Mean-preserving spread

A mean-preserving spread is a transformation of a probability distribution that increases its riskiness—typically measured by greater variance or dispersion—while preserving the expected value, or mean, of the distribution. This concept allows for the comparison of lotteries or random variables with identical means but differing degrees of uncertainty, where the spread-out distribution is deemed riskier.^[1]^[2] The notion of mean-preserving spread was formalized by economists Michael Rothschild and Joseph E. Stiglitz in their seminal 1970 paper, providing a rigorous framework for analyzing increases in risk under uncertainty. They defined one distribution G as a mean-preserving spread of another distribution F if G can be obtained from F by adding noise with conditional mean zero, ensuring the overall mean remains unchanged while the second moments (related to variance) increase. Equivalently, F second-order stochastically dominates G, meaning the integral of the cumulative distribution function of G from the lower bound to any point exceeds that of F, with equality at the upper bound, implying that all risk-averse decision-makers prefer outcomes from F over G.^[3]^[1]

Conceptual Foundations

Informal Definition

A mean-preserving spread (MPS) is a transformation applied to a probability distribution of a random variable, which disperses the possible outcomes more widely around the central tendency while ensuring that the expected value—the long-run average outcome—remains unchanged.90038-4) This concept, introduced by Rothschild and Stiglitz, provides a way to compare the riskiness of different distributions that share the same mean, emphasizing how variability can intensify without shifting the overall average payoff.90038-4) At its core, an MPS heightens uncertainty by reallocating probability mass toward more extreme values, either higher or lower, at the expense of probabilities near the mean, thereby increasing the perceived risk without any compensatory gain in expectation.90038-4) Probability distributions outline the relative frequencies of outcomes for a random variable, such as returns on an investment or income levels, while the expected value quantifies the distribution's central location under repeated realizations.90038-4) This preservation of the mean distinguishes MPS from other changes, like those that might simultaneously alter both mean and variance. Intuitively, an MPS can be visualized as shaking a bag of marbles—each representing an outcome—such that they end up more scattered in position but with the same total weight, amplifying the unpredictability of drawing any single marble.^[4] This notion underpins broader analyses of risk in decision-making, connecting informally to ideas like second-order stochastic dominance, where one distribution is preferred by all risk-averse agents over another with equal mean but greater spread.90038-4)

Intuition and Motivation

In economics, the concept of a mean-preserving spread (MPS) provides a framework for analyzing how the addition of noise or variability to economic outcomes influences decision-making while keeping the expected value unchanged, thereby isolating the effects of risk from changes in anticipated returns.^[5] This approach is particularly useful in modeling scenarios where agents face uncertain prospects, such as investment returns or income fluctuations, allowing researchers to examine pure increases in uncertainty without confounding shifts in the mean. The intuition underlying MPS lies in its connection to risk preferences: risk-averse individuals or firms typically prefer distributions with less spread (lower risk) when the mean outcome remains the same, as greater variability amplifies the potential for unfavorable realizations despite the unchanged expectation.^[6] This preference reflects a fundamental aversion to uncertainty, where the downside of heightened dispersion outweighs any symmetric upside, leading agents to value stability in outcomes like wealth or profits.^[7] In expected utility theory, such spreads are systematically disliked by concave utility functions, underscoring the role of MPS in characterizing risk-averse behavior.^[8] Historically, the formalization of MPS emerged from efforts to rigorously define "more risky" prospects in economic analysis, building on earlier work in portfolio theory by Harry Markowitz, who emphasized variance as a measure of risk in mean-variance optimization. Markowitz's 1952 framework laid the groundwork by treating increased variance at fixed means as heightened risk, but it was Michael Rothschild and Joseph Stiglitz who introduced MPS in 1970 to provide a distribution-based ordering that captures intuitive notions of risk augmentation beyond simple variance.^[3] Their definition enabled precise comparisons of lotteries or investments solely on riskiness, distinguishing pure risk escalation from alterations in expected payoffs. This distinction is crucial in economic theory and practice, as it forms the basis for evaluating choices under uncertainty, such as insurance decisions or asset allocation, where policymakers and investors seek to mitigate risk without altering baseline expectations.^[9] By focusing on spreads that preserve the mean, MPS has become foundational for assessing the welfare implications of risk in diverse contexts, from financial markets to public policy.^[6]

Mathematical Formulations

Definition via Stochastic Dominance

A mean-preserving spread formalizes the notion of increased riskiness while preserving the expected value. Specifically, a random variable with cumulative distribution function (CDF) F is a mean-preserving spread of another random variable with CDF G if both have the same mean and G second-order stochastically dominates F.^[3] This dominance implies that G is preferred by all risk-averse decision makers to F, reflecting the added dispersion in F without altering the mean.^[3] Second-order stochastic dominance (SSD) provides the mathematical foundation for this concept. A distribution G second-order stochastically dominates F if, for all x,

\int_{-\infty}^{x} [F(t) - G(t)] \, dt \geq 0,

with equality as x \to \infty when the means are equal.^[3] This condition ensures that the accumulated probability mass up to any point x under F is no smaller than under G in an integrated sense, capturing a mean-equivalent increase in spread for F. The non-negativity of the integral reflects less "downside risk" in G compared to F, as the area between the CDFs—where F places more weight on extreme outcomes—is accounted for cumulatively.^[3] This framework builds on first-order stochastic dominance, which serves as a prerequisite for understanding SSD. First-order stochastic dominance occurs when G first-order dominates F if G(x) \leq F(x) for all x, meaning G assigns no higher probability to outcomes below any threshold than F does.^[3] SSD extends this by integrating the CDF differences, allowing for crossings in the CDFs themselves but penalizing greater variability in the tails, which aligns directly with the mean-preserving property.^[3]

Integral and Moment Conditions

A distribution F represents a mean-preserving spread (MPS) of another distribution G if the following integral condition holds:

\int_{-\infty}^{x} F(t) \, dt \geq \int_{-\infty}^{x} G(t) \, dt \quad \text{for all } x \in \mathbb{R},

with equality in the limit as x \to \infty. This formulation captures the notion that F exhibits greater dispersion than G while preserving the mean, as the limiting equality ensures \mathbb{E}[X_F] = \mathbb{E}[X_G].^[1]^[2] An equivalent characterization arises through moments: the first moments (means) of F and G are identical, and the second central moment (variance) of F is at least as large as that of G, reflecting increased riskiness. Specifically, \mathrm{Var}(X_F) \geq \mathrm{Var}(X_G).^[1]^[2] To derive these from the second-order stochastic dominance (SSD) condition—where G SSD F implies the integral inequality—the limiting equality follows directly from the expression for the mean, \mathbb{E}[X] = \int_{0}^{\infty} (1 - F(x)) \, dx - \int_{-\infty}^{0} F(x) \, dx, ensuring equal expectations. The variance inequality then emerges via integration by parts applied to the SSD integral, yielding \mathrm{Var}(X_F) - \mathrm{Var}(X_G) = 2 \int_{-\infty}^{\infty} (x - \mu) [F(x) - G(x)] \, dx \geq 0, where \mu is the common mean.^[1]^[2] For distributions with finite means, the integral and moment conditions are fully equivalent to the SSD-based definition of MPS.^[1]^[2]

Examples and Illustrations

Discrete Case

In the discrete case, mean-preserving spreads are illustrated using finite probability mass functions (PMFs) over countable outcomes, allowing for straightforward computation of means and variances to demonstrate increased risk without altering the expected value. A simple example begins with a degenerate distribution where the outcome is $100 with probability 1, which has mean $100 and variance 0. A mean-preserving spread of this distribution is one with 50% chance of $50 and 50% chance of $150, preserving the mean at $100 = 0.5 \times 50 + 0.5 \times 150 while increasing the variance to 2500 = 0.5 \times (50 - 100)^2 + 0.5 \times (150 - 100)^2.^[10] Another illustrative binary example shifts from an initial distribution with 50% chance of $0 and 50% chance of $200 (mean $100 = 0.5 \times 0 + 0.5 \times 200, variance 10000 = 0.5 \times (0 - 100)^2 + 0.5 \times (200 - 100)^2) to a mean-preserving spread with 50% chance of -$50 and 50% chance of $250 (mean $100 = 0.5 \times (-50) + 0.5 \times 250, variance 22500 = 0.5 \times (-50 - 100)^2 + 0.5 \times (250 - 100)^2). This transformation spreads the outcomes further while maintaining the same mean, resulting in higher variance that quantifies the added risk.^[10] To visualize these discrete distributions, bar charts of the PMFs highlight the spread. For the first example:

Outcome	Original PMF	Spread PMF
$50	0	0.5
$100	1	0
$150	0	0.5

The original shows a single bar at $100 with height 1, while the spread replaces it with equal bars at $50 and $150, each of height 0.5, illustrating the dispersion. Similarly, for the second example, the initial bars at $0 and $200 (each height 0.5) shift outward to bars at -$50 and $250 (each height 0.5), emphasizing the mean-preserving increase in spread.^[10]

Continuous Case

In the continuous case, mean-preserving spreads are illustrated using probability density functions over infinite or unbounded supports, contrasting with discrete scenarios that rely on finite point masses. A classic example involves uniform distributions. Consider an initial distribution F that is uniform on the interval [90, 110], with density f(x) = \frac{1}{20} for $90 \leq x \leq 110 and zero elsewhere; its mean is \int_{90}^{110} x \cdot \frac{1}{20} \, dx = 100. A mean-preserving spread G is uniform on [50, 150], with density g(x) = \frac{1}{100} for $50 \leq x \leq 150 and zero elsewhere, yielding the same mean \int_{50}^{150} x \cdot \frac{1}{100} \, dx = 100 but greater dispersion due to the expanded support. Another representative example uses normal distributions, which are symmetric and unbounded. The distribution N(\mu, \sigma^2) has mean \mu and variance \sigma^2. A mean-preserving spread is N(\mu, k \sigma^2) for k > 1, preserving the mean \mu while increasing the variance to k \sigma^2, which disperses the density further into the tails. To verify the mean-preserving spread property, the second-order stochastic dominance (SSD) condition must hold: the less risky distribution F (narrower uniform or smaller-variance normal) satisfies \int_{-\infty}^z F(t) \, dt \leq \int_{-\infty}^z G(t) \, dt for all z, with equality as z \to \infty (due to equal means) and strict inequality over some interval. For the uniform example, the cumulative distribution functions are F(x) = 0 for x < 90, F(x) = \frac{x-90}{20} for $90 \leq x \leq 110, and F(x) = 1 for x > 110; similarly, G(x) = 0 for x < 50, G(x) = \frac{x-50}{100} for $50 \leq x \leq 150, and G(x) = 1 for x > 150. The CDFs cross once at x = 100, with G(x) > F(x) for x < 100 and G(x) < F(x) for x > 100. Integrating these yields, for instance, at z = 100, \int_{-\infty}^{100} F(t) \, dt = \int_{90}^{100} \frac{t-90}{20} \, dt = 2.5 < \int_{50}^{100} \frac{t-50}{100} \, dt = 12.5 = \int_{-\infty}^{100} G(t) \, dt, confirming the inequality holds (computations follow from antiderivatives of piecewise linear CDFs). For normals, the condition follows analogously, as increasing the scale parameter spreads the CDF outward while preserving the mean. Visualization aids understanding: plotting the probability density functions shows the narrower uniform concentrating mass near 100, while the wider one flattens and extends tails; similarly, normal PDFs exhibit taller central peaks for smaller \sigma^2 versus broader, lower peaks for larger k \sigma^2. The CDF plots reveal the single crossing, with the more spread distribution lying above the less spread one on the left (indicating higher probability of low outcomes) and below on the right (higher probability of high outcomes).

Properties and Characterizations

Increase in Riskiness

A mean-preserving spread (MPS) represents an increase in risk by spreading the probability mass of a distribution while preserving its mean, which necessarily results in a higher variance compared to the original distribution.^[10] This elevation in variance quantifies the greater dispersion around the mean, making the spread distribution riskier without altering the expected outcome.^[10] The Rothschild-Stiglitz theorem characterizes this risk increase by showing that if one distribution is an MPS of another, then for any concave utility function—representing risk-averse preferences—the expected utility under the spread distribution is strictly less than under the original.^[10] This equivalence holds because the integral of the utility function over the spread distribution satisfies the condition for second-order stochastic dominance, where the original distribution dominates the spread one.^[10] An MPS implies no first-order stochastic dominance between the distributions, as their equal means prevent one cumulative distribution from being uniformly below the other, but it establishes strict second-order stochastic dominance of the original over the spread distribution.^[10] For a fixed mean, the minimum possible variance is zero, achieved by a degenerate distribution concentrated at the mean; any MPS moves the distribution away from this point, strictly increasing the variance and thus the risk.^[10]

Relation to Dispersion Measures

A mean-preserving spread (MPS) of a random variable strictly increases its variance when the original distribution is non-degenerate.^[3] This follows from the second-order stochastic dominance characterization, where the more spread distribution has higher second moments while preserving the mean.^[1] MPS is also compatible with increases in other dispersion measures, such as the standard deviation, which scales directly with the square root of variance.^[3] Similarly, it leads to higher values of the Gini coefficient, a common inequality metric, as spreading the distribution while fixing the mean amplifies relative differences between outcomes.^[11] Entropy-based measures of variability likewise rise under MPS, reflecting greater uncertainty in the distribution.^[12] However, MPS does not always increase the range or interquartile range; counterexamples exist where the spread concentrates outcomes within certain quantiles, reducing these interval-based metrics despite overall higher risk. A key characterization theorem states that one distribution is less risky than another (with the same mean) if and only if the latter cannot be obtained from the former via an MPS, providing a foundational ordering for comparing dispersion across probability distributions.^[3] Unlike more general notions of spread, such as those allowing for mean shifts (e.g., in location-scale families), MPS specifically maintains the expected value unchanged, isolating the effect of added noise or variability.

Applications in Economics and Decision Theory

Expected Utility and Risk Aversion

In the framework of von Neumann-Morgenstern expected utility theory, a mean-preserving spread (MPS) of a random variable X to X_{\text{MPS}} implies a reduction in expected utility for any risk-averse decision maker whose utility function u is increasing and concave. Specifically, \mathbb{E}[u(X_{\text{MPS}})] \leq \mathbb{E}[u(X)], with equality holding only if X is degenerate (i.e., constant with probability 1).^[3] This result establishes that an MPS represents an unambiguous increase in risk, as all risk-averse agents prefer the original distribution X over X_{\text{MPS}}.^[3] The proof follows from the equivalence between an MPS and second-order stochastic dominance (SSD), where X second-order stochastically dominates X_{\text{MPS}}. To see this, integrate by parts twice to express the difference in expected utility:

\mathbb{E}[u(X_{\text{MPS}})] - \mathbb{E}[u(X)] = \int_{-\infty}^{\infty} u''(t) \int_{-\infty}^{t} [F_{X_{\text{MPS}}}(s) - F_X(s)] \, ds \, dt,

where F denotes the cumulative distribution function. Since u''(t) \leq 0 for concave u and the inner integral \int_{-\infty}^{t} [F_{X_{\text{MPS}}}(s) - F_X(s)] \, ds \geq 0 for all t by the SSD condition, the overall expression is non-positive.^[3] The magnitude of this utility loss relates to the decision maker's degree of risk aversion, as measured by the Arrow-Pratt coefficients. For small MPS—approximating the addition of mean-zero noise with variance \sigma^2—the associated risk premium \pi (the amount the agent would pay to avoid the spread) satisfies \pi \approx \frac{1}{2} r_A(w) \sigma^2, where r_A(w) = -u''(w)/u'(w) is the absolute risk aversion at wealth w, or proportionally to relative risk aversion r_R(w) = w r_A(w) for proportional spreads.^[13] Thus, more risk-averse individuals perceive a larger increase in risk from the same MPS, reinforcing their strict preference for the less spread distribution.^[13]

Implications for Insurance and Portfolios

In the context of insurance, a mean-preserving spread (MPS) represents an increase in the uncertainty of loss outcomes, such as accidents or damages, without altering the expected loss amount. For risk-averse individuals, who prefer less risky prospects as defined by aversion to MPS, purchasing actuarially fair insurance effectively reverses this spread by transferring the risk to the insurer, thereby increasing expected utility.^[3] Full coverage at fair premiums eliminates the variability in final wealth introduced by the initial risk, making it a dominant choice for such agents.^[14] In portfolio management, diversification serves as a mean-preserving contraction, reducing the overall risk of returns without changing the expected return, aligning with the principles of modern portfolio theory. This process mitigates the variance associated with individual assets, akin to undoing an MPS in asset returns, and allows investors to achieve higher utility by lowering exposure to idiosyncratic risks.^[15] Within the Markowitz framework, optimal portfolios on the efficient frontier balance this risk reduction against return maximization, emphasizing that uncorrelated assets enable such contractions without mean loss.^[15] Adverse selection in insurance markets arises when outcomes for different agent types represent increases in risk under asymmetric information, leading to inefficient equilibria. In the Rothschild-Stiglitz model, high-risk individuals have loss distributions that first-order stochastically dominate those of low-risk ones (higher probability of the same loss amount), causing insurers to offer screening contracts that ration coverage and prevent pooling, often resulting in market failure where no equilibrium exists.^[14] This separation increases overall risk exposure compared to full-information settings, as low-risk agents receive suboptimal partial coverage.^[14] Public policies like progressive taxation or targeted subsidies can act as mechanisms to counteract MPS in income or wealth distributions across populations, preserving aggregate means while reducing inequality-induced risk. For instance, such interventions implement a mean-preserving contraction in the income distribution, enhancing welfare for risk-averse households by stabilizing outcomes without net resource transfer.^[16] In fiscal policy under uncertainty, a mean-preserving spread in tax instruments can amplify distortions, underscoring the role of subsidies in mitigating these effects to support efficient resource allocation.^[17]