Impact of Regional Differences in Risk Attitude on the Power Law at the Urban Scale

Abstract

Internal mechanisms and laws exist in the evolution of cities, and the power law is widely applied in multiple areas in the real world. It is crucial to optimize the urban-scale systems through explanation studies of the urban-scale distribution pattern from the perspective of regional differences in risk attitudes. Based on computer simulation technologies, this study explores the influence of regional differences in risk attitudes of micro decision-makers on the power law through setting scenarios of same attitudes with quantitative differences and mixed multi-attitudes. In this case, we selected six provinces in China to verify the scale characteristic of the real world. The results show that the settlement scale is heavily influenced by risk attitudes with a larger slope, which are more pronounced in the mixed multi-attitudes scenario. The increase in the mixed-scale benefits less affects the utility of risk attitudes, where the slope value of the aversion attitudes has smaller variation. The averse model has a larger primary ratio than the others. However, the primary ratio does not reveal a significant bias towards large and small in the mixed multi-attitude scenario. In the six provinces, the advantageous areas with higher economic and cultural levels show larger-scale agglomeration characteristics similar to the impact of seeking attitudes. The primacy ratio increases with the variation degree in urban scales, especially in economically disadvantaged areas.

1. Introduction

The power law is ubiquitously used in the fields of physics, earth, and planetary science, biology, finance [1], as well as in urban planning. A city is a complex system with bottom-up growth characteristics. The scale distribution of cities requires appropriate structures to obtain proper urban populations and numbers of towns. The power law plays a significant role in characterizing the laws of urban development. Thus, the exploration of the effects on the power law of the urban scale deepens research about the law of urban self-organization, improves current urban-scale system and balances development of regional cities.

Since Auerbach firstly applied the power rate to the study of urban scales, the impact studies of the distribution law of urban scales have been continuously enriched [2]. Brakman [3] introduced a negative feedback mechanism model to explore the impact of economic factors, such as transportation and congestion costs, economical levels of industrial activities and returned to scale on the power law at the urban scale. The increase in industrialization and the decrease in transportation costs would lead to faster growth and agglomeration in big cities’ population, with a corresponding increase in the power index. The diversity of economic scales among industries resulted in the diversity of industrial locations, which was transformed into a diversity of urban scale through the spatial coordination of industries, and then resulting in a certain difference in the power index [4]. Additionally, regional household registration policies, population size, transportation level, and urban output also had effects on the power index [5,6,7]. For example, Devadoss et al. [8] explored the power law of Indian rural and urban areas and found that the difference in regional higher power indices stems from larger population sizes. Empirical studies of Chinese cities had also found that a better spatial allocation of economic resources and a higher degree of population agglomeration led to a higher power index [9,10,11]. Therefore, policies, economic and social aspects can have an impact on the power law of urban scale. Moreover, the impact on the power law at urban scale is generally defined from the macro perspective, while research from the perspective of micro decision-making behavior is yet to be further analyzed.

The relationship between interactive decision-making and the dynamic changes in land use is complex, which determines the possibility of regional land development and distribution characteristics at the urban scale. Ligmann-Zielinska [12] showed that, in the same area, various types of decision-makers had different competitions and choices for land. Bacon et al. [13] simulated the decision-making process of land managers through BN models, and Chen et al. [14] constructed an RLC-REP model to determine the land use selection preferences of residential areas for use in the decision-making process. Parker et al. [15] also found that interactions among residential agents would trigger developers to pursue the area, followed by an increase in the development of land. Furthermore, the developers were more likely to choose downtown sites due to the reputation effect since they had higher probabilities to develop [16]. In addition, the government at multiple levels played huge roles in decision-making in urban land management. The demarcation of urban construction boundaries would not only affect the psychological value and preference of developers but promote the occurrence of out-of-boundary development for impacts on the urban scales [17]. The increased uncertainty of government tenure [18] and regional competition for land use [19] also influenced the decision-making behavior in land development. As a result, the combination of diverse tactics gave the gradual exploration of land use a variety of chances depending on how well the “personal preferences” of individual decision-makers were met [20]. That is, land developments will be affected by micro decision-makers, who influence and lead to uncertainty in the laws of urban development.

Meanwhile, different decision-makers were not completely rational, and their cognitive systems were limited [21,22]. Due to the subjective judgments of utility and probability of uncertain exogenous factors, subjects would behave differently [23]. However, judging was risky. While meeting uncertain risks, decision-makers had varied risk attitudes, which influence their decisions [24]. Their aversion tendencies in risk attitudes intensify when they regard the risks of benefit-taking was less than their responsibility-taking [25]. For example, when decision-makers faced a strong livelihood risk, their attitudes generally showed a tendency toward avoidance [26], which finally affected their decisions. Some scholars had constructed the attitude utility functions (AUFs) to quantify risk attitudes. They pointed out that land development was closely related to risk attitudes by evaluating the spatial impact of land use decisions [12]. Based on the previous research, Lu et al. [27] explored the influencing factors of urban expansion patterns and found that reckless decision-makers were more likely to lead the urban sprawl. Han et al. [28] explored the formation of urban settlements through analyzing the decision-makers’ risk perceptions and found that it was highly correlated with the power law. That was shown that the scilicet risk attitude had a certain influence on the power law. Although several studies revealed that risk attitudes influence the distribution of urban scales, these studies were based on the overall regional rule. Moreover, Wegmann et al. [29] researched households in urbanized areas with different incomes. They revealed that people with more assets tend to have lower risk aversion, and risk attitudes vary with spatial distribution of asset owners. However, the study was limited to the relationship between assets and owners and did not discuss from the perspective of urban scale.

In summary, the power law has been widely applied to the study of the internal influence mechanism of urban scale worldwide. However, the existing studies have not deeply considered the impact of regional differences in internal risk attitudes, particularly from the perspective of micro-decision-making behaviors. Since regional differences in risk attitudes of various decision-makers affect regional urban development, this study sets regional differences in risk attitudes (while using computer simulation technology), yields to explore the influence of the power law at urban scale. Added to that, the study contributes to deepening the interpretive study at the urban scale distribution and promotes regional sustainable development.

In the following sections, Section 2 shows the theoretical presentation of the power law and the risk decision. Section 3 delivers the system modeling. Section 4 reveals the simulation results of the modeling and the discussions will be generated in Section 5. Finally, Section 6 summarizes the conclusion and proposes suggestions for future research.

2. Theoretical Basis

2.1. The Power Law

The power law exists in the self-organizing system to explore the relationship between the scale and the frequency of objects’ occurrence. Its probability density function is y = c x-a, which represents the proportional relationship -a between scale y and frequency x [30]. Various power law characteristics exist in the real world, such as the distributions of drainage structure [31], population wealth [32,33], paper frequency [34], and the urban system. The rank-size rule from Zipf in 1949 proved that the urban scale and their ranks had linear characteristics under double logarithmic values. In other words, large-scale things often appear with a low frequency, small-scale things are on the opposite with a high frequency, which phenomenon is highly similar to the power law. Hence, we regard the power law can benefit in explaining the distribution law at the urban scale.

2.2. Risk Decision

Decision-makers have self-perceptions about the probabilities of various events. Therefore, their cognitive differences caused the different cognitive models, value orientations, and the preferences [35]. The “rational people” hypothesis meant that when people were in a limited resource environment, they will make every effort to make the best decisions while pursuing the maximization of interests. The studies on behavioral decision-making in uncertain environment had also been further expanded since Simon proposed the “bounded rationality” [36]. It meant that human behavior is rationality within the limits of a given environment. Due to the differences in the sensitivities to the losses and gains from different decision-makers, there was a continuous characteristic of risk preference hierarchy [37] (Figure 1), from risk-averse (cautious), acquiring through risk-neutral (unbiased), and reaching risk-seeking (reckless). Moreover, the risk-averse is not sensitive to gain, risk-seeking is sensitive to gains, and risk-neutral is in the middle. It can also be seen as fully rational (Figure 2). Therefore, it is known that different decision-makers self-perceptions judgments of losses and gains will be limited by bounded rationality, because of the existence of risk, which will have a deep effect on the urban land development.

3. Model Construction

3.1. Basic Ideas

This paper is based on Netlogo and sets the simulation space to be a homogeneous plane, then divides it equally into several areas while applying two rules (regional quantitative differences in the same attitudes and mixed multi-attitudes), to represent regional different risk reference simulation scenarios. In the model, this study set the same time steps to record different development forms. Then count the settlement scale of them and rank it. Afterward, through the linear regression, the results are recorded to find the changing law of the urban scale’s power law. Finally, a comparative analysis is conducted with the distribution law of urban scale in the real world, and the impact of regional differences in risk attitudes is studied.

3.2. Computer Simulation Design

In the section, the step of the system modeling will be proposed:

(1)

The plane space of the study is made up of 200 rows by 200 columns, using the concept of cellular automata, making each development unit a cell, which indicated the smallest land development unit. There are 32,400 cells, making every development area with 60 rows by 60 columns through nine equal area divisions. In addition, each area has the same possibilities to be developed. The cells have two states of development: developed and undeveloped, assigned 0 or 1, respectively. Moreover, the neighbors’ effect and scale-mixing modes’ attraction affect the development of one cell. Each cell’s developing possibilities are determined by a potential value as:

$$Q=\sqrt{N\times S\times T},$$

(1)

where Q is the initial potential value of the cell, N is the agglomeration strength coefficient, S is the number of developed cells in the neighborhood, and T denotes the largest neighbor’s settlement scale of the nearby target cell. The product of S and T is the hybrid mode attraction. If one cell has three developed cells in its neighbors, and the largest neighbors’ settlement scale is eight; next, the potential values are $Q=\sqrt{N\times 3\times 8}$ (Figure 3). The development principle follows the location of cells with higher development potential value has a higher probability to be selected.

(2)

According to the risk attitude to have the new potential of land development, the attitude utility functions (AUFs) are numerically approximated as:

$$\mathrm{Reckless}: y=\alpha \times x{ }^{\alpha },$$

(2)

$$\mathrm{Cautious}: y=x,$$

(3)

$$\mathrm{Neutral}: y=(x∕\alpha {)}^{1/\alpha },$$

(4)

where y is the utility based on risk attitudes, α is a curving coefficient driving the shape of the AUF, can measure the different effects on the development potential caused by the same risk attitude.

In this study, α is equal to 1.2 and 1.4 in the above approximation, and x is the original value.

In terms of regional division, the largest number of the prefecture-level city in different provinces is 21, and the others are around 10 in China. Thus, it is equivalent to reality to divide the area into nine equal spaces. In addition, nine areas are more convenient to simulate, and the rules of model design include two scenarios (

Table 1. Model design scenario.

Regional Quantitative Differences in the Same Attitudes	Mixed Multi-Attitudes
Regions of 3 seeking or aversion + 6 neutral	Main risk-seeking	Regions of 5 seeking + 2 aversion+ 2 neutral
Regions of 6 seeking or aversion + 3 neutral	Main risk-neutral,	Regions of 5 neutral + 2 seeking + 2 aversion
Regions of 9 seeking or aversion	Main risk-averse	Regions of 5 aversion + 2 seeking + 2 neutral
*	equality	Regions of 3 neutral + 3 seeking + 3 aversion

Note: *: There is no scenario setting here.

):

The first one is the regional quantitative differences in the same attitudes: setting one-third, two-thirds, and all of the plane space as seeking or aversion attitudes in turn. The increasing percentage can be interpreted as the intensity of a shift in a particular attitude. The other area is created with a neutral attitude based on the environment, which is completely logical and has no risk effect. Therefore, the same attitudes apply to this circumstance.

The second one is the mixed multi-attitudes: it includes four types of settings: main risk-seeking, main risk-neutral, main risk-averse, and equality. In the first three settings, the ratio of main risk attitudes with the other two is 5:2:2, while the last one is 3:3:3.

(3)

All the settings are choosing 15 as the time step, in this circumstance, and 15% of the cells have been developed in the whole space. At this time, the settlement form is clearer and can present obvious urban hierarchical features. Thus, it has a better representation effect and avoids the phenomenon of large settlements connecting with large settlements caused by the increase of time steps. In addition, each group takes the mean value five times, and finally counts the scale of the development of land in the region.

4. Simulation Results

In the simulation space, red lines represent the development boundary, and the white parts represent the developed cells. The essential manifestation of the order scale in the power law is fractal, which is the inherent harmony embodiment of the urban system’s hierarchical structure. Complexity has a high degree of uncertainty that means the interaction among attributes such as self-organization, feedback and feedback loops, and dynamic behavior [38,39]. Moreover, the power law itself is exactly a quantitative representation of complexity, producing complex collective behavior through simple operating rules [40]. The power law can be characterized by R², slope, and primacy ratio. Among them, R² is used to test the robustness of the power law. The slope represents the average gap and trend of change in cities of different urban scales. The primacy index is the ratio of the urban scale of the largest and the second largest city, ensuring the agglomeration of urban scales. While the risk is the bounded rationality of dealing with uncertainty. This study is in order to reveal the influence of uncertainty in the self-organizing system by the results of the hierarchical structure of the regional urban system, which are affected by the risk decision-makers in the process of urban land development.

4.1. Results Analysies of the Same Attitudes with Regional Quantitative Differences

As shown in Figure 4, some scenarios that show obvious developed settlement forms. The regional development of the quantity with the same attitude has a greater impact. The settlement forms in every region have a clear trend of expansion. This scenario exhibits a clearer agglomeration state of the settlement scale due to the obvious changes in the risk intensity component of the risk-seeking model. In addition, the risk-neutral model shows more settlement developments than the risk-averse model indicates.

4.1.1. Analyses of the Power Law Conservatism

The fitting simulation results can be obtained through the function (Equation (1)): each simulation fits good, and its R² is almost 97% (

Table 2. Comparison of the R² values.

Types of Risk Attitudes	Risk Intensity Factor	Regional Quantity Differences	Agglomeration Intensity Factor
			2	5	10	20	30	50	80
Averse model	α = 1.2	1/3	0.9996	0.9938	0.9692	0.9668	0.9532	0.9612	0.9996
		2/3	0.9900	0.9982	0.9804	0.9712	0.9734	0.9766	0.9760
		3/3	0.9989	0.9832	0.9918	0.9806	0.9602	0.9386	0.9088
	α = 1.4	1/3	0.9936	0.9876	0.9726	0.9594	0.9562	0.9480	0.9604
		2/3	0.9934	0.9928	0.9914	0.9728	0.9784	0.9818	0.9840
		3/3	0.9940	0.9956	0.9926	0.9894	0.9784	0.9634	0.9446
Neutral model		3/3	0.9900	0.9794	0.9784	0.9398	0.9392	0.9214	0.8958
Seeking model	α = 1.2	1/3	0.9968	0.9846	0.9830	0.9768	0.9786	*	*
		2/3	0.9972	0.9864	0.9612	0.9618	0.9575	*	*
		3/3	0.9964	0.9888	0.9480	0.9408	0.9128	*	*
	α = 1.4	1/3	0.9856	0.9742	0.9700	0.9654	*	*	*
		2/3	0.9936	0.9852	0.9834	0.9586	*	*	*
		3/3	0.9612	0.9704	0.9648	0.8600	*	*	*

Note: *: It is the result of the non-recording mentioned above.

). In the experiments, based on the existence of risk attitudes and mixed attraction, the neighbors’ development possibilities will have exponential to grow in the phenomenon of several large settlements connected into one super settlement. To ensure the reliability of the simulation and the real-world reality, the results are not recorded when the primacy scale is significantly larger than the scale of other settlements, or the connection scale accounts for 40% in the development cells of a certain area.

4.1.2. Analyses of the Form and the Changing Trend of Settlements

It can be seen from

Table 3. Comparison of the slopes of different scenes.

Types of Risk Attitudes	Risk Intensity Factor	Regional Quantity Differences	Agglomeration Intensity Factor
			2	5	10	20	30	50	80
Averse model	α = 1.2	1/3	−0.3074	−0.3950	−0.4926	−0.7086	−0.9018	−1.2774	−1.8264
		2/3	−0.4452	−0.4046	−0.4462	−0.5998	−0.8104	−1.1742	−1.4074
		3/3	−0.2876	−0.3346	−0.4420	−0.4824	−0.5750	−0.6932	−0.8518
	α = 1.4	1/3	−0.3928	−0.4396	−0.5012	−0.7388	−0.9076	−1.2688	−1.8950
		2/3	−0.3612	−0.4474	−0.4552	−0.6966	−0.8310	−1.1274	−1.5236
		3/3	−0.3730	−0.3340	−0.3772	−0.3934	−0.4898	−0.5516	−0.5974
Neutral model		3/3	−0.3632	−0.4260	−0.5366	−0.7660	−0.9374	−1.2752	−1.9156
Seeking model	α = 1.2	1/3	−0.3854	−0.5068	−0.6616	−1.0696	−1.4982	*	*
		2/3	−0.3194	−0.4722	−0.7308	−1.2594	−1.7886	*	*
		3/3	−0.3178	−0.5272	−0.7410	−1.3400	−1.9536	*	*
	α = 1.4	1/3	−0.4386	−0.5678	−1.0338	−1.6474	*	*	*
		2/3	−0.3650	−0.6382	−1.2978	−2.2952	*	*	*
		3/3	−0.4550	−0.6716	−1.3550	−2.3884	*	*	*

Note: *: It is the result of the non-recording mentioned above.

, with the same regional attitude and quantity number, a large value of α makes the distribution of the averse model settlements more uniform. The development probability of the seeking model is higher than the other two attitudes, presenting a large settlement with high agglomeration and large scale. In regards to the affection of neighbors’ development probability, the absolute value of the averse model’ overall slope is growing. In addition, when settlements are smaller, the impact of risk attitudes is greater than the mixed benefits of scale, and the slope change of the averse model change is small. With the growth of the N, the impact of risk attitude is relatively low, moreover, but the impact degree of the seeking model is the slowest.

Furthermore, with the same risk intensity factor, with the regional quantities of risk attitude evolving from 1/3 to 3/3, the effects of the power law become larger. The phenomena of inhibiting development in the averse model results in a higher degree of dispersion of settlement forms. The averse model has more regional development options to have more large settlement scales. In addition, the middle one is the neutral model (Figure 4). The results show the slope of the averse model gradually declines when α = 1.2 or 1.4. This is due to the increase in the influence degree of the averse attitude allowing to gradually reduce the disparity. Biased towards conservative development, and having the main form of small and medium settlements. The seeking model’s slope is just the opposite. The differences in size among settlements are increasing, and the slope becomes steeper. However, the increase rate does not show a significant positive correlation to the “convergence”, which indicates the development of large settlements has a certain upper limit. Added to that, the neutral model is based on a rational state due to the neighbors’ and scale effects’ influence. Its slope also shows a steady growth trend and a sharp larger change. Moreover, the overall slope value is between the complete averse and the averse model, showing its settlement scale pattern has both the agglomeration of large settlements and the dispersion of small settlements. Therefore, the scales of settlements are balanced.

4.1.3. Analyses of settlement primacy

There are two urban indexes (S₂), four urban indexes (S₄), and eleven urban indexes (S₁₁). Index (S₄) was used in this study:

$$S_4=P_1/\left(P_2+P_3+P_4\right)$$

(5)

where P₁, P₂, P₃, and P₄ are the scale of the settlement which is under a certain rank.

Table 4. Comparison of the primacy ratios.

Types of Risk Attitudes	Risk Intensity Factor	Regional Quantity Differences	Agglomeration Intensity Factor
			2	5	10	20	30	50	80
Averse model	α = 1.2	1/3	0.3857	0.4292	0.4323	0.4898	0.6030	0.6927	0.5677
		2/3	0.4556	0.4205	0.4578	0.4972	0.8521	0.6830	0.6723
		3/3	0.3667	0.3524	0.4292	0.4381	0.4244	0.5150	0.4568
	α = 1.4	1/3	0.5000	0.4467	0.5990	0.5142	0.4601	0.6166	0.7592
		2/3	0.4083	0.4285	0.5781	0.4825	0.6188	0.7315	0.7862
		3/3	0.3940	0.3829	0.5079	0.3972	0.4576	0.4255	0.4770
Neutral model		3/3	0.5190	0.4022	0.4458	0.4137	0.4495	0.5326	0.5442
Seeking model	α = 1.2	1/3	0.4333	0.4442	0.4963	0.8318	0.7258	*	*
		2/3	0.4333	0.4832	0.6192	0.6003	0.7342	*	*
		3/3	0.4417	0.4788	0.4099	0.4424	0.4828	*	*
	α = 1.4	1/3	0.4282	0.5720	1.7316	0.8938	*	*	*
		2/3	0.4055	0.4971	0.7152	0.5999	*	*	*
		3/3	0.6572	0.4888	0.6451	0.4621	*	*	*

Note: *: It is the result of the non-recording mentioned above.

demonstrates that, while the predominance of various attitudes is steady and largely consistent, it becomes more erratic as the advantages of agglomeration grow. Thus, the benefits of the large-settlement-scale agglomeration become gradually stable, but different attitudes still have different influences. With the increase of the aversion regional quantities, the aversion scenario has a greater risk degree and is less sensitive to the cells’ neighbor development, and the curve is flatter. The primacy degree is basically at the lowest state especially after being completely in the aversive state.

The seeking model is more sensitive to mixed benefits due to its more adventurous development, therefore, it is easier to form a large settlement easier. In the overall seeking model, large settlements lead to a decrease in the difference after scale sorting, and the primacy ratio is also in a lower state. Therefore, in regional quantitative differences while keeping the same attitudes, the primacy ratio does not have a significant difference. However, the primacy ratio of seeking attitude is still high.

4.2. Analyses of Regional Mixed Multi-Attitudes Simulation

Figure 5 shows that many large settlements have no obvious difference in the mainly seeking model. The mainly neutral model has obvious differences in the scale and the level of settlements. The mainly averse model shows more minor differences in small settlements, while the equality has a similar development to the neutral model.

4.2.1. Analyses of Power Law Universality

Table 5. Simulation results of multi-attitude models.

Agglomeration Intensity Factor	Risk on Intensity Factor	Mainly Seeking Model			Mainly Neutral Model			Mainly Averse Model			Equality
		R2	Slope	Primacy Ratio	R2	Slope	Primacy Ratio	R2	Slope	Primacy Ratio	R2	Slope	Primacy Ratio
2	α = 1.2	0.996	−0.393	0.442	0.991	−0.401	0.442	0.991	−0.323	0.486	0.991	−0.451	0.518
	α = 1.4	0.995	−0.448	0.471	0.991	−0.453	0.523	0.994	−0.405	0.529	0.994	−0.433	0.416
5	α = 1.2	0.970	−0.468	0.408	0.985	−0.473	0.580	0.989	−0.460	0.421	0.977	−0.501	0.477
	α = 1.4	0.968	−0.690	0.473	0.993	−0.393	0.445	0.986	−0.613	0.776	0.989	−0.613	0.580
10	α = 1.2	0.974	−0.694	0.489	0.981	−0.593	0.516	0.970	−0.614	0.718	0.973	−0.684	0.486
	α = 1.4	0.982	−1.253	0.726	0.973	−0.968	1.249	0.974	−1.105	1.967	0.986	−1.172	0.940
20	α = 1.2	0.982	−1.349	1.066	0.989	−1.002	1.120	0.981	−0.977	1.181	0.982	−1.099	0.542
	α = 1.4	0.938	−2.239	0.640	0.955	−1.497	0.971	0.933	−1.692	0.744	0.957	−1.821	0.916
30	α = 1.2	*	*	*	0.984	−1.366	1.000	*	*	*	0.985	−1.448	0.783
	α = 1.4	*	*	*	*	*	*	*	*	*	*	*	*
50	α = 1.2	*	*	*	*	*	*	*	*	*	0.991	−1.783	1.120
	α = 1.4	*	*	*	*	*	*	*	*	*	*	*	*

Note: *: It is the result of the non-recording mentioned above.

shows that different regional quantitative differences in attitudes have high goodness of fit, between 0.933 and 0.994. In the same attitude scenario, the increase of the cells’ development possibility is related to the increase of the agglomerated benefits or degree of risk. However, R² doesn’t vary linearly. This shows that the power phenomenon has obvious stability and does not deviate from the power law. In addition, the four average values of R² dominated by different main attitudes reach nearly 0.980. Therefore, we deduce that the setting of regional risk attitudes of the mainly seeking model, neutral model, averse model or equality will not affect the power law, which has a strong generality.

4.2.2. Analyses of Settlement Scale Differences and Growth Rate

In computer simulation, the absolute value of the slope gradually increases in all the simulated scenarios, which is most significantly affected by the seeking attitudes. The absolute value of the overall slope in the mainly seeking model is larger than that of the mainly neutral and the mainly averse model, and its agglomeration of settlements is more obviously impacted. In addition, the equality simulation results are similar to the results from the mainly averse model. We deduce that the influence of the risk-averse attitude is larger, but the agglomeration effect of the initial large settlement still exists.

In addition, Table 5 shows that the overall change of slope only varies with the increase of the risk degree. When N < 10, the slope variations are small in all the scenarios. However, because of risk attitudes, the impact of lower mixed attractiveness results in the original development potential value with little change. Furthermore, a risk-seeking attitude induces a fast increase in the possibility of cell development that tends from large-scale settlements and leads to a large increase in the slope and allows an easily stable development of settlements. When N > 10, the overall variation in the slope of each scenario is significantly faster than in the mainly neutral model, reflecting the influence of large settlements. This shows that the influence of mixed scale is increasing. When N = 20, the scenarios of the mainly seeking and equality model have the largest variation in slope change, while the slope is the steepest and up to 0.8. It is hampered by the averse attitude, which is more pronounced in the difference in settlement scale, and indicates that urban settlements have a “polarization” trend.

Thus, the influence of different risk attitudes among regions is heterogeneous, which has various effects on the power law. Slope change is presented by the scale variation. In addition, slope change in the mainly seeking model tends to be steeper, while the agglomeration benefit scale expansion of the neutral model has an upper limit, and therefore the range of change is gradually decreased. The scales of the averse and the equality models have clear differences between the large and small settlements, and the slope changes significantly.

4.2.3. Analysis of Settlement Primacy Ratio

The experimental results in Table 5 reveal that the primacy ratios of different main risks attitude models have different volatilities. Nevertheless, the overall primacy ratio gradually grows with the increase of aversion attitude regions quantities and the gap between large and small settlements. On the one hand, it is affected by the averse attitude, which reduces the probability of development and thus inhibits the development of the cells. On the other hand, it highlights the settlement agglomeration in seeking attitudes.

In addition, the scale of settlements will not expand indefinitely. With the increase of risk degree, the differences in the risk primacy attitudes in each region reaches a peak when the agglomeration intensity coefficient reaches 10. Currently, the top developed settlements have a larger scales and lead to a large gap with the scales of the second, third, and fourth ones, and then tend to be flat. When a super-large settlement is reached, both the later large-scale settlements and small-scale settlements gradually develop, the scale gap between the first and later settlements gradually narrows, and therefore the neighbor and the scale effects are limited to a certain extent.

4.3. Difference Analysis of Two Scenarios

(1)

The R² values of the two scenarios show high values, indicating that any simulated scenarios have good goodness of fit, which does not only form a stable settlement scale, but also conform to the power law. In addition, each value not showing clear regularity with the risk attitudes, has a strong universality.

(2)

The previous model acknowledged that the power law expresses the linear relationship between scale and bit order, and the most prominent characteristic is the slope change. The slope of the seeking attitude influence is always at a high level. Compared to the scenario of same attitudes with regional quantitative differences, the change of the slope is more obvious in the mixed multi-attitude scenario, showing a larger value of the seeking attitude. Moreover, the mainly seeking and the mainly averse models show the scale form of a large settlement, defining a large-scale gap with small settlements, which reflects the leading role of seeking attitudes. The equality is within three other types of dominant attitudes, and its slope does not show a clear tendency to be too large or too small.

(3)

As far as the primacy ratio is concerned, there is no discernible relationship between risk attitudes and settlement priority of aversion or neutrality scenarios under the same attitudes with regional quantitative differences, each of which has its impacts. However, based on the influence of seeking attitude, the phenomenon appears to be on the higher side. The results of the multi-attitude scenario simulation are different. They are affected by the main attitudes with following features: the more spaces the plane takes in the aversion development setting, the greater its primacy will be. Therefore, the simulation scenario of mixed multi-attitudes is more significantly affected by the three other scenarios of attitudes than the scenario of the same attitudes with regional quantitative differences, with a greater difference between large and small settlements. In addition, it can be deduced that the scale of settlements has an upper limit of growth and limited agglomeration benefits.

5. Discussion

The exponent of the power law is basically kept around −1 when explaining the internal laws of cities [40], which means the scale system is more reasonable. However, this research shows that the slope gradually approaches or breaks −1, which is related to the high possibility of exploitation brought by agglomeration. In addition, the overall R² of the model is greater than 0.940, which indicates that it is close to the real world. Meanwhile, decision-making subjects with different risk preferences have a different tendency to distinct areas. The more favorable geographical conditions are [41], the higher the level of economic development [42] and the degree of cultural openness will be [43], and the more strongly the risk-taking subjects pursue profit in advantageous areas will be [44], for example, the southeast coast of China. Therefore, for exploring the laws of the real world under the influences of risk attitudes, this study considers six provinces in China as targeted regions: Shandong, Zhejiang, Hunan, Jiangxi, Gansu, and Guizhou. Data were mainly obtained from the “China Urban Construction Statistical Yearbook” [45]. We took the prefecture-level cities as the research unit and selected the urban built-up area data from 2012 to 2020 are generated for statistical analysis. Referring to

Table 6. Urban scale of some provinces in China.

Province	City Quantity	Results	2012	2013	2014	2015	2016	2017	2018	2019	2020
Shandong Province	16	R2	0.955	0.976	0.979	0.989	0.985	0.985	0.986	0.956	0.958
		slope	−0.979	−0.989	−0.990	−0.995	−0.993	−0.993	−0.994	−0.979	−0.980
		Primacy ratio	0.428	0.523	0.510	0.576	0.572	0.596	0.647	0.564	0.550
Zhajiang Province	11	R2	0.945	0.927	0.928	0.927	0.924	0.918	0.924	0.927	0.930
		slope	−0.975	−0.966	−0.967	−0.967	−0.965	−0.963	−0.965	−0.967	−0.968
		Primacy ratio	0.742	0.663	0.657	0.666	0.698	0.717	0.734	0.750	0.734
Hunan Province	13	R2	0.919	0.936	0.934	0.913	0.906	0.912	0.905	0.926	0.686
		slope	−0.962	−0.970	−0.969	−0.959	−0.956	−0.959	−0.955	−0.966	−0.844
		Primacy ratio	0.870	0.858	0.837	0.896	0.901	0.942	0.945	0.950	1.066
Jiangxi Province	11	R2	0.958	0.936	0.970	0.944	0.969	0.964	0.968	0.974	0.970
		slope	−0.981	−0.971	−0.987	−0.974	−0.986	−0.984	−0.986	−0.988	−0.987
		Primacy ratio	0.814	0.911	0.823	0.943	0.884	0.832	0.791	0.796	0.789
Gansu Province	12	R2	0.845	0.851	0.860	0.843	0.859	0.865	0.856	0.857	0.855
		slope	−0.927	−0.930	−0.934	−0.926	−0.934	−0.937	−0.932	−0.933	−0.932
		Primacy ratio	1.183	1.086	1.385	1.564	1.628	1.707	1.710	1.619	1.657
Guizhou Province	6	R2	0.837	0.812	0.831	0.864	0.959	0.972	0.947	0.937	0.933
		slope	−0.933	−0.922	−0.930	−0.944	−0.983	−0.989	−0.979	−0.975	−0.973
		Primacy ratio	1.519	2.074	1.952	1.466	1.241	1.308	1.351	1.321	1.307

, the slope and the primacy of different regions showed their respective size differences and changing trends, which reflect the power distribution characteristics within a region.

The absolute value of the slope in Shandong and Zhejiang Provinces is at a relatively high level, reaching almost 0.98. This phenomenon is similar to the averse model, which reflects the result that the seeking attitude causes a large slope value. But the primacy ratio is not larger, which is contrary to the simulation results. Since the Reform and Opening-up In China, these two provinces have a rapid economic development as large economic provinces in the coastal areas. Especially the priority development in their large cities, such as Jinan in Shandong and Hangzhou in Zhejiang, which have gathered many brands, resources, and industries. Since they have large differences in urban scales from regional small cities such as Liaocheng, Zaozhuang, Quzhou, and Lishui. In addition, the open environment is more likely to attract entrepreneurs and developers with active thinking, who will have a higher level of awareness and a reckless attitude toward performance and profitability [22]. Therefore, policy innovation and land use development are more likely to be promoted with easier breakthroughs in operational boundary constraints. Moreover, Shandong province has two major economic construction cores cities: Qingdao and Jinan. The variation of built-up area of between two cities is only 35.49 m² in 2020. Neither of the scales of the two cities showed its potential as a capital city in the province. Therefore, the overall primacy ratio is low. The change flatter magnitude of the change represents indicates, because the urban planners in Zhejiang Province paid more attention to the development of small and medium cities, and the primacy ratio is also relatively small.

The slope and the primacy ratios of Hunan, Jiangxi Province are at the medium level, which is similar to the mainly neutral model. Data show that the monopoly position of the large city is not strong. The slope is lower than the whole southeast coastal area, which proves the impact of mixed various risk attitudes. On the one hand, the unique geographical location enables the two provinces to accept the influence of the eastern region and stimulates the potential for innovation and reform. On the other hand, the two provinces undertake the influence of the conservative culture of the western region, and the overall risk level is relatively neutral. Changsha-Zhuzhou-Xiangtan is the “one city” integration in Hunan Province, and its urban space development is far greater than that of other cities, while that of Western Hunan is the lowest [46]. The implementation of the central China development policy has led to an increase in the indicators of construction land, based on the government’s “rational choice”, it has a tendency towards the dominant area [43]. The introduction of the dot axis model is also manifested in more risky land development. Thus, the larger gaps between large, medium, and small cities have led to a significant increase in slope, which is consistent with the impact of reckless and averse attitudes in this model. The Jiangxi Province regional built-up areas in the north and south urban areas can be nearly seven times different in urban scales, which manifested as the “polarization effect”. The inclination of policy indicators attracts foreign investment and industrial agglomeration. The increase of investment of various reckless subjects occurs in advantageous areas, such as Gannan, and Yingtan industrial clusters. Though they are small-scale cities, there still have the attractive areas which have large-scale areas in the area.

As for Gansu and Guizhou Provinces, they are located inland with a low economic level. The phenomenon of the two provinces is similar to the mainly averse model, thus, the overall slope is at a relatively low level, but the change range is clear with a larger primacy ratio. Lanzhou is the development core in Gansu Province, with a Gross Domestic Product (GDP) of 283.7 billion (10 times that of the lowest city). The size of regional built-up areas of the city is 23 times different from Longnan Province. In addition, the urban scale development in areas such as Tianshui, Wuwei, and Pinglh havelarge differences. There is no radiation effect of a large metropolis. The built-up areas within the region of Guizhou Province are at most about 7 times different in urban scales. The GDP of Guiyang City reached 404 billion. It can be seen that the rapid development of the primary cities has led to large gaps in urban scales with other areas in these two provinces, where gather more reckless subjects. Furthermore, these two provinces are deeply affected by several factors such as regional historical and geographical environment, farming culture, resources, and environmental pressure. This situation fundamentally shaped the averse preferences of different agents such as urban managers and corporate investors [43], particularly for small cities. Thus, the two provinces have a few large urban-scale cities, affected by aversion and the attitude of seeking.

Therefore, the various change characteristics of slope and primacy ratio among regions are mainly due to the differences in agglomeration benefits and varying degrees of risk attitudes affect. Core cities have higher economic levels and the better resource configuration. They have larger urban scales since they have clear labor force agglomeration, the policy preference, and the cultural openness. The power law at the urban scale also resembles to the phenomenon of the simulation scenario: the large settlements are concentrated but hold fewer numbers; the small settlements are scattered but with a larger number. When a dominant area has greater land development intensity, the impact of the seeking attitude is stronger. The underdeveloped areas that are less attractive can highlight differences in urban scales between large and small cities, which is similar to the mixed multi-attitudes. Added to that, the regional differences in risk attitudes render the urban scale having an agglomeration and a dispersion effects. The different degrees of influence make the power law change, which complies with the laws of the real world. We deduce that the social and economic functions undertaken by cities in the regional urban system are closely related to their urban scales. The development of urban lands in different regions in China is affected by risks. The advancement of reasonable spillover of risk-taking subjects benefits in promoting the layout of productivity and enterprise enrichment among regions, provinces and cities to form reasonable urban scale systems.

6. Conclusions and Prospects

To explore the influence of the differences in risk attitudes within regions in China, this study considers the increase of the returns and the AUF through computer simulation technologies to analyze the changing trends of urban settlements. As a result, the R² values are all around 0.97, indicating that the regional risk difference setting still conforms to the power phenomenon. Based on the above, we concluded that:

(1)

The seeking attitude in the scenarios of the same attitudes with regional quantitative differences and mixed multi-attitudes has a greater impact on the scales and levels of settlements. Precisely, the increased numbers of seeking attitude settings indicate the obvious phenomenon of large and small numbers of settlements. It is more prominent in the second scenario with the characteristics of significant agglomeration gains and rapid growth. The overall primacy ratio of the seeking model is larger in the first scenario. The agglomeration effect of the primary city plays a crucial role. However, in the second scenario, as the degree of aversion increases, the influence of risk attitudes becomes strong, and the absolute value of the slope tends to increase synchronously.

(2)

Empirical studies of six provinces in China have found that the size and number of urban scales are highly related to the law of situational settlement size. The results show a phenomenon of large-scale cities holding small numbers, while the opposite occurs in small-scale cities. Areas with higher levels of economic development with opened policies correspond to higher slopes, which is relevant to the influences of seeking attitudes. Culturally conservative and resource-stressed areas are similar to the lower slopes of the effects of aversion attitudes; the excessive area remains at a moderate level. The primacy ratio is more similar to the mixed attitude scenario, with some volatility in each region. Economies play more role in core areas than the backward regions.

(3)

Risk attitude exists in any decision. The uncertain development of land use leads to the phenomenon of uneven resource allocation and slow urbanization development at the urban scale during urban development. In the past, the concept of “Garden City Theory” in the urban system regards multiple regions as a whole. The ideas of centralization, metropolis, and border town system constantly emphasize integrated urban development. When the centralized provinces (districts) are formulated through policymaking, the polarization center in the region can be guided; the development preferences of risk-taking subjects for small cities can be stimulated; moreover, the “diffusion effect” and “ trickle-down effect” in economically developed areas will be strengthened for promoting the balanced development of regions. As for the decentralized provinces (districts), it will be beneficial to promote the development of regional economies, and cultural exchanges through improving urban economic activities’ degree of agglomeration, and guiding the agglomeration of population and capital.

This study aims at enriching the research on the influence of the power law on an urban scale. It provides a reference for new urbanization construction from the perspective of risk attitude control, in order to promote a complete urban structural system. However, in the procedure of the introduction in risk attitudes, this study does not consider the game mechanism among the government, the developers, or the residents. The simulated scenarios are also based on idealized states, without considering the heterogeneity of the real world, which requires further research.