by Jess Behrens
© 2005-2018 Jess Behrens, All Rights Reserved
Every March, when you sit down to fill out your one of 40 million brackets, the first question is: which 12 seed are you going to pick to beat a five seed? There's definitely some statistical credibility to the notion that the 5 seeds are jinxed. But why? I believe that evolutionary game theory can tell us a little bit about what is going on, and I plan to spend the next few paragraphs doing just that!
Of course, everything I will show in this post was derived from my NCAA database & index methodology. The steps, which I've outlined before but will do so again, involve:
Gather the appropriate season statistics for each invited team & use the method I developed to standardize them.
Transform the standardized statistics into index values & rank the teams 1-68 in each index.
Convert the index 'vector' into a network as described in Chapter 3, export the network, and calculate all centralities in Gephi.
Calculate the Key Player metric and it use it, along with other network centralities, to classify the teams into one of 4 groups: Hawks, Owls, Dove-Owls, or Doves.
Perform 1000 iterations of an evolutionary game theory Monte Carlo simulation in Python as described in Chapter 4.
Use those results in exploratory analysis to determine a possible explanation for why the five seeds seem to be jinxed.
Those simulation results seem to suggest that the fate of the 5 seeds is related to two specific things that affect them in very different ways.
When 5 Seeds don't lose in the First Round
It seems odd when it happens, but there are years where all 4 Five seeds win in the first round. Over the 14 tournaments included in my analysis, this happened in 2007, 2015, & 2018. If you assume that the 21 five seeds who have lost over the 14 years of my analysis are spread equally across each of those years, the Poisson statistic for 0 five seeds losing in three years is p<0.02. I took the simulation results from point 5 (pun intended) above and separated them out into a series of competition plots just like the ones I've been using throughout these posts. There were some curious things that emerged when I did this.
Figure 1 shows the Hawk vs. Doves & Dove-Owl population competition plots in years where no five seeds lose. Figure 2 shows the same relationships, but in years where at least 1 five seed loses
Figure 1. Hawk vs. Doves & Dove-Owl Competition Plot, Years w/ no 5/12 Upsets
to a 12 seed. As you can see there is a stark difference. In the years where no five seeds lose, the Hawks & Dove-Owl simulation results demonstrate a mutually beneficial relationship. The best fit regression line goes from lower left to upper right, which means that when one of them does well,
Figure 2. Hawk vs. Doves & Dove-Owl Competition Plots, Years w/ at least One 5/12 Upset
the other also tends to do well. In contrast, figure 1 shows a negative, or competitive, relationship between Hawks & Doves in these same years. Of note is the fact that the Pearson for both of these years is the same, 0.46, but negative with respect to the Doves & Positive in the case of the Dove-Owls.
The mutualistic relationship between Hawks & Dove-Owls can be found in other groupings of years, but this grouping of three years is when that relationship is at a maximum. A Pearson of 0.46 is not that strong, admittedly, but is a significant regression (p<0.001). It also marks a stark departure from the same relationship in years with at least one 5/12 upset, which is shown in figure 2. As you can see, in years where at least one 5/12 upset occurs, the relationship is slight, but negative, which indicates a degree of competition between the two species.
Does the mutualism seen in Figure 1 mean, then, that the Dove-Owls are actually, physically helping the Hawks in their games? No, but it does appear to indicate that in these years the Owl strategy of sharing with Dove-Owls isn't working when measured at a population level. It may seem strange to say this but the mutualism between Hawks & Dove-Owls in figure 1 is highly significant. Evolutionary Game Theory is something that has a deep & long history within the scientific literature. It's role in research is growing as well, becoming the subject of examination by economists as well. Thus, the math I'm using to generate these results has been validated by multiple researchers; I didn't make it up. And that math is based on the idea that the Owl is able to out compete the more dominant hawk by sharing some of its resources with Doves (& in the case of my simulation, Dove-Owls). So, the goal of the Owl strategy is to increase pressure on the more dominant Hawks from the lesser species by increasing their fitness through sharing. The enemy of my enemy is my friend, essentially. Thus, the fact that Dove-Owls & Hawks are indirectly benefiting one another indicates that the Owl's strategy is backfiring.
But how does all of this impact whether or not a five seed loses a game? I don't have a certain, specific mechanism yet. And their may not be one at an individual game level. The results presented here come from comparisons of population, not individual (re: game by game) fitness. They speak to potential rather than realized results. However, if you look at the number of 12 seeds classified as Hawks, there may be a clue. In the years where no 5/12 upsets occur, Figure 1, the number of 12 seeded Hawks is normal according to the Poisson (p<0.3). That value doesn't gain significance when you move to years with at least one 5/12 upset (p<0.45). However, it's still worth noting these Poisson values because if the number of Hawks in years with a large number of 5/12 upsets is significant, it would be an indication that the mutualism scene in Figure 1 is the result of the Hawk populations inability to control Dove-Owls population.
What do the tools of data science tell us? Are the relationships outlined in Figures 1 & 2 strong enough to be noticed by a cluster detection algorithm? Figure 3 shows the cluster map for Hawk/Dove-Owl population fitness and uses an identical process to what was described in Chapter 11. To reiterate, I run regression analyses for all possible combinations of 2 years in the 14 year data set, or 182 individual regression analyses. I then construct the results as if they were going to be used as a network,
Figure 3. Hawk vs. Dove-Owl Cluster Map, Tournament Year Pair Regression Analysis, Slope & R-Value
where there are 4 nodes (one for each year) and the links between those nodes is either the r-value or the slope of the regression. This vector is then fed into Seaborn/Matplotlib's cluster map function. As you can see, on the bottom right corner of Figure 3 along the right side, the three years where no five seeds lose have been grouped together into an index cluster, which is validated by the heirarchical cluster lines on the left. The mutualistic relationship between Hawks & Dove-Owls is strong enough to show up in a cluster map.
When a lot of 5 Seeds lose in the First Round
Figure 4 was created using a method identical to the cluster map in figure 3, however it considers the R-Value relationship between Hawks & Doves over all possible pairs of tournament years. On the right
Figure 4. Hawk vs. Dove Cluster Map, Tournament Year Pair Regression Analysis, R-Value
half of figure 4 is a cluster that includes 2013, 2009, 2014, 2006, & 2012. The heirarchical lines along the top of figure 4 separate these 5 years into a cluster. The Poisson significance of these 5 years sit opposite of the years where no 5/12 upsets occur, but are also significant (p<0.05). Figure 5 shows the
Figure 5. Hawk vs. Doves & Dove-Owl Competition Plots, 5/12 Upset Cluster Years
Hawk vs. Dove & Dove-Owl competition plots for this clustering of tournaments. Like figure 2, which includes years where at least one 5/12 upset occurs, the slope for both Hawk/Dove & Hawk/Dove-Owl are negative, which is expected. Also of note & expected is that the number of 12 seeded Hawks is significantly high in these years (p<0.05).
Figure 6 shows the same Hawk/Dove & Hawk/Dove-Owl competition plots, but for all tournament years not included in figure 5. Like figure 1, the number of 5/12 upsets in years not included in the upset
Figure 6. Hawk vs. Doves & Dove-Owl Competition Plots, Years other than 5/12 Upset Cluster Years
cluster is low, albeit in this case not quite significant (p<0.1) & the Hawk/Dove-Owl competition plot has a slightly positive regression slope. Furthermore, the number of 12 seeded Hawks is also low, but not quite significant (p<0.1).
All of this is congruent with the idea that the positive Hawk/Dove-Owl regression slope is an indication of 5 seed strength that results from a mismatch between the Dove-Owl & Hawk populations. Furthermore, the fact that the number of 12 seeded Hawks is significantly high in 5/12 upset cluster years suggests that the absence of these Hawks in other years is the reason that Dove-Owl populations are appear to be in a mutualistic relationship with the Hawks.
Fitness Balance: Why the Population Plots work to Explain the Championship & Upsets
Everybody knows that the higher seeds in the tournament should win their first game and that the very high seeds should win the entire thing most of the time. By and large, this is precisely what happens. However, as my project has shown in other champters, one can use fitness plots as an indicator for when upsets are going to occur and to determine which species type is favored to win the championship.
But why do the fitness plots, which measure population more than individual success, work? What it comes down to is this: some of the higher seeded teams (1-6) are in fact Doves when they are supposed to be Dove-Owls, Owls, or Hawks. Their exposure to risk, as manifested in the Loss network Key Player metric, has overcome their exposure to success (win network Key Player Score). This reclassification of a team that should be a hawk to a dove (or owl to dove or dove-owl to dove) changes the proportion of each species type in the total population, which is apparent in the fitness simulation results.
As was shown above, the five seeds are a perfect example of how this happens. Of the 21 five seeds that lost in the first round between 2005 & 2018, 20 of them were doves. The only Dove-Owl among them was the Wichita State team that lost to VCU in 2012. In fact, if you look at all 73 teams seeded 1-6 who lost in the first round, 64 are Doves, 3 are Owls, & 6 are Dove-Owls. Of the 73 teams who upset those higher seeds, 23 were Hawks, 38 were Owls, 9 were Dove-Owls, & 3 were Doves.
Since in the case of most upsets, the higher seed has been reclassified as a Dove, it ends up being the species distribution of the lower seeds who are capable of winning a game that determines the balance of power in any given tournament year. It should be self-evident, even though I haven't spelled out the math for how I standardize the statistics or form the indexes, that it is difficult for a lower seed to reach Hawk/Owl status, & only slightly easier to qualify as a Dove-Owl. Playing in smaller conferences means playing teams that are not as strong from a talent standpoint, which leads to a lower strength of schedule. Thus, since it is possible for higher seeded teams to end up as Doves, what happens in those small conferences is what effectively determines the distribution of species overall, the resulting upsets, & the tournament champion. It is these variations in fitness that drive the competition & energy plots, allowing them to send 'signals' about the tournament's structure, despite not actually simulating the tournament as it has been seeded.
Tables 1 & 2 show the breakdown & Poisson significance of team success by round & species. Table 1 shows the results for years where there are no five seed upsets. Thus, the data from these years goes
Table 1. Poisson Significance of Tournament Results by Round & Species Type, No 5/12 Upsets
with figures 1-3. The mutualistic relationship between Hawks & Dove-Owls in these years is a function of the fact that the Dove-Owls are especially weak. Table 1 shows a very insignificant Poisson, one that is essentially 50/50. What has happened is that the five seeds in these years are not doves, which alters the balance of each species overall percentage within the total population.
Table 2, which covers the years in the cluster identified in figure 4 (2006, 2012, 2013, 2014, & 2009) on the other hand, tells a much different story. As you can see, the Poisson for the dove-owls has increased substantially (p<0.13), but not quite to the point of significance. In these years, which
Table 2. Poisson Significance of Tournament Results by Round & Species Type, Figure 4 5/12 Cluster
correspond to the plots in figures 5 & 6, most of the five seeds have been classified as doves. The classification of the five seeds to a dove has changed the balance of fitness for the entire tournament, which shows up in the fitness plots.
Finally, it's important to note that the process at work in these years, while related to first round upsets, is significantly different from the same process which leads to multiple upsets of very highly seeded teams. As the linear/less linear differentiation showed, the relationship between Hawks & Owls is the driving force behind these spectactular upsets. I mention this because it addresses the complex nature of the NCAA tournament and speaks to why the number of upsets in any given tournament & which teams lose in those games seems so random. It's not at all random. It's just the result of different population level relationships among the species.
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