Aims of the EU (Expected Utility) and How Far Have They Been Achieved | 期望效用（EU）的目标及其实现程度

📚 Aims of the EU (Expected Utility) and How Far Have They Been Achieved | 期望效用（EU）的目标及其实现程度

In decision mathematics and probability, the abbreviation EU often stands for Expected Utility, a central concept in rational choice theory. This article explores the original aims of expected utility theory, how it models decision-making under risk, and the extent to which those aims have been achieved in both theoretical and applied contexts.

在决策数学与概率论中，缩写 EU 通常指期望效用（Expected Utility），它是理性选择理论的核心概念。本文探讨期望效用理论的初始目标、它如何在风险条件下建模决策，以及这些目标在理论与应用中的实现程度。

1. What Is Expected Utility? | 什么是期望效用？

Expected utility is a weighted average of the utilities of all possible outcomes, where the weights are the probabilities of those outcomes. For a discrete set of outcomes x₁, x₂, …, xₙ with probabilities p₁, p₂, …, pₙ, the expected utility is given by:

期望效用是所有可能结果效用的加权平均值，权重为这些结果发生的概率。对于具有概率 p₁, p₂, …, pₙ 的离散结果集 x₁, x₂, …, xₙ，期望效用由下式给出：

EU = Σ p_i u(x_i)

Here u(xᵢ) is the utility assigned to outcome xᵢ, and the sum runs over all possible outcomes. The utility function u captures an individual’s subjective satisfaction or preference intensity, while the probabilities reflect the likelihood of each outcome under risk.

这里 u(xᵢ) 是赋予结果 xᵢ 的效用，求和遍历所有可能结果。效用函数 u 捕捉了个体主观满意度或偏好强度，而概率反映了风险条件下每个结果发生的可能性。

2. Historical Origins of EU Theory | 期望效用理论的历史起源

The concept of expected utility was first introduced by Daniel Bernoulli in 1738 to resolve the St Petersburg paradox. In that paradox, a fair coin is tossed until heads appears, and the payoff doubles each time: $2, $4, $8, and so on. The expected monetary value is infinite, yet most people would pay only a small amount to play the game.

期望效用的概念最早由丹尼尔·伯努利于 1738 年提出，用以解决圣彼得堡悖论。在该悖论中，一枚公平硬币被抛掷直到出现正面，每次抛掷的收益翻倍：2 美元、4 美元、8 美元，依此类推。期望货币价值为无穷大，但大多数人只愿意支付很少的钱来参与游戏。

Bernoulli argued that people do not value money linearly; instead they assign a diminishing marginal utility to additional wealth. By replacing monetary value with a concave utility function, such as the logarithmic function u(x) = ln(x), the expected utility becomes finite, explaining the low willingness to pay. This marked the birth of expected utility as a descriptive and normative tool.

伯努利认为，人们对金钱的估值并非线性；相反，他们对额外财富赋予递减的边际效用。通过用凹效用函数（例如对数函数 u(x) = ln(x)）替代货币价值，期望效用变为有限值，从而解释了人们较低支付意愿。这标志着期望效用作为描述性和规范性工具的诞生。

3. Core Aims of Expected Utility Theory | 期望效用理论的核心目标

Expected utility theory has three main aims. First, it seeks to provide a normative model of rational choice under risk: a decision-maker should choose the option that maximises expected utility. Second, it aims to quantify risk preferences in a unified way, distinguishing between risk-averse, risk-neutral, and risk-seeking behaviour. Third, it attempts to explain and predict actual choices by assuming individuals act as if they maximise expected utility.

期望效用理论有三个主要目标。第一，它旨在提供风险条件下理性选择的规范模型：决策者应选择使期望效用最大化的选项。第二，它旨在以统一的方式量化风险偏好，区分风险厌恶、风险中性和风险追求行为。第三，它试图通过假设个体的行为如同在最大化期望效用来解释和预测实际选择。

In A-level Mathematics, these aims connect to the statistical concept of expectation and to decision-making contexts where probabilities and payoffs are known. The theory provides a mathematical framework for comparing lotteries and gambles.

在 A-level 数学中，这些目标与统计中的期望概念以及概率和收益已知的决策情境相联系。该理论为比较彩票和赌博提供了数学框架。

4. Axiomatic Foundations | 公理基础

In 1944, John von Neumann and Oskar Morgenstern formalised expected utility theory by establishing a set of axioms. If a decision-maker’s preferences satisfy these axioms, then there exists a utility function such that choices can be represented by maximising expected utility. The key axioms include:

1944 年，约翰·冯·诺伊曼和奥斯卡·摩根斯特恩通过建立一组公理将期望效用理论形式化。如果决策者的偏好满足这些公理，则存在一个效用函数，使得选择可以用最大化期望效用来表示。关键公理包括：

Completeness: For any two lotteries A and B, either A is preferred to B, B is preferred to A, or the decision-maker is indifferent.

完备性：对于任意两个彩票 A 和 B，要么 A 优于 B，要么 B 优于 A，要么决策者无差异。
Transitivity: If A is preferred to B and B is preferred to C, then A is preferred to C.

传递性：若 A 优于 B 且 B 优于 C，则 A 优于 C。
Continuity: For any lotteries A, B, C with A preferred to B preferred to C, there exists a probability p such that the decision-maker is indifferent between B and a lottery giving A with probability p and C with probability 1 − p.

连续性：对于满足 A 优于 B 优于 C 的任意彩票 A、B、C，存在概率 p 使得决策者在 B 与一个以概率 p 得 A、以概率 1 − p 得 C 的彩票之间无差异。
Independence: If A is preferred to B, then for any lottery C and any probability p, the lottery pA + (1 − p)C is preferred to pB + (1 − p)C.

独立性：若 A 优于 B，则对于任意彩票 C 和任意概率 p，彩票 pA + (1 − p)C 优于 pB + (1 − p)C。

These axioms are the mathematical foundation of the theory, and their violation in experiments is central to assessing how far the aims have been achieved.

这些公理是该理论的数学基础，而实验中这些公理被违背的情况是评估其目标实现程度的核心。

5. Mathematical Formulation and Properties | 数学表述与性质

The expected utility representation has several important mathematical properties. One is linearity in probabilities: the utility of a compound lottery is the probability-weighted average of the utilities of its components. Another is that any positive affine transformation of the utility function, such as v(x) = a u(x) + b with a > 0, represents the same preferences as u(x).

期望效用表示具有几个重要的数学性质。其一是概率上的线性：复合彩票的效用是其各组成部分效用的概率加权平均。另一个是效用函数的任意正仿射变换，例如 v(x) = a u(x) + b（其中 a > 0），所表示的偏好与 u(x) 相同。

Moreover, risk attitudes are encoded in the curvature of the utility function. A concave utility function (u″(x) < 0) corresponds to risk aversion, a linear function (u″(x) = 0) to risk neutrality, and a convex function (u″(x) > 0) to risk seeking. This property allows economists to classify decision-makers using only the shape of u.

此外，风险态度被编码在效用函数的曲率中。凹效用函数（u″(x) < 0）对应风险厌恶，线性函数（u″(x) = 0）对应风险中性，凸函数（u″(x) > 0）对应风险追求。这一性质使经济学家仅通过 u 的形状即可对决策者进行分类。

6. Quantifying Risk Attitudes | 量化风险态度

To measure the degree of risk aversion, economists use the Arrow-Pratt coefficients. The coefficient of absolute risk aversion is defined as:

为了度量风险厌恶程度，经济学家使用阿罗-普拉特系数。绝对风险厌恶系数定义为：

A(x) = −u″(x) / u′(x)

and the coefficient of relative risk aversion is:

相对风险厌恶系数为：

R(x) = −x u″(x) / u′(x)

These coefficients are invariant under positive affine transformations and provide a local measure of how much an individual dislikes risk at a given wealth level. In A-level contexts, students may encounter simpler measures such as comparing expected monetary value and certainty equivalent.

这些系数在正仿射变换下保持不变，并提供了在给定财富水平下个体厌恶风险程度的局部度量。在 A-level 情境中，学生可能会遇到更简单的度量，如比较期望货币值与确定性等价值。

7. Applications in Economics and Finance | 在经济学与金融学中的应用

Expected utility theory has been widely applied in insurance, investment, and game theory. In insurance markets, a risk-averse individual with a concave utility function is willing to pay a premium to avoid risk, even if the expected monetary loss is lower than the premium. This explains the existence and pricing of insurance contracts.

期望效用理论已广泛应用于保险、投资和博弈论。在保险市场中，具有凹效用函数的风险厌恶个体愿意支付保费以避免风险，即使期望货币损失低于保费。这解释了保险合同的存在和定价。

In portfolio choice, investors maximise expected utility subject to constraints, leading to the mean-variance analysis of Markowitz. In auction theory, bidders’ strategies are derived under the assumption of expected utility maximisation. These applications demonstrate the theory’s power as a modelling tool.

在投资组合选择中，投资者在约束条件下最大化期望效用，从而产生了马科维茨的均值-方差分析。在拍卖理论中，投标人的策略是在期望效用最大化的假设下推导出来的。这些应用证明了该理论作为建模工具的强大能力。

8. Achievements of EU Theory | 期望效用理论的成就

The expected utility framework has achieved several important goals. First, it provides a rigorous normative benchmark for rational decision-making under risk. Second, it offers a unified language for comparing risk preferences across individuals and contexts. Third, it has become the standard model in economics, finance, and political science, underpinning countless theoretical and empirical studies.

期望效用框架已经实现了若干重要目标。第一，它为风险条件下的理性决策提供了严格的规范基准。第二，它为比较不同个体和情境下的风险偏好提供了统一语言。第三，它已成为经济学、金融学和政治学的标准模型，支撑了无数理论和实证研究。

In terms of explanatory power, EU theory successfully predicts many real-world behaviours, such as the purchase of insurance, the demand for diversified portfolios, and the preference for certain outcomes over gambles with the same expected value. These successes indicate that the core aims have been substantially achieved in normative and applied domains.

就解释力而言，期望效用理论成功预测了许多现实行为，如购买保险、对多元化投资组合的需求，以及对具有相同期望值的确定性结果的偏好优于赌博。这些成功表明其核心目标在规范和应用领域已大体实现。

9. Empirical Challenges: Allais Paradox | 经验挑战：阿莱悖论

Despite its successes, expected utility theory faces serious descriptive challenges. The most famous is the Allais paradox, presented by Maurice Allais in 1953. Consider two choice problems. In Problem 1, choose between A: a certain £1 million, and B: a 10% chance of £5 million, 89% chance of £1 million, and 1% chance of nothing. Most people choose A. In Problem 2, choose between C: an 11% chance of £1 million and 89% chance of nothing, and D: a 10% chance of £5 million and 90% chance of nothing. Most people choose D.

尽管取得了成功，期望效用理论面临严重的描述性挑战。最著名的是莫里斯·阿莱于 1953 年提出的阿莱悖论。考虑两个选择问题。问题 1 中，在 A：确定获得 100 万英镑，与 B：10% 机会获得 500 万英镑、89% 机会获得 100 万英镑、1% 机会一无所有之间选择。大多数人选择 A。问题 2 中，在 C：11% 机会获得 100 万英镑和 89% 机会一无所有，与 D：10% 机会获得 500 万英镑和 90% 机会一无所有之间选择。大多数人选择 D。

This pattern violates the independence axiom. Under expected utility, choosing A over B implies u(1m) > 0.1 u(5m) + 0.89 u(1m) + 0.01 u(0), which simplifies to 0.11 u(1m) > 0.1 u(5m) + 0.01 u(0). But this is exactly the condition for preferring C over D, so a consistent EU maximiser cannot choose A in Problem 1 and D in Problem 2. The Allais paradox thus shows that real choices systematically deviate from EU predictions.

这一模式违背了独立性公理。在期望效用下，选择 A 优于 B 意味着 u(1m) > 0.1 u(5m) + 0.89 u(1m) + 0.01 u(0)，简化后为 0.11 u(1m) > 0.1 u(5m) + 0.01 u(0)。但这恰好是偏好 C 优于 D 的条件，因此一致的期望效用最大化者不可能在问题 1 中选择 A 而在问题 2 中选择 D。阿莱悖论因此表明实际选择系统性地偏离了期望效用预测。

10. Further Challenges: Ellsberg Paradox and Framing Effects | 进一步挑战：埃尔斯伯格悖论与框架效应

Another violation comes from the Ellsberg paradox, which concerns ambiguity. Suppose an urn contains 90 balls: 30 red and 60 either black or yellow in unknown proportions. People typically prefer a bet that pays if a red ball is drawn over a bet that pays if a black ball is drawn, and also prefer a bet on black or yellow over a bet on red or yellow. This pattern cannot be represented by any single probability distribution over the three colours, because it reflects ambiguity aversion.

另一个违背来自埃尔斯伯格悖论，它涉及模糊性。假设一个瓮中有 90 个球：30 个红球和 60 个黑球或黄球，比例未知。人们通常偏好押红球出现的赌注，而不是押黑球出现的赌注；同时偏好押黑球或黄球的赌注，而不是押红球或黄球的赌注。这种模式无法用三种颜色上的任何单一概率分布表示，因为它反映了模糊厌恶。

Framing effects, identified by Kahneman and Tversky, show that choices depend on whether outcomes are described as gains or losses relative to a reference point. For example, when a disease outbreak is framed in terms of lives saved rather than lives lost, decision-makers make systematically different choices, even though the underlying outcomes are identical. These findings undermine the descriptive aim of EU theory.

卡尼曼和特沃斯基发现的框架效应表明，选择取决于结果被描述为相对于参照点的收益还是损失。例如，当疾病爆发以挽救的生命数而不是死亡的生命数来表述时，决策者会做出系统性不同的选择，尽管潜在结果是相同的。这些发现削弱了期望效用理论的描述性目标。

11. Extensions and Alternatives | 扩展与替代理论

In response to these empirical challenges, several alternatives have been proposed. Prospect theory, developed by Kahneman and Tversky in 1979, replaces the utility function with a value function defined over gains and losses, and replaces probabilities with decision weights that overweight small probabilities and underweight large ones. This model can explain the Allais paradox, the Ellsberg paradox, and framing effects.

为应对这些经验挑战，人们提出了若干替代理论。卡尼曼和特沃斯基在 1979 年提出的前景理论，用定义在收益和损失上的价值函数取代效用函数，并用高估小概率、低估大概率的决策权重取代概率。该模型能解释阿莱悖论、埃尔斯伯格悖论和框架效应。

Other extensions include rank-dependent utility, which applies probability weighting to cumulative probabilities, and regret theory, which incorporates the anticipation of regret into choices. These models retain the mathematical structure of weighted sums but relax the linear probability assumption, bringing theory closer to observed behaviour.

其他扩展包括等级依赖效用，它将概率加权应用于累积概率；以及遗憾理论，它将预期遗憾纳入选择。这些模型保留了加权和的数学结构，但放宽了线性概率假设，使理论更接近观察到的行为。

12. How Far Have the Aims Been Achieved? | 目标实现的程度如何？

Overall, the aims of expected utility theory have been partially achieved. As a normative model, it remains highly successful: it provides a clear, consistent standard for rational choice, and its axioms are compelling from a prescriptive standpoint. In applied fields such as insurance, finance, and auction design, EU theory continues to be the workhorse model.

总体而言，期望效用理论的目标已部分实现。作为规范模型，它仍然非常成功：它为理性选择提供了清晰、一致的标准，其公理从规定性角度看具有说服力。在保险、金融和拍卖设计等应用领域，期望效用理论仍是主力模型。

As a descriptive model, however, its achievement is limited. Systematic violations such as the Allais and Ellsberg paradoxes and framing effects show that people do not always behave as EU maximisers. Nevertheless, these failures have stimulated the development of richer models, and the core idea of weighting outcomes by their likelihood remains central to decision science. Thus, while the original aim of accurately describing all choices has not been fully met, the theory has achieved its aim of providing a foundational framework for rational decision-making under risk.

然而，作为描述性模型，其实现程度有限。阿莱悖论、埃尔斯伯格悖论和框架效应等系统性违背表明，人们并非总是像期望效用最大化者那样行动。尽管如此，这些失败激发了更丰富模型的发展，而按概率加权结果的核心思想仍然是决策科学的核心。因此，虽然准确描述所有选择的原始目标尚未完全实现，但该理论已经实现了为风险条件下的理性决策提供基础框架的目标。

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Aims of the EU (Expected Utility) and How Far Have They Been Achieved | 期望效用（EU）的目标及其实现程度