How do you decide how many draws a Monte Carlo simulation needs?

## Draws are a purchase, not a default The most common weak answer to "how many runs" is a round number chosen by habit: ten thousand because it sounds serious, a million because the machine can. The professional answer names a tolerance and derives the count. ## The rate that prices everything A simulation estimate is an average of `n` independent runs of the same random experiment. Its typical error is proportional to `1/sqrt(n)`, with the constant of proportionality set by how variable a single run is. Everything about simulation budgeting follows from that square root. - Doubling accuracy costs 4x the runs. - Ten times the accuracy costs 100x the runs. - Going from a 1% margin to a 0.1% margin on an overnight job turns it into a hundred-night job. This is why simulation is described as easy to start and expensive to finish. A crude answer arrives in seconds; a precise one may not be affordable at all, which is a genuine reason to look for an analytic shortcut or a variance-reduction technique instead of more compute. ## Working the arithmetic For a simulated probability, the per-run quantity is a 0/1 indicator with variance `p(1-p)`, so the simulation error is `sqrt(p(1-p)/n)`. Suppose a strategy simulation returns a win probability near 0.62. At `n = 10,000`: sqrt(0.62 * 0.38 / 10000) = sqrt(0.0000236) = 0.0049 So roughly half a percentage point, and a rough 95% interval of about plus or minus one point. If the decision needs a quarter-point margin, that is four times the runs, 160,000. If it needs a tenth of a point, that is 100 times the original, a million runs. Note that `p(1-p)` is largest at `p = 0.5` and small near 0 or 1, so rare-event probabilities have a small absolute error but a terrible relative error: estimating a probability of 0.001 well enough to know it within 10% needs on the order of a hundred thousand runs before a single meaningful event pattern emerges. For a simulated average that is not a probability, run a short pilot of a few thousand iterations, measure the spread of the per-run values, and solve `spread / sqrt(n) <= tolerance` for `n`. Pilot-then-size is the standard workflow and costs almost nothing. ## Sequential stopping An alternative to fixing `n` up front is to monitor the running Monte Carlo margin and stop when it drops below tolerance. That is efficient, and it is legitimate here in a way it is not when peeking at experimental data, because the quantity being estimated is a fixed property of a known model and more draws only ever tighten the same target. Two cautions: the margin estimate is itself noisy early on, so impose a minimum run count before allowing a stop, and the stopping decision must be based on precision, never on the estimate having reached a number someone hoped for. ## What more draws cannot fix This is the point that separates a middle answer from a senior one. Draws shrink only the simulation's own sampling noise around the answer that the model implies. They do not touch: - **Model error.** If the simulated rules leave out a real mechanism, a billion runs converge precisely on the wrong number. - **Input error.** Parameters estimated from data carry their own uncertainty. A win rate simulated from a hit probability that is itself only known to plus or minus three points cannot be trusted to half a point, no matter how many runs are made. Propagating input uncertainty means re-running the simulation across plausible parameter values, not adding iterations at one fixed value. - **Bugs.** A miscoded rule is systematic, and averaging never removes a systematic error. A good habit is to report two numbers: the simulation margin, which you control with draws, and the sensitivity of the answer to the parameters you are least sure of, which you control only by better inputs. ## Reproducibility and reporting Record the seed, the number of runs, and the version of the model alongside the result. A simulated figure that cannot be regenerated is not evidence, and a figure quoted without its run count invites a reader to treat noise as signal. ## What to say out loud Name the tolerance the decision needs, quote the `1/sqrt(n)` rate and the 4x/100x consequences, size `n` from a pilot, report the margin with the number, and finish by separating simulation noise from model and input error.