Reinforcement-learning models commonly predict complete future states, observations, or feature occupancies, even though action selection depends only on differences between the consequences of candidate actions. As a result, these models may devote substantial statistical and representational capacity to high-dimensional phenomena that evolve independently of the agent's current choice. We introduce the Counterfactual Quotient Model, which treats action-conditioned futures as equivalent when they differ only by a component shared across actions. Its canonical centered representation removes this common component while preserving every pairwise action comparison expressible by the modeled reward family. The implemented model learns these action-dependent effects directly from synchronized counterfactual rollouts, so shared stochastic dynamics cancel before function approximation rather than after complete futures have been predicted. We establish the decision sufficiency, identifiability, common-mode invariance, approximation behavior, and regret properties of the resulting representation. Controlled experiments in physics-based environments provide initial evidence for these properties: direct effect learning suppresses action-independent variation, supports previously unseen reward queries, and improves action ranking relative to models trained to predict absolute futures.
Figures & tables
Domain
CQM -B
World
SF
Direct value
Task-TD
COCOA
Cartpole
0.9998±0.0003
0.2521±0.1389
0.4236±0.0840
0.9904±0.0084
0.8407±0.0766
0.6445±0.0791
Reacher
0.9553±0.0290
0.2598±0.0592
0.3456±0.0647
0.8534±0.0974
0.6769±0.0892
0.6316±0.1331
Cheetah
0.6411±0.0118
0.2693±0.0901
0.3713±0.0362
0.6115±0.0523
0.5338±0.0501
0.5957±0.0398
Walker
0.3134±0.0101
0.2989±0.0114
0.2719±0.0099
0.2665±0.0375
0.2900±0.0171
0.2268±0.0357
Table 1: Held-out action accuracy over five seeds (mean ± approximate 95% confidence interval). Bold is the best learned method. Chance is 1/3 on Cartpole and 1/5 elsewhere.
Domain
CQM -B
World
SF
Direct value
Task-TD
COCOA
Cartpole
0.0000001±0.0000002
0.1909±0.0598
0.1412±0.0208
0.00007±0.00007
0.0139±0.0086
0.0850±0.0310
Reacher
0.0011±0.0005
0.1561±0.0262
0.1095±0.0349
0.0039±0.0027
0.0167±0.0082
0.0192±0.0094
Cheetah
0.0542±0.0041
0.2044±0.0327
0.1450±0.0121
0.0606±0.0143
0.0894±0.0216
0.0627±0.0078
Walker
0.2427±0.0209
0.2536±0.0199
0.2716±0.0182
0.2847±0.0332
0.2604±0.0206
0.3125±0.0099
Table 2: Held-out normalized regret (lower is better). The oracle has zero regret. Bold is the best learned method.
Domain
CQM -B
World
SF
Cartpole
0.000007
1.2182
1.0658
Reacher
0.0385
1.0369
1.0125
Cheetah
0.4503
1.0228
1.0113
Walker
1.0453
1.6460
1.1585
Table 3: Effect NMSE for the three matched vector predictors. Unlike absolute-future NMSE, this metric isolates action-dependent prediction error.
Figure 1: Held-out action-score alignment for one representative Reacher seed. Each hexagon aggregates (s,a,g) triples. The horizontal coordinate is the oracle advantage and the vertical coordinate is the model advantage, each centered over actions and normalized within (s,g) . The red diagonal is perfect agreement. CQM concentrates tightly around the diagonal (correlation 0.986 ), while the absolute world model ( 0.236 ) and successor features ( 0.542 ) show weak or fragmented action geometry. Reward-aware methods are strong but less tightly aligned.
Figure 2: CDF of held-out normalized regret for the same Reacher run. Curves farther left and higher are better. CQM places almost all queries at zero or negligible regret; the world model has a long tail, and successor features improves on the world model but remains substantially worse than the quotient and reward-aware critics.