Evolution of Family Systems and Resultant Socio-Economic Structures Kenji Itao1 and Kunihiko Kaneko1, 2, *

Evolution of family systems and resultant socio-economic structures Kenji Itao1 and Kunihiko Kaneko1, 2, *

1Department of Basic Science, Graduate School of Arts and Sciences, University of Tokyo, Komaba 3-8-1, Meguro-ku, Tokyo 153-8902, Japan. 2Research Center for Complex Systems Biology, University of Tokyo, Komaba 3-8-1, Meguro-ku, Tokyo 153-8902, Japan. *[email protected]

ABSTRACT

Family systems form the basis of society and correlate with distinct socio-economic structures. In nuclear families, children build families of their own after marriage, whereas in extended families, children stay at the parents’ home. The inheritance of wealth among siblings is whether equal or unequal. These two dichotomies shape the four basic family systems. However, their origin remains unknown. In this study, we theoretically simulated a model of preindustrial peasant societies consisting of families. By introducing family-level and society-level competition for their growth, we demonstrated that the four family systems emerge depending on environmental conditions characterizing the land scarcity and the external perturbations that damage society. The commencement of agriculture and the area location explain the geographical distribution of family systems across the world. Analyses on the wealth distribution among families demonstrate the connection between family systems and socio-economic structures quantitatively. This connection explains the development of distinct modern ideologies.

Family systems provide the basis of a social structure, in- Isolated and self-sufficient peasant societies prevailed in cluding its sociological, ideological, and economic character- the preindustrial world9–12 and provided a basis for modern istics1–3. Generally, family systems are based on parent-child political systems1. Family system determines residence and and inter-sibling relationships. A parent-child relationship inheritance patterns in land usage1,8 . Children may cultivate refers to children whether living together at their parents’ lands of their own or work together on their parents’ land. home, which is termed an extended family, or building homes One heir may inherit the land and most of the property, and of their own, which is termed a nuclear family. An inter- the other children inherit only a little13, 14 or the land and the sibling relationship refers to siblings whether being equal or property may be divided equally among family members14. unequal, especially in the distribution of inheritance. These It is unclear why various family systems exist in different ar- characteristics last in area over a long period and have at- eas. Researchers have tried to explain the appearance of each tracted significant attention in the study of history4–6, histori- 7 1,2,8 system as a result of cultural transmission and the adaptation cal demography , and anthropology , among others. to environmental conditions, such as land scarcity, farmland 1,2 management system, and frequency of famine among oth- Family systems are broadly classified into the four types . 8, 15, 16 (1) Absolute nuclear families, which are nuclear families with ers . However, the argument requires further scrutiny. The discussion regarding the relationship between family sys- unequal inheritance, are dominant in England and the Nether- 2 lands. (2) Egalitarian nuclear families, which are nuclear tems and ideologies remains largely psychoanalytical . Theo- families with equal inheritance, are dominant in France (es- retical studies for unveiling the conditions under which each pecially in Paris) and central Spain. (3) Stem families, which family system evolves, and for connecting family systems with socio-economic structures need to be conducted. In such are extended families with unequal inheritance, are dominant 17 in Japan and Germany. (4) Community families, which are studies, mathematical modelling would be relevant . Theo- extended families with equal inheritance, are dominant in retical studies to date have involved modelling the influence

arXiv:2009.11035v1 [physics.soc-ph] 23 Sep 2020 of demographic condition and marriage rule on household China, Russia, and Northern India. Links exist between each 18 19 of these family systems and modern social ideologies. Liberal- size or the spatial pattern of paternal/maternal inheritance . ism, liberal egalitarianism, social democracy, and communism We adopt a framework involving multi-level selection for a are dominant in the regions with absolute nuclear, egalitarian hierarchical system. We note that families constitute society, nuclear, stem, and community families, respectively1,2 .A whereas society provides the environment for families. The more detailed classification has been presented based on pa- multi-level selection was originally introduced to explain the ternal/maternal inheritance and the external/internal marriage evolution of cooperative behaviour among eusocial insects rule8. However, in this study, we ignore such details and focus by examining the conflict between the fitness of an individ- only on the conditions under which each of the four family ual and that of a group20, 21. This framework is generally systems mentioned above emerges and the socio-economic applied to the evolution of hierarchical systems and is used to structure that corresponds to each. discuss the public good or social structure in biological and a b Nuclear Extended members live and work together. Children build families of their own by inheriting their parents’ wealth. Wealth w is accumulated through production, which in turn increases the level of production and the population by taking demographic reports into account7. Each society splits into two when the number of families therein doubles its initial value Nf , and each family is randomly assigned to one of the two daughter societies. At this time, another society is removed at random Time so that the number of societies in the entire system remains fixed to Ns. Therefore, societies that grow faster replace others, resulting in the society-level selection. This multi-level Fig. 1. Schematic of the model. a, The multi-level evolutionary selection of families and societies follows the hierarchical process. Societies (green) consist of families (blue) in which family Moran process22–24. members (black) live together. The grey frame represents a single Capacity c N , which represents the amount of available generation. When the population of a society reaches twice the × f initial population, a society splits and another society is removed land resources in a society, and ε, which is the loss of wealth from the system at random to keep the number of societies fixed at the time children build families of their own for surviving — i.e., we adopt the hierarchical Moran process. b, The separation external threats are given as environmental parameters. If the strategy of the family and the life cycle. Black squares, circles, and wealth w at that moment is less than ε, the child dies without triangles represent members of different generations. The size of the reproducing. blue circle reflects the amount of wealth that each family possesses. Families have population and wealth, as well as two strategy Dashed arrows show the separation of siblings and arrows show the production of wealth through family labour. Each family has parameters: λ, which represents the parameter characterizing a separation strategy s. With the probability of s, siblings produce the inequality in the inheritance of wealth among siblings, together, after which they become independent and reproduce the and s, which represents the probability of children staying at next generation, as in an extended family. With the probability of their parents’ home to produce together. The mth child’s share 1 s, siblings are separated, after which they produce independently of the inheritance is proportional to exp( λm). Therefore, − − and reproduce the next generation, as in a nuclear family. λ = 0 represents an equal division of inheritance, whereas a larger λ represents the first child inheriting more. The life cycle of the families depends on the separation social evolution22–27. We have previously applied the frame- strategy s. With the probability of 1 s, siblings are sepa- work to construct a mathematical model for the evolution of − rated, i.e., inherit some property determined by λ, lose ε of kinship structures in clan societies22. Diverse kinship struc- wealth, and build families of their own to produce indepen- tures emerge depending only on the environmental parameters dently. In contrast, with a probability of s, siblings remain in that characterize the degree of cooperation and competition their parents’ family to produce together, after which they are resulting from marriage. separated. According to the law of diminishing return, pro- In this study, we investigate the evolution of family systems ductivity increases sub-linearly with labour force input28–30. and social structures by introducing an agent-based multi- Thus, the total output of siblings is always higher when each level evolutionary model of preindustrial peasant societies. produces independently to form a nuclear family than when Competition is considered in two levels: that of family, which they concentrate their labour in an extended family. Under is the individual agent of the model, and society, which is a a limited capacity of available land resources, however, the group of families. Families produce wealth through family total output of the society consisting of nuclear families will labour and reproduce. They possess two strategy parameters be lower than that of extended families because of the in- concerning the time children leave their parents’ home and the efficiency in land usage. In other words, there is a conflict distribution of inheritance among them, which are transmitted between society and family levels. A nuclear family is an with slight mutations in each generation. Evolutionary simula- advantageous strategy at the individual family level. However, tions show that the four family systems emerge depending on an extended family is advantageous at the society level, under environmental parameters that characterize the land scarcity conditions of limited capacity. and the external perturbations that damage society. We then describe the characteristics of social structure in terms of the After sibling separation and the production of wealth, each distribution of wealth in society and relate them to family family reproduces. The number of children in families follows systems. the Poisson distribution with a mean of b + f w, where b and f represent the minimal birth rate and the feedback ratio, respectively. s and λ are culturally transmitted over generations, Results with a slight variation through “mutation”31, 32 with the rate Evolution of Family Systems of µ. At the time of altering generations, each society splits if A schematic representation of the model is shown in Fig. the number of families reaches 2Nf . See Methods for details. 1a,b. Society consists of families. In each family, individual Simulations are performed for 2,000 time steps by changing

2/8 al 1. Table 4. Fig. in varied hs iga ffml ytm.Hr,w lsiytefamily if the extended classify as we the Here, systems shows systems. 3c family of Fig. diagram phase respectively. extended evolve, cultivation, families for nuclear room or sufﬁcient or limited is there siblings among When or inheritance respectively. small the evolves, is in survival equality for or required inequality wealth large, the when the is shows that 3a,b large, Fig. steps. of 1,000 param- last dependence strategy the the in averaged families of and eters condition each for times in steps region. 1,000 parameter every approximately within converges strategy the of of values the on pending strategy inheritance the and red) given strategy each separation The to adapted condition. environmental value speciﬁc are but a society, around each concentrated within 2. diverge not Fig. do in the strategies shown of Family are series parameters time strategy The family of 1. Tableevolution in shown values parameter the with conditions environmental for respectively, ε blue, and red in of values the of evolution 2. Fig. 0.7 σ ε c N d N b Symbol µ = f 3 ecnutdmlilvleouinr iuain100 simulation evolutionary multi-level conducted We s

f c 0 . 0001 iesre ffml taeyprmtr.Tetemporal The parameters. strategy family of series Time aaeesue ntemodel. the in used Parameters .010.01 0.0001 , 0 . tegho h os nproductivity in noise the of Strength independence for wealth of Loss and capacity of Ratio for rate system Mutation a in societies society of a Number in families of number Initial wealth of rate rate birth Decay to wealth from rate Feedback rate birth Minimal Explanation 1and 01 ∗∗ s and ersnstevralsvre nFg S1. Fig. in varied variables the represents c = λ s s 0 ≥ . and upon 7 c , 0 and 3. . λ 5 c s n sncerif nuclear as and ffmle nasceyaeplotted are society a in families of c ssalo ag,ta swhen is that large, or small is ε ε and λ epciey h evolution The respectively. , and N potdi le vlede- evolve blue) in (plotted λ f ε ∗ When . ersnstevariables the represents s ε < s ssalor small is potdin (plotted 0 . 5 Simi- . Variable Variable 0 0 0 Value . 1 . . 03 50 30 0 5 3 . . 5 ∗ ∗ ∗ ∗ 1 ∗ ∗ ∗ ∗ ∗ When hntemnmlbrhrate birth minimal the when , hw htif that shows 4a,b h hs iga ffml ytm.Se,cmuiy absolute community, Stem, systems. family of diagram phase The ai aiysseseov eedn ntetoenviron- two the on four depending the evolve Therefore, systems children. family preceding basic the smaller by is that children of subsequent half the than by inheritance the of share if if equal inheritance as unequal as them classify we larly, λ if if family family extended an classified of are red, those strategies as green, Separation in respectively. plotted orange, and are blue, families nuclear egalitarian and nuclear, of values average the show graphs λ The values. parameter mental 3. Fig. eedneo h hs iga nteparameters the on diagram phase the of Dependence large relatively of because parame- regions larger ter in evolves is distribution wealth equal less and Therefore, accumulated, decreases. share each and increases, for small regions. parameter most in evolves family nuclear the that small for evolves that strategy family selfish nuclear a a choosing is Therefore, if usage limited. land is in capacity inefficiency the of the that of but because independently, decreases work society they if increases siblings of when even evolve Fig. selection. society-level of result a as evolves behaviour small of because dominant is competition society-level selfish the and when Conversely, important, evolves. less behaviour becomes There- society of society-level. benefit of the that fore, than rather dominant becomes small selection multi-level on of effect dependence the Generally, show quan- exist. However, trends values. titative parameter other the of Phase independent S1. Fig. and against 4 plotted Fig. diagrams in parameters other upon systems evolves orange) in both (plotted if family nuclear egalitarian an if and evolves blue) in (plotted family if evolves red) if evolves parameters mental c b a f ≥ r hw nFg S1. Fig. in shown are ( swat siiilyaccumulated, initially is wealth As family of diagrams phase the of dependence the show We a and ) o2ada qa if equal as and log2 N N c eedneo aiysrtg aaeesuo environ- upon parameters strategy family of Dependence s s s and utmtl,if (ultimately, s w = < c ( λ b n cuuainwudb ceeae.However, accelerated. be would accumulation and and 0 1 o aiisi h at100seso simulation. of steps 1,000 last the in families for ) ε . < h oit-ee tesi o osdrdso considered not is fitness society-level the , 5 neiac taeisaecasfida nqa if unequal as classified are strategies Inheritance . r large. are c log2 ε ssaland small is r ml.Acmuiyfml potdin (plotted family community A small. are N c c f ssal ealta h oa production total the that Recall small. is hscieini ie ywehrthe whether by given is criterion This . and slreor large is λ N < s c ε = tmfml potdi green) in (plotted family stem A . log2. and b 1 shg,tenme foffspring of number the high, is ε ,tefml-ee competition family-level the ), s ε slre naslt nuclear absolute An large. is ≥ N r ulttvl outand robust qualitatively are s N 0 ssal ula families nuclear small, is . c 5 f slreand large is rlarge or n stoeo nuclear a of those as and ε ε 22–24 4c. Fig. in shown as , ol erelatively be would N o large For . s N λ n large and s cooperative , ≥ N ε s log2 µ ssmall, is and , d N and , f and 3/8 N N or c, f f . a We fitted the poor and rich tails of the wealth distribution using the power-law wα and exponential distribution exp( βw), − respectively. The fitted values of α and β are averaged over 100 trials for each environmental parameter c and ε in Fig. 6a,b. The heaviness of the tail on the poor side is increased for smaller values of c and alpha, while the heaviness of the 1 30 100 Ns tail on the rich side is increased for smaller values of ε and β. b These results suggest the characteristics of the wealth distribution in the four corresponding family systems. However, it remains unclear whether they result from the environmental conditions or family systems. To confirm the relevance of family systems, we computed the dependence of the tails on 10 50 100 family systems by sampling each family system using fixed Nf environmental parameters c and ε near the boundary of the c four phases of family systems, where the values of s and λ are distributed to cover all four family systems. The average values of α,β for each family system, which were sampled for the fixed environmental parameters, are shown in Fig.6c,d. They demonstrate the trend that the 1.0 1.2 2.0 poor tail is heavier for an extended family, and the rich tail is b heavier for unequal inheritance. By comparing these results (plotted in black) with the results averaged over all environ- Fig. 4. Dependence of the phase diagrams of family systems on parameters with in Table1( a, N b, N c, b). Unless the value is mental parameters (plotted in red), it is shown that the above ∗ s f shown on the axis, the parameter values are fixed to those in Table1. trend depends on each family system and is further intensified by environmental parameter values.

Wealth Distribution and Evolution of Social Struc- Discussion ture By simulating the multi-level evolution model of family sys- Subsequently, we investigated the wealth distribution of fami- tems using the environmental parameters for the capacity of lies for each society after evolution. Note that data from the available land resources c and the amount of wealth required wealth distribution of the majority (people except elites) in for survival ε, we demonstrated that an extended or nuclear modern society suggest an exponential-type tail for the rich family evolves under small or large c, respectively, and that 33, 34 side, which decays slower than the normal distribution unequal or equal inheritance among siblings evolves under 35 36 (say a log-normal distribution or gamma distribution ) and small or large ε, respectively. Therefore, the four basic family 37 a power distribution for the poor side (say a gamma distri- systems are represented as “phases” depending on c and ε. 36 bution ). The gamma distribution is obtained by assuming Additionally, we clariﬁed the characteristics of wealth dis- the growth of wealth with positive feedback and nonlinear tribution determined by the dominant family systems within saturation, as well as a multiplicative stochastic process (see societies. The tail of the poor side is heavier for extended Supplementary Text). families and that of the rich side is heavier for families with Fig.5a shows the frequency distribution of the wealth unequal inheritance. of families within each society. In every parameter region, From this dependence of family systems on environmental the distribution approximately follows a power-law on the conditions, one can discuss their geographic conﬁguration, poor side and an exponential tail on the rich side, which is as in Fig.7. Demographics report that the death rate caused consistent with recent data33–37. Here, the heaviness of tails by war, famine, and starvation, decreases in the following depends on the environmental parameters. When c and ε order: the centre of the continent, peripheral, and island re- are smaller, the poor and rich tail are heavier, respectively. gions7, 38, 39, which suggests that ε decreases in the same order. Because inherited wealth depends on birth order, we also Areas in Western Europe, particularly Spain, Italy, and France, plotted the distribution of wealth by distinguishing the eldest were frequently attacked by foreign people, such as Muslims, siblings from the others in Fig.5b. With decreasing ε and Magyars, and Vikings, which resulted in a large ε therein40. increasing inheritance inequality, the distribution of wealth The period since the onset of agriculture gives a measure for between siblings separates further. Increasing land scarcity capacity. In areas where agriculture started early, population (smaller c) and the evolution of extended families result in growth resulted in the exhaustion of available land and labour- poorer younger siblings, whereas greater land availability intensive farming developed, as seen in China11, 41, Russia42, (larger c) and the evolution of nuclear families give rise to and Japan43. In Western Europe, especially Holland, the Paris wealthier younger siblings Basin, Southern England, and central Spain, the capacity was

4/8 where , r ossetwt h bevto fncerfamilies nuclear of observation the with consistent are 4a,b aa,Grayadmn at frrlWsenEurope Western rural of parts many and Germany Japan, ugs htEgadadteNtelnshdlarge had Netherlands the small and England that suggest those Europe especially Western studies, on historical to socio-economic referring by and structures systems family between 4c. relationships Fig. in one the to the similar in tendency distribution a equal demonstrates and latter regions former the in in inheritance especially Europe, Western and The Japan England latter. in the low in was families rate extended birth and Fig. regions in former the trends in The Germany. and Japan, Russia, China, in whereas and Spain, small and is France, Netherlands, the England, in system. rate society a within families where small. India, Northern and Russia, where Italy, and Spain, where Netherlands, and the small absolute and region: England each in families in nuclear systems family the explain can one on early plagues and wars religious and second gathering sons subsequent by cultivation medieval land in virgin progress enabled agricultural times reasons: following the of developed was farming labour-saving and large distribution. frequency the of (blue). plot siblings log-log other the the show Insets fitting. distribution exponential with conditions environmental 5. Fig. a 0.7 et eeaietevldt forrslsrgrigthe regarding results our of validity the examine we Next, of number the conditions, environmental these from Apart above, discussed reports geohistorical and 7 Fig. From 3 b c r lorlvn aaeesfrdtriigtefamily the determining for parameters relevant also are β itiuin fwat neovdsocieties. evolved in wealth of Distributions ε ned elh amr rsee n employed and prospered farmers wealthy Indeed, . 11 N 7 and n ihi Russia in high and , . f c slrefrlresaeln aaeeta seen as management land large-scale for large is a ag;eaiainncerfmle nFrance, in families nuclear egalitarian large; was N c s eesal n omnt aiisi China, in families community and small; were 0.0001 slrei aiymngmn am sseen as farms management family in large is 45 , 44 30, 9, 12 1, 7 and 6 Figs. data. rich have which , ε N h ouainsantdbcueof because stagnated population the , and f 30 ubro societies of number , ε n ooiswr established were colonies and , 42 = c h bevto funequal of observation The . 0 eelre tmfmle in families stem large; were . ε 0001 ε , a ag and large was 0 . 01 and a, 0.01 c 41 N h rpsso h rqec itiuino aiywat ntels ,0 tp for steps 1,000 last the in wealth family of distribution frequency the show graphs The = s saresult a as n birth and , 0 . 7 , α c ε 3 h rneln hw h oe-a tig n h upeln hw the shows line purple the and fitting, power-law the shows line orange The . was was and N 46 f , b a,tewat itiuinotie normdlconnects model our in obtained distribution wealth the way, stratified weakly h ocuinhr sidpneto h eal ftepresent the of details the of independent is here conclusion The n miia rasuisars h ol ihquantitative necessary. with be world will the indicators across studies area research empirical theoretical and between collaboration a structures, and social systems, family relation- conditions, the environmental of among exploration ships further and For family considered. between is selection society multi-level with and sub-linearly input, increases production labour the as long as model, general. rather be to expected are structures socio-economic respectively. people, privileged and vulnerable of presence power or inequality respectively to imbalance, response threats a to as exposed egalitarianism individuals oppose many and of presence the authoritar- in support ianism people of that study shows the which with ideology, political consistent are results our his- Additionaly, socio-economic tory. studies society-level with systems family sub- and poor ordinate uniformly were peasants and sparse significantly small have to gested order of distinctions acceptance class the and in resulted which advanced, stratification society the of and peasants exploiting by prospered small farmers had that Japan suggest results and Our Sweden, equality. Germany, and freedom of values the large have majority the liberty ism of individual force of development labour the independent explains an and standards capital living of regions better other had in who those than labourers as majority the 0.7 n a lodsusln-emcagsin changes long-term discuss also can One the and systems family the regarding results our that Note 3

11 4–6, c 12 nEgad rneadSanwr ugse to suggested were Spain and France England. in hc rvddro o communism for room provided which , α and .010.01 0.0001 49 12, 11, b, β hr,pol eels ifrnitdand differentiated less were people There, . h elhdsrbto fteeds rd and (red) eldest the of distribution wealth The α 50–52 , 43 12, 1, n large and hc loe epet appreciate to people allowed which , 55 eas small because , 48 , 47 15, , 11 7, 1, β ε usaadCiawr sug- were China and Russia . ned h idecasis class middle the Indeed, . h accumulation The . α α c and and and 1 54 53, β β n capital- and ε Wealthy . ml the imply n their and , nthis In . 5/8 a b late England France Netherlands Spain

Russia c d Germany Japan India

Onset of agriculture of Onset early China

island peripheral central Fig. 6. The dependencies of the heaviness of the tails of the wealth distribution upon environmental parameters and family systems. a, Geographic location Heaviness of the poor tail α given by the power-law fitting wα . b, Heaviness of the exponential tail β given by the exponential fitting Fig. 7. Phase diagram of family systems against environmental exp( βw). Smaller α and β show the heavier tails for the poor − conditions; the capacity of land resources c, and the amount of wealth and rich sides, respectively. c, d, Averaged value of α (c) and β required for survival ε, associated with the period since the onset of (d) classified by the dominant family system (stem, community, agriculture and geographic location, respectively. Stem, community, absolute nuclear, and egalitarian nuclear families) in each society, absolute nuclear, and egalitarian nuclear families are plotted in green, with error bars. Values for the environmental parameters fixed to red, blue, and orange, respectively. Small or large c and small or ε = 0.0003 and c = 1.5 are plotted in black, whereas those averaged large ε are explained by the early or late onset of agriculture and over ε = 0.0001,0.0003,0.001,0.003 and c = 1.0,1.2,1.5,2.0 are peripheral or central geographic location, respectively. plotted in red.

issues. Second, the present model did not consider marriage. influence on family systems. At times of cultivation, a nuclear Marriage combines the wealth of two families to build a new family evolves because of sufficient capacity. As the popu- family. Such horizontal transmissions of wealth introduce lation increases and capacity become limited, an extended another form of interaction between families. The effect of family would replace it. Additionally, the risks of invasion marriage-related interaction on social structures should be ex- from surrounding areas, infectious disease, and famine, and amined. Third, marriage age, external/internal marriage, and accordingly ε would increase gradually because of the dense paternal/maternal inheritance are considered to further clas- population. This scenario explains the historical change of sify family systems8 and are out of the scope of the present family systems from a primitive nuclear family (which is an model. Finally, in the modern world, peasant societies should intermediate form of a nuclear family and the inheritance no longer be regarded as isolated systems, but as components in which siblings temporary living together with parents af- of a world-system11. A new model needs to be developed to ter marriage and the distribution of wealth is weakly biased, consider the interaction between towns and peasant societies, as observed in some peasant societies, hunter-gatherers, and as well as international, political, and commercial networks. 8 nomads) to a stem family and then to a community family . To summarize, we presented a multi-level evolution model Here, environmental factors change gradually as a result to account for the emergence of the four basic family sys- of the interaction between the society and the environment. tems depending on environmental conditions and resultant However, the change in environmental factors, in turn, alters socio-economic structures, as is consistent with family-level family systems and social structures. Such historical dynamics studies and economic history at a society-level. In this study, of society were discussed by Braudel as the interaction of the microscopic characteristics of family systems determine historical factors on different time scales4–6. the macroscopic social structure of the distribution of wealth, The present model has some limitations. First, the dif- which forms the basis for the development of societies. This ferentiation of people between the elites and the majority study allows an explanation of the universal evolutionary con- was not considered. As society becomes stratified, people straint that human societies satisfy. start to rent land from the elites. This results in the divergence of environmental factors and different family systems Methods amongst the elites and the majority, as has been ethnograph- ically observed8. A model is needed for dealing with social We adopted the following algorithm for changes in the wealth stratification and the interaction of classes to discuss broader and population of families. Families lose a proportion d of

6/8 their wealth when the generation is altered as a result of aging conflict between family and society level preferences for nu- equipment, disaster, or taxation, for example. With probability clear versus extended families under conditions of the limited 1 s, siblings leave their parents’ home before production, and capacity. − the inheritance is distributed with a wealth loss of ε. Families We assumed that the mth child’s share of the inheritance produce through the labour provided by the family members. is proportional to exp( λm). In some societies, however, − Here, the production of wealth is inversely proportional to the last child inherits the most instead of the first8. If nec- the number of families in society if capacity is exceeded, essary, the order of children could be arranged in reverse to proportional to the logarithm of labour, increases with linear include such a case. Although an exponential distribution feedback from wealth, and is perturbed by noise resulting from is adopted in this study, the results described below do not internal and environmental fluctuations following a Gaussian change qualitatively if a linear distribution is considered. distribution. With probability s, siblings work together at their We carried out a multi-level evolutionary simulation using parents’ home and build families of their own after production. the settings mentioned above. The initial values of strategies Finally, people produce offspring and the generation is altered. are s = 0.5,λ = log2 for all families. In other words, the The birth rate increases linearly with wealth. Here, families family is differentiated to neither nuclear nor extended and reproduce without considering marriage explicitly. neither equal nor strongly biased inheritance. However, no For the parent family i and its jth child’s family i, j, the pop- qualitative changes are observed under other initial conditions. ulation N and the amount of wealth w at time t are expressed The source codes are available online. The notations and as follows: parameter values adopted in the simulations are summarized in Table1. with probability 1 st , − i Nt = 1 (1 j Nt ) (1) References i, j ≤ ≤ i t 1. Todd, E. L’invention de l’Europe (Editions du Seuil Paris, 1990). Ni t t t t 1 λi j λi k 2. Todd, E. La diversité du monde: structures familales et modernité (Seuil, wi,∗j = (1 d) wi− e− / ∑ e− ε (2) − · · k=1 − 1999). t t t t 3. Laslett, P. Family, kinship and collectivity as systems of support in pre- w = w ∗ + r (1 + η)(1 + w ∗ )log(1 + N ) (3) i, j i, j · i, j i, j industrial europe: a consideration of the ‘nuclear-hardship’hypothesis. unless, Continuity change 3, 153–175 (1988). t t 1 4. Braudel, F. Civilization and capitalism, 15th-18th Century, Vol. I: The wi∗ = (1 d) wi− (4) structure of everyday life, vol. 1 (Univ. of California Press, 1992). t t − · t t w = w ∗ + r (1 + η)(1 + w ∗)log(1 + N ) (5) 5. Braudel, F. Civilization and capitalism, 15th-18th century, vol. II: The i i · i i t wheels of commerce, vol. 2 (Univ. of California Press, 1992). Ni t t 6. Braudel, F. Civilization and capitalism, 15th-18th century, vol. III: The t t λi j λi k t wi, j = wi e− / ∑ e− ε (1 j Ni ) (6) perspective of the world, vol. 3 (Univ. of California Press, 1992). · k=1 − ≤ ≤ 7. Macfarlane, A. The savage wars of peace: England, Japan and the then, Malthusian trap (Springer, 2002). Nt+1 = Poisson(birth + f wt ) (7) 8. Todd, E. L’origine des systèmes familiaux, vol. 1 (Gallimard Paris, i, j · i, j 2011). t+1 t t t si, j = si + ζ, λi, j = λi + ζ (8) 9. Cameron, R. E. et al. A concise economic history of the world: from Paleolithic times to the present (Oxford University Press, USA, 1993). where, 10. North, D. C. Structure and change in economic history (Norton, 1981). r = min(1,capacity / # families) (9) 11. Wallerstein, I. The modern world-system I: Capitalist agriculture and the origins of the European world-economy in the sixteenth century, η N(0,σ 2), ζ N(0, µ2). (10) ∼ ∼ vol. 1 (Univ. of California Press, 2011). 12. Rösener, W. Die Bauern in der europäischen Geschichte (CH Beck, Following a study on preindustrial farming56, we assume 1993). that production increases in proportion to the logarithm of 13. Berkner, L. K. Inheritance, land tenure and peasant family structure: a labour input, and is perturbed by Gaussian noise with a mean German regional comparison in Family and Inheritance; Rural Society 2 in Western Europe 1200-1800, ed. J. R. Goody, J. Thirsk, E. P. Thompson of 0 and a variance of σ , resulting from internal and environ- (Cambridge Univ. Press, 1976). mental fluctuations. Therefore, the total output of siblings is 14. Kaser, K. Power and inheritance: Male domination, property, and family always higher when each produces independently to form a in eastern europe, 1500–1900. The Hist. Fam. 7, 375–395 (2002). nuclear family than when they concentrate their labor in an 15. Laslett, P. The world we have lost: further explored (Routledge, 2015). extended family (the productivities of N members are N log2 16. Goldschmidt, W. & Kunkel, E. J. The structure of the peasant family. and log(N + 1), for nuclear and extended families respec- Am. Anthropol. 73, 1058–1076 (1971). tively). However, under a limited capacity of available land 17. Axelrod, R. Advancing the art of simulation in the social sciences. In Simulating social phenomena, 21–40 (Springer, 1997). resources, the total output of the society consisting of nu- 18. Wachter, K. W., Hammel, E. A. & Laslett, P. Statistical studies of clear families will be lower than that of extended families historical social structure (Elsevier, 2013). 1 (the productivities are N N log2 and log(N + 1), for nuclear 19. Siess, V. A mathematical modelling of emmanuel todd’s origin of family and extended families respectively). In other words, there is a systems with a diffusion reaction system (2019).

7/8 20. Wilson, D. S. Altruism and organism: Disentangling the themes of 49. Dupeux, G. La société française, 1789-1970 (FeniXX, 1972). multilevel selection theory. The Am. Nat. 150, s122–S134 (1997). 50. Kastner, K. Aus der Chronik des Kirchspiels Hohenkirch Kr [eis] 21. Wilson, D. S. & Wilson, E. O. Rethinking the theoretical foundation of Briesen (Westpr [eussen]).: Seine Landgemeinden und Gutsbezirke sociobiology. The Q. review biology 82, 327–348 (2007). (Truso-Verlag, 1978). 22. Itao, K. & Kaneko, K. Evolution of kinship structures driven by marriage 51. Mager, W. Haushalt und Familie in protoindustrieller Gesellschaft: tie and competition. Proc. Natl. Acad. Sci. 117, 2378–2384 (2020). Spenge (Ravensberg) während der ersten Hälfte des 19. Jahrhunderts. 23. Traulsen, A. & Nowak, M. A. Evolution of cooperation by multilevel Eine Fallstudie (1981). selection. Proc. Natl. Acad. Sci. 103, 10952–10955 (2006). 52. Hayami, A. & Kurosu, S. Regional diversity in demographic and family 24. Takeuchi, N., Hogeweg, P. & Kaneko, K. The origin of a primordial patterns in preindustrial japan. J. Jpn. Stud. 27, 295–321 (2001). genome through spontaneous symmetry breaking. Nat. communications 53. Weber, M. The Russian Revolutions (Cornell University Press, 1995). 8, 1–11 (2017). 54. Thaxton, R. Salt of the earth: The political origins of peasant protest 25. Spencer, C. S. & Redmond, E. M. Multilevel selection and political and communist revolution in China (Univ. of California Press, 1997). evolution in the valley of oaxaca, 500–100 bc. J. Anthropol. Archaeol. 55. Claessens, S., Fischer, K., Chaudhuri, A., Sibley, C. G. & Atkinson, 20, 195–229 (2001). Q. D. The dual evolutionary foundations of political ideology. Nat. Hum. 26. Turchin, P. & Gavrilets, S. Evolution of complex hierarchical societies. Behav. 1–10 (2020). Soc. Evol. Hist. 8, 167–198 (2009). 56. Evenson, R. E. & Mwabu, G. The effect of agricultural extension on 27. Turchin, P. Warfare and the evolution of social complexity: A multilevel- farm yields in kenya. Afr. Dev. Rev. 13, 1–23 (2001). selection approach. Struct. Dyn. 4 (2010). 28. Ricardo, D. Principles of political economy and taxation (G. Bell and sons, 1891). Acknowledgements 29. Malthus, T. R. An essay on the principle of population as it affects the The authors thank Tetsuhiro S. Hatakeyama, Yuma Fujimoto, and future improvement of society, with remarks on the speculations of Mr Kenji Okubo for a stimulating discussion, and Yasuo Ihara for illu- Godwin, M (1798). minating comments. This study was partially supported by a Grant- Bacci, M. L. A concise history of world population (John Wiley & Sons, 30. in-Aid for Scientific Research on Innovative Areas (17H06386) from 2017). the Ministry of Education, Culture, Sports, Science, and Technology 31. Cavalli-Sforza, L. L. & Feldman, M. W. Cultural transmission and evolution: A quantitative approach. 16 (Princeton University Press, (MEXT) of Japan. 1981). 32. Creanza, N., Kolodny, O. & Feldman, M. W. Cultural evolutionary Author contributions statement theory: How culture evolves and why it matters. Proc. Natl. Acad. Sci. 114, 7782–7789 (2017). K.I. and K.K. designed the model; K.I. conducted the simulations; 33. Chakrabarti, B. K., Chakraborti, A., Chakravarty, S. R. & Chatterjee, A. K.I. and K.K. analysed the data; and K.I. and K.K. wrote the paper. Econophysics of income and wealth distributions (Cambridge University Press, 2013). Additional information 34. Tao, Y. et al. Exponential structure of income inequality: evidence from 67 countries. J. Econ. Interact. Coord. 14, 345–376 (2019). The authors declare no conflict of interest. 35. Gibrat, R. Les inégalits économiques (Librairie du Recueil Sirey, Paris, Source codes for the model can be found here: https:// 1931). github.com/KenjiItao/family_system.git 36. Chakraborti, A. & Patriarca, M. Gamma-distribution and wealth inequality. Pramana 71, 233–243 (2008). 37. Reed, W. J. The pareto law of incomes—an explanation and an extension. Phys. A: Stat. Mech. its Appl. 319, 469–486 (2003). 38. Khazanov, A. & Wink, A. Nomads in the Sedentary World (Taylor & Francis, 2012). 39. Umesao, T. An ecological view of history: Japanese civilization in the world context (ISBS, 2003). 40. Bloch, M. & Fossier, R. La société féodale, vol. 7 (Albin Michel Paris, 1968). 41. Pomeranz, K. The great divergence: China, Europe, and the making of the modern world economy (Princeton University Press, 2000). 42. Hizen, E. The demographic background of the land problem in russia (1880’s-1920’s). Jpn. Slav. East Eur. Stud. 15, 1–25 (1994). 43. Hayami, A. Japan’s Industrious Revolution: Economic and Social Transformations in the Early Modern Period (Springer, 2015). 44. Grigg, D. B. Population growth and agrarian change: an historical perspective (CUP Archive, 1980). 45. Pirenne, H. Economic and social history of medieval Europe, vol. 60 (Houghton Mifflin Harcourt, 1956). 46. Berkner, L. K. The stem family and the developmental cycle of the peasant household: An eighteenth-century austrian example. The Am. Hist. Rev. 77, 398–418 (1972). 47. Tawney, R. H. The agrarian problem in the sixteenth century. 13 (Longmans, Green and Company, 1912). 48. Shaw-Taylor, L. The rise of agrarian capitalism and the decline of family farming in england 1. The Econ. history review 65, 26–60 (2012).

8/8 Supplementary Information for Evolution of family systems and resultant socio-economic structures 1 * KENJI ITAO AND KUNIHIKO KANEKO

*[email protected]

SUPPLEMENTARY TEXT Deriving the gamma distribution from the stochastic process In the wealth production of our model, we consider the linear feedback from wealth and that from the logarithm of the labour force, as well as the multiplicative noise. Additionally, through the decay of wealth and the division of the inheritance, the amount of wealth for each family is saturated. Here, the growth of wealth with positive feedback and nonlinear saturation with a multiplicative stochastic process is assumed. The production of wealth increases with wealth w, which is generally saturated for large w. A simple example is given for the linear growth and the nonlinear saturation as w˙ = bw aw2, whereas the production rate ﬂuctuates through the noise. Therefore, the simplest example− of wealth dynamics is expressed as follows:

w˙ = aw2 + bw + cwη, (S1) − where η follows a normal distribution. The stationary distribution that corresponds to Eq. (S1) is expressed as follows:

1 aw2 + bw P(w) = 2 exp(2 − 2 dw) (S2) (cw) Z (cw) 1 2a 2b ∝ exp( w + log w) (S3) c2w2 − c2 c2 (2b 1)/c2 2a = w − exp( w). (S4) − c2 This deﬁnes the gamma distribution. Our model includes a complicated process for the distribution of wealth to siblings as well as feedback from the labour force to wealth production, which cannot be written using simple forms, as shown in Eq. (S1). However, one may expect linear feedback, nonlinear saturation, and multiplicative stochasticity. Therefore, the simple model Eq. (S1) would capture the dynamics to a certain degree. arXiv:2009.11035v1 [physics.soc-ph] 23 Sep 2020 SUPPLEMENTARY FIGURE a

0.003 0.01 0.1 μ b

0.1 0.5 1.0 f c

0.3 0.4 0.7 d

Fig. S1. Dependence of phase diagrams of family systems on the parameters µ, f , and d. Stem, community, absolute nuclear, and egalitarian nuclear families are plotted in green, red, blue, and orange, respectively. Unless the value is shown on the axis, the parameter values are ﬁxed to those in Table 2. a, The dependence on mutation rate µ. As µ becomes larger, regions with nuclear families shown in blue and orange are enlarged. b, The dependence on feedback rate f . A large f accelerates wealth accumulation, which results in the development of unequal inheritance. Additionally, a large f results in many children. Therefore, family-level competition is intensiﬁed, and the regions of a nuclear family are enlarged. c, The dependence on the decay rate d. A large d prevents wealth accumulation, which results in the development of equal inheritance. When d is small, unequal inheritance develops. If it is too small, the nonlinear saturation of wealth accumulation works only weakly so that the wealth of stem families can reach w (1000). At that time, the feedback from the wealth to the birth rate should be saturated in reality, and∼ our O model assuming linear feedback would be invalid. There should be a biological upper limit to the number of offspring. For families with more wealth, higher negative feedback terms that are not considered here should keep the fertility rate constant. Therefore, this model is only valid for w (10). As long as d 0.4, the nonlinear saturation works to keep w (10) and the phase diagrams are qualitatively∼ O robust. ≥ ∼ O