Simpson's paradox in broker fill rates

Question

You route orders through two brokers and track fill rates, split by order size:

	Small orders	Large orders	Overall
Broker A	90/100 (90%)	180/900 (20%)	270/1000 (27%)
Broker B	850/1000 (85%)	15/100 (15%)	865/1100 (78.6%)

Broker A fills a higher fraction of small orders (90% vs 85%) and of large orders (20% vs 15%), yet B's overall fill rate dwarfs A's (78.6% vs 27%).

Explain the paradox. Which broker is actually better?

Accepted Answer

What's happening. This is Simpson's paradox: A leads in every subgroup yet loses overall. The driver is a confounder tied to group membership, namely order size, which hugely affects fill rate (small orders fill around 85 to 90\%, large orders only 15 to 20\%), combined with opposite mixes: - Broker A handled mostly large orders (900 of its 1000), the hard-to-fill kind. - Broker B handled mostly small orders (1000 of its 1100), the easy kind. The overall rates are size-weighted averages: 	ext{A: } 	frac{100}{1000}(0.90) + 	frac{900}{1000}(0.20) = 0.27, \qquad 	ext{B: } 	frac{1000}{1100}(…

Simpson's paradox in broker fill rates

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