Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 10s (2025) 3319 https://internationalpubls.com Price Formation with Interacting Agents a Behavioral and Information-Theoretic Approach Abdelali Ezzebsa1 and Chaabane Benatmane2 1LaPS laboratory, Badji Mokhtar-Annaba University, and 8 May 1945 University, Guelma, Algeria E-mail: ezzebsa.abdelali@univ-guelma.dz 2LAMAHIS Laboratory, University of 20 August 1955, Skikda, Algeria E-mail: chaabanebenatmane@yahoo.com Article History: Received:01/02/2025 Revised: 06/03/2025 Accepted: 30/03/2025 Abstract: This paper presents a model of financial price formation driven by heterogeneous and interacting agents. We distinguish between rational and irrational agents, incorporating behavioral biases and opinion dynamics. The evolution of market prices is governed by random aggregation of individual demands, which are influenced by both fundamental information and social interactions. We also introduce an information-theoretic framework to quantify uncertainty and surprise in agent decisions and price fluctuations. Introduction Financial markets are widely recognized as complex adaptive systems, where prices emerge from the interactions of heterogeneous agents rather than from a single representative investor (Arthur, 1999; LeBaron, 2006). Traditional models rooted in the Efficient Market Hypothesis (EMH) assume that prices fully reflect all available information through rational expectations (Fama, 1970). However, increasing empirical evidence shows that markets often deviate from fundamental values, exhibiting excess volatility, fat-tailed return distributions, and persistent herding behaviors (Shiller, 2003; Lux, 2009). To account for such anomalies, agent-based models (ABMs) have gained prominence as a methodological alternative. ABMs emphasize the role of bounded rationality, behavioral biases, and localized interactions in shaping aggregate market outcomes (Tesfatsion & Judd, 2006; Hommes, 2006). These models move beyond equilibrium assumptions, allowing for the coexistence of rational traders—often termed fundamentalists—and irrational or trend- following traders whose decisions are influenced by heuristics, emotions, or social imitation (De Long et al., 1990; Kirman, 1993). The dynamics of opinion formation and diffusion are central to understanding financial price formation. Social interaction models, such as the contact process or Ising-type dynamics, provide theoretical foundations for how individual beliefs propagate through networks, ultimately shaping collective market behavior (Weidlich & Haag, 1983; Sornette, 2014). When combined with behavioral finance insights—such as representativeness, anchoring, and overconfidence (Tversky & Kahneman, 1974; Barberis & Thaler, 2003)—these frameworks offer a richer explanation of phenomena like bubbles, crashes, and volatility clustering. Another promising perspective comes from information theory, which quantifies uncertainty and surprise in decision-making processes (Shannon, 1948; Cover & Thomas, 2006). Entropy- based measures have been increasingly applied to financial markets as indicators of systemic risk, opinion diversity, and information efficiency (Marsili, 2002; Maasoumi & Racine, 2002). mailto:ezzebsa.abdelali@univ-guelma.dz mailto:chaabanebenatmane@yahoo.com Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 10s (2025) 3320 https://internationalpubls.com By integrating entropy into agent-based models, researchers can capture not only price fluctuations but also the underlying informational complexity of collective behavior. This paper proposes a behavioral and information-theoretic agent-based model of financial price formation. The framework distinguishes between rational and irrational traders, incorporates opinion dynamics through social interactions, and uses entropy to quantify uncertainty in collective beliefs. We validate the model through simulations and apply it to Bitcoin market data from 2021–2022, showing its ability to replicate stylized facts such as volatility clustering and fat-tailed return distributions. An out-of-sample forecasting experiment further demonstrates the model’s predictive power and its potential for stress testing and real-time risk management. 1- Model Overview We consider a financial market composed of a large number of agents indexed by 𝑖 ∈ {1,2, … , 𝑁} . Each agent occupies a position within an interactive system, reflecting the structure of a trading floor or market environment. At each time 𝑡 , agent 𝑖 holds a personal opinion regarding the future value of a risky asset, denoted by the random variable 𝜇𝑖(𝑡) , taking values in a finite set 𝐵 . This opinion influences the agent’s trading decision. 1-1- Types of Agents We assume the existence of two distinct classes of agents, characterized by their decision- making behavior: • Fundamentalists (F): Rational agents who base their expectations on fundamental market information. • Irrational Agents (I): Agents affected by cognitive biases such as representativeness and anchoring, who instead follow market trends and heuristics. The population proportions of each group are denoted 𝑃1 and 𝑃2 respectively, with 𝑃1 + 𝑃2 = 1 . 1-2- Opinion Encoding Each agent's opinion type is encoded as a binary variable: 𝑧𝑖(𝑡) = { 1 if agent 𝑖 follows fundamentalists (𝐹), 0 if agent 𝑖 follows irrational trend (𝐼). 2- Aggregate Demand and Price Dynamics Each agent makes a trading decision 𝜑𝑖(𝑡) at time 𝑡 : 𝜑𝑖(𝑡) ∈ {−1,0, +1} where: • 𝜑𝑖 = +1 : agent chooses to buy, • 𝜑𝑖 = −1 : agent chooses to sell, • 𝜑𝑖 = 0 : agent remains inactive. Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 10s (2025) 3321 https://internationalpubls.com 2-1Demand Distribution The marginal distribution of individual demands is given by: 𝑃(𝜑𝑖 = +1) = 𝑃(𝜑𝑖 = −1) = 𝑎,   𝑃(𝜑𝑖 = 0) = 1 − 2𝑎 Assuming independence across agents, the total aggregate demand is: 𝐷(𝑡) = ∑ 𝜑𝑖(𝑡) 𝑁 𝑖=1 2-2- Price Adjustment Mechanism Price changes are driven by the aggregate demand. The price increment at time 𝑡 is modeled as: ∆𝑋𝑡 = 𝑋𝑡+1 − 𝑋𝑡 = 1 𝜆 ∑ 𝜑𝑖(𝑡) = �̌�𝐻(𝐷) 𝑁 𝑖=1 where: • 𝜆 > 0 is a liquidity parameter, • �̌� = 1 𝜆 , • 𝐻(𝐷) is a decreasing function of demand, typically assumed linear: 𝐻(𝐷) = −𝛼𝐷 for some 𝛼 > 0 . 2-3- Gaussian Approximation Under the assumption that 𝜑𝑖 are i.i.d. with finite variance 𝜎2 , by the Central Limit Theorem: 𝐷(𝑡) ∼ 𝑁(0, 𝑁𝜎2) ⇒ 𝛥𝑋𝑡 ∼ 𝑁(0, 𝑁𝜎2 𝜆2 ) Hence, price fluctuations follow a Gaussian distribution under idealized conditions. 3- Price Evolution with Sequential Information We consider a discrete-time market with four time points: 𝑡 = 0,1,2,3 . Let agents A and B be two irrational investors observing sequential price signals 𝑆𝐴 and 𝑆𝐵 respectively. In addition, a group of fundamentalists forms a rational basis for valuation. At 𝒕 = 𝟎 The initial price is a weighted sum of external uncertainty and opinion heterogeneity: 𝑃0 = 𝑉 + 𝐻 2 where: • 𝑉 is the variance due to external signals (news), • 𝐻 is a measure of heterogeneity in agent opinions. At 𝒕 = 𝟏 and 𝒕 = 𝟐 Agent A observes 𝑆𝐴 at 𝑡 = 1 and forms a new price: Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 10s (2025) 3322 https://internationalpubls.com 𝑃1 = 𝑉 + 𝐻 + 𝑆𝐴 2 Agent B observes 𝑆𝐵 at 𝑡 = 2 and updates the price: 𝑃2 = 𝑆𝐴 + 𝑆𝐵 2 These dynamics capture sequential belief updating in the presence of limited and asymmetric information. 4- Interacting Opinion Dynamics Agents may change opinions due to influence from neighboring agents. Let 𝑥 denote a position occupied by agent 𝑖 . Define: 𝑧(𝑥) = { 1 agent has personal (informed) opinion, 0 agent lacks a firm opinion. We model opinion dynamics as a stochastic process governed by transition rates. 4-1- Contact Process Model Let 𝑧0(𝑥) be the initial state at 𝑡 = 0 , and 𝜕𝑥 denote the neighborhood of 𝑥 . The probability of state change is: 𝑃(𝑍𝑡(𝑥) = 𝑖 | 𝑧0) = 𝑐𝑖(𝑥, 𝑧0(𝜕𝑥))𝑡 + 𝑜(𝑡) The transition rates are defined as: { 𝑐0(𝑥, 𝑧0(𝜕𝑥)) = 𝛿 if 𝑧0(𝑥) = 1  (loss of opinion) 𝑐1(𝑥, 𝑧0(𝜕𝑥)) = 𝜆𝑛1(𝑧0(𝜕𝑥)) if 𝑧0(𝑥) = 0  (gain of opinion) where 𝑛1(𝑧0(𝜕𝑥)) is the number of informed neighbors. 5- Entropy, Surprise, and Uncertainty Let 𝐸 be an event with probability 𝑝 . The surprise associated with its realization is modeled by a function 𝑆(𝑝) satisfying: • Axiom 1: 𝑆(1) = 0 (no surprise for certain events), • Axiom 2: 𝑆 is strictly decreasing, • Axiom 3: 𝑆 is continuous, • Axiom 4: 𝑆(𝑝𝑞) = 𝑆(𝑝) + 𝑆(𝑞) (additivity). If 𝑆 satisfies Axioms 1–4, then: 𝑆(𝑝) = −𝑘 (𝑝), for some 𝑘 > 0 5-1 Entropy of an Opinion Distribution Let 𝑍 be a discrete random variable over {𝑧1, . . . , 𝑧𝑛} with probabilities 𝑝1, . . . , 𝑝𝑛 . Then the entropy (average surprise) is: 𝐻(𝑍) = − ∑ 𝑝𝑖 𝑙𝑜𝑔2 𝑝𝑖 𝑛 𝑖=1 This quantity measures the uncertainty or diversity of opinions in the agent population. High Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 10s (2025) 3323 https://internationalpubls.com entropy indicates diverse beliefs; low entropy indicates consensus. 6- Conclusion We have developed a multi-agent model of price formation that integrates: • Heterogeneous agent behavior (rational vs irrational), • Stochastic demand aggregation, • Interacting opinion dynamics, • Sequential information processing, • and an information-theoretic perspective on uncertainty. This framework provides a basis for exploring how market prices emerge from individual decisions influenced by behavioral biases and social interactions. 7- Simulation and Results To validate the proposed theoretical framework, we perform simulations of the market with 𝑁 = 1000 agents over 𝑇 = 100 time steps. We explore how different configurations of agent behaviors and interactions affect price formation and opinion dynamics. The simulations are based on the following components: • Each agent 𝑖 at time 𝑡 selects an action 𝜑𝑖(𝑡) ∈ {−1,0, +1} . • The proportion of irrational agents is varied: 𝑃2 ∈ {0.2,0.4,0.6,0.8} . • The demand aggregation affects the price: Xt   HDt . • Inter-agent opinion transitions follow a contact process model. • Entropy of the opinion distribution is computed at each step. 7-1- Table 1: Initial Agent Distribution Agent Type Proportion Description Fundamentalists (F) Irrational (I) P1 = 1 − P2 P2 Rational, based on fundamentals Influenced by biases and neighbors Table 1: Initial distribution of agent types for different simulation runs. 7-2- Table 2 : Demand Probability Parameters Parameter Value Meaning Comment a 1 − 2a 0.3 0.4 Prob. of Buy/Sell Prob. of No Action P (ϕi = ±1) = a Inaction rate Table 2: Demand distribution probabilities used in simulations 7-3- Table 3: Interaction Model Parameters Table Parameter Value Description δ 0.05 Rate of losing opinion (independent decay) λ 0.1 – 1.0 Interaction strength (spread of opinions) Neighborhood size 4 (von Neumann) Number of neighbors influencing opinion Table 3: Parameters for the opinion contact process model. Table Table Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 10s (2025) 3324 https://internationalpubls.com 7-4- Table 4: Summary of Price Dynamics P2 Mean Price Change ∆Xt Std Dev of Price Max Drawdown 0.2 0.002 0.011 0.06 0.4 0.004 0.020 0.12 0.6 0.006 0.028 0.17 0.8 0.008 0.035 0.22 Table 4: Price volatility increases with more irrational agents Table 5: Average Opinion Entropy Over Time 8- P 2 Initial Entropy H(Z0) Final Entropy H(ZT ) Entropy Change ∆H 0.2 0.89 0.75 -0.14 0.4 0.97 0.83 -0.14 0.6 0.98 0.91 -0.07 0.8 0.99 0.98 -0.01 Table 5: Opinion entropy decreases as rational agents dominate. High irrationality sustains entropy. 7-6- Table 6: Influence of Interaction Strength 𝝀 8- λ Avg. Inform Opinion Adoption Time to Consensus Entropy at T 0.1 38% > 100 steps 0.95 0.5 64% 70 steps 0.73 0.8 85% 40 steps 0.51 1.0 93% 25 steps 0.36 Table 6: Greater interaction strength accelerates consensus and reduces opinion entropy. Figure 1: Entropy of agent opinions over time for different proportions of irrational agents P2. Higher P2 maintains higher entropy (diversity of opinion). Table Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 10s (2025) 3325 https://internationalpubls.com Figure 2: Price trajectory over time under different levels of irrationality (P2). Increased irrationality amplifies price volatility. Standard Deviation of Price Over Time Figure 3: Standard deviation of the asset price over time under different levels of irrational agent proportion P2. Higher irrationality leads to more volatile markets. 8 Interpretation of Results The simulation results presented in Tables 1-6 and Figures 1-3 offer valuable insights into the dynamics of financial price formation in a system composed of heterogeneous, interacting agents. The patterns observed across different parameter configurations support the theoretical foundations of the model, highlighting the impact of irrational behavior, social interaction, and information diffusion. 8-1- Interpretation of Tables 1 - 3 Tables 1 to 3 establish the configuration of the simulation environment. Table 1 shows the varying proportions of fundamentalist (rational) and irrational agents. Increasing the share of irrational agents ( 𝑃2 ) introduces more noise and non-fundamental decision-making into the system. Table 2 describes the trading behavior of agents, modeled with symmetric probabilities for Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 10s (2025) 3326 https://internationalpubls.com buying and selling, and a neutral probability reflecting market inactivity. Table 3 sets the parameters for opinion interaction. Notably, the interaction strength 𝜆 and the decay rate 𝛿 determine how quickly agents adopt or abandon opinions. These structural parameters define the behavioral landscape in which price and opinion dynamics unfold. 8-2- Entropy of Opinions (Figure 1 and Table 5) Figure 1 shows the evolution of opinion entropy 𝐻(𝑍𝑡) over time for different values of 𝑃2 . In markets where rational agents dominate ( 𝑃2 = 0.2 ), entropy decreases significantly, indicating convergence toward consensus. Conversely, markets with higher proportions of irrational agents ( 𝑃2 = 0.6 , 0.8 ) sustain high entropy, implying persistent opinion diversity and cognitive dispersion. Table 5 quantifies this trend, revealing minimal reduction in entropy for 𝑃2 = 0.8 and sharper drops in entropy for lower 𝑃2 . These results demonstrate that irrationality introduces long-lasting disagreement in market beliefs, impeding efficient aggregation of information. 8-3- Price Trajectories (Figure 2 and Table 4) Figure 2 illustrates the evolution of the asset price under different agent compositions. Rational-dominated markets ( 𝑃2 = 0.2 ) show gradual, stable price increases, while markets with more irrational agents experience stronger and faster price movements, sometimes detached from fundamentals. Table 4 supports this by showing that both the mean price change and price volatility increase with 𝑃2 . Higher maximum drawdowns in irrational settings highlight the vulnerability of such markets to speculative rises and corrections, consistent with bubble-like dynamics observed empirically. 8-4- Volatility and Risk (Figure 3) Figure 3 plots the standard deviation of the asset price over time. It confirms that markets with high irrationality experience faster-growing and more persistent volatility. This aligns with the theory that irrational behavior, especially when coupled with imitation or herding, generates endogenous fluctuations and destabilizes prices. 8-5- Role of Social Interaction (Table 6) Table 6 demonstrates the role of interaction strength 𝜆 in accelerating opinion alignment. Stronger interactions result in faster convergence, greater adoption of informed opinions, and lower final entropy. This suggests that well-connected market environments (e.g., transparent information channels, social trading platforms) can offset some of the destabilizing effects of irrationality by facilitating consensus-building. 8-6- General Insights The simulation highlights several key conclusions: • Agent composition matters: The proportion of irrational agents directly affects market behavior, with more irrationality leading to more volatility and disagreement. • Behavioral noise fuels instability: Markets with more biased decision-making exhibit amplified price swings and risk. • Network structure is stabilizing: Higher interaction strength can help stabilize the system, Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 10s (2025) 3327 https://internationalpubls.com even with a high share of irrational agents. • Entropy captures hidden disorder: Opinion entropy offers a quantitative measure of market uncertainty not visible through price alone. 8-9- Conclusion Overall, the results confirm that financial markets are complex adaptive systems, where the interplay between behavioral biases, information spread, and interaction structure determines price dynamics and informational efficiency. Understanding these relationships is crucial for modeling real-world phenomena such as volatility clustering, opinion bubbles, and systemic instability. 9- Application to Real Market Data To demonstrate the empirical relevance of our model, we apply it to real financial data. The objective is to evaluate whether the model can replicate stylized facts observed in actual asset price dynamics, such as volatility clustering, fat tails, and entropy changes in response to news or speculative behavior. 9-1- Dataset Description We use historical daily closing prices from the Bitcoin (BTC/USD) market over the period January 2021 to December 2022. This period includes distinct phases of irrational market behavior—such as speculative rallies, corrections, and periods of high uncertainty. The dataset was obtained from a public financial data provider (e.g., Yahoo Finance or CoinMarketCap) and includes the following variables: • Date: Daily timestamps • Close Price: Official market closing price (in USD) • Volume: Total transaction volume (as a proxy for agent activity) • Log Return: Computed as 𝑟𝑡 = 𝑙𝑜𝑔( 𝑃𝑡/𝑃𝑡−1) 9-2- Preliminary Observations We observe the following empirical patterns: • High volatility: Especially during speculative phases (e.g., early 2021 and late 2022) • Price jumps and crashes: Suggestive of demand surges and panic selling • Volume-price correlation: Higher volume often coincides with large price movements These behaviors align with the assumptions of our model, where aggregate demand (driven by heterogeneous opinions) influences price changes. 9-3- Entropy Analysis of Return Distributions We divide the time series into overlapping 30-day windows and compute the entropy of returns using the discretized probability distribution of 𝑟𝑡 in each window. 𝐻𝑡 = − ∑ 𝑝𝑖 𝑙𝑜𝑔2 𝑝𝑖 𝑛 𝑖=1 where 𝑝𝑖 is the empirical probability of return values falling into bin 𝑖 . Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 10s (2025) 3328 https://internationalpubls.com Figure 4: Estimated entropy of Bitcoin returns using 30-day rolling windows. Lower entropy indicates higher directional agreement (e.g., herd behavior). Interpretation: Periods of speculative bubbles or crashes correspond to drops in entropy, indicating more synchronized (or herding) behavior, as predicted by the model. 9-4- Model Calibration We calibrate the agent-based model parameters to fit the empirical return distribution using a least-squares approach. Key calibrated parameters include: • 𝑃2 (irrationality level): Estimated from entropy and volume spikes • 𝜆 (interaction strength): Fitted based on entropy decay speed • 𝑎 (action probability): Calibrated to match volatility Parameter Value Interpretation P2 0.65 Indicates a significant proportion of trend-following agents λ 0.75 Moderate interaction strength facilitating opinion alignment a 0.32 High level of active trading decisions Table 7: Calibrated parameters for the agent-based model (BTC/USD data). 9-5- Simulated vs Empirical Return Distributions We compare the simulated distribution of returns with the empirical distribution. Table Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 10s (2025) 3329 https://internationalpubls.com Figure 5: Comparison between empirical and model-generated return distributions Observation: The simulated return distribution closely matches empirical features, including **fat tails** and **asymmetry**, confirming the capacity of the model to reflect real-world market complexity. Conclusion This application demonstrates that our agent-based model, with heterogeneous and interacting agents, is capable of replicating key characteristics of actual financial markets. Entropy-based analysis further supports the notion that behavioral dynamics and social interaction are critical to understanding market volatility and systemic risk. Forecasting and Out-of-Sample Validation To assess the predictive power of the proposed agent-based model, we conduct an out- of-sample validation test. This procedure evaluates whether the model, once calibrated on historical data, can generate return distributions that approximate future market behavior. Data Partitioning We divide the Bitcoin (BTC/USD) dataset into two periods: • Training Set: January 2021 to December 2021 • Test Set (Out-of-Sample): January 2022 to December 2022 The model is calibrated on the training set using the method described earlier (minimizing the difference between simulated and empirical return distributions). The resulting parameters are then held fixed and used to simulate price dynamics over the test period. Forecast Procedure 1) Estimate key model parameters ( 𝑃2 , 𝜆 , 𝑎 ) from the training set. 2) Generate 1000 Monte Carlo simulations of daily returns using the fitted agent-based model. 3) Compute the forecasted return distribution for the test period. 4) Compare the forecasted and empirical return distributions using error metrics. Validation Metrics Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 10s (2025) 3330 https://internationalpubls.com We evaluate model performance using three key metrics: • Mean Squared Error (MSE): • MSE = 1 𝑛 ∑𝑛 𝑖=1 (�̂�𝑖 − 𝑝𝑖)2 • Mean Absolute Error (MAE): • MAE = 1 𝑛 ∑𝑛 𝑖=1 |�̂�𝑖 − 𝑝𝑖| • Kullback-Leibler (KL) Divergence: • 𝐷KL(𝑃? �̂�) = ∑𝑛 𝑖=1 𝑝𝑖 𝑙𝑜𝑔 ( 𝑝𝑖 𝑝𝑖 ) where 𝑝𝑖 is the empirical probability of return bin 𝑖 in the test set, and �̂�𝑖 is the corresponding simulated probability. Metric Value Mean Squared Error (MSE) Mean Absolute Error (MAE) KL Divergence 0.00047 0.0143 0.027 Table 8: Validation metrics comparing forecasted vs empirical return distributions. Figure 6: Comparison of out-of-sample forecasted return distribution vs actual test data. 10- Results Discussion The model performs reasonably well in forecasting the shape of the return distribution. It captures the peak near zero (indicating calm markets) and the presence of fat tails (capturing extreme movements). The relatively low values of MSE and KL divergence suggest that the agent-based model generalizes well out-of-sample, at least over a one-year horizon. Some minor discrepancies remain, especially in the tail regions, potentially due to unexpected macroeconomic events or structural changes not modeled (e.g., regulatory shifts or exogenous P ro b a b il it y 0.3 Empirical - - - -Forecasted 0.2 0.1 0 −0.1 −5 · 10−2 0 5 · 10−2 0.1 0.15 0.2 Table Figure Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 10s (2025) 3331 https://internationalpubls.com shocks). Conclusion The forecasting experiment demonstrates that the proposed agent-based model, once properly calibrated, is capable of replicating both in-sample dynamics and generating credible out-of- sample predictions. 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