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Groups as the Most Natural and Useful Level of Analysis

In my view, debates concerning whether kin selection, individual selection, group selection or multilevel selection are ‘true’ are ill-conceived.  These are merely different frameworks for thinking about evolutionary change.   Since any two of these frameworks, independently brought to bear on the same question, can result in similar answers, it is perhaps more productive to ask which level or levels of selection are the more useful or natural to think about specific problems.

One problem that has interested a great many researchers, including the entire field of international relations (IR), is how to explain the dynamics of interaction between sovereign territorial states (i.e., ‘countries’).  Here I first argue that the group, specifically the sovereign territorial state, is the most natural level of analysis in IR which has, in fact, long been the conventional wisdom in the field.  Second I argue that developing a more formal theory of selection at this level in IR can add much to the “conventional historian’s account” of the field.

What is the most useful unit of analysis in international relations?  In answering this question I will look at the works of three prominent IR scholars, Kenneth Waltz, Alexander Wendt, and Robert Keohane.  These scholars are not-cherry-picked, but are the researchers most associated with the three most prominent paradigms in modern international relations theory and were named, in a recent survey of over 1,500 IR scholars, the field’s three most influential scholars of the last 20 years[1].  Their approaches to international relations can thus be considered roughly representative of the field.

The most thorough examination of the proper unit of analysis in IR is Waltz’s seminal 1959 book,Man, The State and War [2].  He asks whether international warfare is best understood by appeals to individual psychology, appeals to domestic political institutions, or appeals to states’ interactions with each other.  Although acknowledging that “some combination of the three… may be required for an accurate understanding of international relations” [3], he finds that these levels of analysis are unequal partners.

Waltz, importantly, finds little utility in appeals to evolved psychology in IR, writing:

While human nature no doubt plays a role in bringing about war, it cannot by itself explain both war and peace, except by the simple statement that man’s nature is such that sometimes he fights and sometimes he does not. [4].

He goes on to argue that since human nature should be relatively constant and states are not always at war or always at peace,one must look above the level of the individual to explain the conditions under which war occurs and the conditions under which it does not.   He eventually settles on the interactions between states, especially the modern territorial states, as the most appropriate level of analysis.

Sovereign territorial states have many properties that lend themselves naturally to the logic of multilevel selection [5, 6].   First, they are territorial, meaning that an individual’s group membership can be more-or-less defined by looking where he or she lives on a map.  Second, they are sovereign, meaning that they have a government with sole legal authority over the affairs within its territory and the legal authority to negotiate with others outside of it.  This allows for an easier demarcation of internal and external dynamics of the group. Third, as political scientist John Mueller put is, once the major powers signed the UN charter “they essentially declared international boundaries sacrosanct no matter how illogical or unjust some of them might seem to interested parties.” [7] This allows us to assume low rates of what Samir Okasha calls “multilevel-selection 2” (MLS2), where selection occurs as groups dies off and other groups are born.  Instead we can restrict our analysis primarily to MLS1, where groups are not removed from the system, but selection acts to change the frequency of traits both within and between them [8].

Waltz argues, in his 1979 book, Theory of International Politics, that states are most usefully thought of as “unitary actors” with their own preferences and motivations.  It is not that Waltz thinks states actually lack internal divisions, but that for most questions in IR the unitary actor model does just fine without appeals to lower levels of analysis [9].  This line of reasoning has been very productive in IR theory, and although I lack space for a complete review, the unitary actor assumption is commonly made in theoretical models of state behavior, including James Fearon’s influential bargaining models of war [10], and rarely has to be justified to a political science audience.

Waltz’s theory of international politics is explicitly evolutionary.  Noticing that the death rate of states is low (thus, as pointed out above, limiting the importance of MLS2), Waltz’s selection operates through success-biased emulation.  He writes that “the theory says simply that if some [states] do relatively well, others will emulate them or fall by the wayside.” [11]

Wendt’s 1999 Social Theory of International Relationsis often taken to be a critique of Waltz’s book.  But in it he not only defends the unitary actor assumption, but considers it “essential to both the explanatory and political aspects of the states systemic project.” [12] However, Wendt goes much further than Waltz in explaining the evolutionary mechanisms underlying international change. To Wendt, important questions in IR cannot be understood without understanding the “distribution of ideas in the system,”[13](where ‘ideas’ are roughly equivalent to Dawkins’ memes or Boyd and Richerson’s cultural traits). Wendt stresses that ideas change through a process of “cultural selection” which operates through states’ imitation of successful states.  Since he also accepts that states do not often die, Wendt’s theory is ultimately one of MLS1 selection – where selection operates by changing the frequency of ideas within and between groups.

Robert Keohane is the unitary actor assumption’s most prominent critic [14].  However, even he stresses the primacy of groups, just ones operating at other levels of organization.  In addition to states he argues for the importance of international organizations (such as the United Nations), transnational organizations (such as terrorist groups and multinational corporations) and domestic institutions (such as legislatures). His theoretical framework is one of institutions.  Although his theory is the least explicitly evolutionary, others have argued that institutions are difficult to explain without evolutionary principles.  For example, Elinor Ostrom, in a recent review of her institutional analysis framework, finds that “developing formal methods for examining diverse processes of structure change over time is an important next step for scholars who have mastered some of the new techniques derived from the study of genetics.” [15]

In my handful of presentations to political science audiences I have never once had to justify my choice of accounting for the payoffs to state behavior at the level of groups. It is easy to see how scholars in IR have settled on groups as the proper and most useful unit of analysis.  Say one is interested in the spread of democratic forms of governments or interstate war.  While it is easy to see bicameral legislatures or conscription armies as properties of groups, it is more difficult, at least for me, to see how their costs and benefits can be easily partitioned down to the level of individual selection.

Using IR’s three most influential scholars as examples, I have argued that groups are the most natural level of analysis for answering questions in IR and, specifically, MLS1 on cultural traits or ‘ideas’ is the most important evolutionary framework.  But, as Pinker asks, “what does ‘natural selection’ add to the historian’s commonplace that some groups have traits that cause them to grow more populous, or wealthier, or more powerful, or to conquer more territory, than others?”

At first this seems a silly question, akin to asking “what does ‘natural selection’ add to the naturalists’ commonplace observation that some organisms have traits that cause them to have more offspring, control more resources, or live longer than others?”  It is as if evolutionary theory should have both begun and ended with Darwin.

There are many suggestions in the writings of the above scholars, that a more formal theory of selection amongst groups has a lot to offer theories of international relations.  However, since few scholars are trained in both international relations and evolutionary theory, the utility of natural selection to understanding state behavior is still somewhat of an open question.  Perhaps the most prominent work in this vein, thus far, is a series of models and empirical analyses by Lars-Erik Cederman and colleagues [16-19] addressing the spread of democracy through what is known as the “democratic peace.” The democratic peace is a robust empirical finding that democracies tend not to go to war with each other. Cederman’s models show the conditions under which this property of democracies can allow them to spread at the expense of non-democracies. These are interesting models and important first steps to an understanding of the dynamics of democracy and peace in the international system.  However, a greater understanding of the findings of evolutionary game theory can make them more useful.

For example, in one of Cederman’s models, democracies are ‘tagged’ so that they can recognize each other and cooperate, out-competing non-cooperative non-democratic states under a variety of conditions [17].  At first this seems a ‘commonplace’ application of a robust empirical observation. However, as Cederman acknowledges, his model relies on the assumption that tagged states will not defect on these arraignments.  In essence, a democratic tag is a perfect predictor of cooperation, what is known as a ‘green beard’ in evolutionary biology [20].

What evolutionary theory adds to this analysis is the non-obvious recognition that ‘green beards’ are unlikely to last long.  The problem is that in the system Cederman describes selection for defection should act the most strongly on democratic tagged states, because they are more likely to be the benefactors of cooperation.

For illustration, suppose two democracies cooperate by mutually leaving their mutual border undefended so they can save on the cost of defending them.  Meanwhile, their borders with nondemocracies are well defended.  Now suppose one of these states is in need of resources that it cannot easily get through other means (such as trade).  Which neighboring state is the least costly for it to invade?  All things equal, it is the one with undefended borders – the other democratic state.  Thus, in order for strategic tagging to work there must be some force maintaining the correlation between the tag and altruistic behavior. This is a general analytic finding in evolutionary theory [21], but perhaps not immediately obvious to an historian or political scientist [22].

In summary, the science of international relations has long recognized human groups as the most natural and useful unit of analysis in understanding its important questions.  In fact, it is difficult to see how the selection on traits easily thought of as properties of groups, such as systems of government and foreign policies can be reexamined as the result of selection process operating primarily at the level of individuals.  Even if it was possible, it is unclear why one would want to go through the mental and mathematical gymnastics required.  I further disagree with Pinker that the understanding of selection on cultural traits has nothing to add to international relations theory and give an example where non-trivial applications of selection models can lead to novel insight.

1. Avey, Paul C., Michael C. Desch, James D. Long, Daniel Maliniak, Susan Peterson, Michael J. Tierney. 2012.  The Ivory Tower Survey How IR scholars see the world. Foreign Policy Magazine. (http://www.foreignpolicy.com/articles/2012/01/03/the_ivory_tower)

2. Waltz, Kenneth N.  1959. Man, the State, and War. New York, NY: Columbia University Press.

3. ibid. p. 14

4. ibid. p. 29

5. Tilly, Charles. 1992. Coercion, Capital and European States. Wiley-Blackwell.

6. Spruyt, Hendrik. 1996. The Sovereign State and its Competitors. Princeton, NJ: Princeton University Press.

7. Mueller, John E. 2004. The Remnants of War. Ithaca, NY: Cornell University Press. (p. 67)

8. For a discussion of MLS1 and MLS2, see Okasha, Samir. 2006. Evolution and the Levels of Selection. Oxford University Press.

9. Waltz, Kenneth N. 1979. Theory of International Politics. McGraw-Hill.

10. Fearon, James D. 1995. Rationalist Explanations for War. International Organization, 49:379-414.

11. Waltz. Theory of International Politics.(p. 117)

12. Wendt, Alexander. 1999. Social Theory of International Politics. Cambridge, MA: Cambridge University Press. (p.243)

13. ibid. (pp. 5, 41, 96, 138)

14. Keohane, Robert O.  1984. After Hegemony: Cooperation and Discord in the World Political Economy.Princeton University Press.

15. Ostrom, Elinor. 2011. Background on the Institutional Analysis and Development Framework. Policy Studies Journal 39:7-27.

16. Cederman, Lars-Erik. 2001. Back to Kant: Reinterpreting the Democratic Peace as a Macrohistorical Learning Process.” American Political Science Review 95:15-31.

17. Cederman, Lars-Erik. 2001. Modeling the Democratic Peace as a Kantian Selection Process.”Journal of Conflict Resolution 45:470-502.

18. Cederman, Lars-Erik and Kristian S. Gleditsch. 2004. Conquest and Regime Change: An Evolutionary Model of the Spread of Democracy and Peace.” International Studies Quarterly 48:603-629.

19. Cederman, Lars-Erik and Mohan PenubartiRao. 2001. Exploring the Dynamics of the Democratic Peace. Journal of Conflict Resolution 45:818-833.

20. This is called a green beard, from a hypothetical example by Richard Dawkins where agents with green beards always cooperate with other green-bearded agents; Dawkins, Richard. 1976. The Selfish Gene. Oxford: Oxford University Press.

21. See Grafen, Alan. 1990. Do Animals Really Recognize Kin?” Animal Behaviour 39:42-54.

22. For a similar example see:

Riolo, Rick L., Michael D. Cohen and Robert Axelrod. 2001. Evolution of Cooperation Without Reciprocity. Nature 414:441-443.

Roberts, Gilbert and Thomas N. Sherratt. 2002. Does Similarity Breed Cooperation?  Nature 418:499-500.

Published On: July 6, 2012

Matthew R. Zimmerman

Matthew R. Zimmerman

Matthew R. Zimmerman is a PhD student in the Cultural Evolution Lab at the University of California Davis.

12 Comments

  • hgintis says:

    This is a beautiful contribution. One small comment: I don’t think the question of gene-level, individual level and group level evolution is “ill conceived.” It is true that one can do the accounting on any level and get the same answer. But meotic drive is gene level, not individual or group level. Inclusive fitness explains lots of things on the individual level that have no causal significance on the gene level and for which group level causation is absent. And group level effects are causally strong in many cases, as in the example of International Relations. Do you really think you can restate your argument on the level of individuals or genes? That’s just pretty unlikely.

    • Thank you for the response. I agree that restating the argument at the level of individuals or genes would be incredibly hard, or impossible, for me to do.

      I intended my point to be that the way the debate is currently being carried out, at least by the more well-known biologists, is ill-conceived. E.O. Wilson, Dawkins, Pinker and Jerry Coyne are all weighing in on whether individual-level selection or group-level selection is right and are sowing confusion. The Nowak, Tarnita and Wilson paper mistake a simple model (Hamilton’s Rule) for a general theory (inclusive fitness).

      What we should be debating is the utility of specific models and modeling assumptions for specific questions of interest. For example this paper by Boyd, Richerson and Henrich could have made some sort of individual vs. group selection argument. But instead of getting in that debate, they took issue with the weak selection assumption Lehmann et. al used to do the accounting. To me, this is a more productive approach.

      http://www.sscnet.ucla.edu/anthro/faculty/boyd/BoydRichersonHenrichBES11ContraLehmann.pdf

  • Bret Beheim says:

    Herb,

    The only thing preventing us from describing something like the Cold War entirely in terms of individuals is that we are finite beings of limited intelligence. This small issue aside, it’s just a matter of seeing how the actions of others affected each focal individual, and vis versa.

  • This is a great posting in re states and politics, but the state is only sometimes relevant to the question of biological evolution. States are not usually units of actual genetic selection. To be such, they would have to be genetically uniform enough that when one state exterminated the population of another state, it would make a major genetic difference in the population, AND it would be necessary that one state exterminate the other state or almost do so. It does sometimes happen that a state acts as a unit and almost totally exterminates a rival population that is really genetically different, leading to genetic change in the overall population; the clearest case is Australia, where the Anglo-Australians almost eliminated the Aboriginal population. The American genocide of Native Americans has also had a major population effect, but extermination has rarely occurred and the genes are still around though the population structure is very different. However, even in cases of genocide, it only rarely happens that the victims are genetically fairly uniform, different from the victors, AND mostly annihilated. If, however, you look at smaller groups, especially “tribes” (if that word means anything much), near-extermination of one group by another was pretty common in the past, and must have had significant genetic effects. Worldwide, demonstrable genetic changes in the human animal over the last 5000 years (i.e. since the state was invented) have had more to do with tolerating fresh milk, resisting diseases, capturing vitamin D while preventing skin cancer, and other such matters–the state has something to do with these things, but not necessarily very much. Still, it is not totally irrelevant. I suppose the US subsidizing of milk may have helped the milk digestion gene (for lifelong production of lactase) get commoner.
    best wishes, Gene Anderson

    • I agree that states are unlikely to be an important component of genetic selection. I meant my comment to apply primarily to selection on cultural traits – or maybe political institutions depending on how one wants to parse such things. States seem to explain much more cultural variation than genetic information. Adrian Bell and colleagues did some rough estimates:

      Bell, Adrian V., Peter J. Richerson, and Richard McElreath. 2009. “Culture Rather than Genes Provides Greater Scope for the Evolution of Large-Scale Human Prosociality” Proceedings of the National Academy of Sciences 106(42):17671-17674.
      https://sites.google.com/site/evolusone/PNAS-2009-Bell.pdf?attredirects=0

      Choi and Bowles have a model of genetic group selection and warfare. They assume that everyone in a group dies when they lose a battle but, even then, group-level selection becomes pretty weak above a few dozen individuals.

  • Matt,

    You seem to switch back and forth between the level of analysis and the unit of analysis; however, in IR these are not interchangeable. I see no evidence that IR scholars in general treat the state as “the most natural level of analysis.” Structural realists, who follow Waltz, accept that the most useful level of analysis is the international system as a whole, while the units of analysis that are examined are individuals states. These realists often utilize the “billiard ball” analogy when explaining international interactions – functionally similar states (as units) whose interactions are governed by the ordering principles of the system (level of analysis) or international environment in which they find themselves. Neoliberals in IR have tended to object, preferring to examine the interaction of two levels of analysis in tandem – the state level and the international level. Furthermore, whether the preferred UNIT of analysis in IR is the state has no bearing upon the question of whether MLS has relevance for understanding international politics. To see why, consider an analogy with the discipline of psychology. If you ask psychologists what the most useful or “natural” unit of analysis is for them, they will probably tell you “individuals.” Does this suggest that either genetic group selection, cultural group selection, or MLS has nothing to contribute on the question of human psychology? Of course not. Whether researchers find groups or individuals most “useful” has very little to do – on its own – with whether MLS or other levels of selection are relevant, useful, or valid.

    That being said, I am a serious believer in the relevance of cultural group selection for IR, and I believe it’s contributions in this area are potentially great, especially in the realm of what IR scholars call the Diffusion of Democracy and the related themes you discuss. But I see no reason why this should trigger pessimism regarding the relevance of individual level selection for illuminating coalitional dynamics. For example, cultural evolutionists essentially model learning processes and cultural diffusion over time in a given population. But the “content” of these learning processes is provided in part by evolved mechanisms in the brain that make learning possible and operate according to privileged hypotheses about the world (e.g. see Sperber’s “Explaining Culture”). These systems may make certain lessons harder or easier to learn and operate to endow particular social forms and institutional arrangements with unique intuitive appeal. In short, selection at the individual level matters because without it our analysis is reduced to a population of black-box learners, which is a massive step backwards. This wouldn’t be evolutionary at all, and it already exists within the realm of social constructivism in IR. (e.g. see Adler’s concept of “cognitive evolution”). Quite tellingly, in fact, one of the most promising mergers in international relations today is between psychologists and constructivists (see the recent edited volume by Shannon and Kowert). In short, analysis at the level of the individual is indeed useful for understanding coalitional politics, AND an understanding of the evolved psychology that individuals possess that has been shaped by individual level selection is useful for explaining the structure of adaptations in the brain that 1) regulate and respond to coalitional behavior, and 2) shape the character of cultural diffusion.

    Even if we accept that groups such as states are the most useful units of analysis, Waltz himself has acknowledged that if we want a *full* explanation of a given phenomenon, we must utilize various levels of analysis (i.e. individual level, state level, international level) in combination. Others in IR go further by arguing that such methodological eclecticism should be a regular practice (see the edited volume by Lake and Powell 1999).

    Lastly, I’m surprised that the Waltz 1959 passage is used here to defend the position that the individual level of analysis is not useful. Waltz in this book relies on an outdated and false notion of biological constitution, which views human nature and its products as constants. This is the classically false argument that human nature cannot explain behavioral variation because the former is a constant and the latter is a variable. I hope that we in this forum have not retreated to this seriously false notion of biological adaptation. Waltz’s claims about human nature are as wrong as they are irrelevant for the topic at hand.

    • Anthony,

      Thank you for your comments. I have a manuscript in the works that address some of the issues you raise and I would appreciate your comments on if you are willing to look it over. There is a lot to unpack in your comment, so I’ll do my best to address your concerns.

      I take your point on “levels” vs “units.” I should have written the state is the “most natural ‘unit’ of analysis.” Thanks for the catch. I hope this does not cause too much confusion in relation to my large point that structural realism does not invoke individuals as particularily useful units of analysis.

      I also did not mean to imply that individual level selection has no possible bearing on any question in international relations. This comment is a response to Pinker’s essay which casts doubt on the utility of thinking about group level selection, and my goal was to give examples where thinking about groups is the most useful. I think the modeling approach one takes should depend on the question of interest.

      For the record, I think that individual-level analysis is important, but for many questions in IR, it probably has to be a level in a MLS framework.

      A related point is that you can’t endogenize everything. At some point you have to “black box.” For example, sometimes black boxing learners is fine, and sometimes it might not be. If you want to model the evolution of social learning or social learning biases, you do not want to black box them (see classic models by Boyd and Richerson and Alan Rogers). But sometimes it makes sense to take these biases as given and see what happens to the system. For another example, we aren’t updating Hamilton’s rule to account for the Higgs Boson.

      I am not defending Waltz’s views of human nature – whatever they are. My intent was to point out that he did not consider analysis of individual humans as particularly important for his questions of interest. That being said, I did not interpret Waltz’s statement as a comment on where behavioral variation comes from. I read his point as the observation that genetic change is likely to happen at a much slower rate than changes in international politics. Therefore, changes in political behavior, is unlikely to be due to genetic change. This seems reasonable to me, especially for large groups.

  • Thanks Matt,

    The point that structural realism does not invoke individuals as particularly useful units of analysis is I think mostly right, but I think Waltz was more nuanced than this. In other words, the question is: useful for what? Waltz, as you may know, famously drew an analytic line between theories of foreign policy and theories of international politics. According to Waltz, a “systemic” theory of international politics was one that explained aggregate patterns of state behavior with reference only to attributes that were features of the system, not of states. However, when we try to explain the individual actions of states (i.e. when we try to develop a theory of foreign policy), we need a broader theoretical framework; namely, we need to consider state-level domestic institutions such as bureaucratic frameworks and regime type (i.e. democracy, authoritarian…), alongside individual-level variables such as leadership, in conjunction with systemic pressures. I’m not saying Waltz was right, and you can disagree about whether the foreign policy/international politics distinction is useful or valid, but if we are debating what Waltz meant, I don’t think Waltz meant to disregard the individual level as *generally* not useful. It seems he felt it was not useful specifically for a theory of international politics. This, of course, supports your point. However, given that Waltz also felt that the domestic and individual levels of analysis were in fact useful and indeed necessary for explanations of foreign policy, this actually ends up contradicting your point. Ultimately, to the extent that everything comes down to “what did Waltz mean?” this may not be the most effective hook for your argument.

    Can’t endogenize everything, couldn’t agree more. But my point is less about the scope of endogeneity and more about the quality of it. Specifically, if our models rely on unrealistic assumptions that we know to be problematic or false, the burden of argument is to explain why, or to correct them. When confronted with the fact that rational choice models rest on unrealistic assumptions regarding psychology and decision-making, rationalists have made similar arguments appealing to the utility of black boxing certain variables, either for the sake of parsimony or prediction (or both). From a modeling perspective, especially when prediction is your major concern, this may make sense. But if we are trying to explain the world, it becomes more problematic. But let’s face it, if we accepted the rationalist counter-arguments, you and I would not be doing the kind of research we are doing. 🙂 A related worry of mine is that cultural evolution models in IR are simply social constructivism by another name, and at the end of the day all we have done is give “systemic change” a new name. The last thing IR needs is terminological explosion! I’m not saying there can be no gains from exchange; rather, I echo your sentiment that IR scholarship has yet to take advantage of these gains in a meaningful way.

    I appreciate your thoughts and look forward to your work in this area. I was especially intrigued by your thoughts on green beards and democracy. I’d of course be happy to look at other work of yours, and I hope we could exchange thoughts further. Thanks again~

  • Thanks, Anthony

    I think there is a lot of overlap between Wendt-style constructivism and themes in the cultural evolution literature. Right now the literature is a terminological muddle, different researchers use the same words with different meanings (e.g. “learning”) and use many different words to describe the same thing (e.g., “emulation”, “imitation”, “construction”, “bounded learning” – though these do not always mean the same thing depending on the authors) and I agree that importing terms from gene-culture evolution may not be helpful.

    Thus far, my strategy has been to adopt whatever I think is the most common usage in the IR and/or policy diffusion literature and list common synonyms or alternate definitions in footnotes. At the end of the day, I am more concerned with formally modeling the relevant processes and want to use the words that best represent to the target audience what is going on with the math.

  • Peter Turchin says:

    Matt, I want to add my voice to the praise – a very nice and well-argued post. Other thoughts about the discussion.

    I have a rule of modeling, which I formulated when I was still an ecologist. I call it the ‘rule of two levels.’ It means that you never want to model more than two levels in a hierarchical organization. So, for example, if you are interested in IR or (as many of my friends are, in the World-Systems analysis), you take the polities as given (‘black boxes’ in your terminology) and use the model to predict the dynamics at the level of the system of polities.

    It’s OK and, in fact, very important to try to also understand the inner workings of the polities. So one should build another model in which polity-level dynamics are predicted on the basis of various social groups and strata that are its constituent elements. At that level of analysis, the groups within polity are the elements (‘black boxes’), while the polity is the ‘system.’

    What you don’t do is attempt to model simultaneously three levels (social groups, polities, the world-system). Experience shows that this is a self-defeating theoretical strategy. Instead you need to ‘modularize’: run one model to understand what makes polities work, then take it as a given to understand the dynamics of systems of polities. Under some conditions it is even possible to prove mathematically that you can do it (if time scales of processes at differnet levels are sufficiently distinct).

    • Thanks, Peter, for the praise!

      I wonder where the standard Price-1972-style multilevel selection models fit into this framework. Selection happens at two levels – within group and between group – and the result is the mean change in the trait at the third, systemic level. In principle you can add as many levels as you want. Of course, a major limitation is that this is only really feasible with one time-step unless there is a *really* simple model of the change in variance between levels.

      • Peter Turchin says:

        The Price equation is great as a semi-quantitative tool for training intuition, but the concensus is, and I agree, that further progress will be made by building specific models that are dynamically complete (unlike the Price equation).

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