The best bid is 50.00 with 800 lots resting and the best ask is 50.02. Joining the bid puts you behind those 800 lots with a 25% chance of filling before the price moves; improving to 50.01 puts you first in a new queue with a 70% chance. A passive buy at 50.00 earns 1.2 cents against your fair value, at 50.01 it earns 0.2 cents. Which order do you send?

The best bid is 50.00 with 800 lots resting and the best ask is 50.02. Joining the bid puts you behind those 800 lots with a 25% chance of filling before the price moves; improving to 50.01 puts you first in a new queue with a 70% chance. A passive buy at 50.00 earns 1.2 cents against your fair value, at 50.01 it earns 0.2 cents. Which order do you send?

Approach: Compute expected edge per lot as fill probability times the edge conditional on that fill, compare the two orders, then ask what the 800 lots ahead of you are worth.

Join the bid at 50.00. Joining gives 0.25 * 1.2, which is 0.30 cents of expected edge per lot; improving to 50.01 gives 0.70 * 0.2, which is 0.14 cents, so joining is worth more than twice as much. Improving buys fill probability under price-time priority by giving up 1 cent of edge, and here 45 extra percentage points of fill probability cost 83% of the edge. The 800 lots ahead of you are also the reason the deeper queue keeps its edge: they absorb uninformed flow first, so your fill is more likely to arrive when the price is about to move through you. The calculation has to use the edge conditional on filling rather than the unconditional edge.

Follow-up: If a hidden midpoint order can fill you at 50.01 with a 40% probability, how does that change the comparison?

Key concepts: queue position, price-time priority, expected edge, fill probability.