Tensions in Practice: A point probability in tension with protection across a plausible range¶
A toy choice with linear outcome values
A gamble pays 10 on success and 0 otherwise; a certain option pays 4. Both arrangements value units linearly. Using a defended success probability of 1/2 ranks the gamble at 5. If the decision instead protects against any success probability from 1/4 to 3/4, the gamble’s worst mean is 2.5 and the certain option wins. The change is the policy for uncertain probabilities, not curvature in the value function.
Use a defended probability estimate
Rank prospects under one justified probability assignment.
Protect across ambiguity
Avoid relying on one probability within a declared plausible range.
Why these aims pull against each other
A point estimate permits a sharper ranking; a worst-case range policy can forgo gains when the pessimistic endpoint does not occur.
Choose an arrangement to see what changes and what remains difficult.
Payoffs remain fixed. The score column changes from an expectation at p = 1/2 to the smallest expectation over the declared range; shading marks the chosen option.
What this choice protects
What it costs
When it fits
Compare the arrangements
Use the point estimate
Use success probability 1/2 and maximize mean units.
| If success | If failure | Score | |
|---|---|---|---|
| Gamble | 10 | 0 | 5Mean |
| Certain | 4 | 4 | 4Not chosen |
- What it protects
- The gamble’s expected 5 exceeds the certain 4 under that estimate.
- What it costs
- The choice depends on trusting this single probability assignment.
- When it fits
- Fits when the point estimate is adequately defended for the decision.
Illustration note: The values are linear; no risk-aversion curve is being estimated.
Protect across the range
Use the lowest expected value over p in [1/4,3/4].
| If success | If failure | Score | |
|---|---|---|---|
| Gamble | 10 | 0 | 2.5Worst case |
| Certain | 4 | 4 | 4Chosen |
- What it protects
- The certain option has value 4 throughout the range, above the gamble’s worst mean 2.5.
- What it costs
- It forgoes the gamble’s higher mean when p exceeds 0.4.
- When it fits
- Fits when the range is credible and a worst-case policy is substantively justified.
Illustration note: A plausible range alone does not mandate maximin; the conservative rule is an additional declared choice.
What this illustration does—and does not—establish
The source supplies the tension. The invented setting, alternatives and any numbers illustrate a limited comparison; each arrangement retains its stated costs and conditions.
- The probability and range are invented assumptions, not confidence intervals or empirical estimates.
- The comparison does not assert that every cautious choice is risk aversion or that every ambiguity policy is maximin.
- The certain option’s stipulated certainty is part of the toy; model error outside the range remains possible.
Source entries
Risk Aversion
This source passage supplies the contextual tension. The concrete arrangements and schematic examples are editorial illustrations, not measured findings.
Risk vs. Ambiguity (Ellsberg Distinction)
Standard risk-aversion theory models risk but not ambiguity; ambiguity aversion requires additional mechanisms (maxmin expected utility, robust preferences).
Risk and ambiguity boundary
Not identical to ambiguity aversion: Ellsberg's paradox (1961) and subsequent work show that people often distinguish risk (known probabilities) from ambiguity (unknown probabilities) and are additionally averse to ambiguity. Ambiguity aversion is not captured by concavity of U alone.