Mathematical models of complex historical processes – an introduction

Sergey GavriletsDepartment of Ecology and Evolutionary Biology,

Department of Mathematics, National Institute for Mathematical and Biological

Synthesis, University of Tennessee, Knoxville

• Descriptive approaches• Experimental/manipulative approaches• Theoretical approaches

• Models and modeling– Verbal, graphical– Statistical– Mathematical (e.g. dynamical systems)

• analytical• numerical

Alternative approaches to Alternative approaches to scientific investigationscientific investigation

What can mathematical models What can mathematical models do?do?

• explain complex interactions between multiple forces and factors (nonlinear feedbacks, etc)

• identify crucial parameters and factors, evaluate relevant temporal and spatial scales, identify general patterns

• guide analysis of data (e.g. by providing null hypotheses and tools for testing them)

• guide empirical research, point to the gaps in knowledge• train our intuition, provide simple and intuitive tools and metaphors for

thinking about complex phenomena• predict consequences of various actions/interventions (e.g., if

experimental approaches are impossible/impractical)• generate new knowledge

Evolutionary biology:Evolutionary biology:Theoretical population genetics and “Modern Theoretical population genetics and “Modern

Evolutionary Synthesis”Evolutionary Synthesis”• The “Modern Synthesis” of the 1930s and 1940s became possible only after the

development of theoretical population genetics in the 1920-1940s– “a great impetus to experimental work on the genetics of populations” (Sheppard, 1954) – a “guiding light for rigorous quantitative experimentation and observation”

(Dobzhansky, 1955)

• Provine (1978): “the work of Fisher, Wright, and Haldane had significant influence on evolutionary thinking in at least four ways”

– showed that the processes of selection, mutation, drift, and migration were largely sufficient to account for microevolution

– showed that some directions explored by biologists were not fruitful– complemented and lent greater significance to particular results of field and laboratory

research– stimulated and provided framework for later empirical research.

The role of mathematical The role of mathematical modelingmodeling

• Mathematical models are the most appropriate tool for synthesis in most sciences. The level of maturity of a science can probably be measured by the degree of its mathematical sophistication.

• To really understand a complex process often means not only to have an appropriate verbal explanation or a metaphorical picture but also a corresponding mathematical model.

• A common wisdom is that a picture is worth a thousand words. In the exact sciences, an equation is worth a thousand pictures.

• Haldane’s attitude (from “A defense of beanbag genetics”, 1964)– Mistrust of verbal arguments where algebraic arguments are possible– Skepticism when not enough facts are known to permit of algebraic arguments

History to mathematicsHistory to mathematics• “Problems of history may still turn out to be as inspirational to mathematicians, as

problems of physics have been, and as problems of biology are bound to become.” Nicolas Rashevsky (1954)

• Nicolas Rashevsky (1899-1972)

• Pioneer of mathematical/theoretical biology and mathematical biophysics• Pioneer of mathematical modeling of social and cultural evolution– “Mathematical theory of human relations” (1947)– “Mathematical biology of social behavior” (1950)– “Looking at history through mathematics” (1968)

Modeling (the transition to) large-Modeling (the transition to) large-scale socialityscale sociality

• Scaling up models of small-scale sociality– Cultural evolution– Warfare and group selection– Coevolution of behavior and norms– Collective action and cooperation

• Models capturing new and emergent properties of large-scale sociality

Theoretical methodsTheoretical methods• Chemistry

– Coagulation-fragmentation theory

• Economics– Game theory (cooperation, altruism, public goods)– Contest theory (individual and group-level)– Coalition formation theory (cooperative and non-cooperative)

• Evolutionary and population biology models– Evolutionary game theory (cooperation, altruism, public goods)– Group/multilevel selection– Cultural evolution– Ecological models

Modeling must be empirically Modeling must be empirically based!based!

• Traits in SESHAT database– Polities

• Scale of societies (size, territory, etc)• Hierarchical structure and complexity• Horizontal complexity (bureaucracy, division of labor, taxation,

punishment, etc), cultural complexity

– Warfare (technologies, animals, armor, naval, fortification, intensity, military organization, cultural distance, etc)

– Ritualism and religion– Resources

• Models of territorial growth– Ordinary differential equation (Rashevsky 1953ab;Dacey

1969,1974) – Agent-based simulation (Bremer&Mihalka 1977, Cusack and

Stoll 1990, Cederman 1997)

• Hierarchical structure and complexity– Agent-based simulation (Gavrilets et al 2010, Cederman and

Girardin 2010, Griffin 2011)– Behavioral biology models

• Theories of hierarchy formation (Dugatkin, Bonabeau, others) • Theories of coalition and alliance formation (Mesterton-Gibbons et al.

2011)

• Ethnic markers and the emergence of cultural boundaries– Continuously varying traits (Boyd&Richerson 1987)– Discrete traits (McElreath et al 2003)– Mathematical theory of speciation (Gavrilets 2004)

can be adapted for cultural and ethnic “species”• Reinforcement• Ecological speciation• Genetic incompatibilities• Speciation by drift

• Ecology-inspired models– Commoners-elite, competing polities, workers-owners-police

(Spencer 1998, Turchin 2003)

• Division of labor and social stratification– Henrich&Boyd (2008)– Models of the division of labor in biology

• Multicellularity• Insect societies

– Size and complexity• Diversity of agent-based models designed for describing

specific processes in specific geographic areas during specific historical periods

Three examplesThree examples

• 1. Emergence of chiefdoms and states– Gavrilets, Anderson, and Turchin (Cliodynamics

2010)

• 2. Emergence and spread of empires– Turchin, Currie, Turner, and Gavrilets (PNAS 2012)

• 3. Collective action problem

Robert Carneiro

(1970 Science 169, 733–738)

Carneiro’s (1970) argument•Good agricultural land was limited•Independent farming villages emerged •Population pressure led to conflict and war between these villages•Successful warlords became chieftains•Chiefdoms competing with one another led to state formation

Prerequisites of state formation (Carneiro, Weber, Flannery, Wright and others)• Warfare between different polities starting with villages• Circumscription

– environmental, due to the resource concentration– social, due to the presence of other human groups nearby

• Political subordination to the victor and intensification of production as a price of defeat

• The existence of agricultural potential capable of generating surpluses • Significant variation in productive and/or demographic potential

between local communities

• The ability to delegate power and the invention of hierarchically structured control mechanisms in which each superior directly controlled only a limited number of subordinates

“Scalar stress” and hierarchical nature of control mechanisms

• “Scalar stress” is a decrease in the ability of leaders to process information and maintain efficient control over subordinates as their number increased

• The hierarchical nature of the organizing principle allows, in theory, for an unlimited growth in the size and complexity of societies.

Questions• Do models support for verbal arguments

and generalizations from data?• What are

• the levels of complexity that can be achieved, • its dynamic patterns, • timescales, and • the qualitative and quantitative effects of

various parameters and factors.

• To what extent is the hierarchical organization of early societies reflected in archaeological data?• settlement hierarchy, burials, religious

monumentsThe core area of the Monte Ablan site (300-100 BC)

Model: Gavrilets, Anderson and Turchin (2010)

• An array of autonomous local communities (“villages”)– Circumscription!

• Each community is a part of a polity • Polities have a hierarchical structure

• The complexity of the polity c is the number of levels of control above the level of individual villages.

• Tribute flow

Model (cont)

• The polities are engaged in warfare. • The polities grow, decrease in size, or disappear as a result

of conquest with the polity-winner absorbing (a part of) the polity-loser.

• The new polities appear and old polities decrease in size when a subordinate community secedes from its polity taking with itself all of its subordinates.

Model captures

• Warfare&circumpscription (Carneiro)• Variation in productive/demographic potential

(Webster)• Scalar stress (Johnston)• Ability to delegate power

– But no internal specialization: secession is relatively easy (Wright)

• Reasons for collapse: military defeat and rebellion

General observations• a rapid formation of polities of various size and complexity as a result of warfare • quick (within 50-100 years) approach to an equilibrium in which the focal characteristics

are maintained at approximately constant values• this equilibrium is stochastic and is characterized by the dynamic instability of the

individual polities, with quick collapse of the chiefdoms reaching relatively large size and complexity.

=1 vs. =2

Settlement patternsSettlement patterns

Distributions of rank and power

“Chiefly cycles”

• The dynamics generated by the model resembles the “chiefly cycles” observed prior to sustained Western contact in eastern North America, southern Central America and northern South America, Oceania, southeast Asia and the Philippines, and across large parts of sub-Saharan Africa

=1 =2

Maximum size (S=4,=1) Maximum size (S=4,=2)

Mean complexity (S=5,=1) Mean complexity (S=5,=2)

Effects of parameters• Most important parameters:

– and , with higher values strongly promoting the existence of larger, more complex, and more stable polities.

• crucial importance of the existence of well-defined and accepted succession mechanisms for the stability of polities

• stronger dependence of the outcome of the conflict on the polities' power (wealth); predictability

ji

iij FF

FP

No qualitative differences between simple and complex cheifdoms

• During each individual run, the number of control levels is not stable but changes dynamically and therefore cannot by itself serve as an indicator of the presence of ``true'' states.

• Carneiro (1981): ``The emergence of chiefdoms was a qualitative step. Everything that followed, including the rise of states and empires, was, in a sense merely quantitative''

Instability of large and complex polities

• Polities that include a significant part of (or all) villages do emerge but are relatively short-lived. – A major reason: a relative ease of rebellion and autonomous

existence of polities.

• In our model, it is explicitly assumed that any ``internal specialization'' is absent and that all mechanisms for the autonomous existence of a rebellious part are already in place and can be readily used if rebellion is successful.

General conclusions• The model

– supports the classic verbal arguments about the formation of complex polities (“a predictable response to certain specific cultural, demographic and ecological conditions” Carneiro 1970)

– provides strong support for the generality of “chiefly cycles” (Anderson; Marcus)

– illustrates the fluid nature of many characteristics of polities– identifies the most important parameters and time scales– provides a theoretical tool for further exploration of the

evolution of political complexity in early societies

2. Turchin, Currie, Turner and 2. Turchin, Currie, Turner and Gavrilets (2012)Gavrilets (2012)

• A greatly simplified version of Gavrilets et al. (2010) but with addition of– cultural traits affecting power and cohesion of

polities (cultural multi-level selection)– realistic geographic landscape (Afro-Eurasian map,

agriculture, effects of elevation)

• Testing against historical data

• Which prediction to test?– General patterns rather than unique events– “empire density” (average over time): the model

does pretty well (Tom Currie’s talk)

• Which prediction to test?– General patterns rather than unique events– “empire density” (average over time): the model does

pretty well (Tom Currie’s talk)

• Why does our model “work”?– Multiple “signals” (ultra-sociality traits, military

technology, landscape, geographic position, etc)– Distance to the steppe as the strongest “signal” in

data; the model shows the robustness of its effects to other factors

Modeling simplificationsModeling simplifications

– Gavrilets et al. 2010: • conquest is beneficial if costs of war are low• each polity acts as a whole but• sub-chiefs always prefer to be independent; they rebel

if rebellion is feasible

– Turchin et al. 2012: • conquest is always beneficial• each polity acts as a whole but• complete disintegration of polities happens with a

certain probability

Modeling simplificationsModeling simplifications

• Largely disregarded are – conflicting interests of different individuals and

factions comprising competing societies; – various costs and benefits of cooperation and

competition

3. Collective action theory3. Collective action theory

Collective action problemCollective action problem

• If a group member benefits from the action of group-mates and individual effort is costly, then there is an incentive to ‘free ride’. However, if individuals follow this logic, the public good is not produced and all group members suffer.

Collective action problemCollective action problem

• Generic for situations requiring cooperation (altruism, public goods)

• Overcoming it is a major challenge faced by animal and human groups

• Possible solutions (in evolutionary biology): – Kin selection, reciprocity, reputation, punishment,

group selection

• Within-group conflict/inequality

ModelModel

Dynamics in hierarchical groupsDynamics in hierarchical groups

Efforts and fitnessesEfforts and fitnesses

“Altruistic bullies”

Scaling upScaling up

• From groups composed of individuals to societies composed of factions

• Needed:– Modeling competition between multifarious

societies (facing the collective action problem) in spatially explicit context

Towards empirically-based Towards empirically-based modelingmodeling

• Traits in SESHAT database– Polities

• Scale of societies (size, territory, etc)• Hierarchical structure and complexity• Horizontal complexity (bureaucracy, division of labor, taxation,

punishment, etc), cultural complexity

– Warfare (technologies, animals, armor, naval, fortification, intensity, military organization, cultural distance, etc)

– Ritualism and religion– Resources