Sensitivity and model error

Ask how much a decision changes when an uncertain input moves.

12 minutes · intermediate · Original Poker4Quant lesson

What you will learn

Evaluate call EV over a probability range and distinguish arithmetic from estimation error.

Prerequisites: Ties and shared pots · Risk budgets in a toy model

A linear model has a transparent slope

For a no-tie final call, EV(q)=q(P+C)−C. Changing q by Δq changes EV by (P+C)Δq when P and C are fixed. Write percentage changes as decimals: one percentage point is 0.01.

With P=150 and C=50, the slope is 200 chips per probability unit, or 2 chips per percentage point. The threshold remains 25%. Inputs close to the threshold can make small estimation errors decision-relevant even when the arithmetic is exact.

A plausible interval is not automatically a confidence interval

A stated range of plausible probabilities is a scenario assumption unless it comes from a justified statistical procedure. Evaluate endpoints to see whether the EV sign is robust across that range, but do not invent a probability that q lies inside it.

Holding every other input fixed isolates one source of sensitivity. In a real model, probability, bet sizing, fees and future behavior may move together. A clean derivative or endpoint check does not validate those relationships; it helps identify which assumptions deserve more attention.

Worked example

P=150 and C=50. Estimated q=30%, with a plausible scenario range 20%–35%. What does the call model imply?

EV(0.30)=0.30×200−50=10.
EV(0.20)=40−50=−10.
EV(0.35)=70−50=20.
The range crosses the 25% threshold, so the sign is sensitive to q.

The point estimate gives +10, but the stated scenarios span −10 to +20. Better arithmetic cannot resolve uncertainty about the input estimate.

Common mistakes

Using 1 instead of 0.01 for a one-percentage-point change.
Labeling an arbitrary scenario range a statistical confidence interval.
Treating one-input sensitivity as a complete joint model validation.

Keep these ideas

Expose the input-to-payoff relationship.
Check endpoints against the decision threshold.
Separate computation precision from model evidence.