Abstract

This revision retains the mathematical framework of the August working paper and adds completed paired comparisons from 15 October 2026. Counterfactual regret, represented information sets, action agreement and paired utility estimates are treated as distinct quantities. Their interpretation does not rely on infrastructure scale or publication status.

Mathematical definitions

vᵢ(σ) = ∑ₕ∈H π^σ(h) uᵢ(h)
Expected utility on terminal histories.
Rᵢᵀ(I,a) = ∑ₜ₌₁ᵀ [vᵢ(σᵗ, I, a) − vᵢ(σᵗ, I)]
Cumulative counterfactual regret; action values include counterfactual reach weighting.
Δ̂ = (1/n) ∑ᵢ Dᵢ; SE(Δ̂) = sD / √n
Mean paired difference and its standard error for independent pairs.
|Iₜ₊₁| > |Iₜ| ⇏ ε(σₜ₊₁) < ε(σₜ)
Coverage growth alone does not establish lower exploitability.

Notation and scope

Let H be terminal histories, I an information set, a a legal action, σ a strategy profile and uᵢ(h) the terminal utility of agent i. Reach probability π^σ(h) gives the expected utility vᵢ(σ). Counterfactual action values at I weight histories by the reach of other agents and chance. The notation follows the established extensive-form literature [1]; it does not specify a private implementation.

Cumulative counterfactual regret

Cumulative regret compares the counterfactual value of action a with the value of the strategy actually used at I over T updates. It is a learning quantity, not an observed evaluation return. Results established for two-player zero-sum games under their assumptions cannot be asserted for the six-agent experiment merely because regret notation is used.

Indexed coverage and strategic quality

A count of represented information sets measures stored coverage. It does not weight states by visitation probability and does not estimate best-response exploitability. An increase in this count therefore does not imply a decrease in exploitability. Likewise, agreement with a recorded reference is descriptive and cannot substitute for a utility comparison.

Paired estimates and uncertainty

Dᵢ is the role-balanced utility difference for scenario pair i. The mean uses n independent pairs. The reported confidence limits are taken from the completed evaluation records, not inferred from rounded means or from the number of actions. The October comparisons share scenarios; their estimates are correlated. No cross-study pooled effect is computed.

Revised empirical results

The revised R3 policy was compared with Original and R2 in two completed evaluations. Each has 60,000 pairs and 120,000 interactions, authenticated inputs and an independent complete-record audit. Both point estimates are positive and both intervals span zero. Historical positive results from other policy lineages or completion rules are not merged into these comparisons.

Conclusion

The current data support a completed, internally consistent comparison and quantify uncertainty. They do not establish a positive effect at the reported interval level, equivalence, convergence, cross-domain generalization or real-time deployment benefit. Further evaluation should preserve the pair as the sampling unit and distinguish exploratory comparisons from confirmatory analyses.

Completed comparisons: 15 October 2026

Paired mean utility differences · U per 100 policy-role interactions
ComparisonPairsΔ̂95% CI
E20261015-A
R3 − Original		60,000+5.5004[-1.7253, +12.7261]
E20261015-B
R3 − R2		60,000+2.6058[-0.0998, +5.3115]
R3 − OriginalR3 − R20510
Points: paired mean differences. Lines: reported 95% confidence intervals. Dashed line: zero.

Both intervals contain zero; improvement is not established. The comparisons share scenarios and are not independent replications. Intervals are pointwise, without adjustment for multiple comparisons.

References and data

[1] Zinkevich, M., Johanson, M., Bowling, M. & Piccione, C. (2007). Regret Minimization in Games with Incomplete Information. Advances in Neural Information Processing Systems 20.

Rhythmlag Lab. Revised paired evaluations, 15 October 2026: E20261015-A and E20261015-B. The data page records units, denominators, completion counts and result digests.

Data and measurement definitions