# Talk:Price equation

WikiProject Genetics
This article is within the scope of WikiProject Genetics, a collaborative effort to improve the coverage of Genetics on Wikipedia. If you would like to participate, please visit the project page, where you can join the discussion and see a list of open tasks.

Whow. This page has suddenly changed a lot. I like a lot of the new contents of this page as it suddenly adds to the understanding of Price's equation, however, I think it has lost a lot of its encyclopedic value since the huge edit by PAR on 23 Nov 2004. A couple of reasons for this comment are:

• The Price equation itself (not the small one, but the full one) doesn't appear in the article until somewhere half way through the article, after some examples have already been given. A better structure would be to give the Price equation, then give its variations (and derivation), and then (maybe even in a seperate page, like Price equation (examples)) give some examples
• The article is way too long (see also my comment on moving the examples to a seperate page)
• What is the use of defining expectation, variance and covariance in this page? It would be better to just link to their pages, without introducing them again.
• My edits on the page before the big update seem to have been lost because of this complete rewrite. As an example, there's no longer a link to Fischer's fundamental theorem of population genetics. Maybe it's just a personal view, but I like edits that are atomic much more than I like monolithic rewrites of pages.

I assume there are different views possible on these comments.

--Anthony Liekens 11:02, 30 Nov 2004 (UTC)

You've just listed a bunch of misgivings I've had myself. I thought well, I'll just do it and see what people think. I would not be offended by any of the rewrites you suggest, and I think I will undertake some of them myself unless you beat me to it. Let's make it a good page. The objections I have to your list are:

• I did try to include everything in the previous page. Fisher's theorem is still there, near the end of the simple Price equation section. Maybe it needs to be put up front as part of an introductory overview.
• I looked at the pages on expected value and covariance and it seemed to me that, for the expected value anyway, there are a number of definitions for discrete, continuous, etc, using a different notation, and there would be a lot of head-scratching unless it was explicitly defined as it is being used in the article.

Another thing I would like to include how kin selection is handled by the Price equation, but I don't feel like I understand it well enough right now to write anything.

Paul Reiser 15:17, 30 Nov 2004 (UTC)

To comment on the old discussion here, it may be useful to define expectation and covariance here because the definitions used here are not the actual definitions of expectation and covariance. The Price equation uses the expectation operator and the covariance function for simplicity; however, the actual definitions of these things are based on estimates from a population. That is, the Price equation really relates ARITHMETIC MEANS from one population to the immediate next population. It says nothing about "expectation" and "covariance." In order to make a statement about expectation and covariance, one would have to introduce probability. However, Price equation has nothing to do with probability. At best, it is a statistical statement.

It would be nice to get rid of ALL references to expectation and covariance and just speak of averages. However, due to rampant misuse over the years, it is expected that expectation and covariance be used with the Price equation. It's unfortunate.

--TedPavlic 17:07, 18 February 2007 (UTC)

Added a technical template. This thing badly needs a summary to explain what this equation is in layman's terms, instead of just diving right into the hard math. SineSwiper (talk) 05:17, 1 February 2014 (UTC)

## Expectation and Covariance

It makes no sense to me that this article defines expectation and covariance both as means of some property of a finite deterministic population. It seems to me like expectation should simply be mean or average. However, that means that covariance should be replaced with something like mean covariate, where covariate would mean the product of two variates. This actually is similar to the terminology used in ANCOVA.

That being said, it was Price himself who started this whole mess by using these terms. Is it right to reformulate on a Wikipedia page?

I at least think that some extra notes need to be made. Presently using the terms expectation and covariance confuses what the equation actually means. I'm still not sure which terms should be used. Maybe $C_{xy}$ could be used instead of $\operatorname{cov}(x,y)$ as a compromise?

--TedPavlic 11:05, 22 February 2007 (UTC)

I think we should stick to the terms used in the literature, whatever they are, and if they are not the most appropriate, we should make a clear note of that in the introduction. I am in favor of people not having to "shift gears" when they go from the Wikipedia article to the literature and back, but they should be very aware of the problem. The Wikipedia article might then serve as a small push in the right direction without staking out non-peer-reviewed territory. PAR 15:57, 22 February 2007 (UTC)

I agree that the article should use the terms given in the literature. For the statistical pedants, the expectation and covariance can be recovered if we consider random draws (with replacement) of entities from the population. The expectation for a random draw is then equal to the population mean, etc. —Preceding unsigned comment added by 129.215.37.184 (talk) 17:20, 4 January 2008 (UTC)

## Out of pure curiosity

"The Price equὈplications in economics." <-- Does anyone know what these applications are? The Price equation certainly looks like something that would be used somewhere in economics but I don't recall actually seeing it anywhere (and assuming this isn't some kind of confusion with "price" as in cost of a good). Thanks!radek (talk) 21:30, 14 May 2009 (UTC)

Pure conjecture here, but I would imagine that if stock portfolio strategies are chosen to proliferate based on past performance, then you might get population-type models that the Price equation would be appropriate to model. Likewise, there is a similar genotypic–phenotypic map from stock holding (i.e., 0's and 1's based on whether you hold certain stocks) to, say, level of risk. You may be able to describe selection for riskiness using the Price equation (and infer things from or about the underlying stock "genotype"). So portfolio management might be a good place to look. Regardless, someone should mark that statement is needing sources.—TedPavlic (talk/contrib/@) 16:39, 9 May 2011 (UTC)

It has no applications in economics, simply because economics know the difference between probability theory and statistics. It also has no applications in physics, for the same reasons. See evolutionandgames.com/price — Preceding unsigned comment added by 141.5.9.33 (talk) 11:09, 9 July 2012 (UTC)

## Which reference?

It might be useful if the article actually informs the reader which paper of Price's should be consulted for the Price equation. It would appear to be his 1972 solo author one, but this isn't exactly clear from the text. Can someone who knows fix it? Cheers, --PLUMBAGO 12:57, 16 March 2010 (UTC)

## Price Equation is a theorem, not a tautology

The price equation is not formally a tautology. Rather, it is a mathematical theorem. The bird example was also not a tautology. See the section "Verifying tautologies" at the Tautology page. Ironically the article linked to the Tautology page.

--Douglas Theobald (talk) 20:57, 22 March 2010 (UTC)

Ok, good. The important message we have to impart is that the Price equation is not a scientific law, its not like Maxwell's equations, or E=mc^2. The bird example is not trivial if it creates an understanding in the reader that no new information is imparted by the Price equation, but rather provides a very intuitively useful way of dealing with the assumptions that have been made. PAR (talk) 16:55, 23 March 2010 (UTC)

No, the Price equation is pretty firmly a tautology. A mathematical theorem usually shows (with proof) that given certain assumptions, certain conclusions are guaranteed to be true. For example, given that a two real numbers are positive, their product is also positive. There are no "givens" in Price's equation. It is a simple regrouping of variables. It is identical with starting with the tautology "x + x = x + x" and, after manipulating only the right-hand side, ending up in "x = x = 2*x". So I think it is reasonable to use the stronger term; however, it should not be used in a way that somehow promotes the Price equation. On the contrary, the fact that the Price equation is a tautology should speak to the simplicity of the Price equation. Likewise, criticisms of the framework on which the Price equation is built (e.g., it is a tool for population modelling that uses synchronous generational clock) could be added to help emphasize that the Price equation is not a physical law but rather a simple tautological statement about variables often used in particular population models (note: you could reformulate the Price equation as a mathematical theorem by saying, "Given that a population replicates in this way, then it must be the case that the average from one generation to the next acts this way."). —TedPavlic (talk/contrib/@) 23:31, 8 May 2011 (UTC)
Following "Verifying tautologies", take (x+x=x+x) to be proposition A, (x+x=2x) to be proposition B then a truth table is as follows:
 A B $A \Leftrightarrow B$ T T T T F F F T F F F T
Since all values on the right are not "True", it is not a tautology. I would say it is a theorem, the "givens" being that X is a real number. PAR (talk) 03:54, 9 May 2011 (UTC)
That's why I said "after manipulating only the right-hand side", which implies an algebra inherited from the reals. I also said, '(note: you could reformulate the Price equation as a mathematical theorem by saying, "Given that a population replicates in this way, then it must be the case that the average from one generation to the next acts this way."),' which is your point. However, the article at the moment does not introduce Price equation in a way amenable to this description. In the context of interest of the Price equation, the givens are taken for granted for all such models. The statement isn't that "for a given subset of the worlds considered, it is the case that..." Instead, the statement is that "for all worlds considered, it is the case that...". That is, unlike the theory of relativity which can be disproved if you show that some of the assumptions of the theorem are not met by physical reality, there is nothing available to disprove the Price equation. —TedPavlic (talk/contrib/@) 16:28, 9 May 2011 (UTC)
I guess I could qualify that statement. Again, you could criticize the population model (e.g., synchronous generations, mean-field/deterministic dynamics, linearity, etc.), or you could reformulate the Price equation as a theorem following from such a population model (thus allowing for the possibility that in other population models, "w-delta-zed" doesn't have this relationship), but it seems like this would be a novel criticism (or a criticism placed in the wrong article) that would be inappropriate for Wikipedia, as Wikipedia is not a primary source. The Price equation, as presented in the literature, is a simple mathematical consequence of things taken for granted by population geneticists. It is not presented as a theorem with assumptions or conclusions to be tested. Instead, it is taken as an alternative view of a conventional population model. Perhaps that does not fit the strict mathematical definition of a logical tautology; however, it fits a stronger definition of a conventional theorem that you would find in theoretical biology. —TedPavlic (talk/contrib/@) 16:35, 9 May 2011 (UTC)
I think we agree, the Price equation is just a mathematically true statement, given a bunch of real variables with certain constraints on their values, and certain definitions of new variables from the given variables. It does not depend upon identifying these variables with anything in the real world. In particular, it is not a law, like Newtons law of gravitation or something. My main concern is that this is conveyed in the article. So I think your example $x+x=x+x \Leftrightarrow x+x=2x$ is a good example, its the same sort of thing, only simpler. Maybe we could come up with a statement that conveys this fact without trying to find a one-name label for it. If we cannot easily agree on that label, chances are it won't mean much to a new reader. PAR (talk) 02:55, 10 May 2011 (UTC)

## WΔZ, aka The Killing Gene

Watching The Killing Gene, and freezing the playback when the detectives are reviewing the textbook, "To die for: Theories of ALTRUISM", the text in the book seems to be an earlier version of the "Price Equation" Wikipedia article, stripped of it's blue text links. Probably not worthy of an added note in a Popular Culture section, but another interesting bit of art imitating Wikipedia. :) 24.235.163.41 (talk) 01:23, 23 June 2010 (UTC)

## Agent based model of Price Equation and A Question about z

First of all thanks to everyone who contributed to this. I am a working at understanding mathematical formalisms of evolution and currently at work on a simple agent based simulation (NetLogo) that should illustrate this equation nicely--that is, IF i understand the equation correctly. Here are some questions: Covariance usually has units, right? But the wi term is dimensionless because it is a ratio of populations. WHAT kind of number is zi? Or to put the question another way, is the Covariance term in the Price equation insensitive to how you measure traits? Thanks. 67.139.71.34 (talk) 15:56, 6 May 2011 (UTC)

This question was also sent to me by e-mail. I've responded in a blog post as well as elsewere on the web and in an e-mail response to the original asker. In short (regarding units; for meaning, see one of the two links):
• The $w_i$ terms are dimensionless. They represent a "fitness." In particular, they are the number of offspring produced from one parent with level $z_i$ of the trait. In an agent-based population model, this is one of the most important parameters. Each $z_i$ should have a corresponding $w_i$ at each generation (and this mapping may change from generation to generation), and the $w_i$ variables govern the number of offspring in the next generation.
• So the $cov(w_i, z_i)$ covariance has units of trait level (i.e., the same units as the product quantity $w_i z_i$). Somewhat roughly, it represents the contribution of the increase in the level of the trait to the average population (as determined by the differential fitness – the relative weight of the fitness of one level to the fitness of the other levels).
• The $z_i$ has application-specific units. For example, if the trait is "height" then $z_3=5$ represents that the third group all have a height of 5 (which might be in "feet"). In the case of Fisher's fundamental theorem of natural selection, $z_i$ has the same units as $w_i$ (i.e., they are both "unitless") because $z_i=w_i$ by definition (i.e., they both are the fitness; high DIFFERENTIAL fitness begets higher fitness and low DIFFERENTIAL fitness begets low fitness).
I hope that helps. —TedPavlic (talk/contrib/@) 23:29, 6 May 2011 (UTC)
I agree with the above, except that $cov(w_i,z_i)$ represents the degree to which the trait (zi) contributes to fitness (wi). (Actually it correlates, but the additional assumption is that zi causes wi). In the simple Price equation $cov(w_i,z_i)=w\Delta z$, this just says that the correlation results in an increase in the prevalence of the trait if it is beneficial, decrease if it is not. In other words it is the term on the right that represents the increase in the level of the trait to the average population.

I am at a bit of a loss for words on how to comment. By way of standing, I am a PhD statistician in biological applications from UCB with post-doctoral work in population genetics under L L Cavalli-Sforza. The statistical language used in this article is quite loose, and the evolution/genetics/fitness stuff is only a little better. I could rewrite it, but it would be a big effort and I am not certain that it would be accepted. So I will content myself with suggesting that you get some really top-notch folks to help you with revising this article stylistically. Feel free to contact me privately if there is some way I can help. dave1 at wcf dot com. Gomberg (talk) 19:45, 29 January 2012 (UTC)