The Haselbauer-Dickheiser Test can be found at http://matrix67.com/iqtest/.
In this blog post, I will study Question 25 from this test.
The question is about population dynamics in Biology.
Below I present the original question for your convenience:
Please do not read the rest of this article, if you want to attempt to solve this question on your own. The rest of this article describes my attempt at solving this question and you should not read it, unless you want to or you do not mind coming across relevant ideas, spoilers, hints, solutions, and strong opinions concerning this test.
You have been warned and I now consider that you continue to read knowing that what you come across for the rest of this article may forever spoil things for you and/or present strong opinions against this test.
Last warning: please do not read this blog post, unless you are certain that you know what you are doing. If you are not sure, then it would be best if you stopped reading at this point.
OK. If you are here, it means that you want to know my opinion. Well, ok then!
The first thing I would like to say is that you do not need to read my analysis. You can simply go to the small course on Behavioral Ecology on sparknotes.com and find the answer to this question there. I will give you the links:
The first link contains links to the whole of the small course. The last link contains answers to questions as well as the answer to the specific question we are discussing here. In the list of links, I have included all links that are relevant to the question and answer. The first four links correpsond to the underlying theory and the last link corresponds to questions and their answers. The reason I gave you the five links instead of only the first link, is that there is a slight problem in the sparknotes.com site, and the link navigation does not work perfectly, because some links are incorrect. You will also find that some images are missing. That does not matter that much though, because you can use the links I provide as guidance. Also, the missing images from the theoretical pages do not present that much hinderance. In addition, the answer provided in the last link is correct.
So, everything is settled, but I would recommend that you read my analysis, that I present below. In my analysis, I will explain many apsects as of to why the answer is this and not some other and I will provide a comprehensive overview of the subject.
My only inhibition is that my analysis will be a bit lengthy. I though of ways of compacting it, but then I would not be really giving you the full picture. And this is a subject that demands that every detail is mentioned. No shortcuts. Actually, my analysis is a story which has many beginnings. And the most difficult thing for me was to decide which beginning to choose. After a lot of consideration, I decided to start from *my* beginning.
And so I begin.
I am definitely not a biologist. I would know if I were! But when I was studying Signal Processing, Signals and Systems, and Systems Analysis, Design and Control, I was known in my class for my fascination about the study of biological systems. Here I was, studying Electronic Systems, and realizing that the same mathematic formalism, the same concepts, the same linear and non-linear modeling could describe population dynamics, intraspecific competition, interspecific competition, and predator-prey dynamics as well.
Population dynamics, intraspecific competition (competition among a species) and interspecific competition (competition between two or more species), are modelled extensively all over the world’s institutions. For these studies, the concepts involved are: the Logistic Function and Equation, Carrying Capacity, r/K selection theory, the Lotka–Volterra model and other related topics.
Of course, I knew enough about these subjects, so I could not fall for some misconceptions that other people seemed to have when they tried to tackle this question. A few people wrote to me, asking about the answer to this question, offering their opinions. Unfortunatlely, they all seem to misunderstand this question. And there were a lot of misundertsandings. If I could pick the most prominent of all, I would pick the “maximization of offspiring” fallacy. To me, it was obvious from the get-go that this question has nothing to do with “maximization of offspring”. The goal is not the “maximization of offsping”. The goal is the “stability of the ecosystem”. While each species may want to maximize their offsping number, this is irrelevant to what we are trying to model. We are trying to model a system that, first and foremost, is stable. And I would also not fall to the “insufficient parameters” fallacy. On the contrary, the question was well posed and no information was missing. Perhaps some parameters were mising because they were simply not needed in order for us to construct our model, or they would cancel out eventually. Same thing. But I wasn’t entirely sure about this second fallacy yet, that is, until I constructed a model and found that everyting I needed was indeed in place.
The approach I was contemplating was that of the predator-prey dynamics. We are talking about Furbles and to help you establish a visual representation, imagine a Furble being something like if Hillary Clinton and Nancy Pelosi had a child together. Although we are talking about Furbles, which is only one species, we have Dominator Furbles and Sharer Furbles. And Dominators are way different than Sharers in their behavior. This difference made the problem resemble the predator-prey population dynamics.
And what better model for predator-prey dynamics than the Lotka–Volterra equations! Actually, hmmm… they say that the Lotka–Volterra model is a bad model but profoundly important. Anyway, a very wise person once said that all models are wrong, but some of them are useful. And I say that the Lotka–Volterra model is up there with the most useful models of all.
Using the Lotka–Volterra models… this *is* the way to approach these problems. This *is* the way to model predetor-prey dynamics. So, for someone like me, coming from a systems background, the Lotka–Volterra formalism was the road that would lead to the answer to Question 25 from the Haselbauer-Dickheiser Test.
If you know a bit about the Lotka–Volterra model, you realize that for two competing species, we have a system of two differential equations. Now you might ask, how was I contemplating differential equations in a test that assumes no such advanced mathematical knowledge. Oh, but I had all things figured out! You see, once you construct the two differential equations, and since you are looking for a stable state, you set the derivatives to be equal to zero. By doing so, you end up with the equations of two lines. These are called the isoclines of the Lotka-Volterra model. If your model is correct for this particular question, you expect these two lines to intersect. The attributes of the intersection point will give us the answer. And the equations of the two lines, even though they are deduced from a purely mathematical standpoint, they can also come from pure thinking. I would have only been able to produce them from the differential equations, but someone more clever than me (vitrually everyone else) could have certainly been able to produce them just by logical thinking.
Ok, so I had everything figured out. But talk is cheap. “Come up with the model, Dimitrios! And show us the results!”, I think I heard you say. Well, here is the thing… I did not have everything figured out. There were some “details” that bothered me.
One such “detail”, although ever so subtle, was the phrase “Individual Furbles cannot switch strategies.” This phrase bothered me to no end. Well, of course a Furble cannot switch strategy. Or, could it? The Lotka-Volterra models do not model behavioral switching. Although this is a thing in Biology, Lotka-Volterra models are a different cup of tea. But here, the most common thing occurs: no switching in behavior. Foxes are foxes, hares are hares. Predators are Predetors, Prey is Prey, Shareres are Shareres. Why discuss about behavior switching? You might argue that the examiner wanted to be as specific as possible. But here is the thing: The whole Haselbauer-Dickheiser test is so lacking in explanations, that adding such an explanation, which was really not needed, looked very strange to me.
And what is the deal with the word “strategy”? Strategies? We do not discuss about strategies as far as the Lotka-Volterra model is concerned. We discuss behaviors. And this was someting else that was puzzling me.
The last thing that bothered me was that, for the Question 25 scenario, I could not really create a model that would make me content. Somehow, the data that were given, were not matching those of other scenarios that I had encounter. In other words, I could not see how I could create robust parameters for the Lotka-Volterra model from the values given in Equation 25. And so I kept struggling.
Little did I know that there is another approach to population dynamics, and it does not have to do with such modeling. This other approach has to do with Game Theory. And this is the key to understanding how we should approach this question. This question uses Game Theory jargon and requires Game Theory formalism for its solution. This is why we have the use of the word “strategy” and this is why we need the added assurance that “Individual Furbles cannot switch strategies.”
So, yes, this question is about populations dynamics, but there are two different formalisms to approach these problems. One is the Lotka-Volterra equations, the other is Game Theory. Obviously, Question 25 is of the second kind. We need to think in terms of Game Theory in order to utilize the givens and find the answer to the question.
John Maynard Smith and George Robert Price published their seminal paper titled “The Logic of Animal Conflict” on November 2, 1973. This was the beginning of the application of Game Theory to the study of population dynamics in Biology. This new branch of science, which combines together Game Theory and population dynamics into something new, is now called “Evolutionary Game Theory”. The particular piece of the theory we need to know to solve Question 25 is called the Hawk-Dove game.
Here are some interesting links:
But the most useful link for the purposes of our discussion is the third one, which discusses the Evolutionary Game Theory. This Wikipedia article provides an example of the Hawk-Dove game that is closer to Question 25 that any other analysis I have come across. So it is best to begin from this example, because in this case we will have to make the least of adaptations and modifications in order to get to our answer.
Below, I provide a screenshot of the particular piece of the theory we are going to use:
And so, we can now begin to solve Question 25.
Here, the Hawks are the Dominator Furbles and the Doves are the Sharer Furbles.
We will construct the so called “payoff matrix” from the numbers given in the question.
If a Dominator encounters a Dominator, one Domintator gets to have 10 children, the other Dominator suffers the cost of 10 children. The net result is zero for each Dominator.
If a Dominator encounters a Sharer, the Dominator gets to have 10 children, the Sharer gets to have 3 children.
If a Sharer encounters a Dominator, the Dominator gets to have 10 children, the Sharer gets to have 3 children.
If a Sharer encounters a Sharer, each Sharer gets to have 5 children.
From the payoff matrix we can produce two equation with two unknowns, the population dencities. Let D be the population density of the Dominators and S be the population Density of the Sharers. The following two equations hold:
D + S = 1
0 * D + 10 * S = 3 * D + 5 * S
The first equation is deduced from the fact that we have population dencities. Each of the two populations is a fraction of the whole population. The whole population is equal to 1, so the population of the Dominators as well as the population of the Sharers is a fraction (more than 0 and less than 1) and the sum of these two populations is 1.
The second equation is deduced from the fact that the payoff for each encounter is proportional to the population density of the one that we will encounter, i.e. the chance, the posssibility that this encounter has in order to be materialized. So, the left part of the equation gives us the total payoff for the encounters of a Dominator (i.e. the first row of the payoff matrix). And the right part of the equation gives us the total payoff for the encounters of a Sharer (i.e. the second row of the payoff matrix). This equation holds because we assume a stable system, i.e. one with equal total payoffs.
This is a system of two equations than can easily be solved. From the firat equation we have D = 1 – S. Then from the second equation we have 0 + 10 * S = 3 * (1 – S) + 5 * S => 10 * S = 3 – 3 * S + 5 * S => 10 * S + 3 * S – 5 * S = 3 => 8 * S = 3 => S = 3/8.
Thus, the population densities are S = 3/8 and D = 5/8. So, when the total population is 1, the individual populations are 5/8 Dominators and 3/8 Sharers. Now that the total populationn is 2000 Furbles, the Dominator Furbles are (5/8) * 2000 = 1250 and the Sharer Furbles are (3/8) * 2000 = 750.
Thus, the answer is that we expect 1250 Dominator Furbles.
Perhaps you have calculated other values for the payoff matrix. Be careful though. Before you ask if your values are correct, try to solve the system of the two equations and see what results you get. Remember, the population densities that you obtain must each be from 0 to 1. So, each population density must be a positive number less than 1. Otherwise, it has no biological meaning.
Let me give you an example. Suppose you calculate the payoff of a Dominator-Dominator encounter. And suppose you forget or do not want to include the cost, but only the gain. So, instead of (10-10)/2=0 you might enter (10+0)/2=5 as the payoff. Let us see what we get in this case: 5 * D + 10 * S = 3 * D + 5 * S => 5 * (1 – S) + 10 * S = 3 * (1 – S) + 5 * S => 5 – 5 * S + 10 * S = 3 – 3 * S + 5 * S => – 5 * S + 10 * S + 3 * S – 5 * S = 3 – 5 => 3 * S = -2 => S = -2/3. So, S is negative and this is an absurd result, biologically speaking.
To recapitulate, the question is correct and well posed. It is a question about population dynamics, and it is meant to be solved as a Hawk-Dove game. Actually, the question is so well posed, that I would not be surprised if the authors of the test asked John Maynard Smith himself to author the question. Anyway, although the question is an excellent question on the Hawk-Dove game, it seems to me that it has no place in an IQ test.