95% CI	Raw $p$	Holm $p$
Ordinary head	0.03493	$[0.02383,0.04602]$	0.0004883	0.0009766
Relational head	0.03843	$[0.02763,0.04922]$	0.0004883	0.0009766

Appendix

Table 21: The separate frozen hard-UC primary family. Differences are STE-hard minus native heads at selective

n=24

. Both comparisons retain all twelve collection effects. Intervals are individual paired

t_{11}

intervals; Holm covers the two exhaustive sign-flip tests.

$n$	System / readout	All F1	Selective F1	Exact (%)	Selected	Reference	Pair MSE
24	Pair / hard	0.5131	0.5865	1.45	16.72	9.57	0.0407
24	Ordinary / native	0.5902	0.6327	2.02	16.34	9.57	0.0445
24	Relational / native	0.5832	0.6292	2.53	16.43	9.57	0.0398
24	STE / hard	0.6130	0.6676	1.16	10.75	9.57	0.0354
24	Ordinary / hard	0.5231	0.5972	1.41	15.87	9.57	0.0386
24	Relational / hard	0.5621	0.6237	1.44	14.21	9.57	0.0402

Appendix

Table 22: All six independently timed systems. All-class F1 uses the natural mixture; remaining metrics use selective non-singleton cores. Exact is complete-set recovery (%). Pair MSE uses latent simulated truth unavailable to the predictor. Means average graphs, initializations, and twelve collections.

n=48

and the common-hard controls are descriptive.

Learned relations and readouts
Method / readout	F1	Exact (%)	Precision	Recall	Selected	Reference
Ordinary / native head	0.7982	64.51	0.7851	0.8573	1.165	1.122
Ordinary / hard UC	0.8913	85.37	0.9042	0.8997	1.136	1.122
Ordinary / soft UC	0.8564	72.84	0.8396	0.9211	1.327	1.122
Pair / hard UC	0.8955	85.94	0.9087	0.9067	1.145	1.122
Pair / soft UC	0.8863	77.71	0.8724	0.9454	1.337	1.122

Appendix

Table 23: Whole-source held-out human-profile reconstruction at 10% records. F1, exact recovery (%), precision, recall, and cardinalities are equally weighted source macro means. All 36 references are included, of which 33 are singletons. Hard uses a majority decoder; soft/head/posterior inclusion cutoffs are development-selected; posterior-half and GFM are native decisions. No formal tests are performed.

Profile	Method	F1	Exact (%)	$\|\widehat{S}\|$
ERS A	Ordinary / native	0.4444	0.00	1.62
	Relational / native	0.4367	4.17	1.62
	STE / soft UC	0.5456	0.00	2.54
	STE / hard UC	0.4583	0.00	1.58
	Count / hard UC	0.4958	0.00	2.00
	Posterior / half	0.5488	0.00	2.75

Appendix

Table 24: All three selective human references at 10% records: ERS A (

n=11

), ERS B (

n=4

), and Oakland (

n=8

), each with reference size three. Rows average eight shared resamples; learned readouts additionally average three initializations. These descriptive profile-level outcomes are not independent population replications. All methods and other fractions remain in the retained outputs.

Source	Profiles	Alternatives	Records	Selective references
ERS	9	4–16	134–1,123	2
Oakland	7	4–11	11,266–144,229	1
San Francisco	3	4–7	24,109–34,931	0
San Leandro	2	4–5	23,236–25,355	0
CSES	10	9–9	937–2,338	0
Voter Autrement	5	11–12	743–11,219	0

Appendix

Table 25: Actual human-profile inventory for the posterior-control extension. The displayed order also defines cyclic source folds. Each full reference is strict; subsample relations can be incomplete or tied. CSES construction and provider references are retained in Appendix E .

System	F1	Exact	$\|S\|$	Size MAE	Correct size	UC Brier
Jeffreys–GFM	0.8742	0.7656	1.352	0.2713	0.8254	0.0437
Empirical Bayes–GFM	0.8705	0.7522	1.369	0.2825	0.8119	0.0447
Learned edges–GFM	0.8704	0.7614	1.217	0.2485	0.8183	0.0400
Learned mixture–GFM	0.8366	0.7301	1.185	0.2300	0.8257	0.0486
Raw-count UC	0.8481	0.7950	1.194	0.1658	0.9038	—
Count soft plug-in †	0.8462	0.7679	1.233	0.2117	0.8635	0.0507

Appendix

Table 26: Native 10%-record source macro results. Exact and correct size are frequencies, not percentages; size MAE is

\mathbb{E}\lvert|S|-|C|\rvert

. UC Brier is the mean squared error of membership scores against finite-reference indicators; lower values establish neither calibration nor better set decisions. † Prespecified secondary soft plug-in control.

Source	J	E	L	M	R	S †	Rel	Ord
ERS	0.719/0.444	0.714/0.431	0.702/0.435	0.680/0.426	0.638/0.431	0.651/0.444	0.618/0.343	0.516/0.296
Oakland	0.976/0.929	0.981/0.929	0.941/0.851	0.944/0.863	0.982/0.964	0.979/0.946	0.979/0.911	0.824/0.810
San Francisco	0.910/0.833	0.896/0.792	0.968/0.903	0.829/0.708	0.924/0.875	0.924/0.792	0.792/0.514	0.880/0.736
San Leandro	0.833/0.750	0.833/0.750	0.861/0.750	0.812/0.729	0.750/0.750	0.750/0.750	0.750/0.750	0.750/0.750
CSES	0.882/0.787	0.882/0.787	0.886/0.804	0.895/0.838	0.865/0.850	0.858/0.850	0.893/0.854	0.854/0.846
Voter Autrement	0.926/0.850	0.917/0.825	0.865/0.825	0.859/0.817	0.930/0.900	0.915/0.825	0.735/0.325	0.580/0.000

Appendix

Table 27: Source-specific 10%-record F1/exact recovery, with profiles weighted equally within source. Variation across these six purposively assembled series is descriptive and does not justify a six-source population test.

Profile	$n$	$c$	J	E	L	M	R	S †	Rel	Ord
ERS 10	6	1	0.429/0.250	0.425/0.250	0.653/0.375	0.597/0.375	0.458/0.250	0.396/0.250	0.569/0.250	0.444/0.333
ERS 33	16	1	0.958/0.875	0.958/0.875	0.958/0.875	0.958/0.875	1.000/1.000	1.000/1.000	0.764/0.292	0.596/0.042
ERS 68	11	3	0.652/0.000	0.652/0.000	0.670/0.000	0.671/0.000	0.539/0.000	0.646/0.000	0.658/0.000	0.635/0.000
ERS 69	4	1	0.571/0.375	0.571/0.375	0.599/0.417	0.589/0.417	0.583/0.500	0.583/0.500	0.513/0.458	0.500/0.500
ERS 70	4	1	0.792/0.375	0.792/0.375	0.764/0.458	0.653/0.458	0.375/0.375	0.542/0.375	0.500/0.333	0.375/0.375
ERS 71	7	1	0.661/0.250	0.619/0.125	0.547/0.250	0.590/0.250	0.675/0.375	0.717/0.500	0.686/0.542	0.625/0.375

Appendix

Table 28: All 36 profile results at 10% records. A profile label gives its source and catalogue identifier, not a new dataset;

n

is alternative count and

c

reference UC size. Every method cell is F1/exact recovery, averaged over the shared resamples and initializations.

Profile ( $n$ )	System	F1	$\|S\|$	Correct size	Size MAE
ERS 68 (11)	Jeffreys–GFM	0.6521	3.125	0.7500	0.3750
	Empirical Bayes–GFM	0.6521	3.125	0.7500	0.3750
	Learned edges–GFM	0.6704	1.875	0.0833	1.2083
	Learned mixture–GFM	0.6708	1.750	0.1250	1.2500
	Raw-count UC	0.5387	2.500	0.3750	1.0000
	Count soft plug-in †	0.6464	2.375	0.3750	0.8750

Appendix

Table 29: All three selective reference profiles at 10% records, each with true UC size three. Eight shared inputs and three initializations are averaged within profile. Exact-set and correct-size frequencies answer different questions.

	5% records		25% records
System	F1	Exact	F1	Exact
Jeffreys–GFM	0.8199	0.6662	0.9448	0.8626
Empirical Bayes–GFM	0.8097	0.6553	0.9443	0.8598
Learned edges–GFM	0.8203	0.7091	0.9421	0.8626
Learned mixture–GFM	0.8136	0.6865	0.9172	0.8169
Raw-count UC	0.8220	0.7093	0.9458	0.9144

Appendix

Table 30: Secondary nested-fraction source macro results using the unchanged native decisions and profile/source weighting.

System	Fits	Allocated s	Charged s	Fit range (s)	Updates
Empirical Bayes–GFM	18	2160	1.50	0.02–0.14	0
Ordinary head	18	2160	1513.45	84.03–84.13	403,880
Relational head	18	2160	1515.35	84.04–85.74	323,530
Learned edges–GFM	18	2160	1515.09	84.05–84.31	349,842
Learned mixture–GFM	18	2160	1518.67	84.11–84.62	208,304

Appendix

Table 31: Actual fitting/development costs. Empirical Bayes uses a fitted scalar optimizer with no neural gradient updates; zero in that column does not mean an unfitted prior. Neural update counts include candidate training and do not measure equivalent statistical information.

System	Calls	Mean ms	Total s	Draws/input
Jeffreys–GFM	2,592	7.415	19.22	4,096
Empirical Bayes–GFM	2,592	7.467	19.36	4,096
Learned edges–GFM	2,592	9.034	23.42	4,096
Learned mixture–GFM	2,592	9.472	24.55	4,096
Raw-count UC	2,592	0.033	0.09	0
Count soft plug-in †	2,592	3.058	7.93	0

Appendix

Table 32: Held-out prediction cost on the L40S, including saved native posterior draws where applicable. Sampling/GFM is more expensive than the raw covering decision; these held-out costs occur outside the fitting cap.

$\delta$	Records	Certified	Abstained	$\|L\|$	$\|U\|$	False in/out
0.01	5%	0.0976	0.9024	0.048	6.553	0/0
0.01	10%	0.1698	0.8302	0.088	6.131	0/0
0.01	25%	0.3923	0.6077	0.295	4.891	0/0
0.05	5%	0.1151	0.8849	0.061	6.425	0/0
0.05	10%	0.2069	0.7931	0.130	5.992	0/0
0.05	25%	0.4593	0.5407	0.369	4.444	0/0

Appendix

Table 33: Finite-cohort certificate diagnostics with no human-profile coverage guarantee.

L/U

are inner/outer membership bounds under the calculated edge restrictions. “Certified” is the fraction of alternative statuses decided; it is not the fraction of exactly recovered complete sets.

Symbol / implementation	Meaning
$P,\widehat{P}$	Latent and estimated reciprocal pair probabilities; diagonal $1/2$ .
$W,T$	Decisive wins and observed tied counts; masks retain missingness.
$D_{\tau}$	Zero-diagonal sigmoid edge strength; soft surrogate.
$Q^{(k)},R,t$	Exact-length soft walks, aggregated reachability, TC score.
$v,\operatorname{cov},u$	Covering witness, cover strength, UC score.
$\tau,\gamma,\gamma_{c}$	Edge, path and covering temperatures.

Appendix

Table 34: Notation and implementation distinctions. Scores are not calibrated membership probabilities; posterior outputs are conditional on their model.

Pairwise Preference Learning

Pairwise preference learning infers rankings, rewards, or policies from judgments that compare two alternatives at a time.

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Isotonic Bradley-Terry Model for Paired Comparison Data
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Aug 3, 2026J/K move · Enter open · S save

Isotonic Bradley-Terry Model for Paired Comparison Data

Ryoya Yamasaki

Hitotsubashi Institute for Advanced Study, Hitotsubashi University, 2-1 Naka, Kunitachi, 186-8601, Tokyo, Japan.

Pairwise Preference Learning Pairwise Comparison Pairwise Preference Evaluation Pairwise Ranking Bradley–Terry Model Isotonic Regression

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In this paper, we study prediction problems for paired comparison data, for example, predicting the win probability between two unmatched players and ranking all the players according to the order of their strengths by using win probability data between two matched players. Paired comparison data are typically analyzed using Bradley-Terry and Thurstone-Mosteller models. These models predict the win probability by transforming the difference between learned rate parameters, which represent players';strengths, with a pre-specified inverse link function, and employ the order of learned rate parameters for player ranking. However, these models may suffer from model misspecification owing to the selection of a fixed inverse link function. Therefore, in this study, we propose to learn the rate parameters by a (sub-)gradient method and the inverse link function by an isotonic regression technique alternately. The proposed model guarantees monotonic improvement in training error, and is likely to yield an exact tie when the available data is insufficient to establish a strict ranking. We also verified that the proposed model could improve the win probability prediction and ranking performance through numerical experiments with synthetic data and real-world data of football Premier League, baseball MLB, and tennis ATP tour.

Abstract

Pairwise Preference Learning

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