by Jess Behrens

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

In 2007, *Spoon* released its 6th Studio album *Ga Ga Ga Ga Ga,* easily one of the best albums of the past 20 years IMHO. __The Underdog__, one of the best tracks on the album, was __politically motivated____. It __actually was a response to how our President (at that time) and his administration were behaving, given the choices that they made. Of particular emphasis within the song is how if a dominant nation, the US in this case, makes it a point to not care about things that they don't understand, if they don't fear the underdog, that arrogance comes back to haunt them. I wish I could put the lyrics up here, they'd do a much better job of succinctly describing what I'm trying to say. But I'm not going to do that because I don't really want to contend with an angry BMG.

**While it may seem like a stretch and a bit corny to go from music to math, in the lyrics of **__The Underdog__**, Britt Daniel & Spoon more or less nailed the way in which the NCAA Tournament empowers the underdogs. Upsets occur in a number of ways, but they are always related to the stochastic nature of how in touch with reality a dominant team really is about itself and its opponent. One of the primary theories behind this work is that some of the teams, the Hawks specifically, have an extra cost term that is most likely related to this psychological conundrum.**

Thus, if you put Daniel's song into the logic of the project I'm currently describing, President George W. Bush was a Hawk & the extra cost associated with being a Hawk was his assumption that rolling into Baghdad would singularly end the conflict resulting in a victory where we'd be welcomed like conquering heroes. History, of course, told a dramatically different story.

To bring this back to the Tournament, what I'm going to do in this post is describe how one of the structural factors affecting tournament results, the size of the largest cluster of small conference teams within a given tournament, relates evolutionary game theory to traditional game theory, and determines the relative success of the Hawk species type. I don't know that there is a direct correlation to politics or George W. Bush beyond the similarity of the psychology that Daniel so perfectly summarizes. If there is, I'll let you tell me.

One of the most interesting structural factors affecting the NCAA Tournament is how teams at the top ('good' teams) & the bottom (small conference champions) cluster. In fact, these clusters are fundamental to the math I use in creating the Indexes I subsequently rank and convert into the Win & Loss Networks. Truth be told, one of the first patterns I noticed, and that informed this project, were the presence of very large or very small maximum cluster sizes of teams from small conferences in some of the tournaments. The results in those years were very different than those seen in others, many times spectacularly, it seemed.

**But the relationship between maximum cluster size & the Underdogs is not as straight forward as it probably should be. In **__Chapter 7__**, I posted a graphic showing the number of upsets that occur in Less Linear years and how it was very likely related to the more stochastic relationship between Hawks & Owls in those years. **

As I will show, the Less Linear years typically do NOT contain the sort of empowered underdog I'm saying is related to small conference cluster size. In fact, the years with a high number of upsets seem to be a function of the network center's destabilization than the empowerment of the periphery. Still, both types are a result of the mismatch among all four species types in a given year's total population, so like the clustering factor I'm about to describe, the destabilization of the center is also a function of population considerations. It's just that the major upsets are most likely the result of different processes in which the underdog doesn't always look like one, and other times the perceived underdog should actually be favored.

Cluster Sizes & Types

Looking only at the teams who come from the small conferences, there are two primary cluster types that have a direct impact on the tournament. I've listed both of them and the years that they are associated with each below:

Cluster 3-4: 2005, 2007, 2009, 2010, 2012, 2013, 2015, 2016

Years other than 4: 2006, 2008, 2011, 2014, 2017, 2018

Tables 1 & 2 show the results by round for each species type in these two separate groups. As you can see Cluster 3-4 years are Hawk dominant, while the remaining years (those that are not 3-4) produce

Table 1. Poisson Significance of Tournament Results by Round & Species Type, Cluster 3-4 Years

exclusively Owl champions. The fact that no Hawks have won the tournament in years where these small conference teams cluster in totals other than 3 or 4 seems like a bigger deal than it is

Table 2. Poisson Significance of Tournament Results by Round & Species Type, Years Other than 3-4

statistically. As Table 2 demonstrates, that zero is not significantly low. Thus, it wouldn't be a surprise at this point if a Hawk was able to win the tournament in one of the off years.

Figures 1 & 2 show the individual fitness plot for each of these two cluster types. Figure 1 *(All Species p<0.001, r-values: Hawks -0.09, Owls -0.24, Doves 0.49, Dove-Owls 0.20)*, labeled Cluster 4 Years but

Figure 1. Individual Fitness by Species, Cluster 3-4 Years

including years with a cluster of 3 or 4, looks very similar to the Less Linear individual plot, save for the fact that the Owl & Hawk scatters overlap in Figure 1. The telltale signal of Hawk dominance, a Dove-Owl line that crosses both the Owl & Hawk line, is present as well.

Figure 2 *(Hawks p<0.37; Dove-Owls p<0.31; Owls & Doves p<0.001, r-values: Hawks 0.01, Owls -0.22, Doves 0.56, Dove-Owls 0.01)*, showing Tournament years other than 3-4, is not as familiar, however. The

Figure 2. Individual Fitness by Species, Years Other than Cluster 3-4

presence of the different cluster values breaks the Hawks & Dove-Owls. When grouped this way, the Hawks & Dove-Owls have no significant fitness relationship to their percent of the total population.

Figure 3. Individual Fitness by Species, No Dove-Owls, Years Other than Cluster 3-4

Curious about what this meant, I re-ran the Monte Carlo simulations for these 'Other' years without including Dove-Owls. This involved folding the Dove-Owls into the Doves, thus increasing the percentage of Doves in the population, but keeping the energetics described in __Chapter 3__ otherwise the same.

Unfortunately, this experiment didn't resolve the dilemma for the Hawks *(Hawks p<0.36, Doves & Owls p<0.001; r-values: Hawks 0.02, Owls -0.19, Doves 0.46)*. They still lack significance at the individual level. Although not presented as a figure here, of note is that all 4 population types produce significant results in the population fitness plot *(All Species p<0.001)*, so eliminating the Dove-Owls is a test of effect, not theory. As was the case in the individual fitness plots during Linear years, there is no significant relationship between the fitness of the average Hawk & the percentage of total Hawks within the population. In effect, the rare cluster of teams from small conferences effectively breaks the link between Hawk success at a population level and its effect on the individual Hawks.

Upsets & the Real Underdog

First round upsets of highly seeded teams (1-3 Seeds) is not significantly high or low with respect to the small conference team cluster size break. I didn't include those results as a table because of a lack of significance. **However, if we consider the years in which the cluster of small conference teams is different than 3 or 4 (2006, 2008, 2011, 2014, 2017, & 2018), the effect of this structural component should be obvious.** In 2006, a little known team from the Colonial Athletic Association named __George Mason made a phenomenal run to the Final Four__. Perhaps you recall the 2008 Davidson Wildcats, powered by one Steph Curry, and a __painfully close last second three against Kansas.____ Relax Jayhwaks fans. ____You know how the story ended____.__

Or maybe you remember that little team from the Atlantic 10 and their coach....what was their name again? Oh yeah, Virginia Commonwealth. Or was that Butler in 2011? Oh yeah, __both__. It goes on - __Dayton__ & __UConn__ in 2014, __South Carolina__ in 2017, __Xavier__ in 2017, and the Sister Jean powered __Loyola Ramblers__ this past year. Table 3 shows the significance of this pattern. I've included all of the other

Table 3. Poisson Significance of Tournament Results by Round, Seeds 7-11

Tournament Structural elements for comparison purposes. It's not that 7-11 don't make runs in other years. It's that those runs don't come in bunches.

The way that shows up in Table 3, however, is a bit different. Instead of seeing a red spike of significance, which would indicate more than expected 7-11 seeds made it to a given round, Table 3 shows a whole lot of green & gray *(p<0.1)*, which indicates significantly fewer 7-11 seeds made it to a given round. **The absence of significantly low values in the bottom right 'Cluster Other than 3 or 4' & 'Linear Tournament Years' is what stands out. These two structural types, linear years & 'other' cluster years, have a high degree of overlap. The absence of green indicates that the 7-11 seeds aren't being beaten by their higher seeded competitors. Or said another way, the maximum cluster size of the small conference teams has evened out team strength. ** In the absence of being taken seriously, the Underdogs have ganged up.

Traditional Game Theory, Evolutionary Game Theory, & the Small Conference Team Cluster Size

I began to suspect that the number of teams from small conferences that cluster might have a degree of relationship to traditional game theory when I took a class through __Coursera__ last year. The course was taught by Dr. Matt Jackson of Stanford & covered, among other things, a linear quadratic model published in __Econometrica____ (Ballester, Calvo-Armengol, & Zenou 2006)__. The research in this article focused on identifying the 'Key Player' in a network using a linear quadratic model that considers both local and global effects. I know that's a very inadequate description of their work, and I apologize for its brevity, but there you go.

Methodologically, I took this linear quadratic model and modified it to include the ratio of the percent Betweenness Centrality over the percent Clustering Coefficient. I chose these two centralities because of their strong relationship with outcomes in both the win & loss networks. Percents were calculated by taking the sum for both of these centrality measures and dividing each teams small portion of that into this total. Furthermore, percents were calculated for each team in both the win & loss network. This ratio was then included in Ballester, Calvo-Armengol, & Zenou's linear quadratic equation and calculated for both the win & the loss network. A net score was then calculated for each team by subtracting the loss score from the win score. Finally, both the win & loss scores, independently, were used to determine each teams risk, and, therefore, their classification as a Hawk, Owl, Dove-Owl, or Dove. Tables 4 & 5 show the top three teams by tournament year for both cluster types. The the Key Player Metric as defined by Ballster, et. al. as well as my modified version are included.

Table 4. Top Three Teams by Net Key Metric Score & Net Weighted Key Metric Score, Cluster <=4 Years

Table 5. Top Three Teams by Net Key Metric Score & Net Weighted Key Metric Score, Cluster >=5 Years

Although the two are extremely similar, as they should be, in terms of identifying the Tournament winner only the Cluster <=4 years provide good results. Furthermore, the weighted score does significantly well. A bit about how the Poisson was determined: it was assumed that, on average, the Key Player index would be right 50% of the time. So, for cluster <=3 years, that works out to be 4.5 (9 / 2 = 4.5) and is 2.5 for cluster >=5 years. A year is counted as correct if the tournament winner is the top team or if the champion and runner up are the top two teams. The reasoning here is that it should be correct if the metric predicts the final game or the champion.

In some ways, this is like splitting hairs because both methods are so effective at identifying who the 'good' teams are. Table 6 further summarizes Tables 4 & 5 using the Poisson significance of the Key Player metric. In these 9 years, which includes one of the 'Other than 3 or 4' years described

Table 6. Poisson Significance of Key Player Metric & Hawk Strength by Small Conference Cluster Type

above (2008, which had a maximum cluster of 2), 8 out of 9 of them name either the Champion as the top team or the runner up followed immediately by the champion. In contrast, in only one year, 2014, does the Weighted Key Player Metric correctly identify the champion. To reiterate, this break is slightly different than, but definitely related to, the cluster type described earlier. Only 2008, with a maximum cluster size of 2, was re-assigned.

Furthermore, the tournament years with a maximum cluster that is <=4 heavily favor the Hawks. The expected values in the bottom half of Table 6 were developed in a manner identical to that shown in Table 1 & 2, but with 2008 re-assigned. In 6 of those 9 years, the Hawk won. The three years where the Hawk did not win, which includes 2008 which is still an 'Other than 3-4'/Owl favoring year, were decided on the last play of the game or in overtime.

What this seems to suggest is that tradtional game theoretic considerations, such as the linear quadratic model developed by Ballester, Calvo-Armengol, & Zenou, shows up in this evolutionary game theory project vis a vis the maximum small conference team cluster size.

Furthermore, the 'off' cluster years represent a strengthening of the Underdog both in terms of where the Champion falls within the Key Player metric (Table 5) & the relative strength of teams seeded from 7 to 11. The likely explanation for this is psychological and involves the lack of familiarity with these small conference teams & the self-assumed superiority on the part of the dominant teams.

There are still more posts to come. My next topic will be an update on the relationship between the Less Linear years and major upsets.

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