Probability Weighted Utility for Decisions

Classic example:

• Option A: 100% chance of GBP 100,000.
• Option B: 50% chance of GBP 250,000, 50% chance of GBP -50,000.
• Expected value of Option A: GBP 100,000.
• Expected value of Option B: GBP 100,000.
• Same EV - should you be indifferent?

Most people prefer Option A (the certain gain).

Why:

• Losing GBP 50,000 hurts more than gaining GBP 250,000 helps (loss aversion).
• You need GBP 100,000 for tangible plans - actually getting it matters more than the chance of more.
• The utility (subjective value) of GBP 250,000 is NOT 2.5x the utility of GBP 100,000 - it's typically 1.5-2x due to decreasing marginal utility.

This is rational, not biased - your utility function genuinely treats certainty differently. Probability-weighted utility captures this.

Mathematical statement:

• Expected Utility = Σ (probability of outcome × utility of outcome).
• For each option, sum across all possible outcomes.
• Choose the option with highest expected utility.

The challenge:

• 'Utility of outcome' is subjective - depends on YOUR values, situation, time horizon.
• Probability assessment is uncertain - especially for high-stakes infrequent decisions.
• Both inputs need honest + thoughtful assessment.

Simplified procedure:

1. List all options (e.g. take job A, take job B, stay put).
2. For each option, identify possible outcomes (good, neutral, bad).
3. Assign probability to each outcome.
4. Assign utility (subjective value, 0-100 scale typical) to each outcome.
5. Calculate expected utility for each option.
6. Compare; choose highest EU.

Decision: Take a startup job (Option A) or stay at current corporate role (Option B)?

Option A - Startup job:

• Outcome 1: Startup IPOs, 5% probability. Utility: +100 (life-changing).
• Outcome 2: Startup grows steadily, 35% probability. Utility: +60 (great career boost).
• Outcome 3: Startup stagnates, 30% probability. Utility: +10 (decent experience, no big win).
• Outcome 4: Startup fails, 30% probability. Utility: -30 (need to find new role in down market).
• Expected Utility = 0.05 × 100 + 0.35 × 60 + 0.30 × 10 + 0.30 × -30 = 5 + 21 + 3 - 9 = 20.

Option B - Stay at corporate:

• Outcome 1: Get promoted, 30% probability. Utility: +40.
• Outcome 2: Stay at level, 60% probability. Utility: +5 (boring but stable).
• Outcome 3: Layoff during downturn, 10% probability. Utility: -25.
• Expected Utility = 0.30 × 40 + 0.60 × 5 + 0.10 × -25 = 12 + 3 - 2.5 = 12.5.

Decision:

• Option A (startup) has higher expected utility (20 vs 12.5).
• But this answer is highly sensitive to your utility values + probability estimates.
• If you have dependents + can't tolerate the -30 utility of startup failure: probably stay corporate.
• If you're young + can afford the risk: startup looks like the better bet.

Single decisions with non-linear value:

• Career pivot: take risky job vs stay safe.
• Business venture: start company vs continue employment.
• Marriage / family planning: probability + utility weighing.
• Major investment: property purchase, business angel investment.
• Health decisions: surgery options vs continued monitoring.
• Insurance buying: which catastrophic-loss insurance to prioritise.

Repeated decisions with consistent stakes:

• Less useful here - law of large numbers means EV is a better approximation of long-term outcome.
• Probability-weighted utility helpful for understanding why you might still feel anxious about high-EV high-volatility outcomes.

1. Treating all currency units as equal utility: 'GBP 50,000 is GBP 50,000' ignores that the same amount has very different value at different starting points.
2. Overweighting near-term utility: gains/losses in next 12 months feel more important than gains/losses 5+ years out - but they may not actually be.
3. Underweighting long-term utility: especially for retirement, education, health investments.
4. Ignoring opportunity costs: the utility of staying with status quo includes 'foregone opportunity to do something better'.
5. Failure to consider all major outcomes: easy to miss the 5%-probability disaster that dominates expected utility.
6. Anchoring on first option analysed: utility comparisons should be done fresh for each option, not adjusted relative to the first.
7. Treating utility as objective: 'this is worth X to anyone' is wrong - utility is personal.

Practice probability-weighted utility on lower-stakes choices to build intuition:

• Buying a major appliance: short-term cost vs long-term ownership cost vs reliability.
• Choosing a holiday destination: probability of weather + experiences × utility.
• Selecting investments: each major portfolio choice involves EU calculation.
• Hiring decisions: each interview update + your utility for different candidate fits.

Why repeated low-stakes practice matters:

• Builds calibration on probability estimates.
• Develops intuition for utility scales.
• Trains conscious deliberation that becomes automatic over time.
• When high-stakes decisions arrive, you've practiced the framework + can use it well.

Excel / Google Sheets:

• Build decision tree with branching outcomes.
• Cells for probability + utility for each branch.
• Formula calculates expected utility per option.
• Sensitivity analysis: vary key inputs to see how robust the answer is.

Decision-analysis software:

• Causal: UK-built probabilistic modelling tool; freemium.
• TreeAge: classic decision-tree software; expensive but powerful.
• Hivelocity / Lumen: newer cloud-based tools.

Decision journals (manual):

• Track your decisions + outcomes + retrospective utility assessment.
• Reveals systematic biases (over/underestimating probabilities, miscalibrated utility).
• See our decision journal guide.