by Jess Behrens

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

Based on what I've posted so far, there are 4 logical combinations of linear and energetic separation states that can be used to 'type' a given tournament structure:

Linear with Separation

Linear without Separation

Less Linear with Separation

Less Linear without Separation

Of these four types, I've already covered two. In __Chapter 6__, I posted about the linear relationship of competition between Owls & Hawks in the NCAA Tournament. That included a group I labeled 'Less Linear' & included 6 tournaments: 2005, 2010, 2012, 2013, 2015, & 2016. Since all of these years were also separated, these Less Linear years also correspond to #3 in the above list. Less Linear years produce a highly significant number of upsets among seeds 1-3 and heavily favor the Hawk type to win the championship. I've re-posted the results by species type for the Less Linear with Separation Years in Table 1.

Table 1. Poisson Significance of Tournament Results by Round & Species Type, Less Linear w/Separation

Chapter 7 describes how results from the Monte Carlo simulations show that Hawks & Owls separate into energetic 'niches' in some years, while in other years this separation does not occur. The years where this separation occurs again favors the Hawk species type to win the tournament, while the years without separation favor the Owls. Furthermore, the years without separation experience a significantly low number of upsets among the seeds 1-3.

Each of the years without separation, which include 2006, 2009, 2017, & 2018, are also linear years. Thus, they correspond to #2 in the above list. Table 2 contains the results by species type & round for the Linear Years w/out Separation.

Table 2. Poisson Significance of Tournament Results by Round & Species Type, Linear w/out Separation

This leaves two additional 'types' from the above list without description. Point #4 in the list can not, at this point, be described because it did not occur during the years included in this analysis. However, the first point, Linear Years with Separation, have occurred, and it is results from those years, which include 2007, 2008, 2011, & 2014, that I will now describe.

**Given what has already been described about the impact that linear & separation structural factors have on the team that wins the tournament (i.e., separation favors Hawks while Linear favors Owls), a good hypothesis for these Linear Years with Separation would be that neither species type holds a significant advantage over the other when it comes to the tournament championship. ** As Table 3 shows, at least in terms of the Poisson, this hypothesis is correct.

Table 3. Poisson Significance of Tournament Results by Round & Species Type, Linear w/Separation

Every version of Table 3 that I've shown so far has indicated that one species type is favored over the other. Table 3 is the first time the Poisson has shown that neither the Hawks nor the Owls have a statistical advantage in Tournament Champions; furthermore, that lack of advantage occurs in those years that would logically be the most heavily contested. The Owls do have more champions (3) than the Hawks (1) in these years. However, this number is not significant, and is in fact less significant, than the Hawks single champion.

What do the other relationships, i.e. the population fitness plots, from each of these three combination of structural factors look like in? I'm going to re-post these population plots and include the scatter plots to illustrate what I believe are the differences in them that produce the results shown in Tables 1-3.

Figure 1 shows the population fitness plot, or Total Population Energy vs. Percent of Population by

Figure 1. Population Fitness by Species, Linear w/Separation Tournament Years

Species, for the Linear with Separation Tournaments *(All Species p<0.001; r-values: Hawks 0.75, Owls 0.87, Dove-Owls 0.77, Doves 0.82)*. The plot is very similar to the population fitness plot for Separation years that was first posted in Chapter 7 and that has been re-created in Figure 2 *(All Species p<0.001; r-values: Hawks 0.76, Owls 0.88, Dove-Owls 0.85, Doves 0.84)*. The primary difference between the two

Figure 2. Population Fitness by Species, Less Linear w/Separation Tournament Years

is that the point where the Hawk & Owl lines cross is about 3% (45% vs. 48%) less than in Chapter 7's Less Linear w/Separation plot. I've included the scatter distributions to highlight where this linear crossing occurs. In Figure 1, which includes years with no statistical advantage for the Hawks or the Owls, the Owl distribution is centered on that cross over spot and is much more separated from the Hawk scatter.

In Figure 2, only the top edge of the Owl distribution reaches the point where these two regression lines meet. Furthermore, in both Figure 1 & Figure 2, the Hawk scatter is located near the lower left corner and has, effectively, the same range of distributions (they top out at about 20% of the total population). Finally, the Dove-Owl's have a much greater range in Figure 2 than in Figure 1, a point which may be related to the large number of upsets that occur in these years and which I will address in a future post.

Within the Evolutionary Game Theoretic literature, the point where these fitness lines cross is highly significant. Referred to as the __Evolutionary Stable Strategy__, these points represent the percentage at which the population are at equilibrium. It's the equivalent of a __Nash Equilibrium__. **One way to interpret Figures 1 & 2, vis a vis the significant advantage to Hawks conferred by Figure 2 and the lack of any advantage in Figure 1, is that an Owl scatter distribution centered on the ESS point counter balances the energetic niche advantage conferred to Hawks in years with Separation. **Furthermore, the two Owls who won the tournament in Less Linear years were made possible by the fact that in those years, the ESS point is within reach, if infrequently so.

In contrast to Figures 1 & 2, Figure 3 shows the extreme overlap between Hawks & Owls during Linear

Figure 3. Population Fitness by Species, Linear w/out Separation Tournament Years

Years without Separation. The Owl scatter in Figure 3 *(All Species p<0.001; r-values: Hawks 0.84, Owls 0.95, Dove-Owls 0.80, Doves 0.87)* is located at near the center of a much larger and more diffuse (stochastic) Hawk scatter. As mentioned in an earlier post, it is likely that the much less diffuse energy states available to Owls in these years results in their realized advantage over Hawks (Table 2). However, this dominance is not absolute, which is apparent in that one Hawk did win the tournament during these years, because the two so completely overlap.

Individual fitness plots during the linear with separation tournament years confirm the points made earlier, as Figure 4 *(Hawks p<0.05, All Other Species p<0.001; r-values: Hawks -0.04, Owls -0.34, Dove-Owls 0.17, Doves 0.46)* demonstrates. I'm not going to re-post all of the other individual fitness figures

Figure 4. Individual Fitness by Species, Linear w/Separation Tournament Years

because, unlike the population plots I did re-post, the individual fitness plots already contain the species type scatters. Of note in Figure 4 is the fact that the Hawk & Owl lines do not cross in these tournament years, and the Dove-Owl line is much sharper, crossing both Owl & Hawk at between 32% & 38%.

The fact that the Individual Hawk & Owl lines do not cross indicates that there is no evolutionary stable point for these two species types at the individual level. Since the majority of Owl scatter points is beyond the point where the Dove-Owl line crosses both their line & the Hawk line, it would seem to suggest that the Owls would have an advantage. However, the fact that the Dove-Owl line crosses both of these lines has, in other tournament configurations, indicated an advantage for the Hawks. Thus, both species type seem to have an indication of strength in Figure 4, which makes sense given the lack of a statistical advantage for either (Table 1).

Upsets

Congruent with the lack of a statistically significant relationship between energetic separation & upsets of 1-3 seeds, Table 4 shows no significant upsets in linear years with separation. The only surprise in

Table 4. Poisson Significance of First Round Loss by Seeds 1-3, Linear w/Separation Tournaments

Table 4 is the lack of statistical significance in the number of 1-3 seeds that make the final game & win the tournament during linear years with separation. That point will be addressed in a later post.

Predicted Average Energy (Success)

As was done in previous posts, I'm including figures showing the predicted average energy from the three tournament types I've highlighted in this post: Linear with Separation; Linear without Separation; and Less Linear with Separation. To summarize the process again, here's how the data in Tables 5, 6, & 7 were produced:

Use the network to classify teams as Hawk, Owls, Dove-Owls, or Doves

Run the evolutionary game theory simulations using the percent of total population represented by the totals calculated in Step 1

Calculate the linear best fit regression model using the simulations from Step 2 for each species type.

Plug the percentages from Step 1 into the model developed in Step 3 to calculate the predicted total population energy (or success) for each species and divide by the number of individuals within that species for the given tournament year.

Table 5 shows the predicted average energy for Hawks & Owls in Linear Years with Separation; Table 6 contains the same but for Linear Years without Separation; and Table 7 is identical reproduced from

Table 5. Average Predicted Energy, Hawks & Owls, Linear Tournament Years with Separation

Chapter 6 and shows the average energy for Hawks & Owls during Less Linear Years with Separation. In

Table 6. Average Predicted Energy, Hawks & Owls, Linear Years with No Separation

all of these figures, the average Hawk out-competes the average Owl, except for 2018. Thus, the expectations that the average Owl would out-compete the average Hawk in years where Tables 1-3

Table 7. Average Predicted Energy, Hawks & Owls, Less Linear Years with Separation

show that Owls have the advantage is not met. However, what is important to note is that none of the T-Tests comparing the predicted average Hawk & average Owl energy are significant. That was not the case in previous chapters. **The goal of Tables 5, 6, & 7 is to illustrate that the regression lines in these groupings are more likely to be reliable precisely because they do not produce significantly different results when applied to real tournament data.** Thus, the lack of statistical significance likely suggests that these groupings represent important structural 'types' that will be consistent across tournament years.

So, that's it for linear & energy separation considerations. My next post will consider another, equally important factor - the number of teams with low strength of schedule that cluster in a given tournament year and the impact they have on tournament results.

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