The Better Desk

Question

Two desks trade the same product over a quarter containing quiet days and volatile days.

	Quiet days	Volatile days	Overall
Desk A	72 wins / 90 trades (80%)	1 win / 10 trades (10%)	73/100 (73%)
Desk B	9 wins / 10 trades (90%)	18 wins / 90 trades (20%)	27/100 (27%)

Desk B has the higher win rate on quiet days (90% vs 80%) and on volatile days (20% vs 10%), yet Desk A's overall win rate crushes B's (73% vs 27%). Which desk is actually better at trading, and what exactly is going on?

Accepted Answer

This is Simpson's paradox: an association that holds in every subgroup reverses on aggregation, because the groups are mixed with wildly different weights. Each overall rate is a weighted average of the same two per-regime rates: A: 0.9 times 80% + 0.1 times 10% = 73%, qquad B: 0.1 times 90% + 0.9 times 20% = 27%. Desk A took 90% of its trades in the easy regime; Desk B took 90% of its trades in the hard regime. The aggregate comparison is mostly a comparison of trade mixes, not of skill. Conditioning on the regime, which determines difficulty and plausibly drove how often each desk traded. Desk B wins both comparisons: it is better at trading quiet days and better at trading volatile days. The honest conclusion: B is the better trading desk, per trade, in every condition it faces, while A's aggregate number is a base-rate artifact. (A separate, legitimate question is whether A's choice to concentrate in the easy regime was itself skill, if desks freely chose when to trade, regime selection is part of performance. The paradox forces you to articulate which question you are answering.) Why interviewers ask it Aggregated performance numbers, win rates by venue, fill quality by broker

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