by Jess Behrens

*© 2005-2018 Jess Behrens, All Rights Reserved*

Before I present all of the results from the EGT simulations and how tournament population structure is associated with tournament results, I'm going to present the results for All Tournament Years together, regardless of structure. My goal in doing this is to present the reader with the plots, figures, and tables that I will use over the next few posts. All of these figures & plots were done in Seaborn and the simulation itself was run written in python, both using the anaconda distribution with python 3. The upset and other Poisson distribution tables were created in excel.

Competition Figures: One figure type used to show simulation results is the competition plot. Always in blue, this figure type compares the total population energy for two of the four species. The dots represent the total energy earned by the two species (one on the x axis the other on the y axis) in the same iteration of a single 6 round simulation. Each year 2005-2018 has 1000 points on these graphs, for a total of 14,000 individual points. The line represents the best fit linear regression line for those 14,000 points. The Pearson correlation coefficient and its significance are also displayed in the upper right of the figure. When presenting these competition figures, I will also report the significance of this regression line (the p-value). In the case of Figure 1, *p<0.001*.

Figure 1. Hawks vs. Owl, All Tournament Years

The Y axis, labeled 'OT' for 'Owl Total', represents the accumulated energy for all owls in a given iteration of the simulation. Likewise, 'HT' is for 'Hawk Total' and also represents the total accumulated energy for all hawks in the same iteration of the simulation as used for the owls. While it is possible to produce plots for each of the species type relative to one another, that will not be done here as this post is meant to educate the reader about what to expect in the next series of blog posts.

**Plots like those shown in Figure 1 are commonly used in ecology to show the relationship between species in nature.** Furthermore, the correlation coefficient you see in Figure 1 is no different than what is produced in the more 'data science' based correlation heatmap, an example of which you will find in Figure 2. I've chosen to use the regression plots throughout my posts because they are more common within the ecological literature. But the two approaches reference the same tool and have the same goals. When the regression line in Figure 1 moves from upper left to lower right, it means that

Figure 2. Population Fitness Correlation Heatmap, All Species, All Tournament Years

the species are competing with one another. Said another way, when the Owls do well (large Y axis value) the Hawks don't do as well (low X axis value). On the other hand, if the line moved from lower left to upper right, it would indicate a degree of synergism, or cooperation, between the species in that the total energy for both would be increasing. Finally, if the line is flat, it's an indication that there is no relationship between the two species.

It should be apparent to the reader that Figure 1, when considered over the entire time period 2005-2018, Hawks & Owls are very strong competitors in the tournament, which is expected. Furthermore, the relationship is very strongly linear in that if one knows the relative success of the owl population, he or she can predict the probable success of the Hawks in that same year.

Population Fitness Figures: Another type of figure that I will use throughout the next several posts, called a population fitness figure, is shown in Figure 3. These figure types are a little bit different than the competition plot shown in Figure 1. Instead of comparing the total accumulated energy between two species in the same simulation, Figure 3 shows the total accumulated energy (Y axis) of all four species as a function of that species percentage of the total population (X axis) simultaneously.

Figure 3. Species Population Fitness, All Tournament Years

This figure does not include the individual points for each simulation because doing so would confuse the reader. Instead, only the best fit linear regression lines included on the Figure (all are *p<0.001; r-values: Hawks - 0.91, Owls - 0.95, Doves - 0.83, DoveOwls - 0.86)*. There is a legend on the right side of each of these figures that identifies each of the species by the lines color. As with the competition plots, these fitness plots, or figures, are very common in ecology and evolutionary game theory. As Figure 3 shows, the Hawks and Owls regression line crosses at about 0.3, while the Doves and Dove-Owls never cross each other or the Hawks/Owls lines. **Whenever these lines cross, it represents the point of equivalence, or stability, between the species type.** Thus, when compared across all tournament years without consideration of tournament structure (which I will get to in the next posts), Hawks and Owls are Have are evolutionarily stable at about 0.3 percent of the total population. However, as Figure 2 shows, the overall species fitness for each species type increases as their percentage of the population increases.

**Individual Fitness Figures:** Related to the population fitness figures, such as the one shown in Figure 3, are the individual fitness figures. Shown below in Figure 4, the Individual fitness figures show the average energy for a member of one of the species populations as a function of that species percentage of the total population.

Figure 4. Average Fitness by Species, All Tournament Years

Unlike figure 3, Individual fitness figures will always include the scatter points because doing so does not confuse the reader and helps illustrate the degree to which the populations are energetically separate. As in Figure 2, the regression lines are included as well (*All species p<0.001; r-values: Hawks - -0.04, Owls - -0.22, Doves - 0.5, DoveOwls - 0.13*) and the same stability and energetic relationship rule applies to lines that cross. **What Figure 4 demonstrates is that while fitness follows a much stronger and more linear relationship at the population level, the results and distribution of fitness is much more stochastic (random) at the individual level. Furthermore, even though the population fitness figure (Figure 2) shows a robust increase for Hawks and Owls as their percentage of the total population also increases, this same relationship is slightly negative at the individual level.** Said another way, having more Owls or Hawks is good for the total population of these two species, increasing numbers of Hawks & Owls is slightly bad for individual members of those species.

**Population Separation Figures:** Also related to the population fitness figures, such as the one shown in Figure 3, Population Separation Boxplot Figures show the degree to which each species is energetically distinct from its competitors. Figure 5 shows the 95% confidence intervals for the total energy by species type across all 14 tournament years. **One way to think of this is through the ecological concept of a 'niche'. When species are energetically separate from one another, they have a more distinct niche, or strategy, from one another. ** Within the ecological literature on 'niche' is the __competitive exclusion principle,____ which __states that two species can not have identical niches. Furthermore, when 2 species are competing for the exact same resource, one will be pushed to extinction. While the results presented here, and in future posts, contain only the output from simulated Evolutionary Game Theoretic interactions and are not real data, meaning it can't be stated that the competitive exclusion principle definitely applies to the NCAA Tournament, the same principles of competition to extinction & energetic niches applies. Species with overlapping energy scores from the simulations are less like to have distinct strategies. Knowing if separation, or the lack thereof, favors one of the strategies over another would allow for the development of theories about which strategy is strongest and why.

Figure 5. Total Energy by Species Type with Conf. Intervals, All Tournament Years

As Figure 5 shows, there is significant overlap among Hawks, DoveOwls, & Doves as well as between Owls & Hawks. However, there does seem to be separation between Doves & Owls.

**Tournament Upset Tables:** One of the possible tournament features that may be related to evolutionary game theory is the number of first round upsets that occur in a given tournament year. Always a big, and fun, feature of the NCAA Tournament is its capacity to produce David vs. Goliath upsets. The evidence that these upsets may be driven by population factors is signficant, and I will be presenting all of that in future posts. However, for now I just want to show a couple of example figures so that the reader will know what they will see as they go forward.

Table 1. Upsets by Tournament Seed Ranges, All Tournament Years

As Figure 3 in __Chapter 4__ showed, one way I will be presenting the significance of results in future posts is count by seed or species type vs. expected using the Poisson distribution. Table 1 shows the same thing, Poisson significance, of the total number of first round upsets by seed range and year. The expected number of upsets is considered as the total upsets between 2005 & 2018 for a given seed range divided by 14. While 2018 results were not used in constructing the networks, the ability of this method to predict 2018 results is being considered. Thus, the upsets that occurred in 2018 were also considered.

Significant results are highlighted in either green (significantly low results) or red (significantly high results). Grey cells are those that approach significance, but don't quite make it. Thus, from Table 1, we can conclude that when all years are considered without consideration to tournament population structure, which I will be covering over the next few posts, only 2007 has significant results. There were no upsets in seeds 1-5 that year, which was, according to the Poisson, significantly low. We'll see if that holds up as we move on to consider tournament structure, the subject of my next post!

#MonteCarloSimulations #basketball #NetworkAnalysis #MensCollegeBasketball #NCAA #NCAATournament #EvolutionaryGameTheory #SpeciesCompetitionPlots #SpeciesFitnessPlots #niche #Seaborn #Matplotlib