## Figures

## Abstract

In nature and human societies, the effects of homogeneous and heterogeneous characteristics on the evolution of collective behaviors are quite different from each other. By incorporating pair pattern strategies and reference point strategies into an agent-based model, we have investigated the effects of homogeneous and heterogeneous investment strategies and reference points on price movement. In the market flooded with the investors with homogeneous investment strategies or homogeneous reference points, large price fluctuations occur. In the market flooded with the investors with heterogeneous investment strategies or heterogeneous reference points, moderate price fluctuations occur. The coexistence of different kinds of investment strategies can not only refrain from the occurrence of large price fluctuations but also the occurrence of no-trading states. The present model reveals that the coexistence of heterogeneous populations, whether they are the individuals with heterogeneous investment strategies or heterogeneous reference points of stock prices, is an important factor for the stability of the stock market.

**Citation: **Xu W-J, Zhong C-Y, Ren F, Qiu T, Chen R-D, He Y-X, et al. (2023) Evolutionary dynamics in financial markets with heterogeneities in investment strategies and reference points. PLoS ONE 18(7):
e0288277.
https://doi.org/10.1371/journal.pone.0288277

**Editor: **Vygintas Gontis,
Vilnius University, LITHUANIA

**Received: **August 15, 2022; **Accepted: **June 24, 2023; **Published: ** July 17, 2023

**Copyright: ** © 2023 Xu et al. This is an open access article distributed under the terms of the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original author and source are credited.

**Data Availability: **All relevant data are within the paper.

**Funding: **The author(s) received funding from Social Science Foundation of Zhejiang Province (Grant No. 21NDJC098YB), National Social Science Foundation of China (Grant No. 20BJL147), Major Program of National Social Science Foundation of China (Grant No.22&ZD073), Natural Science Foundation of Zhejiang Province (Grant No. LY23G030001), National Natural Science Foundation of China (Grant Nos. 71773105, 11865009, 71871094, 71631005) for this work. The funders had no role in study design, data collection and analysis, decision to publish, or preparation of the manuscript.

**Competing interests: ** The authors have declared that no competing interests exist.

## Introduction

In financial markets, the stock price movement is not only related to good or bad news for a typical company but also to people’s subjective cognitions [1–4]. For a homogeneous population, common beliefs would lead to similar buying-selling actions. Herding behaviors are easy to occur [5–10]. For a heterogeneous population, whether the individuals with heterogeneous investment strategies or heterogeneous price expectations, they would show different buying and selling behaviors. An equilibrium is easy to be reached in the stock market [11–17]. Understanding the effects of homogeneous and heterogeneous populations on the price movement is quite important for the risk management and the construction of an efficient market [18–24].

In exploring the possible strategic effects and psychological effects on stock price movement, a variety of agent-based models have been borrowed to model people’s social and economic behaviors [25–32], among which the minority game and the majority game are mainly used to simulate people’s trading behaviors [33–38]. Depending upon the minority game model, the effects of response time on price movement have been investigated [39]. The delayed response to historic information leads to the lagged price fluctuations. Depending upon the majority game, the effects of imitation on price movement have been investigated [40, 41]. Buying and selling the stocks according to most people’s choices leads to the occurrence of herding effects in the stock market. Depending upon the mixed minority game, the majority game and the dollar-game [42], the effects of sophisticated individuals on the evolution of stock prices have been investigated. Different kinds of trader-trader interactions lead to typical stylized facts in the stock markets [43–48].

In the study of the effects of investment strategies on the evolution of stock prices, the pair pattern strategies and the reference point strategies are two kinds of typical strategies [49–52]. Depending upon the pair pattern strategies, people buy and sell the stocks frequently, which leads to the occurrence of a power-law return distribution similar to that in real stock markets [53–57]. Different from the pair pattern strategies, with which an individual makes a buying or selling decision according to the history of price movement, the reference point strategies are myopic strategies. They depend on people’s subjective cognitions and provide them anchors to simplify their complex decision-making processes [52, 58, 59]. In real trading markets, the reference point strategies help us finish the transactions in a simplified way.

In the present work, we incorporate the investors with pair pattern strategies and reference point strategies into an agent-based model. The role of homogeneous and heterogeneous investment strategies in price fluctuations is investigated. Different from the agent-based models introduced in references [52, 60], in which the effects of market impact on the competitive advantage of investment strategies and the occurrence of self-reinforcing feedback loops have been investigated, in the present model, we have investigated the role of homogeneous and heterogeneous investment strategies and reference points in price fluctuations. Our main findings are as follows.

- (1)The existence of the individuals with heterogeneous investment strategies or heterogeneous reference points of stock prices can effectively suppress the price fluctuations. As nearly all the individuals have similar investment strategies, large fluctuations occur. As the individuals with heterogeneous investment strategies come into the market, the price fluctuations become moderate.
- (2)As compared with the situation where there is only one kind of investment strategy, the coexistence of different kinds of investment strategies not only helps the stock market refrain from large fluctuations but also being stuck in a no-trading state.
- (3)A theoretical analysis indicates that the stability of the stock market is governed by the coexistence of different kinds of investment strategies. The individuals with homogeneous investment strategies or homogeneous reference points drive the system far away from the equilibrium and lose more. The individuals with heterogeneous investment strategies or heterogeneous reference points draw the system back to the equilibrium and gain more. The coexistence of the people with different kinds of investment strategies can promote the inhomogeneity of the investment strategies, which stabilizes the stock prices.

The paper is organized as follows. In section 2, an agent-based model with pair pattern strategies and reference point strategies is introduced. In section 3, the stylized facts in price return distributions are reproduced and the numerical results are presented. In section 4, a theoretical analysis is given. The conclusions are drawn in section 5.

## The model

### Pair pattern strategies and reference point strategies

Similar to that in references [42, 51], a strategy consisting of a pair of patterns or history signals is incorporated into the present model. For a *M*-bit long history of price changes, there are a total of 2^{M} possible patterns or history signals recording the latest price changes. Since a strategy’s buying pattern and selling pattern should not be the same, there are 2* ^{M}*(2

*− 1) possible pairs of patterns in total. Therefore, the pair pattern strategy space consists of a series of buying-and-selling strategy pairs. A buying or selling strategy consists of a*

^{M}*M*-bit long binary array. For example, if 1 represents an increase in prices, 0 represents a decrease in prices, for a typical individual

*i*with

*M*= 3, his buying and selling strategies are and respectively, which means, facing the latest price history ↑↑↓, on condition that the number of shares in individual

*i*’s hand is below the number of maximal shares

*K*

_{max}that he is permitted to have, he would buy a share. Facing the latest price history ↑↓↑, on condition that the number of shares in individual

*i*’s hand is above the minimum number of shares

*K*

_{min}that he is permitted to have, he would sell a share. Or else, individual

*i*would take a hold. Therefore, for an individual with pair pattern strategies, although he owns

*n*

_{S}pairs of buying and selling strategies, he only uses one pair of buying and selling strategies at a typical time. There are three kinds of decisions he could make: buying, selling or taking a hold. Initially, each individual randomly chooses

*n*

_{S}strategies from the full pair pattern strategy space. These strategies are kept fixed throughout the game. Each individual

*i*keeps track of the cumulative performance of all her pair pattern strategies by assigning a virtual score to each strategy. The real buying or selling decision is made according to the strategy with the highest score. The buying and selling actions are symmetric, which means, an individual is permitted to open a position by buying or open a position by selling.

For the pair pattern strategies, the degree of homogeneity or heterogeneity in investment strategies is related to the value of the memory size *M*. For a small *M*, nearly all the individuals have similar strategies. For a large *M*, each individual has his own typical investment strategies which are different from other people’s investment strategies. Therefore, an increase in *M* is similar to an increase in the degree of heterogeneities in the pair pattern strategies.

The reference point strategy space consists of a series of expected prices, called reference points *P*^{ref} in the present model [52]. If the latest price *P*(*t*) is lower than an individual *i*’s reference point, , on condition that the number of shares in his hand is below *K*_{max}, he would buy a share with probability for and *Γ* = 1 for . If the latest price is higher than individual *i*’s reference point, , on condition that the number of shares in his hand is above *K*_{min}, he would sell a share with probability . Or else, individual *i* would take a hold.

Each individual’s reference point is related to the average price in the latest Δ long price history and his risk tolerance *g*_{i}. An individual *i*’s reference point should be within the range of , in which *α* is a constant. Or else, a new reference point will be randomly chosen from the range of .

The degree of homogeneity or heterogeneity in reference point strategies is determined by the maximum value of risk tolerance *g*^{max}. Facing the latest price *P*(*t* − 1), the difference in the number of investors buying and selling the stocks should be
(1)
in which *N*_{Pref>P} and *N*_{Pref<P} are the number of investors buying and selling the stocks respectively. For quite a small *g*^{max}, i.e. *g*^{max} = 1, all the reference points are within a small range
(2)
in which is the averaged value of stock prices in the latest Δ*t* steps. For quite a large *g*^{max}, i.e. *g*^{max} = *N*, all the reference points are within a wide range
(3)

Therefore, for a small *g*^{max}, nearly all the individuals have the same strategy. For a large *g*^{max}, each individual has his own typical investment strategies which are different from other people’s investment strategies. Therefore, an increase in *g*^{max} is similar to an increase in the degree of heterogeneities in the reference point strategies.

Similar to that in references [35, 61, 64], the effects of an individual’s limited wealth in real financial markets are incorporated into the present model. As an individual *i* has made his buying (*a*_{i} = +1), selling (*a*_{i} = -1) or taking a hold (*a*_{i} = 0) decision at time *t*, his position *k*_{i}(*t*) becomes
(4)

In the present model, because of the limited wealth, an individual’s maximum position *K*, i.e. ∣*k*_{i}(*t*)∣ ≤ *K*, is kept fixed throughout the paper. For an individual *i* with *K* = 1, once the maximal number of shares, *K*_{max} = 1, has been reached, the further buying decision would be ignored. Once the minimal number of shares, *K*_{min} = −1, has been reached, the further selling decision would be ignored [35, 64]. For example, for an individual *i* with buying strategy and selling strategy , if the history ↑↑↓ occurs, on condition that *k*_{i}(*t*) = 0 or *k*_{i}(*t*) = −1, individual *i* will buy a share. Or else, individual *i* will do nothing. If the history ↑↓↑ occurs, on condition that *k*_{i}(*t*) = 0 or *k*_{i}(*t*) = 1, individual *i* will sell a share. Or else, individual *i* will do nothing.

### The evolution of stock prices

In the present model, the total population *N* consist of *ρ*_{ref}*N* individuals with reference point strategies and (1 − *ρ*_{ref})*N* individuals with pair pattern strategies, in which *ρ*_{ref} is the ratio of individuals with reference point strategies. Although each individual only owns buying strategies and selling strategies, he has three kinds of decisions which could be made: buying, selling or taking a hold. After all the individuals have made their buying, selling or taking a hold decisions, i.e. *a*_{i} = +1, -1 or 0, the price is updated according to the equation
(5)
in which , *β* is a pre-given constant and *N* is the total population. As time goes on, the stock price evolves continuously.

### The standard deviation of stock prices

The standard deviation *σ*_{P} of stock prices reflects the degree of price fluctuations, which can be calculated according to the equation
(6)
in which Δ*T* is the time interval within which the values of [*P*(*t*) − *P*(*t* − 1)]^{2} are accumulated.

### The predictability of stock prices

In an efficient market, the prices should be unpredictable. However, in a real stock market, the stock prices are usually predictable because of the lagged information or people’s herding behaviors. The predictability of stock prices can be measured according to the differences between the probabilities of an increase in prices and a decrease in prices as some typical price history occurs. Supposing that the price history *χ* occurs with probability *ρ*(*χ*) and the mean value of price changes facing the price history *χ* is 〈Δ*P*|*χ*〉, we can get the predictability *H* [52, 61]
(7)

A larger value of *H* indicates that the stock prices are more predictable. For example, following a given price history ↓ ↓ ↑, more ↑ than ↓ or more ↓ than ↑ would lead to a larger *H*. Or else, the stock prices would be unpredictable and the value of *H* would be *H* ∼ 0.

### The averaged wealth of the population with a typical kind of investment strategies

For an individual *i*, if the initial cash in his hand is *C*_{i}(*t* = 0) = 0 and the number of shares in his hand is *k*_{i}, the accumulated wealth at time *T* is got according to the equation [52, 61]
(8)
in which and . *P*^{sell} and *P*^{buy} are the real prices that individual *i* buys and sells the share. *P*^{instant} is the instant price in the market. The averaged wealth of the population with a typical kind of investment strategy can be obtained
(9)

## Simulation results and discussions

### Reproduction of the stylized facts similar to that in the real financial markets

In the study of the real financial markets, the distributions of price returns are usually investigated, which reflect the typical characteristics of the price movement. For a random price movement, the distribution of price returns is close to a Normal distribution. For an autocorrelated price movement, the distribution of price returns is close to a power-law distribution [1, 2, 62, 63].

*Gontis* *et* *al*. have investigated the distributions of price returns in the NYSE and FOREX markets [1]. They have found that herding interactions can reproduce the power-law properties and the scaling of return intervals found in real financial markets. *Watorek* *et* *al*. have investigated the price return distributions of currency exchange rates, cryptocurrencies, and contracts [2]. They have found that the characteristics of the tails of the return distributions are closely related to the characteristics of the time scales.

In the present work, we firstly examine whether the present model can reproduce the distributions of price returns, such as normal distributions and power-law distributions in real financial markets [1, 2].

Fig 1(a) and 1(b) show that, as all the investors adopt pair pattern strategies, the probability distribution of price returns is like a power-law distribution, the tail of which is satisfied with the probability density function *PR*(|*R*|) ∼ |*R*|^{−γ}. An increase in memory size *M* leads to a decrease in the exponent *γ*. As we plot a fitting line to the data, we find that, for *M* = 3, *γ* ∼ 4.8. For *M* = 5, *γ* ∼ 4.

(a)The probability distribution of price returns *R* for the ratio of investors with reference point strategies *ρ*_{ref} = 0 and the memory size *M* = 3 (circles), 5(squares); (b) the probability distribution of absolute price returns |*R*| for *ρ*_{ref} = 0 and *M* = 3 (circles), 5(squares); (c)The probability distribution of price returns *R* for *M* = 3 and *ρ*_{ref} = 0 (circles), 0.5 (triangles); (d) the probability distribution of absolute price returns |*R*| for *M* = 3 and *ρ*_{ref} = 0 (circles), 0.5 (triangles). Other parameters are: the total population *N* = 5000, the number of strategies for each investor with pair pattern strategies *n*_{S} = 2, the time interval for getting is Δ = 10, the gene for the investors with reference point strategies *g* = 5000, the maximum and minimum number of shares for each investor *K*_{max} = 1 and *K*_{min} = -1, the constant *α* = *β* = 10. The fitting lines in (b) are *PR*(|*R*|) ∼ |*R*|^{−4.8} (green line) and *PR*(|*R*|) ∼ |*R*|^{−4}(orange slash line) respectively. The fitting lines in (d) are *PR*(|*R*|) ∼ |*R*|^{−4.8}(green line) and *PR*(|*R*|) ∼ |*R*|^{−8}(magenta slash dotted line) respectively.

Fig 1(c) and 1(d) show that, for a given memory size *M* = 3, the probability distribution of price returns is closely related to the ratio of the investors with reference point strategies *ρ*_{ref}. An increase in *ρ*_{ref} leads to an increase in the exponent *γ*. As we plot a fitting line to the data, we find that, for *ρ*_{ref} = 0, *γ* ∼ 4.8. For *ρ*_{ref} = 0.5, *γ* ∼ 8.

Such results indicate that the probability distribution of price returns in the present model is closely related to the memory size *M* and the ratio of the investors with reference point strategies *ρ*_{ref}. Compared with the situation where there is a small *M* and a small *ρ*_{ref}, a large *M* and an intermediate *ρ*_{ref} lead to a broader probability distribution of price returns.

### The evolutionary dynamics of stock prices

In the present model, we are especially concerned about how the heterogeneities in investment strategies and reference points affect the price fluctuations. In the following, we firstly examine how the coupling of the memory size *M* of the investors with pair pattern strategies, the maximum gene *g*^{max} of the investors with reference point strategies and the ratio *ρ*_{ref} of the investors with reference point strategies affects the time-dependent behaviors of the stock prices.

Fig 2(a) shows that, as all the investors adopt pair pattern strategies, the price movement is closely related to the memory size *M*. For *M* = 2, the price has a large fluctuation. For *M* = 5, the price fluctuation becomes ease. In the present model, a large *M* indicates that the investors with pair pattern strategies have more opportunities to choose his personal investment strategies. Given a large *M* and *n*_{S} = 2, it is quite possible that all the investors have different investment strategies. The results in Fig 2(a) indicate that the heterogeneities in pair pattern strategies suppress the price fluctuations.

The time-dependent price *P* (a) for the system with only pair pattern strategies and the memory size *M* = 2 (slashes), 5 (slash dotted lines); (b) for the system with only reference point strategies and the maximum gene *g*^{max} = 100 (circles), 1000 (squares); (c) for the system with the coexistence of pair pattern strategies and reference point strategies, *ρ*_{ref} = 0.1(slashes), 0.4(slash dotted lines), *M* = 3, *g*^{max} = 100; (d) for the system with the coexistence of pair pattern strategies and reference point strategies, *ρ*_{ref} = 0.1(slashes), 0.5(slash dotted lines), *M* = 3, *g*^{max} = 1000. Other parameters are: the total population *N* = 1000, the number of strategies for each investor with pair pattern strategies *n*_{S} = 2, the time interval for getting is Δ = 10, the maximum and minimum number of shares for each investor *K*_{max} = 1 and *K*_{min} = -1, the constant *α* = *β* = 10.

Fig 2(b) shows that, as all the investors adopt reference point strategies, the price movement is closely related to the maximum gene *g*^{max}. For *g*^{max} = 100, the price has a zigzag fluctuation. For *g*^{max} = 1000, the price becomes stable. In the present model, a large *g*^{max} indicates that the reference point strategies scatter about a large range of . Given a large *g*^{max} ∼ *N*, it is quite possible that all the investors have different reference points. The results in Fig 2(b) indicate that the heterogeneities in reference points suppress the price fluctuations.

Fig 2(c) and 2(d) show that, as the investors with pair pattern strategies coexist with the investors with reference point strategies, the price movement is closely related to the ratio of the investors with pair pattern strategies and reference point strategies. An increase in the ratio of the investors with reference point strategies leads to a decrease in the price fluctuations. As compared with the situations in Fig 2(a) and 2(b), we find that a large *M* coupled with a large *g*^{max} can not only inhibit the occurrence of large price fluctuations but also the occurrence of no-trading states.

Such results indicate that the heterogeneities in investment strategies and reference points have a great impact on the evolution of stock prices. Homogeneous populations promote price fluctuations while heterogeneous populations suppress price fluctuations. In the heterogeneous environment where the investors with pair pattern strategies coexist with the investors with reference point strategies, the pair pattern strategies drive the system away from the equilibrium while the reference point strategies draw the system back to the equilibrium. Heterogeneities in the present model are beneficial for stabilizing the stock prices.

In order to get a clear view of the relationship between the price fluctuations and the heterogeneities in pair pattern strategies and reference points, in Fig 3(a) and 3(b) we plot the standard deviation *σ*_{P} of stock prices as a function of the memory size *M* and the maximum gene *g*^{max} respectively.

Fig 3(a) shows that, as all the investors adopt pair pattern strategies, the price fluctuations are determined by the memory size *M*. As *M* increases from *M* = 2 to *M* = 10, the standard deviation *σ*_{P} of stock prices decreases from *σ*_{P} ∼ 22 to *σ*_{P} ∼ 0.1. An increase in the ratio *ρ*_{ref} of the investors with reference point strategies leads to an overall decrease of *σ*_{P} within the whole range of 2 ≤ *M* ≤ 10. An increase in *ρ*_{ref} does not affect the changing tendency of *σ*_{P} vs *M*.

The standard deviation *σ*_{P} of stock prices (a) as a function of the memory size *M* for the maximum gene of the investors with reference point strategies *g*^{max} = 1000 and the ratio of the investors with reference point strategies *ρ*_{ref} = 0 (circles), 0.8 (squares); (b) as a function of *g*^{max} for *M* = 3 and *ρ*_{ref} = 1 (circles), 0.8 (squares). Other parameters are: the total population *N* = 1000, the number of strategies for each investor with pair pattern strategy *n*_{S} = 2, the time interval for getting is Δ = 10, the maximum and minimum number of shares for each investor *K*_{max} = 1 and *K*_{min} = -1, the constant *α* = *β* = 10. Final data are obtained by averaging over 100 runs and 10^{4} times after 10^{5} relaxation times in each run.

Fig 3(b) shows that, as all the investors adopt reference point strategies, the price fluctuations are determined by the maximum gene *g*^{max}. There exist two critical points and . As *g*^{max} increases from *g*^{max} = 1 to *g*^{max} = 500, *σ*_{P} keeps a fixed value of 2 × 10^{5}. As *g*^{max} increases from *g*^{max} = 500 to *g*^{max} = 1100, *σ*_{P} drops quickly from *σ*_{P} ∼ 2 × 10^{5} to *σ*_{P} ∼ 0.01. As *g*^{max} increases from *g*^{max} = 1100 to *g*^{max} = 1800, *σ*_{P} keeps the value of *σ*_{P} ∼ 0.01. A decrease in the ratio *ρ*_{ref} of the investors with reference point strategies leads to a decrease in the critical points and . Within the range of , a decrease in *ρ*_{ref} leads to an overall decrease of *σ*_{P}. Within the range of , a decrease in *ρ*_{ref} leads to an overall increase of *σ*_{P}. A decrease in *ρ*_{ref} does not affect the changing tendency of *σ*_{P} vs *g*^{max}.

Such results indicate that both heterogeneous investment strategies and heterogeneous reference points have a great impact on the price movement. For the system with either pair pattern strategies or reference point strategies, the existence of the individuals with heterogeneous investment strategies or heterogeneous reference points suppresses the price fluctuations. For the system with the coexistence of pair pattern strategies and reference point strategies, the price fluctuations may be promoted within some range and may be suppressed within other range.

In a predictable market, people usually make a deal according to the historic information, which may lead to people’s common behaviors. In the present model, even if people have the same historic information, because they have heterogeneous investment strategies or heterogeneous reference points, it is quite possible that their common behaviors may be inhibited. In order to examine whether the reduction of price fluctuations in the present model results from the reduction of the predictability of price movement, in Fig 4(a) and 4(b) we plot the predictability *H* of stock prices as a function of the memory size *M* and the maximum gene *g*^{max} respectively.

The predictability *H* of stock prices (a)as a function of the memory size *M* for the maximum gene of the investors with reference point strategies *g*^{max} = 1000 and the ratio of the investors with reference point strategies *ρ*_{ref} = 0 (circles), 0.8 (squares); (b)as a function of *g*^{max} for *M* = 3 and *ρ*_{ref} = 1 (circles), 0.8 (squares). Other parameters are: the total population *N* = 1000, the number of strategies for each investor with pair pattern strategies *n*_{S} = 2, the time interval for getting is Δ = 10, the maximum and minimum number of shares for each investor *K*_{max} = 1 and *K*_{min} = -1, the constant *α* = *β* = 10. Final data are obtained by averaging over 100 runs and 10^{4} times after 10^{5} relaxation times in each run.

Fig 4(a) shows that, as all the investors adopt pair pattern strategies, the predictability *H* of stock prices is closely related to the memory size *M*. As *M* increases from *M* = 2 to *M* = 10, *H* decreases from *H* ∼ 6.8 to *H* ∼ 0.05. An increase in the ratio *ρ*_{ref} of the investors with reference point strategies leads to an overall decrease in *H* within the whole range of 2 ≤ *M* ≤ 10. An increase in *ρ*_{ref} does not affect the changing tendency of *H* vs *M*. Comparing the results in Fig 4(a) with the results in Fig 3(a), we find that a more predictable market has a larger price fluctuation.

Fig 4(b) shows that, as all the investors adopt pair pattern strategies, the predictability *H* of stock prices is closely related to the maximum gene *g*^{max}. There exist two critical points and . As *g*^{max} increases from *g*^{max} = 1 to *g*^{max} = 500, *H* keeps a fixed value of 1 × 10^{5}. As *g*^{max} increases from *g*^{max} = 500 to *g*^{max} = 1100, *H* drops quickly from *H* ∼ 1 × 10^{5} to *H* ∼ 0.002. As *g*^{max} increases from *g*^{max} = 1100 to *g*^{max} = 1800, *H* keeps the value of *H* ∼ 0.002. A decrease in the ratio *ρ*_{ref} of the investors with reference point strategies leads to a decrease in the critical points and . Within the range of , a decrease in *ρ*_{ref} leads to an overall decrease of *H*. Within the range of , a decrease in *ρ*_{ref} leads to an overall increase of *H*. A decrease in *ρ*_{ref} does not affect the changing tendency of *H* vs *g*^{max}. Comparing the results in Fig 4(b) with the results in Fig 3(b), we find that a more predictable market has a larger price fluctuation.

Such results indicate that the predictability of price movement is closely related to the heterogeneities in investment strategies and reference points. For the system with either pair pattern strategies or reference point strategies, the heterogeneities are quite possible to make the market become unpredictable, which is similar to the situation in an efficient market. For the system with the coexistence of pair pattern strategies and reference point strategies, the price movement may become more predictable within some range and may become more unpredictable within other range.

The price fluctuations are closely related to the predictability of price movement, which can be understood as follows. For a typical price history, a bias in the subsequent price changes can be used as a signal to predict the price movement [36, 61]. In the present model, the levels of such a bias are measured by parameter *H* and the levels of price fluctuations are measured by parameter *σ*_{P}. Similar to that in the minority game mechanism [29, 64–67], in the process of exploiting the predictable information, for a small memory size *M*, the crowd effect leads to a large price fluctuation. For a large *M*, the crowd-anticrowd effect makes the system become stable. In the present model, the crowd effect makes the predictable system become unstable. For example, facing the signal indicating that the price would rise, most of the investors would make a buying decision, which would lead to a quick increase in the prices. Facing the signal indicating that the price would drop, most of the investors would make a selling decision to avoid losses, which would lead to a quick decrease in the prices. Therefore, a higher level of predictability corresponds to a larger value of price fluctuation. For a heterogeneous population, facing the same history signal, the number of the investors making a buying decision is quite possible to be the same as the number of the investors making a selling decision. Therefore, there would be a low level of predictability and a small value of price fluctuation. For a homogeneous population, facing the same history signal, it is quite possible that most of the investors make the same buying or selling decisions. Therefore, there would be a high level of predictability and a large value of price fluctuation.

In order to find the advantage of the coexistence of different kinds of investment strategies in the present model, in Fig 5(a)–5(d) we plot the averaged wealth of the investors with pair pattern strategies and reference point strategies respectively.

(a)The averaged wealth of the investors with pair pattern strategies as a function of the memory size *M* for the maximum gene of the investors with reference point strategies *g*^{max} = 1000 and the ratio of the investors with reference point strategies *ρ*_{ref} = 0 (circles), 0.8 (squares); (b) as a function of *g*^{max} for *M* = 3 and *ρ*_{ref} = 0.8 (squares); (c)the averaged wealth of the investors with reference point strategies as a function of *M* for *g*^{max} = 1000 and *ρ*_{ref} = 0.8 (squares); (d) as a function of *g*^{max} for *M* = 3 and *ρ*_{ref} = 1 (circles), 0.8 (squares). Other parameters are: the total population *N* = 1000, the number of strategies for each investor with pair pattern strategies *n*_{S} = 2, the time interval for getting is Δ = 10, the maximum and minimum number of shares for each investor *K*_{max} = 1 and *K*_{min} = -1, the constant *α* = *β* = 10. Final data are obtained by averaging over 100 runs and 10^{4} times after 10^{5} relaxation times in each run.

Fig 5(a) and 5(c) show that, as all the investors adopt pair pattern strategies, the wealth of the investors with pair pattern strategies is closely related to the memory size *M*. There exists a critical point *M*_{c} = 4. As *M* increases from *M* = 2 to *M* = 4, increases quickly from to . As *M* increases from *M* = 4 to *M* = 10, increases slowly from to . An increase in the ratio of the investors with reference point strategies does not affect the critical point *M*_{c} = 4. As *M* increases from *M* = 2 to *M* = 4, increases from to while the wealth of the investors with reference point strategies firstly increases from to and then drops from to . As *M* increases from *M* = 4 to *M* = 10, increases from to while decreases from to . Within the range of *M* ≤ *M*_{c}, the coexistence of pair pattern strategies and reference point strategies is beneficial for the investors with reference point strategies.

Fig 5(b) and 5(d) show that, as all the investors adopt reference point strategies, the wealth of the investors with reference point strategies is closely related to the maximum gene *g*^{max}. There exist two critical points and . Within the range of *g*^{max} < 500, fluctuates within the range of . As *g*^{max} increases from *g*^{max} = 500 to *g*^{max} = 700, increases from to . As *g*^{max} increases from *g*^{max} = 700 to *g*^{max} = 1800, increases from to .

A decrease in the ratio *ρ*_{ref} of the investors with reference point strategies leads to a decrease in the critical points and . For *ρ*_{ref} = 0.8, as *g*^{max} increases from *g*^{max} = 1 to *g*^{max} = 300, the wealth of the investors with pair pattern strategies fluctuates around while the wealth of the investors with reference point strategies fluctuate within the range of . As *g*^{max} increases from *g*^{max} = 300 to *g*^{max} = 500, decreases from to while increases from to . As *g*^{max} increases from *g*^{max} = 500 to *g*^{max} = 1800, increases from to while increases from to . Within the range of , the coexistence of pair pattern strategies and reference point strategies is beneficial for the investors with pair pattern strategies.

Comparing the results in Fig 5(a) and 5(c) with the results in Fig 5(b) and 5(d), we find that the coexistence of the investors with pair pattern strategies and reference point strategies is not always good for both sides. In some cases the investors with reference point strategies may defeat the investors with pair pattern strategies and in other cases the investors with pair pattern strategies may defeat the investors with reference point strategies.

## Theoretical analysis

### Relationship between the price fluctuations and the heterogeneities in investment strategies and reference points

The price fluctuation is determined by the difference in the numbers of the investors buying and selling the stocks, which is satisfied with the equation (10)

On condition that all the investors adopt pair pattern strategies, for a given population *N*, the heterogeneity in pair pattern strategies is determined by the memory size *M*. For *M* = 1, the total number of pair pattern strategies is *n*_{pair} = 2^{M} = 2, i.e. *s*_{1} = (0,1) and *s*_{2} = (1,0), which means that, facing an increase in the latest price, *s*_{1} tells the investors to sell and *s*_{2} tells the investors to buy. Facing a decrease in the latest price, *s*_{1} tells the investors to buy and *s*_{2} tells the investors to sell. Therefore, facing a typical change in the latest price, the difference in the numbers of the investors buying and selling the stocks should be
(11)
in which and are the numbers of the investors buying and selling the stocks respectively.

For quite a large *M*, i.e. *M* = 10, the number of the combination of *M* latest price changes is 2^{M} = 1024. The total number of pair pattern strategies is . For a given *N* = 1000, facing a typical combination of the latest *M* price changes, the difference in the numbers of the investors buying and selling the stocks should be
(12)

On condition that all the investors adopt reference point strategies, for a given population *N*, the heterogeneity in reference point strategies is determined by the maximum value of risk tolerance *g*^{max}. Facing the latest price *P*(*t* − 1), the difference in the numbers of the investors buying and selling the stocks should be
(13)
in which *N*_{Pref>P} and *N*_{Pref<P} are the numbers of the investors buying and selling the stocks respectively.

For quite a small *g*^{max}, i.e. *g*^{max} = 1, all the reference points are within a small range
(14)
in which is the averaged value of the stock prices in the latest Δ*t* steps. For quite a large *g*^{max}, i.e. *g*^{max} = *N*, all the reference points are within a wide range
(15)

Given a typical Δ*N* = *N*_{Pref>P} − *N*_{Pref<P} = 2, for *g*^{max} = 1, Δ*N* is quite possible to become Δ*N* = *N*_{Pref>P} − *N*_{Pref<P} = -2 in the next step because nearly all the investors are within a small range around *P*(*t* − 1). For *g*^{max} = *N*, Δ*N* is quite possible to become Δ*N* = *N*_{Pref>P} − *N*_{Pref<P} = 1 or Δ*N* = *N*_{Pref>P} − *N*_{Pref<P} = 0 in the next step because the investors scatter within a wide range around *P*(*t* − 1).

On condition that the investors with pair pattern strategies coexist with the investors with reference point strategies, the price fluctuation is determined by the coupling of the heterogeneities in investment strategies and the heterogeneities in reference points. In a heterogeneous population, the existence of the investors with pair pattern strategies helps the investors with reference point strategies away from a no-trading state while the existence of the investors with reference point strategies helps the investors with pair pattern strategies away from a large fluctuation. The price has the characteristics of a random walk which slowly fluctuates around an equilibrium state.

The above analyses indicate that, for a small *M* and a small *g*^{max}, a zigzag price fluctuation is more possible to occur. For a large *M* and a large *g*^{max}, a slow price fluctuation like a random walk is more possible to occur.

### The averaged wealth of pair pattern strategies and reference point strategies

The wealth of the investors with pair pattern strategies and reference point strategies is determined by the difference between the buying price and the selling price. (16)

The price change Δ*P* = *P*(*t*) − *P*(*t* − 1) = Σ*a*_{pair} + Σ*a*_{ref}. If |Σ*a*_{pair}|> > |Σ*a*_{ref}|, the price change is determined by the investors with pair pattern strategies. If |Σ*a*_{pair}| ≪ |Σ*a*_{ref}|, the price change is determined by the investors with reference point strategies.

On condition that only the investors with pair pattern strategies exist, facing a typical combination of *M* latest price changes, if there are more buyers than sellers, the price increases. The number of the investors buying the stock with a higher price is larger than the number of the investors selling the stock with a higher price. Facing another typical combination of *M* latest price changes, if there are more sellers than buyers, the price decreases. The number of the investors selling the stock with a lower price is larger than the number of the investors buying the stock with a lower price. Therefore, buying high and selling low would lead to the negative wealth of the investors with pair pattern strategies.

On condition that only the investors with reference point strategies exist, facing the latest price *P*(*t* − 1), if there are more buyers than sellers, the price increases. The number of the investors buying the stock with a higher price is larger than the number of the investors selling the stock with a higher price. Facing the latest price *P*(*t* − 1), if there are more sellers than buyers, the price decreases. The number of the investors selling the stocks with a lower price is larger than the number of the investors buying the stocks with a lower price. Therefore, buying high and selling low lead to the negative wealth of the investors with reference point strategies.

On condition that the investors with pair pattern strategies coexist with the investors with reference point strategies, the price movement is determined by the strategy governing the moving trend of the price, no matter whether the strategy is pair pattern strategy or reference point strategy. The investors in the minority side gain more than the investors in the majority side.

For a small memory size *M* and a small maximum gene *g*^{max}, the moving patterns is closely related to the ratio of the investors with pair pattern strategies and reference point strategies. For a small and an intermediate *ρ*_{ref}, because a small *g*^{max} has a greater impact on the price movement than a small *M*, a large *ρ*_{pair} has a greater impact on the price movement than a small *ρ*_{ref}. The moving trend of the stock prices is governed by both the investors with pair pattern strategies and reference point strategies. The wealth of the investors with pair pattern strategies is similar to the wealth of the investors with reference point strategies. For a large *ρ*_{ref}, the price movement is governed by the investors with reference point strategies, which means that, if the investors with reference point strategies buy more, the price increases. If the investors with reference point strategies sell more, the price decreases. The majority choice of the investors with reference point strategies determines the moving trend of the price. Therefore, buying high and selling low lead to the negative wealth of the investors with reference point strategies. For the investors with pair pattern strategies, if they have the buying and selling behaviors similar to the investors with reference point strategies, buying high and selling low lead to their negative wealth. If they have the buying and selling behaviors different from the investors with reference point strategies, buying low and selling high lead to their positive wealth. Therefore, compared with the wealth of the investors with reference point strategies, the wealth of the investors with pair pattern strategies is quite possible to be larger than 0.

For a small memory size *M* and a large maximum gene *g*^{max}, because a large *g*^{max} has little impact on the price movement, the moving trend of the price is determined by the investors with pair pattern strategies. If the investors with pair pattern strategies buy more, the price increases. If the investors with pair pattern strategies sell more, the price decreases. Therefore, buying high and selling low lead to the negative wealth of the investors with pair pattern strategies. For the investors with reference point strategies, if they have the buying and selling behaviors similar to the investors with pair pattern strategies, buying high and selling low lead to their negative wealth. If they have the buying and selling behaviors different from the investors with pair pattern strategies, buying low and selling high lead to their positive wealth. Therefore, compared with the wealth of the investors with pair pattern strategies, the wealth of the investors with reference point strategies is quite possible to be larger than 0.

For an intermediate memory size *M* and a small maximum gene *g*^{max}, similar to the situation where there is a small memory size *M* and a small maximum gene *g*^{max}, the moving patterns is closely related to the ratio of the investors with pair pattern strategies and reference point strategies *ρ*_{ref}. For a small and an intermediate *ρ*_{ref}, because a small *g*^{max} has a greater impact on the price movement than an intermediate *M* and a large *ρ*_{pair} has a greater impact on the price movement than a small *ρ*_{ref}, the moving trend of the price is governed by both the investors with pair pattern strategies and reference point strategies. The wealth of the investors with pair pattern strategies is similar to the wealth of the investors with reference point strategies. For a large *ρ*_{ref}, the price movement is governed by the investors with reference point strategies, which means that, if the investors with reference point strategies buy more, the price increases. If the investors with reference point strategies sell more, the price decreases. The majority choice of the investors with reference point strategies determines the moving trend of the price. Therefore, buying high and selling low lead to the negative wealth of the investors with reference point strategies. For the investors with pair pattern strategies, if they have the buying and selling behaviors similar to the investors with reference point strategies, buying high and selling low lead to their negative wealth. If they have the buying and selling behaviors different from the investors with reference point strategies, buying low and selling high lead to their positive wealth. Therefore, compared with the wealth of the investors with reference point strategies, the wealth of the investors with pair pattern strategies is quite possible to be larger than 0.

For an intermediate memory size *M* and a large maximum gene *g*^{max}, similar to the situation where there is a small memory size *M* and a large maximum gene *g*^{max}, because a large *g*^{max} has little impact on the price movement, the moving trend of the price is determined by the investors with pair pattern strategies. If the investors with pair pattern strategies buy more, the price increases. If the investors with pair pattern strategies sell more, the price decreases. Therefore, buying high and selling low lead to the negative wealth of the investors with pair pattern strategies. For the investors with reference point strategies, if they have the buying and selling behaviors similar to the investors with pair pattern strategies, buying high and selling low lead to their negative wealth. If they have the buying and selling behaviors different from the investors with pair pattern strategies, buying low and selling high lead to their positive wealth. Therefore, compared with the wealth of the investors with pair pattern strategies, the wealth of the investors with reference point strategies is quite possible to be larger than 0.

The above analyses indicate that the strategy that drives the system far away from the equilibrium loses more while the strategy that draws the system back to the equilibrium gains more. The theoretical analysis is in accordance with the simulation data in Fig 6.

The averaged wealth of the investors with pair pattern strategies (circles) and reference point strategies (squares) for (a)*M* = 1, *g*^{max} = 100; (b)*M* = 1, *g*^{max} = 1000; (c)*M* = 3, *g*^{max} = 100; (d)*M* = 3, *g*^{max} = 1000. Other parameters are: the total population *N* = 1000, the ratio of the investors with reference point strategies *ρ*_{ref} = 0.5, the number of strategies for each investor with pair pattern strategy *n*_{S} = 2, the time interval for getting is Δ = 10, the maximum and minimum number of shares for each investor *K*_{max} = 1 and *K*_{min} = -1, the constant *α* = *β* = 10. Final data are obtained by averaging over 100 runs and 10^{4} times after 10^{5} relaxation times in each run.

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