Merge bias: why deterministic dates are optimistic

Here is the most important piece of arithmetic in schedule risk — small enough to do on a napkin, big enough to explain why projects with competent planners still finish late.

Picture two parallel workstreams converging on one finish: civils and M&E, say. Each, taken alone, has a 50% chance of hitting day 100. What is the chance the project hits day 100?

The finish needs both paths done, so the probabilities multiply. Two fair coins, both heads. Add a third parallel path and you're at 12.5%. This is merge bias: at every point where parallel paths converge, the schedule silently sheds probability — and the deterministic date, which assumes every path behaves, drifts into the optimistic tail of its own distribution.

It gets worse: the date moves, not just the odds

Make it concrete: path A takes uniformly 80–120 days, path B uniformly 70–130, both "100 days" in the plan. The project's P50 — the date it's actually even odds to make — isn't day 100. It's day 110. That's exactly solvable (the finish CDF is the product of the two path CDFs), and ten days of expected slippage appear from pure geometry — before a single risk, delay event or bad estimate enters the model. The full derivation is on The Maths → Validation, where the same network doubles as our engine's accuracy test: analytic P50 110.000, simulated 110.075.

Why hand methods can't save you

Classic PERT — add the weighted averages along the critical path — sees only one path, so it misses merge bias entirely; MacCrimmon and Ryavec pointed this out in 1964, and it's been quietly ignored at contract signings ever since. Real programmes are lattices of merges: every handover, every "all systems complete" milestone, every commissioning gate multiplies probabilities together. The only honest way to price that is to simulate the whole network — sample every path, re-solve the CPM logic thousands of times, and let the merges do to the model what they will do to the project.

What to do with this on Monday

• Treat any date at a convergence point as optimistic until simulated — the more parallel strands, the more optimistic.
• When a stakeholder asks why the P50 is later than "the plan", draw the two-coin example. It ends the argument politely.
• Watch criticality indices, not just the deterministic critical path: under risk, near-critical merge branches take turns driving the finish.

You can reproduce the numbers above yourself in about two minutes — the network is case study 03, files included, seed printed.