Zero-Intelligence Models

• Zero-Intelligence Models are stochastic, agent-based models where agents act randomly under exogenous constraints without strategic behavior.
• They replicate empirical market phenomena such as heavy-tailed price returns, scale-free networks, and bursty inter-event timings using Poisson processes.
• Extensions like non-Markovian and adaptive variants integrate simple feedback and parameterized control to better align with observed microstructure dynamics.

Zero-intelligence (ZI) models are a class of stochastic agent-based and Markovian models for collective dynamics in markets and allocation systems, characterized by the complete absence of individual rationality, memory, or forward-looking strategy in agents' behaviors. Originally developed to investigate limit order book (LOB) microstructure, ZI models have since been generalized to diverse domains, including labor-market matching, collective allocation, and algorithmic trading. The central feature of ZI models is that agents' actions (submissions, cancellations, applications) are random draws constrained only by exogenous budgets or feasibility restrictions, and not by any form of optimization or learning. Despite their conceptual simplicity, ZI models reproduce many of the empirical regularities—or "stylized facts"—observed in real-world complex systems, including heavy-tailed price returns in financial markets and scale-free patterns in transaction networks.

1. Formal Structure of Zero-Intelligence Models

At the core of the ZI paradigm is the separation of agent decision-making from system-level constraints. In canonical ZI market models, such as the continuous double auction (CDA), each agent submits quotes drawn randomly from an admissible price interval determined by their budget constraint—buyers bid $p_b\sim {\rm Uniform}(0, r_i)$, sellers offer $p_s\sim {\rm Uniform}(c_j, 100)$, with $r_i$ the buyer's redemption value and $c_j$ the seller's cost (Tseng et al., 2010). A trade is executed if $p_b\geq p_s$, with the transaction price set to $\frac{p_b+p_s}{2}$ in midpoint-rule simulations.

In LOB and queueing formulations (Šmíd, 2015, Mariotti et al., 2022), ZI models treat event flows (limit order arrivals, market order submissions, cancellations) as independent Poisson processes parameterized by event-specific intensities. The state of the LOB is updated by Markovian transitions resulting from these primitives, with no dependence on order book history or agent-specific information beyond event feasibility (e.g., nonnegative queue depths).

Extensions such as the Parameterised-Response Zero-Intelligence (PRZI) trader generalize the price sampling distribution to allow continuous control over agent "urgency" or aggression via a real-valued parameter $s$, enabling interpolation between cautious and aggressive quoting without strategic reasoning (Cliff, 2021). Similarly, in models of labor-market matching, ZI agents apply randomly to firms according to a Gibbs distribution induced by maximum entropy principles, without conditioning on previous outcomes or learning from past rejections (Chen et al., 2013).

2. Statistical Properties and Empirical Relevance

ZI models have been shown to reproduce several universal statistical features observed in real-world markets and allocation networks:

• Heavy-tailed return distributions: Simulation of ZI double auctions yields log-returns with collapse across timescales, mirroring the non-Gaussian scaling seen in real financial data (Tseng et al., 2010).
• Scale-free network structure: The degree distribution $P(k)\sim k^{-\alpha}$ ($\alpha\approx0.51\pm0.05$ for ZI) and community size distribution $P(s)\sim s^{-\gamma}$ ($\gamma\approx1.36\pm0.07$ for ZI) observed in transaction networks emerge naturally from random bilateral trades (Tseng et al., 2010).
• Bursty inter-event distributions: Transaction intervals exhibit fat-tailed distributions $P(\Delta t)\sim (\Delta t)^{-\delta}$, with $\delta\approx1.36\pm0.08$, again closely fitting empirical data.

Importantly, when benchmarked against more "intelligent" agent-based models (e.g., ZIP—Zero-Intelligence-Plus, with margin adjustment learning, and GD—Gjerstad-Dickhaut, with Bayesian belief updating), only ZI models retain power-law returns and network exponents over long horizons. The introduction of strategic learning drives the system toward equilibrium, dampening volatility and erasing fat-tailed fluctuations (Tseng et al., 2010).

In empirical work calibrating ZI models to tick-by-tick LOB data for US equities, simple two-parameter constant-intensity ZI models consistently outperform naive baseline predictors for next best-ask/bid moves, while more complex variants (adding power-law tails, non-unit jump sizes, state-dependent rates) generally fail to improve out-of-sample performance for liquid stocks (Šmíd, 2015).

3. Mathematical Formulation: LOB ZI and Generalizations

The continuous-time ZI LOB model is built on a pure-jump Markov process over the count vector of outstanding orders at each price level (Šmíd, 2015). The primitive event intensities are:

• Limit order arrivals (buy/sell): Poisson with rate functions $\kappa(a,b,p)$, $\lambda(a,b,p)$
• Market order arrivals (buy/sell): Poisson with rates $\theta(a,b)$, $\vartheta(a,b)$
• Cancellations: Poisson with rate proportional to queue depth, $\rho(a,b,p)$, $\sigma(a,b,p)$

The time to next event is exponentially distributed, and each event's conditional probability is proportional to its intensity among all possible events. L1-level observables (best quote and queue depth) are Markov in this construction.

The Generalized Zero-Intelligence (GZI) model further includes:

• Non-unit volume orders (market or limit)
• In-spread order placement and quote-shifting events
• Exogenous quote moves (e.g., shifts due to external news)

The transition densities for best-quote changes incorporate both survival of previously open orders and new stochastic arrivals, resulting in closed-form recursive likelihoods for rigorous econometric estimation (Šmíd, 2015). Both ZI and GZI models admit maximum likelihood estimators for their Poisson intensities, shown to be consistent and asymptotically normal under standard regularity conditions.

4. ZI Models in High-Frequency Volatility Estimation and Optimal Execution

In high-frequency trading research, ZI models are deployed as data-generating processes for evaluating the finite-sample properties of volatility estimators (Mariotti et al., 2022). Here, the ZI limit order book serves as a null model generating microstructure noise through discrete-tick asynchronous order matching.

Under simulated ZI dynamics, the best bid/ask queues follow independent birth–death processes, with each queue evolving according to arrivals and removals (cancellations or market order "hits"). The mid-price $p_{\text{mid}}$ undergoes random walks due to queue depletion events.

In benchmarking volatility estimators, empirical tests under ZI and more realistic queue-reactive (QR) models show that while robust estimators (e.g., pre-averaging, Fourier) perform well under ZI noise, QR models—where event intensities depend on queue state and cross-queue interactions—better reproduce empirical Hausman test noise profiles. Nevertheless, even under these enhancements, the strong ZI assumption provides a tractable foundation for systematic study of microstructure-induced bias in volatility and optimal execution cost estimates.

5. Beyond Markets: ZI Agents in Collective Allocation and Learning Curves

ZI models have also been applied outside financial contexts to analyze collective dynamics under bounded rationality. In job-market matching, each agent (job-seeker) applies randomly to one firm per round, allocating applications by a "softmax" over perceived firm utility derived from static ranking and local demand-supply mismatch, but without updating preferences over rounds (Chen et al., 2013).

The accumulation of matches across rounds defines a "learning curve" error $E(n)$ in analogy to unsupervised machine learning. In regimes of constant job-offer ratio or unemployment, $E(n)$ decays exponentially with rate set by stagewise unemployment, with explicit analytic forms in the "scale-invariant" case. The macroscopic dynamics—fast convergence in seller's markets ($\alpha>1$), slow convergence and high mismatch in buyer's markets ($\alpha<1$), and error scaling with diversity constraints—are fully determined by random allocation subject to exogenous diversity (Jaynes-Shannon MaxEnt principle), not by agent memory or strategic adaptation.

6. Recent Developments: Non-Markovian and Adaptive Zero-Intelligence Variants

A major recent advance is the introduction of non-Markovian ZI models that incorporate simple forms of market feedback while retaining non-strategic agent behavior (Ravagnani et al., 7 Mar 2025). In the Non-Markovian Zero-Intelligence (NMZI) model, the probability that a limit order is a buy or sell is determined by a logistic function of the exponentially weighted moving average of past mid-price returns: $$\Pr[\text{sell LO at } t]=1/(1+\exp(-\alpha \bar R_t))$$, with memory parameter $\kappa$ and reaction strength $\alpha$.

This parsimonious modification generates empirically observed market impact dynamics that classical ZI cannot reproduce: concave build-up of price impact during metaorder execution, exponential post-trade reversion, and a nonzero permanent component. Analytical results show that during metaorder execution, immediate impact from individual trades is offset by reversion due to induced order flow imbalance, leading to characteristic nonlinearity in price trajectories. The parameter space controls both the extent and rate of these nonlinear effects, providing a direct mechanistic link between microscopic randomization and macroscopic stylized facts.

Adaptive zero-intelligence frameworks such as PRZI/PRSH supplement random price selection with parameterized control over quoting aggression and simple stochastic adaptation (hill-climbing on profitability), producing rich population dynamics including punctuated equilibria and long-run cycling among strategy modalities (Cliff, 2021).

7. Limitations, Extensions, and Outlook

While basic ZI models account for many high-level empirical regularities, they exhibit systematic discrepancies with real LOB data near the best quotes, and fail to capture state-dependent clustering, informative order flow, or endogenous feedback mechanisms essential to second-order statistics. More sophisticated state-dependent (queue-reactive) and non-Markovian ZI variants provide more realistic microstructure noise, price impact dynamics, and predictive accuracy, but introduce additional complexity and parameters (Šmíd, 2015, Mariotti et al., 2022, Ravagnani et al., 7 Mar 2025).

Despite these limitations, rigorous econometric studies confirm that even simple ZI models, with constant event intensities and no agent-level learning, are statistically significant predictors of short-term market dynamics and matching outcomes, especially in liquid and highly competitive environments (Šmíd, 2015). ZI frameworks remain foundational for both analytical and simulation-based investigation of collective stochastic processes in markets and allocation systems, with continuing research focused on the integration of minimal endogenous feedback, adaptive mechanisms, and empirical calibration of microstructural features.