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Two-Outcome Square-Root Expected Utility

Compare a two-outcome prospect using probability-weighted square-root utility of nonnegative monetary outcomes.

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Calculator inputs

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How to use this calculator

  1. Enter the known values in the units shown. Results update as you type.
  2. Where results are editable, change one to solve backwards. Lock a value to hold it fixed.
  3. Use the worked example to check the method. Reset restores the starting fields.

Use the result with context

Results are estimates for planning and education. Confirm rates, taxes, fees, and legal requirements with the relevant institution or a qualified professional before making a financial decision.

Formula and method

A 50 percent chance of 100 and a 50 percent chance of 400 gives expected square-root utility 15: 0.5 × 10 + 0.5 × 20.

Expected utility = p₁√x₁ + p₂√x₂, with p₁ + p₂ = 1

Worked example

Enter these known values and leave the other values blank.

Probability of outcome one
50 %
Nonnegative amount in outcome one
100 USD
Probability of outcome two
50 %
Nonnegative amount in outcome two
400 USD
Expected square-root utility
15

Assumptions and limitations

  • The two outcomes are mutually exclusive and exhaustive; their probabilities must sum to 100 percent. Enter both probabilities explicitly.
  • The chosen utility function is the square root of a nonnegative monetary amount. This is one risk-preference model, not an observed preference or expected cash payout.
  • Currency entries are normalized to the captured USD reference before taking square roots. The numerical utility scale depends on the monetary unit; compare prospects on one consistent scale.
  • An outcome with zero probability does not affect expected utility, so its unknown amount cannot be uniquely recovered. Squaring an inverse does not make a negative square-root contribution valid.

Common questions

Is utility 15 a cash payout of 15?

No. It is the weighted average of the transformed outcomes. The expected cash payout in the example is 250.

Can I enter negative outcomes?

This real square-root model requires nonnegative amounts. A model of losses needs an appropriate utility function and wealth baseline.

References

Bookify requires the two outcome probabilities to total 100 percent and rejects inverse values below an already specified nonnegative utility contribution.

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