Working paper · PokerOn

Abstract

We study how to rank play-money Texas Hold'em by decision quality rather than by chip outcomes. An interactive Monte Carlo campaign with real cards, legal streets, divergent action paths, seat rotation, and an independent 30,000-hand future-EV oracle shows that duplicate-normalized winnings do not beat raw winnings at the advertised six-table / 600-hand format (Pearson improvement +0.0005, 95% CI −0.0082 to +0.0092; statistical tie). We therefore reject chips - including duplicate chips - as the public Elo. Instead we define skill as negative expected counterfactual regret against a frozen value model Q_F on information sets that exclude future cards. Terminal luck is stored as a separate zero-mean overlay. Stress tests show zero-mean Q noise preserves rank (Spearman 0.983 at 0.15 BB) while skill-correlated bias destroys it (0.631 at 0.50 BB). Publication requires concordance, RD, and cluster-weighted effective decisions. Until a certified multiway solver season passes preregistered gates, unsupported information-set classes abstain rather than invent Q.

Keywords: decision Elo · counterfactual regret · luck residual · information-set purity · duplicate poker · publication gates

1. Introduction and claim boundary

Casual poker leaderboards crown rivers. That is entertaining and statistically contaminated. PokerOn’s product question is narrower: can we publish a skill channel that (i) scores only information available before the act, (ii) peels runout luck into a labelled residual with mean zero, and (iii) refuses to score spots whose Q class is not certified?

This paper consolidates the simulation campaign in validation/, the mathematical model in docs/02_MATHEMATICAL_MODEL.md, and the live season uniform-ev-v1. Numbers below are from those runs. Where a claim failed, we say so - legitimacy here means honest rejection, not marketing gloss.

2. Experiment I - why duplicate chips fail as Elo

2.1 Design

Real 52-card deck and exact Hold'em showdown ranking
Six players, four streets, legal fold / call / raise, zero-sum pots
Mirrored hole cards and boards by seat; six seat rotations
Leave-one-out peer baseline (player never enters own comparison)
Independent oracle: 30,000 fresh hands per player (not inside the score)
200 Monte Carlo repetitions per condition; master seed 20260813

Engine limits (stated for honesty): one raise per street in that campaign, fixed sizing, no rake, synthetic agents. Adversarial relative to additive toys; not a human field trial.

2.2 Main result - six mirrored tables

Hands / player	Raw Pearson	Duplicate Pearson	Raw Spearman	Duplicate Spearman
150	0.173	0.165	0.166	0.152
300	0.263	0.265	0.234	0.238
600	0.346	0.339	0.321	0.311
1,500	0.501	0.493	0.473	0.459
3,000	0.640	0.613	0.600	0.577

Across two independent 200-repetition batches at six tables / 600 hands (400 blocks): duplicate minus raw Pearson +0.0005 (95% CI −0.0082 to +0.0092); duplicate won 53% of repetitions. That is a statistical tie, not evidence of superiority.

2.3 Table-count stress at 600 hands

Mirrored tables	Dup − raw Pearson	95% bootstrap CI	Dup win rate
2	−0.0846	[−0.1181, −0.0500]	35.5%
3	−0.0320	[−0.0517, −0.0120]	37.0%
6	+0.0071	[−0.0047, +0.0191]	55.5%
12	+0.0194	[+0.0117, +0.0270]	65.5%

Two and three tables were worse than raw. Twelve helped a little. Variance cancellation of the shared deal-seat shock is real when ICC exceeds 1/M; interactive poker still injects opponent noise, path divergence, and skill-by-deal interaction. Product implication: do not define public Elo as duplicate-normalized chips.

3. Estimand and identities

For frozen format and value model Q_F:

r[i,k] = max_a Q_F(I[k], a) − Q_F(I[k], chosen[i,k])
θ_decision[i] = −E[r[i,k]]
I[k] ⊆ information legally known before the act
(future board, realized award ∉ I)

Terminal heads-up call / fold (exact given equity assumptions):

Q(fold) = 0
Q(call) = e·(P + C) − C
break_even = C / (P + C)
luck = award − e·pot
E[luck] = 0 over remaining runouts

Equity teaches; it is not Elo. Nonterminal actions need ranges, fold equity, sizing, and future play - equity alone cannot build Q(I,a). Practice may show equity; rated modes must not before lock.

4. Experiment II - Q noise vs skill-correlated bias

Conditional on a true latent skill and 600 decisions, we inject evaluator error into continuous regret recovery:

Error model	Spearman with latent skill	Interpretation
0.15 BB zero-mean noise	0.983	Rank preserved
0.50 BB skill-correlated bias	0.631	Rank collapsed

Wrong Q with persistent bias is more dangerous than noisy Q. That is why we freeze seasons, hash models, and abstain outside certified classes instead of silently transplanting heads-up charts into multiway pots.

5. Elo / Rasch channel and publication gates

X[i,k] = 1[ r[i,k] ≤ ε[k] ]
P(X=1) = 1 / (1 + 10^((D[k]−R[i])/400))
R = 1500 + (400 / ln 10)·θ
RD = (400 / ln 10)·posterior_sd(θ)
cluster weight = 1 / √(1 + n_same_street_hand)

Continuous EV-loss stays a second column - never blended into a hidden hybrid. Live product gates before treating Elo as published:

≥ 60 cluster-weighted effective decisions
RD ≤ 80
Elo Δ and continuous Δ concordance
Not quarantined

Session “confirmed winner” only if a preregistered sequential posterior rule (~90–95%) fires. Equal players may never confirm - honesty, not a bug.

5.1 Visibility of strongest player (synthetic equal-exposure)

Group / gap	24	60	120	300	600
4 players / close	42.7%	54.6%	64.7%	80.1%	89.5%
4 players / medium	67.6%	83.1%	93.0%	99.0%	99.9%
6 players / close	30.9%	39.5%	49.6%	63.1%	75.7%
6 players / medium	51.2%	68.1%	80.1%	92.5%	97.7%

Synthetic model under equal exposure - not a guarantee about any human field.

6. Information-set classes and abstention

A checked flop in heads-up study often implies a capped preflop-raiser range. In a 3-way pot the player not last to act frequently checks an uncapped range. Those are different games:

class_HU_missed_cbet  ≠  class_3way_OOP_probe
if class ∉ certified(Q_F):  r := undefined   (abstain)

PokerOn leaves unsupported classes unscored rather than inventing a probe chart. Nuts advantage ≠ equity advantage; card order on the same ranks can change the class; bet size is part of the action.

7. What this product claims (and refuses)

Supported

Mirrored deals can reduce shared deal-seat variance when ICC is high enough
More hands improve both raw and duplicate estimators
E[luck] = 0 identity for locked HU all-in overlays
Zero-mean Q noise preserves latent skill rank; biased Q does not
Decision estimand excludes future cards by construction

Not supported / refused

“Six tables + 600 hands proven better than raw chips”
“Card luck is removed” (only the surviving shared component shrinks)
Certified full-NLHE GTO Elo
Scoring 3-way probes with heads-up Q
Practice assistance contaminating unaided Elo

8. Reproducibility

Primary artefacts in the repository:

validation/stackrank_validation.py - interactive duplicate campaign
validation/variance_diagnostics.py - ICC / additive sanity
validation/results/FINAL_REPORT.md - quantitative verdict
docs/02_MATHEMATICAL_MODEL.md - estimand and claim boundary
validation/complete_identities.py (where present) - identity suite
Live season hash: uniform-ev-v1 in the game service

PokerOn Labs - Unrated. This working paper is the public evidence spine for the skill channel on the table. It is attached so visitors can audit what we proved, what we rejected, and what we still refuse to claim.