Randomized Utility Model

• Randomized Utility Model is a framework that models choice by assigning stochastic latent utilities to alternatives across diverse agents and settings.
• It employs probabilistic mixtures over deterministic choice functions, enabling the rigorous testing of underlying linear inequalities and revealed preferences.
• Its applications span economics, social choice, and machine learning, with extensions addressing dynamic behavior, identification challenges, and intransitivity.

A randomized utility model (RUM), sometimes referred to as a random utility model or stochastic utility model, is a foundational framework in economics, social choice, and machine learning for modeling discrete choice under uncertainty and heterogeneity. In RUMs, the observed choice behavior is generated by maximizing a latent utility function that is stochastic either across individuals, choice situations, or both. This structure allows RUMs to encode a wide array of phenomena—including classical rationality, random or limited attention, and even certain forms of intransitivity—while permitting precise statistical identification and nonparametric testing.

1. Formal Structure and Canonical Representation

A randomized utility model is defined over a finite set $X=\{x_1, ..., x_K\}$ of alternatives. For any nonempty menu $A\subseteq X$, the observed choice rule is a probability distribution $P(\cdot\mid A)$ on $A$ given by

$$P(x\mid A) = \Pr_{u \sim \mu} \bigl[u(x) \geq u(y)\; \forall y \in A\bigr],$$

where $u$ is a (typically deterministic) utility function drawn at random from a probability measure $\mu$ over some family $\mathcal{U}$ of utility functions or preference orderings (Stoye, 2018, Soufiani et al., 2012). For every menu the agent selects the alternative attaining maximum utility; ties are assumed to occur with probability zero.

This formalism admits several equivalent perspectives. RUMs can be framed as:

• Probabilistic mixtures over deterministic choice types (maps picking a unique alternative from each menu).
• Probability measures over linear orders (i.e., full rankings), with menu restriction yielding top choice selection.
• Structural models with observable deterministic (systematic) utility components and additive random shocks (“random utility maximization”).

A key property is that the observed array $(P(x|A))_{x \in A, A \subseteq X}$ must be compatible with a probability distribution over deterministic rational choice functions; on a finite universe this amounts to a finite set of linear inequalities.

2. Testable Axioms and Geometric Characterization

The foundational revealed-preference criterion for RUM rationalizability is the Axiom of Revealed Stochastic Preference (ARSP). Letting $\Pi$ denote the set of all deterministic (rational) choice types and stacking all observed choice probabilities into a vector $\Pi \in \mathbb{R}^I$, the ARSP states: $$\sum_{m=1}^{M} T_m\cdot \Pi \leq \max_{R\in \mathcal{R}} \sum_{m=1}^{M} T_m\cdot R$$ for every finite sequence of trial vectors $(T_1, ..., T_M)$ that test choices within menus (Stoye, 2018).

The geometrical implication is that the set of all RUM-rationalizable choice probability arrays equals the convex hull of deterministic rational choice types. Hence, necessary and sufficient conditions are encoded as linear inequalities (in particular, nonnegativity of all Block–Marschak polynomials (Turansick, 2021, Kono et al., 2023)). In dynamic or set-valued extensions, the generalized ARSP involves mixtures over time-indexed deterministic types or set-valued deterministic choice functions (Kashaev et al., 2022, Stoye, 2018).

3. Key Model Classes and Their Properties

Several notable families of randomized utility models—including important submodels and extensions—illustrate the general framework’s flexibility and limitations:

Parametric RUMs (e.g., Plackett–Luce, Bradley–Terry–Luce, Thurstone)

• Each alternative is associated with a parameter $\theta_j$, and utility shocks are drawn from location-parameterized distributions (e.g., Gumbel for the Plackett–Luce model), yielding tractable likelihoods and fast inference (Soufiani et al., 2012).
• Under certain noise assumptions (log-concave densities), the log-likelihood function is strictly concave and the set of global maxima is bounded and unique (after appropriate normalization) (Soufiani et al., 2012, Soufiani et al., 2013).

Random Attention/Consideration Models

• Decision makers first stochastically select an “attention set” or consideration subset before utility maximization.
• The key restrictions are set-monotonicity (attention probability decreases weakly as the menu grows) and menu-independent taste distributions (stability) (Kashaev et al., 2021, Aguiar et al., 2018).
• The nonparametric test of such models reduces to linear programming feasibility and is statistically identifiable under limited menu variation (Kashaev et al., 2021).

Dynamic RUM (DRUM)

• Extends the static framework to panel data or repeated cross-sections, modeling agent choices across multiple periods, each with potentially time-dependent preference draws (Kashaev et al., 2022, Kashaev et al., 2023).
• Characterized by convex hull representations over time-indexed deterministic preference types, with empirical testability via conic linear programming (Kashaev et al., 2022).

Distributionally Robust RUM (DRO-RUM)

• The model allows the distribution of utility shocks to depart in $\phi$-divergence from a nominal law, regularizing against mis-specification and introducing endogenous substitution patterns.
• The gradient of the robust social surplus function still yields choice probabilities, extending the Williams–Daly–Zachary theorem (Müller et al., 2023).

General Random Utility Models (GRUMs)

• Admit linear and high-dimensional attribute-based models (e.g., agent- and alternative-dependent covariates) with additive noise, enabling complex preference heterogeneity.
• Inference and preference elicitation are feasible using MC-EM algorithms, with log-concavity and uniqueness under suitable conditions (Soufiani et al., 2013).

4. Theoretical and Empirical Identification

RUMs are not, in general, identified; i.e., the same observed choice data can often be rationalized by multiple distinct distributions over preferences (Turansick, 2021, Caradonna et al., 13 Aug 2024). Turansick (Turansick, 2021) provides two characterizations for uniqueness:

• A graphical “probability-flow” representation: uniqueness holds iff the associated flow diagram has no pairs of supported branching paths.
• A contour-set-based test: uniqueness holds iff no two distinct supporting orders agree on a collection of pairwise comparisons in a way that can generate branching.

Without sufficient restrictions, full-support multinomial logit and related models are never identified for $|X|\geq4$.

Support restrictions or parametric constraints can restore identification, as characterized by Ryser swaps (perturbations moving mass among conjugate squares of rankings) (Caradonna et al., 13 Aug 2024). Identification is then equivalent to the non-existence of nontrivial signed combinations of Ryser swaps that retain all mass on the restricted support.

5. Extensions: Intransitivity, Limited Rationality, and Robustness

Randomized utility models have been generalized to accommodate documented empirical violations of transitivity and rationality:

• Absence of parametric RUMs with concave log-likelihoods that capture intransitive preferences; requires multidimensional majority-vote constructions (e.g., inspired by Condorcet social choice) (Makhijani, 2018).
• Extensions such as the Random Preference Model allow monotonic but non-transitive (binary) preference functions, providing nonparametric testability even in the presence of intransitivity (Youmbi, 20 Jun 2024).
• Further, any RUM with sufficiently uncorrelated preferences can also be generated by a population of agents who do not maximize any preference; thus, observed rationalizable stochastic choices may reflect aggregate irrationality, limiting the falsifiability of the rational interpretation and severely widening welfare bounds (Caliari et al., 15 Mar 2024).

Set-valued and aggregation models—where alternatives represent groups or unobserved sub-alternatives—yield substantially weaker observable implications for RU-rationality and require specific non-overlap or menu-independence assumptions to ensure identification and avoidance of estimation bias (Liao et al., 31 May 2025).

6. Statistical Testing, Empirical Implementation, and Contemporary Directions

The ARSP and its generalizations translate into systems of linear (in)equalities, supporting empirical implementation and statistical testing using quadratic programming and (multivariate) bootstrap methods (Stoye, 2018, Kashaev et al., 2022, Aguiar et al., 2018). When attention, menu, or consumption dynamics are present, suitably refined conic or Möbius-inversion representations enhance computational tractability and data requirements (Turansick, 6 Dec 2024).

Recent progress includes neural-network-based universal approximators for RUM choice probabilities (“RUMnets”), which provide both theoretical universality and practicable generalization bounds, enabling the modeling of complex preference heterogeneity at scale (Aouad et al., 2022).

Randomized utility models continue to play a central role in structural estimation, policy counterfactuals, social choice, and behavioral economics, with ongoing research focusing on robustness, dynamic choice, partial identification, and the proper handling of aggregate or unobservable alternatives.