Back to Details

expected-value

Compare original and translation side by side

🇺🇸

Original

English
🇨🇳

Translation

Chinese

Expected Value

Expected Value

Table of Contents

目录

Purpose

用途

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.
Expected Value (EV) 提供了一套在不确定性下制定理性决策的框架，核心是计算所有可能结果的概率加权平均值。本指南将引导你完成场景识别、概率与收益估算、EV计算以及结果解读的全流程，同时会纳入风险偏好和现实约束的考量。

When to Use

适用场景

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"
在以下场景中使用本方法：
  • 投资决策：我们应该投资高风险高回报的项目A，还是低风险低回报的项目B？
  • 产品押注：是推出 adoption 不确定的功能X，还是聚焦更稳妥的功能Y？
  • 资源分配：在预算有限的情况下，哪些举措的预期回报最高？
  • 启动/终止决策：计入成功/失败概率后，项目启动的期望值是否为正？
  • 定价与谈判：接受和拒绝某份邀约的期望值分别是多少？
  • 保险与对冲：我们应该购买保险（确定的小额损失）还是承担大额损失的风险？
  • A/B测试解读：计入不确定性后，哪个变体的预期转化率更高？
  • 投资组合优化：如何在给定风险承受能力下实现预期回报最大化的多元化配置？
触发关键词："expected value"、"EV calculation"、"risk-adjusted return"、"probability-weighted outcomes"、"decision tree"、"我要不要冒这个险"、"比较高风险选项"

What Is It?

是什么？

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
Expected Value (EV) = Σ (结果发生概率 × 结果价值)
对每个可能的结果，将其发生概率与对应的价值（收益）相乘，再将所有结果的计算值相加，即可得到EV。
核心公式：
EV = (p₁ × v₁) + (p₂ × v₂) + ... + (pₙ × vₙ)

其中：
- p₁, p₂, ..., pₙ 是每个结果的发生概率（总和必须为1.0）
- v₁, v₂, ..., vₙ 是每个结果对应的价值（收益）
快速示例：
场景：推出新产品功能，预估60%概率成功（带来10万美元收入），40%概率失败（沉没成本损失2万美元）。
计算过程：
  • EV = (0.6 × $100k) + (0.4 × -$20k)
  • EV = $60k - $8k = $52k
解读：平均来看，推出该功能可获得5.2万美元收益。EV为正→如果风险承受能力允许，推出是理性选择。
核心优势：
  • 量化对比：将不同类型的选项放在同一维度（预期收益）下对比
  • 显性化不确定性：强制要求估算概率，而非依赖直觉判断
  • 可复用框架：同一方法可应用于投资、产品、招聘等多种场景
  • 风险调整：按发生概率对结果加权，而非仅考虑最好/最坏情况
  • 投资组合思维：长期最优策略是在多次决策中最大化预期值

Workflow

工作流程

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.
复制以下检查清单跟踪进度：
Expected Value 分析进度：
- [ ] 步骤1：明确定决策内容和备选方案
- [ ] 步骤2：识别所有可能的结果
- [ ] 步骤3：估算各结果的发生概率
- [ ] 步骤4：估算各结果的收益（价值）
- [ ] 步骤5：计算预期值EV
- [ ] 步骤6：结合风险偏好解读结果并调整
步骤1：明确定决策内容和备选方案 你需要制定什么决策？有哪些互斥的选项？参考 resources/template.md。
步骤2：识别所有可能的结果 每个备选方案可能产生哪些结果？按最好到最坏的情况列出所有场景。参考 resources/template.md。
步骤3：估算各结果的发生概率 每个结果的发生概率是多少？可使用基准概率、参考类别、专家判断、历史数据。参考 resources/methodology.md。
步骤4：估算各结果的收益（价值） 每个结果对应的价值（收益或损失）是多少？可按美元、时间、效用量化。参考 resources/methodology.md。
步骤5：计算预期值EV 将概率与收益相乘，再对每个备选方案的所有结果求和。参考 resources/template.md。
步骤6：结合风险偏好解读结果并调整 是否选择EV最高的选项？还是需要针对风险厌恶、非货币因素、战略价值进行调整。参考 resources/methodology.md。
使用 resources/evaluators/rubric_expected_value.json 验证分析质量，最低标准：平均得分≥3.5。

Common Patterns

常见模式

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.
模式1：投资决策（离散结果）
  • 结构：启动/终止决策，包含3-5个离散场景（最好、基准、最坏情况）
  • 适用场景：产品发布、是否招聘、接受投资邀约、购买/租赁选择
  • 优势：简单、直观、易于沟通（可通过决策树可视化）
  • 劣势：过度简化连续分布，二元框架可能遗漏细节
  • 示例：推出产品功能（60%成功收益10万美元，40%失败损失2万美元）→ EV=5.2万美元
模式2：投资组合分配（多选项）
  • 结构：在N个项目中分配预算，每个项目有独立的EV和风险特征
  • 适用场景：风险投资组合、研发预算分配、营销支出分配、团队产能分配
  • 优势：多元化降低方差，可针对风险/收益权衡做优化
  • 劣势：需要估算多个变量，结果相关性会产生影响（非独立）
  • 示例：投资3个创业公司（各5万美元），EV分别为[2万美元、1.5万美元、-1万美元]，总EV为2.5万美元，多元化组合相比单投15万美元的项目风险更低
模式3：序列决策（Decision Tree）
  • 结构：一系列跨时间的决策，早期决策的结果会影响后续选项
  • 适用场景：临床试验（I期→II期→III期）、分阶段投资、探索后落地
  • 优势：捕获选择权（早期结果不好可以停止），反向归纳可得出最优策略
  • 劣势：树结构呈指数级增长，需要为所有分支估算概率
  • 示例：药物I期试验（70%通过率，成本100万美元）→ 通过后进入II期（50%通过率，成本500万美元）→ 通过后进入III期（40%获批概率，收入5000万美元），反向计算EV
模式4：连续分布（Monte Carlo）
  • 结构：结果是连续的（收入可能在0-100万美元之间），使用概率分布建模
  • 适用场景：财务建模、项目 timeline 估算、资源规划、敏感性分析
  • 优势：捕获完整的不确定性，避免离散场景偏差，提供置信区间
  • 劣势：需要分布假设，计算量大，沟通难度更高
  • 示例：收入符合正态分布（均值50万美元，标准差10万美元），运行10000次模拟→均值51万美元，90%置信区间为[35万美元，67万美元]
模式5：竞争博弈（收益矩阵）
  • 结构：你的结果取决于竞争对手的选择，构建收益矩阵分析
  • 适用场景：定价策略、产品发布时机、谈判、拍卖竞价
  • 优势：纳入战略互动，可找到Nash均衡
  • 劣势：需要估算竞争对手的概率和收益，博弈论复杂度高
  • 示例：定价高/低，竞争对手定价高/低→2×2矩阵，基于对竞争对手的判断计算每个策略的EV

Guardrails

注意事项

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.
核心要求：
  1. 概率总和必须为1.0：列出的结果必须穷尽所有可能性且互斥（无重叠），检查p₁ + p₂ + ... + pₙ = 1.0。
  2. 针对一次性高风险决策，必须做风险调整后再使用EV：EV是长期平均值，对于罕见、不可逆的决策（押上全部积蓄、关键手术），需要考虑风险厌恶。1%概率获得10亿美元（EV=1000万美元）不代表你应该抵押房产去赌。
  3. 量化不确定性，不要隐藏：概率和收益都是估算值，通常存在不确定性，可使用区间（乐观/悲观）、敏感性分析或分布建模，不要伪造精确性。
  4. 考虑非货币价值：用美元计算EV很方便，但部分结果的效用无法用金钱衡量（声誉、学习收益、选择权、团队士气），可转换为统一维度或使用多属性效用模型。
  5. 概率必须校准：不要使用无依据的直觉概率，参考基准概率、参考类别、数据、专家预测。校验：你认为“70%置信”的预测是否真的有70%的准确率？
  6. 考虑结果相关性：如果结果不是独立的（经济下行会影响所有投资组合公司），相关性会降低多元化收益，需要对依赖关系建模。
  7. 货币时间价值：不同时间点的收益不等价，将未来现金流折现到现值（NPV = Σ CF_t / (1+r)^t），EV计算应使用NPV而非名义值。
  8. 停止规则和期权价值：在序列决策中，反向归纳可得到最优策略，不要忽略提前停止、转型或等待更多信息的选择权。
常见陷阱：
  • ❌ 忽略风险厌恶：EV（50%概率得10万美元）=EV（确定得5万美元），但大多数人偏好确定的5万美元，针对风险厌恶的主体需使用效用函数。
  • ❌ 锚定单一情景：“最好情况能赚100万美元！”但概率只有5%，要聚焦EV而非刻意挑选的场景。
  • ❌ 虚假精确：估算的时候就不要写“概率=67.3%”，使用区间表达不确定性。
  • ❌ 沉没成本谬误：过去的成本已经沉没，不要纳入前瞻EV计算，仅考虑未来的成本和收益。
  • ❌ 忽略尾部风险：低概率高影响事件（0.1%概率损失1000万美元）可能主导EV，不要直接四舍五入为0。
  • ❌ 静态分析：默认你无法更新认知或调整方向，真实决策允许学习和转型。

Quick Reference

快速参考

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
核心公式：
Expected Value：EV = Σ (pᵢ × vᵢ) 其中p=概率，v=价值
期望效用（针对风险厌恶）：EU = Σ (pᵢ × U(vᵢ)) 其中U=效用函数
  • 风险中性：U(x) = x (EV = EU)
  • 风险厌恶：U(x) = √x 或 U(x) = log(x)（凹函数）
  • 风险偏好：U(x) = x²（凸函数）
净现值：NPV = Σ (CF_t / (1+r)^t) 其中CF=现金流，r=折现率，t=时间周期
方差（风险衡量指标）：Var = Σ (pᵢ × (vᵢ - EV)²)
标准差：σ = √Var
变异系数（风险收益比）：CV = σ / EV（数值越低，风险调整后收益越好）
盈亏平衡概率：EV=0时的概率p*，计算等式：p* × 成功收益 + (1-p*) × 失败损失 = 0
决策规则：
  • EV最大化：选择EV最高的选项（风险中性、重复决策场景）
  • 期望效用最大化：选择期望效用最高的选项（风险厌恶、纳入偏好场景）
  • 最小最大遗憾：最小化所有场景下的最大遗憾（保守策略，避免最严重错误）
  • 满意即可：选择第一个达到EV阈值的选项（有限理性场景）
敏感性分析问题：
  • 概率变化多少会反转当前决策？
  • 最好情况和最坏情况的EV分别是多少？哪些变量影响最大？
  • 概率为多少时EV达到盈亏平衡点（EV=0）？
核心资源：
  • resources/template.md：决策框架、结果识别、EV计算模板、敏感性分析
  • resources/methodology.md：概率估算、收益量化、决策树分析、效用函数
  • resources/evaluators/rubric_expected_value.json：质量标准（场景完整性、概率校准、收益量化、EV解读）
所需输入：
  • 决策：你需要在什么选项中做选择？（至少2个互斥备选方案）
  • 结果：每个备选方案可能产生什么结果？（通常3-5个场景）
  • 概率：每个结果的发生概率是多少？（总和为1.0）
  • 收益：每个结果的价值（收益/损失）是多少？（美元、时间、效用）
产出结果：
  • expected-value-analysis.md
    ：决策框架、带概率和收益的结果场景、EV计算、敏感性分析、结合风险考量的建议