Brock–Hommes Framework in Agent-Based Models

• Brock–Hommes framework is an agent-based heterogeneous expectations model that uses random utility and discrete choice to capture macroscopic economic dynamics.
• The model employs continuous-time Markov evolution with Poisson updating, enabling tractable analysis of bounded rational agents under dynamic peer influence.
• It facilitates nonparametric identification of network structures and peer effects from long-panel choice data, eliminating the need for external instruments.

The Brock–Hommes framework refers to a class of agent-based heterogeneous expectations models, widely studied in economics and econophysics, for describing the emergence of macroscopic dynamics from the interactions of boundedly rational, adaptive agents engaged in discrete choice under social or economic feedback. While the original Brock–Hommes model posits discrete-time choices over forecasting rules subject to random utility and evolutionary updating, subsequent research has significantly expanded the paradigm. Notably, the continuous-time peer-effect discrete choice model with endogenous peer selection developed by Kashaev and Lazzati generalizes the core principles with rigorous identification strategies, nonparametric equilibrium characterization, and a direct link to empirical data—thereby offering an analytically tractable yet flexible setting for studying collective choice dynamics under limited attention and history-dependent peer effects (Kashaev et al., 26 Nov 2025).

1. Core Framework: Random Utility and Bounded Rationality

The defining feature of the Brock–Hommes family is the use of discrete choice mechanisms whereby boundedly rational agents select between alternatives with preferences modeled stochastically. In the continuous-time instantiation (Kashaev et al., 26 Nov 2025), a finite set of agents $\mathcal{A} = \{1, 2, \ldots, A\}$ each selects an alternative $y_a(t) \in \mathcal{Y} = \{0, 1, \ldots, Y\}$ at each instant $t \in [0, \infty)$. Agents periodically revise their choices according to independent Poisson processes, introducing inherent randomness in updating times.

The instantaneous utility for agent $a$ considering action $v$ and observing a subset $S$ of peers is:

$$U_a(v; S, y(t), X_a(t)) = \mu_{a,v}(y(t), S, X_a(t)) + \xi_{a, v, S},$$

where $\mu_{a,v}$ parameterizes deterministic mean utility (potentially incorporating own-state, peer states, and covariates $X_a$), and $\xi_{a,v,S}$ is a random taste shock, often modeled as i.i.d. type-I extreme value.

Given an active peer set $S$, the conditional choice probability (CCP) becomes a multinomial logit:

$$R_a(v \mid y, S) = \frac{\exp[\mu_{a,v}(y, S)]}{\sum_{w \in \mathcal{Y}} \exp[\mu_{a,w}(y, S)]}.$$

2. Endogenous Peer Selection and Social Reference Networks

A key innovation over classic discrete-choice or mean-field imitation models is the endogenous, state-dependent mechanism for peer selection. Each agent $a$ possesses a latent "reference set" $\mathcal{N}_a \subset \mathcal{A} \setminus \{a\}$, representing the set of potential peers whose actions might influence $a$'s utility.

At each revision opportunity, agent $a$ samples a random "active set" $S_a \subseteq \mathcal{N}_a$ of peers to attend to, with each $a'\in\mathcal{N}_a$ included independently with probability $Q_a(a' \mid y)$. This selection probability can be a function of both agent and peer types and choices, allowing for homophily and endogenous attention. Under the type-homogeneity assumption, $Q_a(a' \mid y)$ simplifies to $Q_{h(a)}(y_a, y_{a'})$, guaranteeing tractability.

The active set, and hence the realized social influence, becomes time- and state-dependent—capturing history-dependent, bounded attention and providing a mechanism for complex macro-dynamics even in homogeneous agent populations.

3. Continuous-Time Markov Evolution and Stationary Distribution

The system evolves as a continuous-time Markov chain on the space of configurations $y(t) \in \mathcal{Y}^A$. The infinitesimal transition rate from $y$ to $y'$ (if these differ only in the decision of agent $a$) is:

$w(y' \mid y) = \lambda_a P_a(y'_a \mid y),$

where $P_a(v \mid y)$ is the ex ante CCP, marginalizing over realizations of $S_a$:

$$P_a(v \mid y) = \sum_{S \subseteq \mathcal{N}_a} R_a(v \mid y, S) S_a(S \mid y).$$

The overall time evolution is governed by the Kolmogorov forward equation $\dot\pi(t) = \pi(t) W$, where $W$ is the full generator matrix.

Crucially, under mild irreducibility conditions (positive rates and invariant attention probabilities), the process admits a unique invariant distribution $\mu$ on $\mathcal{Y}^A$ with full support. Every configuration appears with strictly positive long-run frequency, ensuring the observability of all relevant transition patterns and providing strong identification leverage (Kashaev et al., 26 Nov 2025).

4. Identification from Long-Panel Choice Data

A principal result in (Kashaev et al., 26 Nov 2025) is the nonparametric identification of the underlying network structure $(\mathcal{N}_a)$, peer selection kernel $(Q_a)$, and random-utility rule $(R_a)$ solely from the distribution of observed choice trajectories—requiring neither exogenous covariate variation nor instrumental variables.

The identification strategy exploits the variability of CCPs across configurations and agent types:

• Reference Set Recovery: For any candidate peer $a'$, a change in $y_{a'}$ will alter $P_a(v \mid y)$ if and only if $a' \in \mathcal{N}_a$, allowing the reference set to be determined by observed CCP patterns.
• Peer Selection and Choice Rule Recovery: By comparing agents of the same type with different reference set cardinalities under controlled peer configurations, the model's linear constraints inductively pin down both the peer selection probabilities $Q_t$ and the action selection rules $R_a$ for all possible active peer set sizes.
• Recursion for Larger Peer Sets: Knowing the behavior for all sets of size up to $k$, the mixture formula for $P_a(v \mid y)$ supports recursive determination of action probabilities for sets of size $k+1$.

This approach ensures model falsifiability: any empirical violation in the rank or monotonicity patterns of CCPs implied by the mixture and independence assumptions is sufficient to reject the framework.

5. Analytical and Empirical Implications

The structure guarantees significant empirical and theoretical content:

• Testable Predictions: The CCPs must comply with mixture patterns derived from the structure of peer selection and bounded consideration; for instance, relative ranks and monotonicity across configurations.
• Equilibrium Polarization: In simple two-agent binary choice cases ($A=2, Y=1$), the long-run correlation of agent decisions can exceed that of standard models, with polarization amplified or reduced according to the sign and strength of the peer effect.
• Sufficiency without Exclusion Restrictions: The model's empirical identification is achieved via endogenous variation in observed peer groups and heterogeneity in reference set sizes, circumventing the need for external instruments or experimental variation.

These properties enable rigorous inference on underlying social processes, bounded rationality, and dynamic feedback from static and dynamic choice data.

6. Connections to Related Approaches

The continuous-time framework with endogenous peer selection subsumes and extends various prominent discrete choice and imitation models in the literature. Traditional discrete choice models (including logit and multinomial logit) correspond to the limiting case where peer selection is either trivial or uniform and where each agent observes all others at all times.

Peer-effect models with exogenous or all-to-all influence (e.g., the noisy voter model, Kirman's ant recruitment, and Föllmer's imitative dynamics) can be mapped to special or degenerate cases where $Q_a$ is constant and $\mathcal{N}_a$ is maximal. Unlike these, the Kashaev–Lazzati model formalizes the process by which agents selectively attend to certain peers according to a stochastic, endogenous mechanism, consistent with bounded rationality and limited attention.

The approach is distinct from models such as the Pairwise Choice Markov Chain (Ragain et al., 2016), which assign Markovian transitions directly to alternatives rather than agents' interactive updating, and from hidden Markov models for longitudinal preference evolution, where the state dynamics are driven by latent class transitions rather than explicit network-structured peer influence (Zarwi et al., 2017).

7. Summary Table: Key Features of the Continuous-Time Peer-Effect Discrete Choice Model

Aspect	Implementation in (Kashaev et al., 26 Nov 2025)	Distinctive Implication
Agent interaction	Endogenous, history-dependent peer selection	Captures homophily and dynamic attention
Choice process	Continuous-time Markov, random utility, bounded consideration	Full-support equilibrium, tractable analysis
Identification	Nonparametric from long panel data, needs no exogenous variation	Robust recovery of network and choice kernel
Empirical testability	Rank and monotonicity constraints on CCPs	Falsifiability from observed variation
Relation to classics	Generalizes Brock–Hommes, Kirman, Föllmer, noisy voter models	Flexible, encompassing traditional benchmarks

The framework thus extends the legacy of the Brock–Hommes paradigm by providing a rigorously identified, behaviorally grounded, fully observable, and empirically testable architecture for the dynamics of discrete choices under endogenous and limited social influence (Kashaev et al., 26 Nov 2025).