A Bertrand model with Brownian motion and behavioral errors

Abstract

This paper extends a random version of Bertrand model with Brownian motion. The model can be used to adjust price with changing economic environment, as it can serve to capture the behaviours of managers who seek to avoid errors in identifying the optimal response in dynamic price settings within their respective markets. The equilibrium of a stochastic game model is further discussed with the solution solved by Hamilton-Jacobi-Bellman equations, and the equilibrium exhibited fluctuations in the numerical simulation, with a constant value being observed. Finally, the model shows that organisations that are less efficient at managing errors, particularly those with a smaller market share, are more susceptible to failure.

Supplementary Information

The online version contains supplementary material available at 10.1038/s41598-025-32158-9.

Introduction

The Bertrand model, a foundational concept in industrial organization and microeconomics¹, elucidates price competition among companies producing homogeneous products. While the classical Bertrand model offers invaluable insight into competitive pricing strategies, it frequently relies on simplifying assumptions that may not fully represent the intricacies of real-world markets.

The objective of this study is to develop a stochastic game model of Bertrand competition, with a particular focus on how companies optimize their pricing strategies in a dynamic and uncertain environment^2,3. The introduction of stochastic elements will be explored determine how it alters the equilibrium outcomes in comparison to the deterministic Bertrand model with numerical simulation. Furthermore, management errors will be analyzed. These behavioral errors are captured by Brownian motion.

The paper is structured as follows: In Section 2, the relevant literature on the Bertrand model, behavioral errors and firm failure, Brownian motion and behavior is reviewed and the gap in the literature addressed by this study is identified. In Section 3, the stochastic dynamics model is constructed on the deterministic Bertrand model, and a duopoly of the stochastic model is discussed. In Section 4, we present detailed numerical simulations to illustrate the behavior of the model. In Section 5, decision errors and company bankruptcy are analyzed with Brownian motion. We conclude and discuss our results in section 6.

This paper contributes to a more nuanced understanding of price competition in dynamic, uncertain markets. A lacuna remains in the literature with regard to the comprehensive integration of stochastic differential game theory with the Bertrand model. The objective of this paper is to address this gap by developing a Brownian motion game model of Bertrand competition, thereby contributing to our understanding of dynamic price competition under uncertainty. The insights gained from this analysis may inform both theoretical developments in industrial organization and practical applications in business strategy and policy-making.

Literature review

Bertrand model

The Bertrand model, proposed by Joseph Bertrand in 1883⁴, provides an explanation for price competition among companies producing homogeneous products. In its most basic formulation, the model posits that in the context of two or more firms, prices will decline to marginal cost, and economic profit will be eliminated⁵. In a two-stage game, Kreps and Scheinkman (1983)⁶ demonstrated that the introduction of capacity constraints can result in Cournot outcomes. These results have been the subject of extensive research and debate among industrial organizations. Models of monopolistic competition, as developed by⁷, incorporate product differentiation to explain price-cost margins in competitive markets. The phenomenon of Bertrand competition in the context of incomplete information concerning rivals’ costs is examined, elucidating the impact of uncertainty on pricing strategies⁸. The equilibrium solution in a Bertrand dynamic game model is contingent on the pricing decisions of all the participating firms⁹. An analysis of dynamic price competition in alternating-move duopoly games is conducted, thereby demonstrating the possibility of tacit collusion¹⁰. Moreover, the equilibrium of the discrete system of the oligopoly model exhibits characteristics of chaos, as demonstrated by^11,12. The results of the Bertrand model are significantly influenced by the presence of differentiated goods or asymmetric information in the market¹³. It has been demonstrated that a source of chaos in a duopoly with price competition arises from either the degree of substitutability or complementarity between different varieties of products¹⁴. Substitution or complementarity can be regulated by the managers of the company. It depends on the type of management rule, and the consistent rational expectation perfect equilibrium of Bertrand was discussed¹⁵. However, in this paper, the equilibrium of Bertrand’s stochastic differential model of is further discussed.

Behavioral errors and firm failure

Behavioral errors play a significant role in firm failure, as is highlighted in the literature. They provide empirical estimates of filtering failure in a bankruptcy system, showing that Type II errors are more likely than Type I errors in Chapter 11¹⁶. Behavioral costs, stemming from cognitive imperfections and emotional influences, can lead to errors in decision-making by managers¹⁷. Deficient behavioral practices, such as failure to communicate, accounted for a significant percentage of complications in the referenced study¹⁸. Many models have been developed to predict Korean lodging firm failures, emphasizing the importance of understanding and predicting such failures¹⁹. The role of imitation and information in overcoming coordination failures has been explored, shedding light on how behavioral factors can impact firm outcomes²⁰ and how middle managers play a crucial role in avoiding corporate omission errors²¹. Hierarchical forecasting processes were investigated through a behavioral lens, identifying judgment biases that can affect forecasting accuracy²². Overconfidence is focused on as a common psychological attribute among entrepreneurs²³, leading to errors in judgement^24–26 that can result in firm failure. Overall, the literature underscores the importance of understanding and addressing behavioral errors to mitigate the risk of firm failure.

Brownian motion and behavior

Brownian motion (BM) is a phenomenon that has been studied extensively in various fields of science, including physics, finance, and biology^27–37. In relation to BM, the stability of colloids was discussed, highlighting the importance of understanding this phenomenon in the context of particle behavior³⁸. Recognition of how BM affects particle size analysis through sedimentation, emphasizes the impact of random motion on the accuracy of measurements³⁹. A nonlinear-drifted BM model with multiple hidden states is presented for predicting the remaining useful life of rechargeable batteries, showcasing the practical applications of this stochastic process in engineering⁴⁰. The analysis of hedging errors is referred to in models involving geometric fractional BM, shedding light on the rates of convergence and limiting distributions of errors in trading strategies⁴¹. An efficient numerical method for the solution of forward-backward stochastic differential equations driven by G-Brownian motion is proposed, offering a practical approach to handling stochastic processes in mathematical finance⁴². Geometric Brownian motion has been utilized for stock price forecasting during the coronavirus outbreak in Indonesia, showcasing the predictive power of this model under volatile market conditions⁴³. The conflation of BM and Levy processes^44,45 is explored in the Ornstein-Uhlenbeck process, highlighting the interplay between different stochastic components in modelling complex systems^46–49. Stochastic characteristics have been applied to a Lighthill-Whitham-Richards model perturbed by BM, demonstrating the versatility of this stochastic process in mathematical modeling⁵⁰. Overall, the literature on BM and its applications in various fields highlights the importance of understanding random motion in modeling dynamic systems and making accurate predictions on the basis of stochastic processes. This paper further discusses the behavioral errors associated BM.

Model analysis

The deterministic bertrand model

There are two companies in one market. They produce homogeneous goods. The demand function is linear: , where is price, a is the market long-term price target and positive constant. t is time, is output at t time. The function of total cost is assumed to be: , where is the marginal cost of price changes, and >0, b is the adjustment speed, and b>0. In this model, company inertia is activated by the cost of regulating its level of price. This regulation and control cost is , where k>0 is the regulation cost parameter. At t=0, the goal of each company is to maximize the profit⁵¹:

1

where r is the discount factor. The profit is subject to r and the initial condition pi(0).

The sufficient conditions for profit maximization are as follows:

2

where v is the company’s long-term price target, and fluctuates with market share. When the market is reaching maturity, the long-term price v will be steady in a steady state. The dynamics of the model can be expressed as follows:

3

The solutions of system (2) is by Matlab (derived see Appendix):

4

5

where is determined by the initial conditions p1(0), is determined by the initial conditions p2(0). When t→ꝏ, a company’s optimal level of price approaches .

The stochastic dynamics model

A price strategy error is defined as a stochastic element that influences the dynamic behavior functions of companies. In this section, we aim to capture behavioral errors using the diffusion term of BM. As a company attempts to identify the best price action in a dynamic setting with Brownian motion, the errors result from boundedly rational decision-makers making quick decisions with time pressure. Adding BM to the dynamic equations (2), we obtain stochastic differential equations:

6

Companies 1 and 2 will regulate their strategy by price for long-term targets. When , the price of the commodity increases. The demand for the product is smaller than the supply. When products are substitutes, because of competitors’ price, the price of the company decreases. When behavioral management errors arise, these errors have an immediate effect on prices. The error distributions are consistent for companies 1 and 2, but one company may be better at handling errors than the other company. This situation is captured by different Brownian motion parameters (i.e., ). When , the stochastic model returns to the deterministic dynamic model. It is clear that managers do not make errors. If decision makers make behavioral errors, and decision makers of company 1 are superior at dealing with these errors, then .

Given the game of companies 1 and 2 described above in the stochastic model, the solutions of and are as follows (derived see Appendix):

7

8

when p₁(0)>p₂(0), the valve of p₁(t) is bigger than p₂(t).

When t→∞, expected value of equation can be derived in the stochastic model:

9

The variance of the stochastic equation can also be derived:

10

We can get the ordinary BM of variable ³:

11

is a random variable. Unlike the deterministic Bertrand model, it does not converge to a constant variable. In the deterministic model, the optimal price of each company converges to . However, in stochastic model with BM caused by management error, the optimal price has a mean value of . Even with market management factor disturbance, the equilibrium is the same as the mean value in the deterministic Bertrand model. The variance deviation reflects the difference in the decision-maker errors. From formula (9), another characteristic of the model is that there is no relationship between the mean value in the random model with BM and the initial locations of the two companies. That is, the mean value is not sensitive to initial value⁵².

As previously assumed, when , company 1 is superior at handling behavior errors, and the limiting distribution variance is smaller than that of company 2. In reality, as described in system (2), the distribution converges to the same limit when time t is infinite. If , the ability to address behavior errors for two companies is equally capable. When this case occurs, the variance of each company is . The mean company price is v in monopoly model. In a duopoly, the price mean is close to . With a single firm, the limiting variance in the single Orstein–Uhlenbeck mean-reverting equation equals , compared with a value of in duopoly, more firms’ variance is smaller with a firm. Therefore, the variance decreases with the interaction effects of the two companies. That is, more companies’ variance is smaller with a company.

Stochastic duopoly model

We consider a duopoly decision in a mature market where the majority of sales are derived from the two companies, such as Company 1 and Company 2. These entities engage in competition with one another for market share through the investment of resources. Denote the market share of company 1 as x(t), and that of company 2 as y(t). Let, x(t)ϵ[0, 1] be the market share of company 1 with x(0)=x₀, and y(t)=1-x(t) be the market share of company 2 with y(0)=1-x₀. The subscript iϵ {1, 2} is used to reference the two companies. u_i(x(t), y(t)), t≥0 is the production rate for company i at time t, and is the expenditure-effectiveness parameter for company i. Let >0 be the market share churn parameter, where k_i>0 is the discount rate for company i, and C(u_i(t)) is the cost, parameterized as c_iu_i(t)², where c_i >0 and σ(x(t),y(t))dB_t is the Brownian motion function, whose mean value is zero. F_i is the value function for company i, and α_i, β_i are components of the value function, the dynamic model is given^53–55,

12

In a duopoly situation, the churn of market share for one company is contingent upon two key factors: cost expenditure, such as advertising expenditure. Each firm’s advertising aims to increase its own market share, while competitor advertising aims to reduce that of other firms. They take actions that maximize their own objectives while also considering their competitors’ actions. Additionally, they will interact dynamically for the foreseeable future. The extent to which products are perceived to be differentiated from one another. For example, a lack of perceived differentiation between brands often leads to a convergence of market shares. In a duopoly scenario, the fluctuation of market share for one firm results in a gain of market share for the other⁵⁶. The market share is defined as white noise, σ(x(t),y(t))dB_t. For dx(t)+dy(t)=0 and x(0)+y(0)=1, it is clear that x(t)+y(t)=1 for all t≥0. Thus, y(t)=1-x(t). Now, the market share of company 1 or company 2 can be used to completely describe the system dynamics. For simplicity, the market share of company 1 is adapted to present the system dynamics. Thus, u_i(x(t), y(t)), i=1,2 and σ(x(t),y(t)) can be rewritten as u_i(x(t)), and σ(x(t)). Thus,

13

With 0≤x(t)≤1. As long as u_i(x(t)), and σ(x(t)) are continuous functions on [0,1] and further that, u_i(x(t))≥0, xϵ[0,1]and, σ(x(t))>0, xϵ(0,1) and σ(0)=σ(1)=0. With these assumptions, there is a strictly positive drift at x=0 and a strictly negative drift at x(t)=1, that is,

14

Let m_i denote the industry sales income for company i. The objective functions are described by

15

Thus, each company seeks to maximize its expected, discounted profit stream subject to market share dynamics. Now, we intend to explore the closed-loop Nash equilibrium strategies, for which, the Hamilton–Jacobi–Bellman (HJB) equation for each company is formed^57,58. That is,

16

17

where, and u^*₁ and u^*₂ denote the competitor’s expenditure policies in equation (16) and (17), respectively. The optimal feedback expending decisions are obtained,

18

Since 0≤ x ≤ 1 and since it is reasonable to expect the expenditure decisions (18) can be reduced to

19

Substituting (19) in equations (16) and (17), the Hamilton-Jacobi equations are obtained as

20

21

Now consider the following particular forms for the value functions F₁=α₁+β₁x(t) and F₂=α₂+β₂(1-x(t)). These value functions are used in equations (20) and (21) to determine the unknown coefficients α₁, α₂, β₁, and β₂. The following system of equations can be solved for the unknown coefficients.

22

23

24

25

A unique solution to these equations, together with the requirements that β₁>0 and β₂ >0 will be shown to exist. For the symmetric case take, , , , , and Then, the four equations in (22–25) reduce to the following two,

26

There are two solutions for β. One is negative, which makes no sense. Thus, the remaining positive solution is the only correct one. This in turn gives the corresponding α. The solution is

27

where . According to the solution for the value function, the strategies specified in equation (18) decrease to those in (19). This finding validates the choice of (19) in deriving the value function. Note that when m = 0, the company makes zero profit, i.e., the value functions F₁=α+βx and F₂=α+β(1−x) are identically zero. In turn, this implies that the coefficients α and β are zero when m = 0. For asymmetric companies, the equations (22-25) are reexpressed in terms of a single variable β₁,

28

29

30

31

where

Once the value of out of the four solutions of the quartic equation (28-31) is obtained, the other coefficients can be obtained by solving for , and then .

A marginal increase in the value of expenditure or a reduction in its cost will result in an increase in the amount of expenditure. However, in contrast to the predictions of a monopoly model of expenditure, the value function decreases. This phenomenon arises because, in this particular market structure, all research and development expenditures are driven by competitive motivations. Indeed, the optimal level of expenditure would be zero if a single company were to possess both identical products. It can be argued that expenditure on research and development does not result in an overall increase in the size of the marketing pie; rather, it merely affects the distribution of resources within it. Consequently, an increase in expenditure results in a reduction in the value function.

Nevertheless, the same rationale does not extend to scenarios where m increases or r decreases. In such instances, it is accurate to conclude that the expenditure on research and development is less efficient, yet it is also accurate to conclude that the overall size of the market has increased. Although it is challenging to anticipate that the latter effect would outweigh the former, it has been demonstrated that an increase in m or a decrease in r enhances the value function.

The churn parameter serves to diminish the intensity of competitive activity. It may therefore be anticipated that an increase in should result in enhanced profitability, achieved through a reduction in expenditure. Indeed, it can be observed that only the constant α component of the value functions undergoes an increase, while the overall behavior of the value functions remains uncertain. It is possible to ascertain the precise circumstances under which an increase or decrease in the value function of a company will occur as a result of an increase in . In the event that a company’s market share is less than 50%, the effect on the company’s value function is invariably positive. However, if the market share of a company is greater than 50%, its value function can decrease because of an increase in if

Is satisfied. The reason is that when a company has a market share advantage over its rival, δ helps the rival unequally by tending to equalize market shares.

Analytical and numerical results

A comprehensive analysis of the stochastic Bertrand model is provided with numerical simulations to explore the effects of Brownian motion on pricing strategies, market dynamics, and company profitability. We first derive key properties of the equilibrium prices and their sensitivity to stochastic volatility. Next, we present detailed numerical simulations to illustrate the model’s behavior under various parameter settings.

Analytical insights

In the absence of stochasticity (σ=0), the model reduces to the classical Bertrand duopoly with differentiated products. Static Nash equilibrium prices are obtained by maximizing each company’s profit function, , with respect to p_i. Solving the first-order conditions:

32

yields the equilibrium prices:

This price reflects the balance between market demand, cost structures, and product substitutability. The corresponding equilibrium quantities and profits are:

When stochasticity is introduced with Brownian motion (σ>0), the price dynamics follow the SDE³:

33

The drift term, , pulls prices toward static equilibrium, whereas the stochastic term, , introduces random fluctuations proportional to the current price. This multiplicative noise structure implies that higher prices experience greater volatility, a realistic feature in markets where price levels amplify the impact of shocks.

To analyze the stochastic equilibrium, we consider the stationary distribution of prices. Assuming that the price process reaches a steady state, the Fokker–Planck equation for the probability density of firm i’s price is as follows:

34

Solving this equation analytically is complex because of multiplicative noise. However, for small σ, we can approximate the expected price and variance. The expected price is approximately equal to the static equilibrium price , as the mean of the BM is zero. The variance of the price, Var(p_i), is influenced by σ and α. Using a linear approximation around , the stationary variance is:

This result indicates that higher volatility σ increases price dispersion, while a faster adjustment speed α dampens fluctuations. The impact of stochasticity on expected profits is also significant. Since profits are nonlinear in prices, Jensen’s inequality suggests that increased price volatility reduces expected profits. Specifically, the profit function is concave in p_i, so:

35

Thus, firms face a profit penalty under uncertainty, with the magnitude depending on σ and the degree of product substitutability .

Finally, we explore the sensitivity of equilibrium outcomes to key parameters. Higher product substitutability intensifies price competition, lowering equilibrium prices and profits, as in the static case. However, in the stochastic setting, high amplifies the impact of price volatility, as shocks to one firm’s price significantly affect the competitor’s demand. Similarly, higher volatility σ increases the risk of price wars, where temporary price drops due to negative shocks trigger aggressive responses from competitors.

Numerical simulations

To quantify the model’s behavior and validate the analytical insights, we conduct numerical simulations using the Euler–Maruyama method to discretize the SDE:

36

where N(0,1) is a standard normal random variable. We simulate price paths over a time horizon of T=100 units, with a time step Δt=0.01. The baseline parameters are set as follows^59–63:

Demand parameters: a=100, b=2, =1.

Cost: c=10.

Stochastic parameters: α=0.5, σϵ{ 0.05,0.2, 0.5, 0.9 }.

Discount rate: .

These parameters ensure positive demand and profits while allowing for meaningful stochastic effects. We run 1,000 simulations to compute average price paths, price distributions, expected profits, and market shares.

The sample price paths for σ=0.05(a), σ=0.2(b), σ=0.5(c), and σ=0.9(d) are shown in Fig. 1. Prices fluctuate around the static equilibrium price (), with larger deviations for higher σ. The simulated price variance increases with σ, which is consistent with the analytical approximation. For σ=0.05, the standard deviation of prices is approximately 1.8; for σ=0.2, it increases to 7.2. The adjustment speed α also plays a critical role: increasing σ to 1.0 reduces the standard deviation by approximately 30, as prices revert more quickly to equilibrium. The expected profits of equilibrium are computed as the time averaged discounted profits over the simulation horizon:

37

Fig. 1.

Price paths σϵ{ 0.05,0.2, 0.5, 0.9 }.

For σ=0.05, the expected profit per firm is approximately 0.95 of the static equilibrium profit. For σ=0.2, this decreases to 0.85, reflecting the profit penalty due to volatility. The decline is more pronounced for higher , as increased substitutability amplifies the impact of price fluctuations on demand.

Market shares, defined as q_i/(q₁+q₂), exhibit greater volatility with higher σ. For σ=0.2, market shares fluctuate between 0.3 and 0.7 over time, indicating frequent shifts in consumer preference driven by stochastic price differences. This volatility can destabilize market positions, particularly in industries with high substitutability.

Sensitivity analysis

We vary key parameters to assess their impact:

Doubling volatility σ from 0.1 to 0.2 increases price variance fourfold and reduces expected profits between 0.1 and 0.15. Companies could respond by adopting more conservative pricing strategies, targeting higher markups to hedge against volatility. Increasing substitutability from 1 to 1.5 lowers equilibrium prices by 0.10 and amplifies the profit penalty from volatility. This suggests that stochasticity exacerbates competitive pressure in markets with closely substitutable products. A higher adjustment speed α reduces price volatility but has a limited effect on expected profits, as the profit function is less sensitive to small price deviations near the equilibrium.

Robustness checks

To ensure robustness, we test alternative volatility structures, such as additive noise ((). The results are qualitatively similar, but multiplicative noise produces more realistic price dynamics, as volatility scales with price levels. We also simulate correlated Brownian motions (correlation coefficient ) to model common market shocks. Correlated noise increases price comovement, reducing the frequency of price wars but amplifying profit losses during negative shocks.

The analytical and numerical results highlight the critical role of stochasticity in shaping competitive outcomes. Volatility increases price dispersion and reduces expected profits, with stronger effects in markets with high product substitutability. Companies may mitigate these effects by adjusting pricing strategies, such as by targeting higher markups or investing in mechanisms to reduce demand uncertainty. The simulations also reveal the dynamic nature of market shares under stochastic conditions, suggesting that firms must remain agile to maintain competitive positions.

Decision errors and company bankruptcy

In a free market, company bankrupt is common. One of the reasons the company exited the market was the lack of experience of its managers. We use BM to illustrate the exit process. When a new market emerges, some companies will enter with an initial pricing. The initial price fluctuates around the long-term optimal price level, as shown in Fig. 1. Assume that companies are dissymmetric, and that company 1 has an the initial strategic advantage, such that p₁(0)>p₂(0)>0. If p_i(t)=0, one company will exit the market. From formulas p₁(t) and p₂ (t), it is clear that the optimal price of company 1 is greater than that of company 2. This also indicates that the probability of exit from the market is high for company 2.

Now, we further analyze the probability that company 2 will exit the market. We assume that . is defined as company 1’s exit period, and is defined as company 2’s exit period. If the initial value of company 1 is greater than that of company 2 in period 0, the probability that > is 1. This means that company 2 always exits the market before company 1 does. However, a chance for company 1 to exit the market also exists, and its probability of exiting from the market weakens with the exit of company 2. When company 2 exists in the market, company 1 rapidly develops toward an equilibrium optimal price. This development helps companies slow their exit from the market.

How to calculate the precise probability that company 2 will exit the market is noteworthy. Assuming that the initial value difference is small enough for company 1 and company 2, such that . When t→ꝏ, the probability of company 2 exiting the market is less than 1. That is, . We further derive the likelihood of company 2 exiting the market by calculating the lower bound on the probability. The lower bound of company 2 exiting the market is . The value of the lower bound is also viewed as a discount factor of company 2. That is , and When v and p₂ (0) decreases, or when increases, the lower bound increases. That is, for company 2, when it is larger in market share, it has a greater chance to survive. Bertrand competition can result in higher industry output, and it can shift production from a less efficient company to a more efficient company⁶⁴. When behavior errors diminish, it indicates that company 2 can do better in a large market. It has been proved that a large company is less likely to fail than a small one^65,66.

Conclusions

A review of the literature reveals that Brownian motion is the first method used in the Bertrand model. In this study, a Bertrand model including behavioral errors of management, which is modeled with BM from the deterministic dynamic model, is developed. Furthermore, the equilibrium of the stochastic Bertrand model is discussed and numerically simulated. After a behavioral error is made, the company can immediately regulate the price to a new equilibrium. The regulation price does not face an additional cost.

The findings from this study make several contributions to the current literature. First, the model indicates that the price level is a stochastic variable in the dynamic Bertrand model. It also indicates that whether a symmetric company or not, the solution with the stochastic model can be solved. Second, the equilibrium of the stochastic Bertrand model fluctuated in the numerical simulation. Finally, the results show that the probability of company failure decreases with larger markets and companies that are larger and more efficient at handling behavioral errors. The increments of the BM model (Equation 12)are independent over time. In real-world scenarios, increments may not be truly independent. For example, the price movements of products are often autocorrelated, meaning that past price movements can influence future movements. The delay of the model will be discussed in the future.

Author contributions

Bingyuan Gao and Xi Gao wrote the main manuscript text. Shaohua He improved the language of the manuscript. All authors reviewed the manuscript.

Funding

This work was supported by Doctor Scientific Research of Yuncheng University under Grant [YQ-2023077] and Shanxi Scholarship Council of China under Grant [20221857].

Data availability

The datasets used and/or analysed during the current study are available from the corresponding author on reasonable request.

Declarations

Competing interests

The authors declare no competing interests.