by Jess Behrens
© 2005-2019 Jess Behrens, All Rights Reserved
Wayyy back in Chapter 13 last year, I described the use of the Lotka-Volterra equations as a tool to understand inter-species dynamics in the Tournament. This is because, in many ways, the NCAA Tournament behaves much like an ecological Mass Mortality event that is being driven by overpopulation. As was the case for the first 2 post 2019 'lessons', these dynamics shifted with the re-definition of each species as well as the re-weighting of the network to emphasize sensitivity over specificity. While the impact of the Lotka-Volterra equations may have changed, they still provide important insights into how the tournament functions.
The figures included below are binary species plots, showing each of the 4 EGT species in an x, y coordinate plane relative to one another. They reflect the traditional method for evaluating Lotka-Volterra competitive interactions, which means that they only consider pairings of species. Since this is a community of species, considering interactions two at a time is not, in fact, completely accurate. Obviously, as far as evolutionary game theory simulation is concerned, all 4 species are thrown into the mix at random, based on the proportion of each species within the overall community. Thus, owls don't interact with hawks independent of both species' interactions with dove-owls & owls. However, examining them 2 at a time fits with convention and can still provide some insight into the mechanics of the entire community.
In order to understand these plots, it's important to first understand some things about the Lotka-Volterra equations & the Evolutionary Game Theory simulations as I'm using them:
1. The results presented here are specific to and based on Evolutionary Game Theory simulation results. Given the population change components of the Lotka-Volterra equations that have no readily identifiable basketball analogy (rates of birth, death, immigration, emmigration, etc.), the tournament doesn't match up with them perfectly. So the results you see presented here are based on the idea that the percent species breakdown in each tournament year could be used to weight random, simulated interactions for a real bird community with the exact same species composition. Furthermore, the assumption is that the output of a traditional analysis of those results can tell us something about the Tournament.
2. The simulation results presented here are based on 20,000 iterations of a 6 step, diminishing opportunity, random simulation.
a. Years 2005-2010 include 65 teams & 2011 - 2019 have 68 teams. This means that 2005-2010 includes one additional interaction above the 34 first round 'games' & 2011-2019 includes 4 of those.
b. No attempt at seeding is made. Each interaction is done by randomly selecting two species' based on the EGT species' breakdown for a given Tournament year. Each iteration runs through the 2005-2019 tournaments sequentially.
c. The first round, including the 1 or 4 additional play-in 'games', results in a reward of 1 energy unit (V in EGT nomenclature) and grows by 1 for each subsequent round. Thus, the second round is worth 2 energy units, the third 3, up to 6 for the last round. As in the tournament, the total number of simulated interactions drops in half for each successive round (32, 16, 8, 4, 2, 1) .
d. There is no carryover from the previous round because there is no 'winner' in each interaction. Each round in a given tournament year & iteration is simulated separately.
e. The only change in species' weighting and community composition involves Doves & Dove-Owls.
1. Doves that interact with a Hawk reduce the total Dove population by one, and all of the species percentages are changed to reflect their new percentage within the now reduced community for the next round. In effect, a hawk/dove interaction 'kills' the dove.
2. Dove-Owls grow in strength in the second round, and behave like Owls for that round only. After the second round, they begin to die off like the doves when they interact with a hawk, and the overall species composition's are updated in the same way as for doves.
3. Community percent species composition resets to the base values after each iteration.
f. Running Energy totals are maintained for each species within a given iteration/tournament year. The totals form the basis of a linear regression analysis in which each species is used as both the dependent and independent variable for every other species. The slope & r-value from those regressions are used to estimate the effect each species has in competing with every other species. Slope is then used as the Lotka-Volterra alpha value for the plots shown below. They are also used to create the Seaborn clustermaps I will show in later posts.
1. The theory behind using slope as the alpha for Lotka-Volterra derives from the fact that the units for V in the EGT calculations is 1 round (see c above). Thus, when a regression slope, all of which are negative because every species inhibits the 'success' of all other species, drop by 1, it is effectively a 'loss' for that species in the Tournament.
3. Carrying Capacity, a variable needed to calculate and interpret the Lotka-Volterra equations, is not a 'real thing' in the Tournament. However, it is possible to use estimate it by using the high end of the 95% confidence interval from the average number of each species selected across all tournament iterations. As such, that is the value, the high end of the 95% confidence interval, I use as 'carrying capacity' to create the binary figures/plots shown below.
With that simple explanation of how the Evolutionary Game Theory simulations presented here are conducted, it is now possible to present some results. The interpretation of these graphs is based directly on a webpage hosted by the University of Tennessee. It is also similarly presented by the Teaching Issues and Experiments in Ecology website.
In the figures shown below, the lines represent isoclines. If you mouse over to either the x or y axis where one of them meets that axis, python/cufflinks will tell you the value for both points on that isocline. The carrying capacity, described below, for each species' isocline represents the high end of the 95% CI, as is described above in the methodology text.
Let's use Figure 1 as an example, where Owls are on the Y axis and Hawks on the X axis. The point where the Owl isocline meets the Y axis is the carrying capacity of Owls when there are no Hawks. Where the Owl isocline meets the X axis represents that same Owl carrying capacity, but expressed in the equivalent number of Hawks. This same process occurs for Hawks, but the carrying capacity of Hawks with no owls is located at the point where the Hawk isocline meets the X axis, whereas the Owl equivalent point is on the Y axis.
The 'dots' work similarly to the isocline lines. They are located at the average for each species included in the plot as calculated using the 20,000 EGT simulations I described earlier. If you mouse over them, cufflinks will tell you the year the dot represents, the x & y value for the dot in species terms, as well as the percent of each species in that year.
Figure 1. Hawks vs. Owls, Lotka-Volterra Equations, Evolutionary Game Theory Simulations
Figure 1 corresponds to scenario 2 as described on the UT website. Every one of the tournament years is, on average, above the Hawk isocline, but below the Owl isocline. This means, from an energetic standpoint, that the Owls are dominating Hawks within the EGT simulations & that the Hawks are being pushed to extinction by the Owls.
Does this match with the tournament results? Generally, yes. Nine of the 15 tournament champions are Owls (2005, 2007, 2008, 2009, 2011, 2012, 2014, 2015, & 2017), which means that 6 are Hawks (2006, 2010, 2013, 2016, 2018, & 2019). A second simulation of 20,000 random tournaments weighted by the total species population distribution (all tournaments combined) was run to see how often a Hawk or Owl won. Both values (9 Owls, 6 Hawks) are significant according to the Poisson, but the 6 Hawks are more significant (p<0.001) than the 9 Owls (p<0.05). So, yes, the Owls are dominating the Hawks at a population level, which seems to have equated to winning more championships than the Hawks. But, the Hawks are surprisingly and more significantly stronger than expected than the Owls.
One other point of note: there have been 14 Hawks & 16 Owls in the final game over the past 15 years. Three of those years (2005, 2009, & 2012) included two Owls in the final game. In two years (2010 & 2019), both final game participants were Hawks. That means that the final game in 10 of the 15 years have included a Hawk & an Owl. In the 20,000 random tournaments described in the previous paragraph (not to be confused with the 20,000 EGT Simulations), the Poisson probability of that happening is p < 0.000001, or 1 in a million. It almost seems as if the tournament 'wants' a Hawk & an Owl to play in the final.
Even more interesting is the fact that there have been 14 Hawks & 16 Owls in the final game over the last 15 years. The likelihood of that happening across the 20,000 simulated tournaments is even more significantly rare (p < 0.0000001). I admit that it is entirely speculative to say what I'm about to say, but perhaps this means that the tournaments are interacting across years? Could it really be just one big tournament? Is it possible that this system will always very nearly have a finahe teams accumulating loss links faster than win & vice versa? How can this be if the Owls are energetically dominating the Hawks?
Figure 2. Hawks vs. Dove-Owls, Lotka-Volterra Equations, Evolutionary Game Theory Simulations
Answers to a couple of the questions posed in the previous section may begin to take form when one considers Figure 2, which shows Hawks vs. Dove-Owls. Figure 2 corresponds to Scenario #1 as it is described on the UT website. Hawks are dominating the Dove-Owls & pushing them to extinction in the EGT Simulations. The average of each species' in each tournament drag across both lines, however.
Three tournaments fall below the carrying capacity for both species (2005, 2012, & 2007) & three fall above the carrying capacity of both species (2010, 2019, & 2017). Of note is the fact that 2 of the three years in which 2 Owls faced off in the final game (2005 & 2012) occur among the three tournaments that are below both species' isoclines. Similarly, the two years where 2 Hawks (2010 & 2019) play in the final game occur among the 3 tournaments whose average falls above both isoclines. As is pointed out on the UT website, in these years both populations would be moving toward the nearest isocline. In the case of the three years above both isoclines, this would signal a die off or collapse of both populations until the Hawk isocline is reached. For the 3 below both isoclines, the two populations would both be growing toward the Dove-Owl isocline.
Interestingly, the third year among the three tournaments with averages above both isoclines, 2017, very nearly put 2 Hawks in the Final Game as well. When North Carolina beat Oregon by 1 point by snagging 2 offensive rebounds after missing 4 free throws with less than 10 seconds remaining, it ensured that a Hawk (Gonzaga) would play an Owl (North Carolina) in the final. Of course I point this out because it was an extremely close game that almost conformed to the pattern identified here. Table 1 shows a breakdown of the score differential in the 2 Final Four games for each of these three groups of Tournaments. As you can see,
Table 1. Final Four Score Difference, Figure 2 Based Tournament Groups
the games were significantly closer in 2010, 2017, & 2019 when compared to the other two tournament groups. My hypothesis is that the Hawks are stronger in years where there are more of them, that they benefit as a group, even though they are being dominated energetically by the Owls at a population level. I'd also suggest that Figure 2 illustrates one of the tools that the Hawks have for sustaining themselves through in the tournament simulations and, by extension, in the tournament itself: they dominate the Dove-Owls.
Also worth pointing out is that all but one of the Dove-Owls who make it to the sweet 16 occur among the 9 years with averages that fall between the two isoclines. The only exception is Duke this past season, who made the Elite 8. I don't yet have an explanation why the years between the two isoclines produce sweet 16 Dove-Owls except to speculate that it is probably merely a function of the fact that there are more Dove-Owls in those years. As the tables showed in my previous post, all sweet 16 Dove-Owls except 2006 West Virginia & 2019 Duke beat other teams in group 3 (the weakest group) to get there. More Dove-Owls may simply mean greater opportunity for a favorable 2nd round matchup.
Figure 3. Hawks vs. Doves, Lotka-Volterra Equations, Evolutionary Game Theory Simulations
Figure 3 provides yet another clue about where the Hawk strength may reside. Corresponding to Scenario 4 as described on the UT website, Figure 3 shows that there is a stable equilibrium between Hawks & Doves. From the perspective of the simulations, which is all about total population energy, any tournament years below the two isoclines will grow toward the nearest isocline, while those above both isoclines will decline. Any tournament years located between the two isoclines will move toward the point where the two meet, which is a stable equilibrium. Thus, from the perspective of population biology, the Hawks & Doves in the tournament stabilize one another, ensuring their continued, long term presence within the overall community.
As was pointed out earlier, though, Lotka-Volterra does not equal basketball. Despite this, Figure 3 points to the same relevant fact as Figure 2: the Hawks are dependent upon Dove-Owls & Doves for survival. They dominate one & are stabilized by the other, despite being dominated within the simulations by the Owls. Figure 3 also points out the fact that aside from 2013, 2012, & 2018, both Doves & Hawks are initially overpopulated. This actually makes sense, even in basketball terms. The first couple of rounds in every tournament are essentially the equivalent of an epidemic where half of the population dies off on consecutive days. From my perspective, that means that what gives the tournament its character is the fact that too many teams are invited, intentionally, which is how it should be: some of the teams that are assumed to be weak, such as those from the small conferences, turn out to be quite good. Said another way, the only way to find a diamond in the rough is to search through, or 'invite' in tournament terms, a lot of rough.
Origninally, I was going to cover all of the Lotka-Volterra plots in one post. As I look at this, it's gotten pretty long, just covering the relationship between Hawks & the rest of the species. I'll stop here and continue with the rest of the plots/figures in another post.
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