From Malthusian stagnation to modern economic growth: a swarm-intelligence perspective

The correlation between decentralized decision-making and swarm intelligence has emerged as a significant subject within self-organization phenomena. Here, we demonstrate that, if an exponential probability distribution of income emerges in a decentralized economic system, then the total income of all agents can be represented by an aggregate production function, in which the technology factor precisely aligns with the information content inherent in the event of decentralized decision-making by all agents. In particular, for sufficiently large population sizes, the emergence of this technology factor enables the income per capita to increase with the population size, akin to a manifestation of swarm intelligence. More importantly, we find that an exponential probability distribution of income can be generated within a peer-to-peer economy governed by specific game rules, characterizing a decentralized-decision economic system. Building upon this discovery, we propose a swarm-intelligence explanation to elucidate the transition from thousands of years of Malthusian stagnation to modern economic growth.


Introduction
Due to its advantage in handling many-body problems, the use of statistical physics is central to research interest in complex systems, including social and economic systems [1][2][3][4][5][6][7][8][9][10][11][12][13][14][15][16][17][18][19][20].Although agents in economic systems differ significantly from those of inanimate particles, the interactions among a large number of agents can lead to typical macroeconomic patterns.For example, it has been empirically observed that income structures in various market-economy countries uniformly exhibit a two-class pattern [21][22][23][24][25][26][27][28][29][30][31]: the majority of the population is described by an exponential distribution, while the remaining top income class is described by a Pareto distribution.Empirical evidence covers more than 66 countries, including those in Europe, Latin America, North America, and Asia [29].As is well known, the Boltzmann distribution in statistical physics is an exponential distribution.Using this distribution, statistical physicists have developed sophisticated mathematical techniques to establish relationships between macroscopic variables in many-particle systems [32], such as the relationship between energy, entropy, and particle number.Motivated by the empirical observation that the exponential income distribution characterizes the majority of the population in market-based economic systems [21][22][23][24][25][26][27][28][29][30][31], by using mathematical techniques in statistical physics it has been shown that, if an economy follows an exponential income distribution, then the GDP Y and the population size N are governed by a partial differential equation [16,31,[33][34][35]: where the emergent variable T is identified as the technology factor [16,31].
For equation (1), it has been proposed that the technology factor T appears to be linked to a type of swarm intelligence in human societies [16,34].Swarm intelligence is defined as a self-organized collective behavior in decentralized systems in which all the individuals obey certain game rules uniformly [16].In recent decades, the correlation between decentralized decision-making and swarm intelligence has emerged as a significant subject within self-organization phenomena [36][37][38][39][40][41][42].In this regard, the methods in statistical physics are widely applied to decentralized social and biological systems, such as the polarization of opinions [15], the movement of animal groups [43][44][45], as well as the emergence of life and intelligence [16,[46][47][48][49][50][51][52].Motivated by this strand of literature, in this paper we study the relationship between swarm intelligence and decentralized decision-making in economic systems.We show that this study may lead to a swarm-intelligence explanation for the transition from thousands of years of Malthusian stagnation to the onset of modern economic growth.
For thousands of years, humans were trapped in an economic situation where the income per capita remained roughly constant [53,54].That is, the increases in available resources were almost always offset by population growth.This economic situation is known as the 'Malthusian Trap' .As shown in figure 1, from 1000 BC to AD 1800, the global per capita income remained close to a low level, approximately equivalent to the per capita income in 1800 [55].However, the industrial revolution that started in the United Kingdom in the 18th century overcame this poverty trap.Since that time, technological innovation has been accelerating, leading to much faster than expected GDP growth, as illustrated in figure 1.To date, it is widely acknowledged that the development of market-based institutions continues to provide the fertile 'soil' necessary for the development of the industrial revolution.In contrast to centrally-planned command economies, market-based economic systems are often regarded as examples of decentralized and self-organized processes.In these systems, although each person makes self-interested and uncoordinated decisions, the resulting collective behavior promotes rapid economic growth.Although economic growth actually benefits from the persistent improvement of technology, there is a lack of understanding of the underlying mechanism that drives technological emergence in decentralized decision-making environments.Our paper addresses this gap.Specifically, we shall show that the emergence of the technology factor T may trigger a type of swarm intelligence among humans, which is characterized by superlinear growth in GDP.
Here, we first outline the motivation for relating the technology factor T to a type of swarm intelligence among humans.Recently, it has been shown that [56], when the marginal technology return1 ∂Y/∂T ⩾ 0, along the competitive equilibrium2 path ∂Y/∂N = 0, the solution of equation ( 1) yields a power law: where c 0 is a positive constant and Equation (3) indicates that γ is a ratio of marginal technology return ∂Y/∂T to income per capita Y/N.Based on this result, it can be seen that, by equation ( 2), the technology factor T has a potential relationship with swarm intelligence among N agents.For example, if ∂Y/∂T > 0 so that γ > 0, by equation ( 2) the income per capita Y/N increases with the population size N.Then, the swarm intelligence is reflected as the following result:

A person would create more wealth per capita to function in a larger colony
In particular, when ∂Y/∂T is comparable to Y/N, equation (2) describes superlinear growth 3 , as depicted in figure 1. Empirically, it has been shown that [56] the power law (2) with γ > 1 exactly agrees with the data from 106 countries across six continents (from 1960 to 2019).However, if the technology factor T disappears, i.e. ∂Y/∂T = 0, by equation ( 2), γ = 0 roughly describes linear growth, i.e. the Malthusian Trap, as shown in figure 1.Then, the growth of income per capita halts, or in other words, the swarm intelligence vanishes.
The discussion above for equation (2) implies that there is a potential relationship between the technology factor T and swarm intelligence, which may provide novel insights into understanding superlinear growth in human societies starting in the 18th-century.In this paper, we formalize this relationship.Specifically, we strictly prove that, if the probability distribution of income acquisition in an economy follows the exponential distribution, and if each agent in this economy independently makes economic decisions, then the technology factor T in equation ( 1) precisely aligns with the information content inherent in the event of decentralized decision-making by all agents.This means that the emergence of the technology factor T is a result of integrating all decentralized decisions.According to equation (2), it may enable the income per capita to increase with the population size.More importantly, we shall demonstrate that an exponential probability distribution of income can be generated within a peer-to-peer economy governed by specific game rules, characterizing a decentralized-decision economic system.This suggests that, as with the swarm intelligence in biotic populations, superlinear growth (2) with γ > 0 is essentially an effect of swarm intelligence emerging in decentralized economic societies.If one observes that market-based economic systems are just examples of decentralized and self-organized processes, our findings suggest that the emergence of swarm intelligence may play an important role in promoting the transition from Malthusian stagnation to modern economic growth.
The remainder of the paper is organized as follows.In section 2, we show that, if the economy's income structure follows an exponential distribution, then the total income of all agents can be represented by an aggregate production function that is a function with respect to the total number of households and the technology factor.In section 3, we prove that, if each agent in this economy independently makes economic decisions, the technology factor in this aggregate production function is exactly equal to the information content contained in the event of all households decentralizing decision-making.In section 4, we distinguish between information and knowledge, and further propose a definition of knowledge.In particular, we demonstrate that the technology factor in the aggregate production function may be interpreted as the knowledge stock in the economy.In section 5, we show that the emergence of such a technology factor enables the income per capita to increase with the population size, which is interpreted as an effect of swarm intelligence.Based on this finding, we offer a swarm-intelligence explanation to elucidate the transition from thousands of years of Malthusian stagnation to modern economic growth.Section 6 concludes the paper.

Exponential probability distribution of income and aggregate production function
There has been a large body of empirical literature [21][22][23][24][25][26][27][28][29][30][31] to support that income structures in various market-economy countries uniformly exhibit a two-class pattern: the majority of the population is described by an exponential distribution and the remaining top income class is described by a Pareto distribution.Based on this empirical observation, an exponential probability distribution of income has been applied to the study of the macro-behavior of a market-based economy [57].As with this strand of literature, we explore collective behaviors in the exponential-distribution region, which dominates an extremely large proportion of the population.To this end, let us first write down the exponential income distribution as follows [3,7,58]: where ε 1 < ε 2 < . . .< ε n , and a j denotes the number of households each obtaining ε j units of income.
It has been observed [59] that, in contrast to an exponential income distribution, an exponential probability distribution of income is a stronger assumption due to its memoryless property.This property implies that an agent's future earnings in an economy with such a distribution are independent of past income accumulation, suggesting an economic network characterized by equal opportunities to earn [59].Such a network, due to the memoryless property, may exhibit features of decentralized-decision systems.In this paper, we explore the collective behaviors within a decentralized economic system, assuming an income structure that follows the exponential distribution (4).In contrast, the Pareto distribution exhibits a memory effect 4 .
According to the economic interpretation of the exponential income distribution (4), the total number of households, denoted as N, and the total income [i.e. the gross domestic product (GDP)] of the economy, denoted as Y, are given by the following equations: To apply the mathematical techniques from statistical physics [32] to equations ( 5) and ( 6), we introduce two new parameters: By using equations ( 7) and (8), equations ( 5) and ( 6) can be rewritten as ε j e α+βε j .(10) Equations ( 9) and (10) show that N and Y are related by the parameters α and β, indicating that the GDP Y is a function of the total number of agents, N. In this paper, we use the GDP Y to denote the aggregate production function.By applying the mathematical techniques from statistical physics to equations ( 9) and (10), it has been shown [5,16,31,[33][34][35] The derivation for equation ( 11) can be found in appendix A.
Let us order 5 : By equation (12), one might question whether T is dependent on N.However, proposition 1, presented below, ensures that if equation ( 1) is solvable, then the GDP Y can be defined as a function that is self-consistent with respect to the variables N and T. Here, 'self-consistent' means that N and T are treated as independent variables for Y. 4 The Pareto distribution, known as the power law, is characterized by the 'Matthew effect' [69,70], which states that the rich get richer and the poor get poorer. 5Equation ( 11) can be rewritten as: dY . Accordingly, we redefine equation ( 12) as T = λ , where λ is a constant.The constant λ serves a role analogous to the Boltzmann constant, as referenced in [5].Here, for simplicity, we select units such that λ = 1.
Proposition 1: Assume that N is defined by equation ( 9), Y by equation (10), and T by equation (12)

. Then, Y is a self-consistent function with respect to the variables N and T if and only if the following partial differential equation holds and is solvable
where equation ( 13) determines a set of 2-dimensional curved surfaces, Ω (Y, N, T); that is, the function Y = Y (N, T) in the set Ω (Y, N, T) is self-consistent with respect to the variables N and T.
Proof.The proof can be found in appendix B or [16,33].
According to proposition 1, from now on we will always select the function Y (N, T) within the set Ω (Y, N, T) to guarantee that N and T are independent variables.Given this configuration, N and T are two well-defined variables.By substituting equations ( 7), (8), and ( 12) into equation (11), we obtain Since N and T are two well-defined variables, equation ( 14) represents a complete differential.Therefore, we have Equations ( 15) and ( 16) show that µ and θ are functions of N and T. Despite this, equation ( 5) does not imply that N depends on T. According to proposition 1, we have selected Y (N, T) from the set Ω (Y, N, T) to ensure the independence between N and T.
By inserting equations ( 15) and ( 16) into equations ( 7) and ( 8), we likewise have and indicating that both α and β are functions of N and T.
To identify economic implications of µ and θ, the variable T has been proposed as the technology factor [5,16,31,34,58].Given this proposal, by equation (15), µ denotes the marginal labor-capital return 7 [58], which indicates the increment of GDP when a new household enters markets, while technology factor T is held constant.Similarly, by equation ( 16), θ denotes the marginal technology return [58].However, unlike N, which denotes the number of agents, T is an emergent variable whose interpretation has not been rigorously justified.In this paper, we formalize the interpretation of the emergent variable T. In the next section, we strictly prove that, if the probability distribution of income acquisition in an economy follows an exponential distribution, and if each agent in this economy independently makes decisions, then T precisely aligns with the information content inherent in the event of decentralized decision-making by all agents.This is the main result of our paper.

Main result
Henceforth, we will refer to 'agent' as 'household' .First, we assume that the probability of income acquisition in an economy is characterized by the exponential distribution.This means that the income structure of the economy follows the exponential distribution, as presented by equation (4).To establish the main result of this paper, we make three assumptions.

Assumption 1:
The GDP, denoted as Y, is a function of collective decisions among N agents, s = (s 1 , s 2 , . . ., s N ); that is, Y = Y (s), where s i represents the economic decision of the i-th agent, with i ranging from 1 to N. 6 According to footnote 5, equation (13), or equation (1), can be generally written as: N

+ (T − λN) ∂Y(N,T) ∂T
= Y (N, T). 7 To identify the economic implication of µ, one needs to assume that households are the owners of labor and capital.This means that the total number of households, N, can be regarded as a function of the labor L and the capital stock K.Thus, a new household entering markets indicates an input of a unit labor and capital.
The economic meaning of assumption 1 is that the GDP of an economy is determined by collective decisions among all social members.Generally, s = Y −1 [Y (s)] may be a multi-valued function of Y (s); that is, different collective decisions among agents may lead to the same level of GDP.In the language of set theory, this means that, if there is a set of collective decisions, H, in which each collective decision leads to the same level of GDP, then, for any two collective decisions s ∈ H and s ′ ∈ H, one has Y (s) = Y (s ′ ).
Here, s = (s 1 , s 2 , . . ., s N ) implies that the number of agents, N, is discrete (or countable).To guarantee that the differential dN in equation ( 14) makes sense, we need to consider a continuum (or uncountable) of agents.As with Aumann's research for a continuum of traders [60], we order s = (s i ) i ∈R with R = [0, N] for a continuum of agents.

Assumption 2 (Harmonious Coexistence):
The number of agents, N, is independent of the collective decision, s, which is defined as s = (s 1 , s 2 , . . ., s N ) for a finite number of agents, and as s = (s i ) i ∈R for a continuum of agents.
The assumption 2 is an intrinsic requirement of proposition 1.If N and T are both functions of s, then N may depend on T via s; however, this contradicts the chosen setting in which we have selected the function Y (N, T) from the set Ω (Y, N, T) to ensure the independence between N and T. Economically, assumption 2 means that collective decisions among agents do not affect the number of agents in the economy.By this assumption, we eliminate the possibility that a certain collective decision will lead to the entry or exit of some agents, thus ensuring economic competition among agents is always moderate, embodying the so-called 'harmonious coexistence' .
According to proposition 1, Y is a function with respect to the bivariate N and T; therefore, by assumptions 1 and 2 we conclude that T is a function of s; that is, As such, because s = Y −1 [Y (s)] may be a multi-valued function of Y (s), we similarly deduce that s = T −1 [T (s)] may also be a multi-valued function of T (s).
To guarantee that the differential dT in equation ( 14) makes sense, we further make the continuity assumption as below: Assumption 3 (Continuity): If there are two collective decisions s and s ′ such that T (s) < T (s ′ ), then we can find a collective decision s ′ ′ that satisfies T (s) < T (s ′ ′ ) < T (s ′ ), where s ̸ = s ′ ̸ = s ′ ′ .
Before we proceed to prove the main result of this paper, we propose a definition for 'decentralized decisions' as follows: Definition 1 (Decentralized Decisions): The collective decision s = (s 1 , s 2 , . . ., s N ) (or s = (s i ) i ∈R ) is a collection of decentralized decisions among N agents (or a continuum of agents) if the probability that each agent i adopts the corresponding economic decision s i to earn is independent, where i = 1, . . ., N (or i ∈ R).
According to definition 1, decentralized decisions mean that each agent in the economy independently makes economic decisions.This definition has been widely used in the literature on decentralized control, e.g.[61,62].
Given assumptions 1-3, we can prove the following three lemmas.
Lemma 1: Assume that the exponential distribution governs the probability of income acquisition in an N-agent economy.For any given number T 0 , if the collective decision s represents a set of decentralized decisions among N agents, the joint probability of N agents making the decentralized decisions s is denoted by where s ∈ {r |T (r) = T 0 }, and the summation s ′ ∈{r ′ |T(r ′ )=T0 } • in equation ( 22) runs over all decentralized decisions s ′ in the set {r ′ |T (r ′ ) = T 0 }.
Proof.The proof can be found in appendix C.
Using equations ( 21) and (22), it is easy to verify that Lemma 2: Assume that Y (N, T (s)) is a differentiable function with respect to N and T (s).For a continuum of agents, if equation (20) holds, then we have where s ∈ {r |T (r) = T 0 }, with T 0 being any given number.
Proof.The proof can be found in appendix D.
Proof.The proof can be found in appendix E.
Using Lemmas 1-3, we prove the main result of this paper as follows.
Proposition 2: Assume that the exponential distribution governs the probability of income acquisition in an N-agent economy.Let the collective decision s be a set of decentralized decisions among N agents.Given that N is defined by equation ( 9), Y by equation (10), and T by equation (12), if equation ( 13) holds and is solvable, then one has: where s ∈ {r |T (r) = T 0 } with T 0 being any given number.
Proof.The proof can be found in appendix F.
Equation (31) is the main result of this paper.Based on this result, equations ( 21) and ( 23) allow us to provide an economic interpretation for the variable T (s).To this end, we need to introduce Shannon's information theory.

The interpretation for emergent variable T (s)
According to Shannon's information theory [63], the information content of an event A, with a probability of occurrence P (A), is given by −lnP (A).This implies that the less likely an event is to occur, the greater the information content it contains.Lemma 1 indicates that P (s) represents the joint probability of N agents making decentralized decisions s.Therefore, by equation (31), the variable T (s) represents the information content of the event where N agents take decentralized decisions s.Since each agent i makes a decision s i based on personal information or dispersed knowledge, and the collection s = (s 1 , s 2 , . . ., s N ) encompasses all agents in the economy, the decision-making process can be viewed as an integration of all personal information.Thus, we refer to T (s) = −lnP (s) as the economy's information stock.Although there are subtle differences between information and knowledge, we propose the following definition to clarify these distinctions.

Definition 2 (Knowledge):
For the information stock T = T (s), if then T = T (s) denotes the knowledge stock.
Inequality (32) means that the increase in knowledge in the economy does not reduce the GDP.By contrast, information comprises both useful and useless parts.An increase in the useless part may reduce the GDP, i.e. ∂Y/∂T < 0. In this sense, we define 'knowledge' as information that has a positive impact on the economy, or in other words, 'useful information' .
For over a century, economists have recognized that capital and labor do not account for all economic growth.This is reflected in the total factor productivity (or the Solow residual) used in the aggregate production function Y (L, K, T * ) that accounts for the contributions of labor L and capital K, yet has an unexplained contributor known as the technology factor T * .Considering that households are the owners of labor and capital, we can define the total number of households N as a function of labor and capital, N (L, K).Given the independence between N and T (s), we deduce from proposition 1 that T * = T (s).As such, the aggregate production function can be written as: which implies that, if inequality (32) holds, the unexplained technology factor T * may be understood as the knowledge stock in the economy.This aligns with the literature of endogenous growth theory [64,65] that identifies the technology factor as the knowledge stock.Next, we further demonstrate that, if inequality (32) holds, the emergence of the technology factor T (s) induces superlinear growth of the GDP.

A swarm-intelligence explanation for superlinear growth
In section 2, we derived the partial differential equation ( 1) by assuming that the economy obeys an exponential probability distribution of income, as shown in proposition 1.In sections 3 and 4, we demonstrated that, if each agent in this economy independently makes economic decisions, then the variable T in equation ( 1) can be interpreted as the information stock in the economy.In particular, when inequality (32) holds, the variable T can be interpreted as the knowledge stock.Here, we further show that, if inequality (32) holds, equation ( 2) is a result of equation ( 1) when the economy reaches competitive equilibria.To this end, we observe that the competitive equilibrium condition can be denoted by8 [56,66]: Competitive equilibrium means that, under the given game rules, N agents reach a consensus for resource allocation.It is a key concept for describing the competitive behaviors among different agents.This indicates that focusing on the outcomes under a competitive equilibrium is of primary interest in economic studies.Therefore, to establish the existence of a competitive equilibrium in an economic system, one must solve the partial differential equation ( 1), ensuring that equation (34) has a solution for any N ⩾ 1.To illustrate this, let us write down a solution of the partial differential equation ( 1) as follows: where a 0 and γ are two positive constants.Mathematically, equation ( 35) is an analytical solution of the partial differential equation ( 1) when the competitive equilibrium condition ( 34) is accounted for [56].To verify this fact, we write down the partial derivative of equation (35) as follows: from which we can deduce that T = [(1 + γ) /γ] N is a solution of the equation ∂Y/∂N = 0 for any N ⩾ 1; that is, the existence of a competitive equilibrium can be guaranteed.However, the partial differential equation ( 1) also has other solutions that cannot ensure the existence of a competitive equilibrium.To illustrate this, we observe that the partial differential equation ( 1) has a solution as follows [33,35]: where b 0 denotes a positive constant.From equation ( 37), it can be easily verified that ∂Y/∂N = b 0 (lnN + 1) ̸ = 0 for any N ⩾ 1.
In fact, the solution ( 37) is related to the second law of thermodynamics in physics [33,35]: If one denotes Y as the internal energy, then T can be interpreted as the entropy.To observe this, let us write equation (37) in the differential form: where and we have used NlnN ≈ lnN! to approximate the term.
Interpreting Y and b 0 in equation ( 38) as 'internal energy' and 'temperature' , respectively, S can be interpreted as 'Clausius entropy' .Thus, T = S − lnN! can be identified as entropy with the Gibbs term lnN!.
The solutions (35) and (37) suggest that the partial differential equation ( 1) has applications in both economic and physical systems.Similar to the income distribution in a market-based economy, it is known that the energy distribution of inanimate particles also follows an exponential pattern 9 , known as the 'Boltzmann distribution' .The existence of a competitive equilibrium, such as equation (34), is a key difference in distinguishing an economic system from a physical system.Unlike a physical system, which consists of inanimate particles, an economic system is composed of self-aware human beings.Self-interested motivations drive these individuals to compete, and they must ultimately reach certain consensuses to reconcile their interests, achieving what is known as competitive equilibrium.In contrast, particles lack self-interested motivations.
Now, let us focus on the solution (35), which describes an economic system.Using equation (36), the competitive equilibrium condition can be written as By plugging equation (40) into equation (35), we obtain the result under the competitive equilibrium: which is identical to equation (2).It is easy to see that for equation (35), ∂Y/∂T ⩾ 0. According to definition 2, this means that T in equation ( 35) denotes the knowledge stock in the economy.When γ = 0, equation (41) indicates a linear growth of the GDP, and the income per capita Y/N remains constant.This scenario describes the Malthusian trap 10 , as illustrated in figure 1.However, by equation ( 40), we observe that γ = 0 makes no sense for N ⩾ 1.Therefore, we conclude 11 γ > 0, indicating superlinear growth of the GDP, so that the income per capita Y/N increases with the population size N.This result suggests that superlinear growth of the GDP can be explained by the emergence of an exponential probability distribution of income in a decentralized competitive economy, because this leads to the emergence of the knowledge stock T, i.e. equation (31).In fact, from equation (35), it can be seen that, if γ = 0, then the knowledge stock T vanishes in the GDP.
Theoretically, it has been shown that an exponential probability distribution of income can emerge in a peer-to-peer type of Arrow-Debreu economy where private property rights are impartially arranged [7,16,31,34,56,58,66].The game rules of this peer-to-peer economy can be summarized as follows [58,66]: (a) The property rights are arranged such that, if an agent uses their goods to produce products, then the products, as new goods, belong to this agent.(b) Each agent is self-interested and is allowed to exchange goods without incurring transaction costs.(c) Each agent's economic decision is independent.(d) Information flows freely.(e) Each agent employs constant returns technologies freely to produce goods.
In microeconomics, the constant returns technology is considered the most sensible long-run production technology, as discussed on Page 356 in [67].From this sense, the rule (e) is thus employed to ensure that the peer-to-peer economy operates as a competitive market over an extended time scale.More importantly, the unrestricted use of this technology leads to zero economic profit for firms, suggesting a reduction in industrial barriers and monopolies.Essentially, given the game rules (a)-(e), the peer-to-peer economy represents an ideal competitive market without friction.Provided these rules are followed, an exponential income distribution, i.e. equation (4), will emerge as the number of agents increases sufficiently.A detailed rigorous derivation of this result is provided in [58].To elucidate this result, a simplified model is presented in [66].Specifically, recognizing that rule (c) signifies decentralized decision-making, equation ( 31) can in principle be derived using rules (a)-(e), as illustrated by proposition 2. Consequently, akin to swarm intelligence in biotic populations [43][44][45]68], swarm intelligence in human societies emerges with a large population size when certain game rules are established.This insight offers a potential swarm-intelligence perspective for understanding the transition from thousands of years of Malthusian stagnation to modern economic growth.
During the past few centuries, the development and improvement of market-based institutions have led to the emergence of a decentralized-decision system, where price signals effectively passed information among people 12 .This has facilitated the evolution and refinement of rules (a)-(d).On the other hand, free competition under these rules over an extended period weakened monopolies.As a result, as these rules developed and were refined, a larger population size began to foster the emergence of human swarm intelligence, characterized by superlinear growth, as indicated by equations ( 2) and (41).Conversely, if the development of these rules is obstructed, an increase in population size could become a burden, with the growth in available resources being consistently negated by the growth in population, a phenomenon known as Malthusian stagnation.

Conclusion
In this paper, we have demonstrated that, if the exponential distribution governs the probability of income acquisition in an N-agent economy, and if each agent makes economic decisions independently, then the total income of all agents can be represented by an aggregate production function, in which the technology factor precisely aligns with the information content inherent in the event of decentralized decision-making by all agents.In particular, for sufficiently large population sizes, the emergence of this technology factor enables the income per capita to increase with the population size, indicating superlinear growth of the GDP.This finding suggests that superlinear GDP growth can be attributed to the emergence of an exponential probability distribution of income in a decentralized economy.In particular, we regard superlinear GDP growth as a manifestation of swarm intelligence among agents, meaning that an agent can generate more wealth per capita within a larger economic community. 10We assume that a0 is a small quantity. 11By equation (35), it is easy to check that γ = ( . Because ∂Y/∂T ⩾ 0, we conclude γ ⩾ 0. 12 The development of information technology [71] has further promoted the enhancement of the price mechanism, thereby improving its efficiency and responsiveness.
In this paper, we also demonstrate that an exponential probability distribution of income can emerge in a peer-to-peer economy with certain game rules, characterizing a decentralized-decision economic system.This suggests that swarm intelligence in human societies, such as superlinear GDP growth, is an emergent phenomenon in a decentralized-decision system with certain game rules, particularly when the population size is large.Based on this finding, we offer a swarm-intelligence explanation to understand the transition from thousands of years of Malthusian stagnation to modern economic growth.In our opinion, over the past few centuries, the development and improvement of market-based institutions have led to the emergence of a decentralized-decision system where price signals effectively facilitate information exchange among humans.This has promoted the development and refinement of endogenous game rules within such a system.As these game rules have gradually been established and perfected, a large population size has begun to foster human swarm intelligence, manifesting as superlinear GDP growth.
Next, we verify the necessity.If equation ( 13) holds and is solvable, then N is obviously independent of T. Furthermore, it is proved that equation ( 13) is solvable [33][34][35].□

Appendix C
Proof of lemma 1 Proof .Since the exponential distribution governs the probability of income acquisition in an N-agent economy, without loss of generality we assume that the income structure follows the exponential distribution (4).As a result, the joint probability distribution among N agents, P * (1, . . ., N) , can be written as To obtain equation (C.1), we rely on the assumption of decentralized decisions, which implies that the probability of each agent acquiring income is independent.By equation ( 5), the probability of obtaining ε j units of income is a j /N.Therefore, under the assumption of decentralized decisions, the joint probability for a j agents, each obtaining ε j units of income, is a j /N a j = a j a j /N a j /N • • • a j /N .Equation (C.1) is derived by applying the assumption of decentralized decisions across all income levels ε 1 < ε 2 < . . .< ε n .

Figure 1 .
Figure 1.From the Malthusian trap to the industrial revolution [55].