Maxmin Expected Utility (MEU) Preferences

Updated 27 August 2025

MEU preferences are defined by evaluating acts based on the lowest expected utility over a set of priors, encapsulating ambiguity aversion.
They are applied in robust optimization, mechanism design, insurance, and asset pricing, offering rigorous models for decision-making under uncertainty.
Recent computational advances, including Bayesian estimation and MILP reformulations, enhance the tractability of MEU models in dynamic and strategic scenarios.

Maxmin Expected Utility (MEU) preferences formalize decision-making under ambiguity by positing that acts are evaluated according to the lowest expected utility over a set of priors or utility functions. This approach encapsulates the notion of ambiguity aversion and stands as a canonical alternative to subjective expected utility (SEU), especially when agents cannot credibly assign a unique probability law or possess incomplete preference information. The MEU principle, also referred to as worst-case expected utility, appears in diverse applications such as persuasion games, robust optimization, insurance contract design, asset pricing, and mechanism design, frequently interacting with computational and information-theoretic constraints.

1. Axiomatic Foundations of MEU Preferences

The MEU framework is commonly established via two related axiomatic bases:

Gilboa-Schmeidler's model represents the MEU preference as:

$U(f) = \min_{P \in \mathcal{P}} \mathbb{E}_P[u(f)]$

where $\mathcal{P}$ is a convex, weak*-compact set of probability measures and $u$ is a von Neumann–Morgenstern utility function.

Multi-utility representations generalize MEU preferences to settings where a family $U$ of utility functions represents the agent’s tastes or risk attitudes. A binary relation over lotteries admits a MEU representation if and only if it is a preorder, satisfies independence, and is sequentially continuous with respect to the weak topology; that is,

$p \succsim q \Longleftrightarrow \mathbb{E}_p[u] \geq \mathbb{E}_q[u] \;\; \forall u \in U$

where $p, q$ are simple lotteries and the set $U$ is essentially unique (Leonetti, 2022).

These axiomatic conditions clarify the precise requirements for MEU preference structures, explicitly distinguishing risk aversion (through the utility function $u$ ) and ambiguity aversion (through minimization over a set of priors or utility functions). When the outcome space is countable, MEU representations exist; the extension to uncountable sets fails without further assumptions.

2. MEU Preferences and Information-Theoretic Duality

A significant conceptual advance is the equivalence between MEU principles and information-theoretic formulations. An axiomatization based on three criteria—real-valuedness, additivity (up to a shift), and monotonicity with respect to probabilities—yields a unique law connecting utilities to probabilities:

$U(A|B) = \alpha \log P(A|B) + \beta$

Here, $\alpha$ reflects resource costs, and $\beta$ is arbitrary. This connection, known as utility–information equivalence, shows that utilities and Shannon information content are linearly related, and optimal decision-making can be cast as a variational problem, maximizing a free utility functional:

$J(P; U) = \sum_x P(x) U(x) - \alpha \sum_x P(x) \log P(x)$

In the limit $\alpha \to 0$ (no resource cost), the maximization reduces to the MEU principle (Ortega et al., 2010).

This dual formulation links MEU to entropy minimization and the Kullback–Leibler divergence as a resource penalty for deviation from a reference distribution. The Gibbs measure (probability proportional to the exponential of utility) emerges as the equilibrium solution.

3. MEU in Robust Optimization and Preference Elicitation

MEU preferences underpin robust optimization strategies wherein decision makers face uncertainty in both outcome probabilities and utility evaluations:

Decision problems with preference ambiguity: Rather than knowing the utility function precisely, the true utility is assumed to reside in an ambiguity set constructed via pairwise comparisons, zeta-ball neighborhoods, or linear moment inequalities (e.g., defined via Lebesgue-Stieltjes integrals) (Wu et al., 2023). The robust (maximin) objective takes the form:

$\max_{z \in Z} \min_{u \in U} \mathbb{E}[u(f(z,\xi))]$

Piecewise linear approximations, support function representations, and mixed-integer programming reformulations provide computational tractability (Haskell et al., 2018, Wu et al., 2023, Liu et al., 2021).

Multistage decision-making: Extending MEU to dynamic, multistage contexts involves time-consistent robust models when the ambiguity set of utility functions is state-dependent and rectangular, supporting recursive BeLLMan-like formulations. When ambiguity is state-independent, dynamic inconsistency generally arises (Liu et al., 2021).
Experimental elicitation: Mixing bets and subjective gambles offer methodologies to empirically assess ambiguity perception and aversion. The sharp predictions of MEU involve full hedging (maximal mixing intensity) within a belief interval, but observed behavior often falls short, indicating moderate ambiguity aversion, as differentiated from pure MEU (Schmidt, 2019, Cassese, 2023).

4. MEU in Strategic Communication and Mechanism Design

Ambiguity aversion modeled via MEU has substantial implications for strategic environments:

Persuasion games: If both sender and receiver adopt MEU preferences, there is no strict advantage for the sender in using ambiguous information structures over standard ones, both without and with prior ambiguity—at least in binary state/binary action environments (Cheng, 2020, Cheng, 26 Aug 2025). This result is established via minimax arguments (Sion’s theorem) ensuring that set-valued information structures don’t expand the sender’s feasible payoff beyond single experiments.
Distributional robustness: Classical MEU guarantees may be fragile—small perturbations to the prior can result in a sharp loss of payoff (lack of robustness). A refinement requires that the guarantee holds approximately under weak (topological) limits of the ambiguity set. This property is characterized behaviorally by continuity axioms: graphical limits of acts must not lead to a strictly worse evaluation. For continuous moment sets and Wasserstein ambiguity sets, robustness is preserved (Ball et al., 29 Aug 2024).

MEU preferences model ambiguity in real-world risk-sharing contexts:

Insurance design: The ambiguity set $\mathcal{C}$ (priors) modifies optimal insurance contracts. Under MEU, the decision maker’s (DM’s) demand for insurance is shaped by worst-case beliefs, which may assign positive probability to events excluded by the insurer’s beliefs, resulting in full insurance on such events. No-sabotage conditions (monotonicity of indemnities) yield essentially layer contracts. Both Wasserstein and Renyi ambiguity sets numerically demonstrate that increased model uncertainty reduces the DM’s certainty equivalent while raising the willingness to pay for insurance (Birghila et al., 2020).
Dynamic asset pricing: The $\alpha$ -maxmin expected utility ( $\alpha$ -MEU) model blends worst-case and best-case evaluation, addressing dynamic inconsistency via intra-personal equilibrium strategies. Equilibrium asset prices, risk-free rates, and risk premia are shown to react to ambiguity and ambiguity aversion parameters, with greater ambiguity resulting in a lower risk-free rate and higher equity premium (Fan et al., 5 Jul 2025).

6. Computational Approaches for MEU Optimization

Recent methodological advances enable efficient solutions to MEU-driven decision problems:

Density-free generative Bayesian computation: MEU optimization is performed via quantile neural estimation, leveraging the identity

$\mathbb{E}[U] = \int_0^1 F^{-1}_U(\tau)\, d\tau$

where $F^{-1}_U$ is the utility quantile function output by a deep neural network trained on simulated data. This approach is likelihood-free and handles high-dimensional, non-linear models, connecting Bayesian learning to dual expected utility theory (Polson et al., 28 Aug 2024).

Piecewise linear and MILP approximations: Uncertain utilities are efficiently represented by grid-based piecewise linear functions, which, together with MILP reformulations, make robust multi-attribute optimization tractable even as the number of attributes increases (Wu et al., 2023).

7. Mathematical Properties and Contrasts with Generalized Expected Utility

MEU preferences contrast sharply with algebraic or generalized expected utility (AEU) criteria:

Property	MEU	AEU
Aggregation	Non-linear min	Linear sum
Dynamic consistency	May fail	Guaranteed
Autoduality	One-sided	Bidirectional
Source of ambiguity	Priors	Semiring

AEU models aggregate utility linearly via semiring operations, guaranteeing dynamic consistency and autoduality. MEU, by contrast, captures ambiguity aversion through minimization, which is inherently non-linear and potentially leads to dynamic inconsistency unless further rectangularity conditions are imposed (Weng, 2012).

Maxmin Expected Utility (MEU) preferences are the foundational tool for modeling robust decision-making under ambiguity. They rigorously separate risk from ambiguity aversion, admit robust multi-utility and multi-prior extensions, and tie intimately to information-theoretic principles and computational methods. Contemporary research explores their ramifications in dynamic settings, strategic communication, and economic design, refining robustness and computational tractability in increasingly complex and uncertain environments.