Expected Value

Table of Contents

• Purpose
• When to Use
• What Is It?
• Workflow
• Common Patterns
• Guardrails
• Quick Reference

Purpose

Expected Value (EV) provides a framework for making rational decisions under uncertainty by calculating the probability-weighted average of all possible outcomes. This skill guides you through identifying scenarios, estimating probabilities and payoffs, computing expected values, and interpreting results while accounting for risk preferences and real-world constraints.

When to Use

Use this skill when:

• Investment decisions: Should we invest in project A (high risk, high return) or project B (low risk, low return)?
• Product bets: Launch feature X (uncertain adoption) or focus on feature Y (safer bet)?
• Resource allocation: Which initiatives have highest expected return given limited budget?
• Go/no-go decisions: Is expected value of launching positive after accounting for probabilities of success/failure?
• Pricing & negotiation: What's expected value of accepting vs. rejecting an offer?
• Insurance & hedging: Should we buy insurance (guaranteed small loss) vs. risk large loss?
• A/B test interpretation: Which variant has higher expected conversion rate accounting for uncertainty?
• Portfolio optimization: Diversify to maximize expected return for given risk tolerance?

Trigger phrases: "expected value", "EV calculation", "risk-adjusted return", "probability-weighted outcomes", "decision tree", "should I take this gamble", "compare risky options"

What Is It?

Expected Value (EV) = Σ (Probability of outcome × Value of outcome)

For each possible outcome, multiply its probability by its value (payoff), then sum across all outcomes.

Core formula:

EV = (p₁ × v₁) + (p₂ × v₂) + ... + (pₙ × vₙ)

where:
- p₁, p₂, ..., pₙ are probabilities of each outcome (must sum to 1.0)
- v₁, v₂, ..., vₙ are values (payoffs) of each outcome

Quick example:

Scenario: Launch new product feature. Estimate 60% chance of success ($100k revenue), 40% chance of failure (-$20k sunk cost).

Calculation:

• EV = (0.6 × $100k) + (0.4 × -$20k)
• EV = $60k - $8k = $52k

Interpretation: On average, launching this feature yields $52k. Positive EV → launch is rational choice (if risk tolerance allows).

Core benefits:

• Quantitative comparison: Compare disparate options on same scale (expected return)
• Explicit uncertainty: Forces estimation of probabilities instead of gut feel
• Repeatable framework: Same method applies to investments, products, hiring, etc.
• Risk-adjusted: Weights outcomes by likelihood, not just best/worst case
• Portfolio thinking: Optimal long-term strategy is maximize expected value over many decisions

Workflow

Copy this checklist and track your progress:

Expected Value Analysis Progress:
- [ ] Step 1: Define decision and alternatives
- [ ] Step 2: Identify possible outcomes
- [ ] Step 3: Estimate probabilities
- [ ] Step 4: Estimate payoffs (values)
- [ ] Step 5: Calculate expected values
- [ ] Step 6: Interpret and adjust for risk preferences

Step 1: Define decision and alternatives

What decision are you making? What are the mutually exclusive options? See resources/template.md.

Step 2: Identify possible outcomes

For each alternative, what could happen? List scenarios from best case to worst case. See resources/template.md.

Step 3: Estimate probabilities

What's the probability of each outcome? Use base rates, reference classes, expert judgment, data. See resources/methodology.md.

Step 4: Estimate payoffs (values)

What's the value (gain or loss) of each outcome? Quantify in dollars, time, utility. See resources/methodology.md.

Step 5: Calculate expected values

Multiply probabilities by payoffs, sum across outcomes for each alternative. See resources/template.md.

Step 6: Interpret and adjust for risk preferences

Choose option with highest EV? Or adjust for risk aversion, non-monetary factors, strategic value. See resources/methodology.md.

Validate using resources/evaluators/rubric_expected_value.json. Minimum standard: Average score ≥ 3.5.

Common Patterns

Pattern 1: Investment Decision (Discrete Outcomes)

• Structure: Go/no-go choice with 3-5 discrete scenarios (best, base, worst case)
• Use case: Product launch, hire vs. not hire, accept investment offer, buy vs. lease
• Pros: Simple, intuitive, easy to communicate (decision tree visualization)
• Cons: Oversimplifies continuous distributions, binary framing may miss nuance
• Example: Launch product feature (60% success $100k, 40% fail -$20k) → EV = $52k

Pattern 2: Portfolio Allocation (Multiple Options)

• Structure: Allocate budget across N projects, each with own EV and risk profile
• Use case: Venture portfolio, R&D budget, marketing spend allocation, team capacity
• Pros: Diversification reduces variance, can optimize for risk/return tradeoff
• Cons: Requires estimates for many variables, correlations matter (not independent)
• Example: Invest in 3 startups ($50k each), EVs = [$20k, $15k, -$10k]. Total EV = $25k. Diversified portfolio reduces risk vs. single $150k bet.

Pattern 3: Sequential Decision (Decision Tree)

• Structure: Series of decisions over time, outcomes of early decisions affect later options
• Use case: Clinical trials (Phase I → II → III), staged investment, explore then exploit
• Pros: Captures optionality (can stop if early results bad), fold-back induction finds optimal strategy
• Cons: Tree grows exponentially, need probabilities for all branches
• Example: Phase I drug trial (70% pass, $1M cost) → if pass, Phase II (50% pass, $5M) → if pass, Phase III (40% approve, $50M revenue). Calculate EV working backwards.

Pattern 4: Continuous Distribution (Monte Carlo)

• Structure: Outcomes are continuous (revenue could be $0-$1M), use probability distributions
• Use case: Financial modeling, project timelines, resource planning, sensitivity analysis
• Pros: Captures full uncertainty, avoids discrete scenario bias, provides confidence intervals
• Cons: Requires distributional assumptions, computationally intensive, harder to communicate
• Example: Revenue ~ Normal($500k, $100k std dev). Run 10,000 simulations → mean = $510k, 90% CI = [$350k, $670k].

Pattern 5: Competitive Game (Payoff Matrix)

• Structure: Your outcome depends on competitor's choice, create payoff matrix
• Use case: Pricing strategy, product launch timing, negotiation, auction bidding
• Pros: Incorporates strategic interaction, finds Nash equilibrium
• Cons: Requires estimating competitor's probabilities and payoffs, game-theoretic complexity
• Example: Price high vs. low, competitor prices high vs. low → 2×2 matrix. Calculate EV for each strategy given beliefs about competitor.

Guardrails

Critical requirements:

1. Probabilities must sum to 1.0: If you list outcomes, their probabilities must be exhaustive (cover all possibilities) and mutually exclusive (no overlap). Check: p₁ + p₂ + ... + pₙ = 1.0.

2. Don't use EV for one-shot, high-stakes decisions without risk adjustment: EV is long-run average. For rare, irreversible decisions (bet life savings, critical surgery), consider risk aversion. A 1% chance of $1B (EV = $10M) doesn't mean you should bet your house.

3. Quantify uncertainty, don't hide it: Probabilities and payoffs are estimates, often uncertain. Use ranges (optimistic/pessimistic), sensitivity analysis, or distributions. Don't pretend false precision.

4. Consider non-monetary value: EV in dollars is convenient, but some outcomes have utility not captured by money (reputation, learning, optionality, morale). Convert to common scale or use multi-attribute utility.

5. Probabilities must be calibrated: Don't use gut-feel probabilities without grounding. Use base rates, reference classes, data, expert forecasts. Test: are your "70% confident" predictions right 70% of the time?

6. Account for correlated outcomes: If outcomes aren't independent (economic downturn affects all portfolio companies), correlation reduces diversification benefit. Model dependencies.

7. Time value of money: Payoffs at different times aren't equivalent. Discount future cash flows to present value (NPV = Σ CF_t / (1+r)^t). EV should use NPV, not nominal values.

8. Stopping rules and option value: In sequential decisions, fold-back induction finds optimal strategy. Don't ignore option to stop early, pivot, or wait for more information.

Common pitfalls:

• ❌ Ignoring risk aversion: EV($100k, 50/50) = EV($50k, certain) but most prefer certain $50k. Use utility functions for risk-averse agents.
• ❌ Anchor on single scenario: "Best case is $1M!" → but probability is 5%. Focus on EV, not cherry-picked scenarios.
• ❌ False precision: "Probability = 67.3%" when you're guessing. Use ranges, express uncertainty.
• ❌ Sunk cost fallacy: Past costs are sunk, don't include in forward-looking EV. Only future costs/benefits matter.
• ❌ Ignoring tail risk: Low-probability, high-impact events (0.1% chance of -$10M) can dominate EV. Don't round to zero.
• ❌ Static analysis: Assume you can't update beliefs or change course. Real decisions allow learning and pivoting.

Quick Reference

Key formulas:

Expected Value: EV = Σ (pᵢ × vᵢ) where p = probability, v = value

Expected Utility (for risk aversion): EU = Σ (pᵢ × U(vᵢ)) where U = utility function

• Risk-neutral: U(x) = x (EV = EU)
• Risk-averse: U(x) = √x or U(x) = log(x) (concave)
• Risk-seeking: U(x) = x² (convex)

Net Present Value: NPV = Σ (CF_t / (1+r)^t) where CF = cash flow, r = discount rate, t = time period

Variance (risk measure): Var = Σ (pᵢ × (vᵢ - EV)²)

Standard Deviation: σ = √Var

Coefficient of Variation (risk/return ratio): CV = σ / EV (lower = better risk-adjusted return)

Breakeven probability: p* where EV = 0. Solve: p* × v_success + (1-p*) × v_failure = 0.

Decision rules:

• Maximize EV: Choose option with highest EV (risk-neutral, repeated decisions)
• Maximize EU: Choose option with highest expected utility (risk-averse, incorporates preferences)
• Minimax regret: Minimize maximum regret across scenarios (conservative, avoid worst mistake)
• Satisficing: Choose first option above threshold EV (bounded rationality)

Sensitivity analysis questions:

• How much do probabilities need to change to flip decision?
• What's EV in best case? Worst case? Which variables have most impact?
• At what probability does EV break even (EV = 0)?

Key resources:

• resources/template.md: Decision framing, outcome identification, EV calculation templates, sensitivity analysis
• resources/methodology.md: Probability estimation, payoff quantification, decision tree analysis, utility functions
• resources/evaluators/rubric_expected_value.json: Quality criteria (scenario completeness, probability calibration, payoff quantification, EV interpretation)

Inputs required:

• Decision: What are you choosing between? (2+ mutually exclusive alternatives)
• Outcomes: For each alternative, what could happen? (3-5 scenarios typical)
• Probabilities: How likely is each outcome? (sum to 1.0)
• Payoffs: What's the value (gain/loss) of each outcome? (dollars, time, utility)

Outputs produced:

• expected-value-analysis.md: Decision framing, outcome scenarios with probabilities and payoffs, EV calculations, sensitivity analysis, recommendation with risk considerations