Strategy A wins 8 of 10 trades in liquid names and 45 of 90 in illiquid names. Strategy B wins 63 of 90 liquid and 4 of 10 illiquid, and B has the higher pooled win rate. Which do you allocate to, and what would have to be true for the pooled number to be the one you use?

Strategy A wins 8 of 10 trades in liquid names and 45 of 90 in illiquid names. Strategy B wins 63 of 90 liquid and 4 of 10 illiquid, and B has the higher pooled win rate. Which do you allocate to, and what would have to be true for the pooled number to be the one you use?

Approach: Compute the four stratum rates and the two pooled rates, then ask what differs between the two strategies besides their skill inside each stratum.

Allocate to A, which wins in both strata. A wins 8/10 = 80% against B's 63/90 = 70% in liquid names and 45/90 = 50% against 4/10 = 40% in illiquid names, while the pooled rates are 53% for A and 67% for B. The reversal is Simpson's paradox: liquidity drives both the win rate and the trade mix, so it is a confounder, and pooling averages each strategy over a different weighting of it, with 90 of A's 100 trades in the harder bucket and 90 of B's in the easier one. The stratified comparison is the decision-relevant one whenever liquidity is a fixed feature of the world you condition on at trade time. The pooled estimate is the right one only if the mix is itself a choice each strategy controls and could sustain at size, in which case you are comparing capacity alongside skill and the honest test equalises the mix.

Follow-up: A splits into three liquidity buckets and beats B in two of them while losing the third. What rule now decides the allocation?

Key concepts: Simpson's paradox, confounding, stratification, pooled estimate.