The Statistics of Sin in Hominid Populations: Checking the Math on De Cruz’s Use of the Price Equation (Part Two)

April 23rd, 2018 by

In our first blog, we summarized Helen De Cruz’s recent presentation of a Transmission Model of original sin. We also looked at a possible problem with her attempt to reappropriate Joseph Henrich’s use of the Price Equation. However, at the end of blog one, we granted her this key move in mirroring Henrich’s use of the equation. Doing so allows us to move forward in her presentation to a bigger problem which we address here.

During Part Five of her presentation, De Cruz explains how she follows Henrich’s use of the Price Equation in attempting to model a Social Transmission account of sin. His study is an attempt to give reasons for the loss of artifact creation abilities by indigenous Tasmanians between the last ice age and the arrival of Europeans. His answer has to do with the fact that the Tasmanians were cut off from the rest of Australia with the rise of sea level. This lowered the number of people that indigenous Tasmanians could interact with in their generation by generation pattern of learning to make advanced tools. As population shrinks, fewer and fewer people are trying to learn to make advanced tools – and as a result – fewer and fewer people are coming up with innovations on tool usage. As population shrinks further, they even lose the ability to make tools they had known how to make in previous generations. This brings us to the second part of how De Cruz uses the Price equation. We will, however, need to explain more of what Henrich does in his paper in order to make clear where De Cruz takes a wrong turn.

De Cruz’s presentation makes clear that she follows Henrich’s steps by simplifying her Price equation from ∆z̅ = Cov(f,z) + E(f∆z)   to  ∆z̅ = zh – z̅ + Δzh  just as he does.  Readers need not worry about the math involved in the simplification above; Henrich gives the details on why he can do this in Appendix A of his article. More importantly, Henrich knew that his study had to account for mistakes that Tasmanians made in their attempts to learn how to make tools from the best exemplars in their community.  While the best tool maker has the highest z score ( represented by zh ) other people in the community do less well and earn lower z scores.  However, once in a while, someone gets lucky and actually comes up with an innovation on the skill they were trying to copy from zh. This person then becomes the new (zh ).That skill gets spread through the community and the culture’s abilities evolve upward over time.

Henrich’s paper captures the influence that mistakes and innovations have on the evolution of population by translating the Price equation into this simpler equation: Δz̅ = – α + β(γ + Ln(N)). De Cruz applies the same exact modification in her talk. Again, don’t worry about the math. (Do note, in passing, that the ability to make this simplification from the original Price equation is based exactly on our granting De Cruz the assumption in Part One of our blog about everyone following the highest exemplar. Without that simplification she can’t move on to this simplified equation.)

While the equation Δz̅ = – α + β(γ + Ln(N) may look intimidating, the concepts at play are easy to understand.  In the equation above, α is a number that represents how difficult it is to learn some skill. It represents mistakes that are made in the learning process. The larger α is, in the equation above, the lower it will push the population’s positive rate of change over time (Δz̅). In plain English, the harder a thing is to learn, the slower that skill will spread through the population over time. That is not all that is at play, however. β represents lucky guesses and innovations people make when they try to learn the same skill. β will raise the rate of change in the average skill of a population over time (Δz̅).  These β changes are rare though. However, the larger a population of skill learners is, the more frequently these lucky changes will emerge. This increase in β (thanks to population growth) will overcome the negative effect of mistakes made in mimicking a skill (α). The size of a population is represented by the N, in the right hand of Henrich’s equation: Δz̅ = – α + β(γ + Ln(N).

At last, we arrive at the other problem with De Cruz’s use of the Price equation. The gist of Henrich’s entire study is that the larger population (N) is, the higher the rate Δz̅ will be or vice versa. His argument was that when sea levels rose, the Tasmanians were cut off from the rest of Australian population of learners who were trying to figure out skills and also coming up with innovations. Their N value in the above equation dropped below 1000 and this helps to explain why their Δz̅ dropped and they lost the ability to make tools over time. Trace your finger along the x axis in “Figure 3” below and note how the shift in population affects the change in the average z̅ of the population (y axis).  Then turn your attention to how De Cruz uses this concept.


De Cruz’s presentation suggests that Δz̅ is the overall increase or decrease in the rate of conformity to the moral norms of a hominid population. For her, α represents the distorting influences of socially transmitted sin.  β represents the ability of individual people to reflect on the moral norms of their community and improve them – as Schleiermacher and Raushenbush taught people could do. So far so good.

Now we come to something perplexing. De Cruz seems to say the equation works in a way opposite to how Henrich does. During her explanation of her use equation she says,

“. . .you will see that as a population size increases (and that’s what sort of cancels this out) that the distorting influences of socially transmitted sin will also increase. So the cultural Price equation predicts that we individual members of the community will inevitably be impacted by sin. That lowers the community’s moral standard. And overly large alpha values… lots of transmitted sinful behaviors will lower the average z value. So what that means is that individuals need to reflect and not critically accept all the moral ideas in their societies. They can make a difference.[8] [Emphasis mine]

If you look back at Henrich’s equation, Δz̅ = – α + β(γ + Ln(N)) you will see that the rise in population (N) is independent of α. Contra De Cruz, as population rises, α does not rise. A rise in population (N) only affects the rate of helpful innovations (β). Recall that according to the order of operations in math, the figures inside the parentheses are first multiplied to β but only later added to α.

What is the point here? The difficulty of doing something (α; the level of an individual’s failure in attempting to conform to a social norm), does not rise with the population size (N); it stays constant. According to Henrich, as population rises and innovation with it (i.e. β) the negative effects of α can actually be overcome. So contrary to De Cruz, the Price equation as used by Henrich does not predict that individual members of the community will inevitably be impacted by sin. Ironically, if we apply Henrich’s use of the model to De Cruz’s hominid population, then the Price equation may predict just the opposite. As population (N) grows, the effects of sin (α) will be offset, because people are innovating better ways  (i.e. β) to accomplish z. In other words, the overall average conformity to the population’s morality (Δz̅ ) will actually increase with their improvements – if the population grows.[9] If this analysis above is correct, then contrary to DeCruz, the Price equation (as modified by Henrich) doesn’t show that the Augustinian-Pelagian dichotomy is a false dichotomy. It instead seems to imply a Pelagian-like ability to overcome the downward pull of sin such that the society gets better and better over time – as a result of population growth.

In conclusion, it should be noted that our two blog responses to De Cruz do not detract from her excellent points on the social aspect of the transmission of sin. At least, they will furnish Dr. De Cruz with examples of the sorts of questions that may get asked about the more statistical aspect of her project. At most, they will require a revisitation of the type of equation that is used to model how sin is transmitted through social groups.


[8] From video around 41 minutes. See:

[9] If none of this makes sense, simply look at Figure 3 from Henrich. As population (x axis) increase, the overall rate of accomplishing z increase. For De Cruz this would mean that as population grows, the ability to conform to the population’s moral norm grows with it.


Jesse Gentile is a new PhD student at Fuller seminary studying systematic theology. He has interests in theological anthropology, epistemology, ethics, technology and pretty much everything else. Jesse is the father of two awesome elementary school kids and is the husband of Ella (who works as a wills and trust attorney). He regularly does itinerant preaching among plymouth brethren assemblies throughout the U.S. Jesse hold’s degrees in Biblical Studies, Philosophy, and Instructional Design.

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