Probability uncertainty and robust decisions
Treat a probability as an interval rather than a point: test whether the edge survives the downside, size stakes from the lower bound, and see why estimation error calls for shrinking Kelly.
Learning objectives
- Express a probability estimate as an interval and compute EV at both ends with a low / central / high EV check.
- Compute Kelly on the lower bound of the interval and compare it with Kelly on the point estimate.
- Explain why noisy estimates make full Kelly over-stake on average, using a shrinkage simulation.
- Reach and defend a ‘no action’ decision when the downside scenario removes the edge.
A probability is an estimate, not a fact
Every probability you use is produced by something fallible: a model with estimated parameters, a small sample of past matches, team news you may have missed, or a structure (such as independent Poisson goals) that is only approximately right. The honest output is therefore an interval: a low, a central and a high value you would defend.
Because EV = p × o − 1 is a straight line in p, its worst and best cases sit at the ends of the interval. Checking the EV at the low end, the estimate and the high end gives one of three verdicts: positive across the range, negative across the range, or dependent on the estimate. Lesson 4, Expected value under uncertainty, introduced the idea; here we turn it into a staking rule.
EV_low = p_low × o − 1 EV_high = p_high × o − 1| Interval | EV at low | EV at high | Verdict |
|---|---|---|---|
| 45%–51% | +3.5% | +17.3% | Positive across the range |
| 40%–46% | −8.0% | +5.8% | Depends on the estimate |
| 38%–42% | −12.6% | −3.4% | Negative across the range |
Kelly on the lower bound
A simple robust rule is to feed the lower bound, not the central estimate, into the Kelly formula. With the quote of 2.30, a central 48% and an interval of 45%–51%, Kelly on the estimate is 8.00% of a simulated bankroll, while Kelly on 45% is 2.69%. Half of that is 1.35%.
Δf_K = Δp × o / (o − 1)That sensitivity is the heart of the problem. Near break-even, a small error in p is a large error in the stake. Lower-bound sizing builds a margin of safety into the stake itself rather than relying on discipline later.
Why estimation error calls for shrinkage
Why does noise favour smaller stakes? The stake is computed from your estimate, but growth is earned at the true probability. When you overestimate, Kelly over-stakes and growth falls steeply; when you underestimate, Kelly under-stakes (or stakes nothing) and growth falls only gently. Averaged over many estimates, the losses from over-staking dominate.
Our Academy calculator illustrates this with a simple normal-noise teaching model: the estimate equals the truth plus Gaussian noise, Kelly is computed on the estimate and multiplied by k, and growth is scored at the truth. It is not Baker and McHale's exact estimator. With a true 50%, odds 2.20 and noise of 5 points (standard deviation 0.05), the best multiplier was 0.5.
| Multiplier k | Mean growth per pick | Share of known-p Kelly growth (0.00415) |
|---|---|---|
| 0.25 | 0.00172 | 41% |
| 0.50 | 0.00235 | 57% |
| 0.75 | 0.00191 | 46% |
| 1.00 (full Kelly on the estimate) | 0.00035 | 8% |
In the same model the best multiplier falls as noise rises: 1.0 with no noise, 0.85 at SD 0.02, 0.7 at SD 0.03, 0.5 at SD 0.05 and 0.35 at SD 0.08.
When the downside removes the edge: no action
Consider a hypothetical quote of 2.10 (break-even 47.62%) and an estimate of 50% with an interval of 46%–54%. At the estimate the EV is +5.0%; at the low end it is −3.4%. The verdict is ‘depends on the estimate’, and Kelly on the lower bound is zero. The robust decision is no action.
Passing is a legitimate research result, not a failure of nerve. In Value 7, where stakes are free virtual Credits, logging the pass together with your interval lets you check later whether your intervals were honest — the habit that real forecasters lack most.
Worked example
Worked example · From a point estimate to a robust stake
A hypothetical quote of 2.20 on Ashbridge to beat Redcastle. Your central estimate is 50%, and you would defend an interval of 45%–55%. Stakes are shares of a simulated virtual-Credit bankroll.
- Break-even
1 / 2.2045.45% - EV across the interval
0.45 × 2.20 − 1; 0.50 × 2.20 − 1; 0.55 × 2.20 − 1−1.0%; +10.0%; +21.0% → depends on the estimate - Kelly on the point estimate
(0.50 × 2.20 − 1) / 1.208.33% - Kelly on the lower bound
0.45 × 2.20 − 1 < 00% → no action - If new evidence narrows the interval to 47%–53%
EV_low = 0.47 × 2.20 − 1; Kelly(0.47) = 0.034 / 1.20+3.4%; 2.83%, half of it 1.42% - Cost of ignoring noise (teaching model, SD 0.05)
Mean growth at k = 1 vs k = 0.5 vs known-p Kelly0.00035 vs 0.00235 vs 0.00415 per pick
With a ±5-point interval the robust answer is to pass, even though the central EV is +10%. With a ±3-point interval a small stake of about 1.4% of the simulated bankroll is defensible. In the noise model, staking full Kelly on noisy estimates kept only about 8% of the growth that perfect knowledge would deliver.
Practice · Does the edge survive the downside?
Open the Fair Odds and EV lab with the preset (odds 2.10, estimate 50%, interval 46%–54%). Read the verdict. Then narrow the interval to 48%–52% and compute, by hand, Kelly on the lower bound and half of it.
Lab 1 · Fair odds and expected value
Synthetic · deterministicNo clear advantage. Your range straddles break-even. Whether the bet has value depends on which end of your estimate is right, so there is no clear advantage. A careful analyst treats this as a pass unless better evidence narrows the range.
Show data table
| True probability (%) | Expected ROI |
|---|---|
| 36% | -24% |
| 37% | -22% |
| 38% | -20% |
| 39% | -18% |
| 40% | -16% |
| 41% | -14% |
| 42% | -12% |
| 43% | -10% |
| 44% | -8% |
| 45% | -5% |
| 46% | -3% |
| 47% | -1% |
| 48% | 1% |
| 49% | 3% |
| 50% | 5% |
| 51% | 7% |
| 52% | 9% |
| 53% | 11% |
| 54% | 13% |
| 55% | 16% |
| 56% | 18% |
| 57% | 20% |
| 58% | 22% |
| 59% | 24% |
| 60% | 26% |
| 61% | 28% |
| 62% | 30% |
| 63% | 32% |
| 64% | 34% |
Educational calculator. Uses only the numbers you enter or seeded synthetic data — no live prices, no staking, no real money.
Show what you should find
Break-even is 47.62%. With 46%–54% the EV is −3.4% at the low end, +5.0% at the estimate and +13.4% at the high end: the verdict depends on the estimate, so the robust decision is no action. With 48%–52% the EV is +0.8% at the low end and +9.2% at the high end (positive across the range). Kelly on 48% is (0.48 × 2.10 − 1) / 1.10 = 0.73% of the simulated bankroll; half of it is about 0.36%. Compare 4.55% for Kelly on the 50% point estimate.
Common mistakes
- ✗ Using the central estimate's EV as ‘the edge’.Check EV at the low end of an honest interval. At 2.20 with 45%–55%, the central +10% hides a −1.0% downside.
- ✗ Assuming small probability errors mean small stake errors.At odds 2.30, Kelly moves 1.77 points per point of probability; 48% gives 8.00% but 45% gives 2.69%.
- ✗ Picking a shrinkage multiplier from a paper or a table.The right multiplier depends on your own estimation error: in the teaching model it ranges from 0.85 (SD 0.02) to 0.35 (SD 0.08). Measure your error out of sample.
- ✗ Treating ‘no action’ as a wasted opportunity.When the downside removes the edge, passing is the correct output of the analysis. Log it and review your intervals later.
When it fails
Supporting research
- Baker & McHale (2013): Optimal Betting Under Parameter Uncertainty: Improving the Kelly CriterionDecision Analysis · Peer-reviewed theoryEstimated probabilities justify shrinking Kelly stakes; the best shrinkage depends on the error model.
- Kelly (1956): A New Interpretation of Information RateBell System Technical Journal · Peer-reviewed theoryThe Kelly rule assumes known probabilities, which is exactly the assumption relaxed here.
- MacLean, Thorp & Ziemba (2010): Long-term capital growth: the good and bad properties of the Kelly and fractional Kelly capital growth criteriaQuantitative Finance · Peer-reviewed theoryFractional Kelly lowers risk at a modest cost in growth, under known return distributions.
Check your understanding
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