by Jess Behrens
© 2005-2019 Jess Behrens, All Rights Reserved
In Lesson 1, I talked about the problem of over-fitting. I'm going to continue on that theme, but focus on how it relates to the Evolutionary Game Theory portion of this project. I spent a lot of time early in this project defining how I use Evolutionary Game Theory (EGT) & what it is, specifically the snowdrift/hawk-dove game that I use to group the teams based on their network attributes. So, I won't rehash all of that.
The overly specific queries I described in Lesson 1, as well as the problematic method for weighting each query, effected the Hawk/Owl/Dove-Owl/Dove populations by making them 'fragile'. Essentially, since I'm always in the process of refining my method where possible to improve it, every time I would add a new'query, the species into which many of the teams were classified would change significantly. Over-fitting, or over-weighting, for specificity led to larger changes in the structure of the network in much the same way over tightening the springs can ruin a trampoline. Tighten one too much, and you have to over compensate on another one so the 'pull' will even out. This approach to 'tightening' the network with over-fitted queries lead to dramatic changes in the regression statistics and resulting seaborn cluster maps as well, which made their interpretation very difficult.
The good news is that, in conjunction with changing the way in which each query is weighted, I also simplified the definitions for each EGT species. While I won't publish the specific equation here, I will generally define what makes each species what it is in the game & how that equates to the math used to define them. Having now tested these definitions quite extensively, I can happily say that the proportions of each species within each tournament year are much more stable.
EGT is about population dynamics, and is a branch of what is traditionally called game theory, which is in turn a branch of behavioral economics. As such, the goal of an evolutionary game theoretic analysis is to use species' 'energy' growth as an indicator of its relative success within the overall population. In the case of the tournament, energy equates to tournament games, with 1 energy unit of EGT = 1 Round in the tournament. Thus, the loss of a unit of energy within a given species' due to competition with the other species' is in effect the loss of a team from that population.
EGT doesn't work in the same way as 'winning & losing' does in basketball, and to understand its potential role in a given system like the NCAA Tournament requires monte carlo simulations with thousands of iterations. Within each iteration, random interactions between species' are simulated; the participants in each interaction are chosen based on the proportion of each species in that system, in this case the Tournament. Each of these interactions results in energy growth or loss for the interacting species' as a population total. The amount each energy total changes depends on which species are selected for the interaction. Species who do well overall - whose energy totals increase - are said to have greater 'fitness' in that they have greater resources to reproduce.
Obviously, reproduction is not a 'thing' in the tournament - teams aren't budding off into new teams & the tournament is set in terms of total participants. Also, 'death', which equates to a tournament loss, is not usually an outcome of an individual EGT simulated interaction, and most simulations involve an arbitrary temporal cutoff. Instead, death occurs as a function of time, as it would in a real population. The goal of EGT is to examine the relative effect each species' has on every other species within the population through simulated interactions, given the percentage of each species within the overall population and normal population changes (birth, death, in-migration, out-migration, etc.). All of this is obviously very different from a single elimination basketball tournament.
So, if EGT doesn't work the same as the NCAA Tournament, what can it possibly tell us? A lot, it turns out. In fact, many of the 'surprise' tournament runs occur in years where the EGT population breakdown suggests a favorable environment for doing so. But more about that later. For now, I'll just dive into how each team is assigned to one of the four species' types and how that decision is made as it relates to each team's network statistics.
Hawks: Because their evolutionary game theory strategy involves an extra expenditure of energy as a show of force when competing for food, basketball teams labeled as Hawks are more strongly connected to the loss network than Owls. In Evolutionary Game Theory, this extra energy cost can have a negative effect on the hawk population in total, especially when the 'food' they compete for provides less new energy than the output of energy they expend in getting it. In this definition of a Hawk, that extra 'cost' is analogous to the strength of a teams connection to the loss network. While Hawks are still among the strongest of teams in terms of their connection to the win network, mathematically speaking, in this definition, they are accruing loss links at a faster rate than win links, when normalized to their tournament year. They also typically do not experience as great of a net benefit from their network position, as expressed in the Key Player metric cost/benefit analysis, as do Owls.
Owls: The ultimate fakers in terms of evolutionary game theory, Owl basketball teams are also among the strongest teams in the tournament. Effectively, they are playing not to lose, rather than to win. Mathematically, they are accruing win links faster than loss links as well as benefiting more from the Key Player cost benefit analysis on average than any other species. Owls are defined by a weak, if not non-existent, connection to the loss network and typically also have a weaker connection to the win network than do hawks.
*EDIT* Hawk/Owl: I'm adding this after I've published this original post. The difference between Hawks & Owls is probably best captured in a battlefield metaphor. Owls, because of their strong relationship to the Net Benefit that is provided by the networks & measured using the Key Player metric, are the equivalent of an army who has the high ground. The structure, and I do mean 'structure' because I use structural equations to create the indexes, of the regular season supports the Owls. The Hawks, on the other hand, are the opposing army who has to charge uphill in order to dislodge their opponents. However, I've made no attempt to quantify or analyze team game strategy, look at how the final minutes of play sort themselves out in each game, etc. I'm making this designation PURELY based on the status of each team within the two networks. The hill, or battlefield, in this metaphor, is simply the accumulation of the season.
Dove-Owls: Stuck in no man's land, Dove-Owls are my creation & are not defined anywhere else in the EGT literature. I conceived of them as the teams who are most unpredictable in the tournament. These are the teams who might play well & might flame out. Energetically speaking, in the EGT simulations, they act like Owls when they face Doves & other Dove-Owls, and behave like Doves when they face Hawks or Owls. In the code I wrote, they actually gain more energy in the second round of simulations, essentially mimicking some teams in every tournament year who play really well for two rounds. Mathematically, these teams have highly variable benefit from the Key Player cost/benefit analysis because they are very poorly connected to either the win or loss network. Many Dove-Owls are simultaneously very well connected to the other network, win or loss, from the one to which they are very poorly connected. In effect, I'm trying to say mathematically that these are the good teams in each tournament whose season statistic's indicate a fatal flaw.
Doves: The weakest and most timid of the 4 EGT species types, Doves are completely non-confrontational in energetic terms & completely dependent for survival on the other EGT species types who 'share' part of their food, which includes other Doves. Mathematically, these teams gain virtually no benefit at all from the Key Player cost/benefit analysis & have poor connections to both the win & loss networks as well.
I've included Figures 1 & 2 to show the breakdown by species type in each tournament from 2005-2019. Figure 2 provides a 3 dimensional look at the 1,002 teams that span 2005-2019 & references 3 of the important metrics I use in breaking the teams into different species'. As you can see, Hawks are actually fairly rare in the tournament system. I'll spend more time talking about the significance of this and what it means in later posts. Also, if you're wondering who the Dove-Owl (orange squares in Figure 2) is with a Key Player score of over 8,000, that's none other than Florida Gulf Coast from 2013.
Figure 1. Evolutionary Game Theory Species Breakdown, Tournament Year
After re-doing both networks using these new methods, the result was very clear - teams in each tournament break out into 3 separate groups - the strongest, relatively strong, & the weakest. Because all of this analysis
was done only on the tournaments that run from 2005-2017, both 2018 & 2019 are real tests for these methods. Since I'm trying to predict first round wins & losses first and foremost, one would expect these results to be 'fitted' to the 2005-2017 tournaments. And they are. However, 2018 & 2019 should fit expectations - the strongest should win the most often, save against themselves, etc. Table 1 shows the strongest & 2nd strongest groups for 2018 & 2019.
Table 1. Group 1 & Group 2, 2018 & 2019 NCAA Tournament
While they aren't perfect, the results shown in Table 1 demonstrate that these methods do a generally good job of dividing teams into logical groups that conform with actual results, despite the fact that only 2005-2017 were used to create the network weights. In these two years, only 2018 Arizona, 2018 NC State, 2019 Utah State, & 2019 Cincinnati lost to teams that were not among the top group. Within this group of 4, Arizona lost to Buffalo, who is in the second group. The others are seeded 7 - 9. I did a quick analysis of this and, while 2005-2017 are not shown here, the 7-10 & 8-9 games are far less predictable than all of the others combined. In fact, if you account for the proportion of each strength group (1-3) within each tournament year, teams seeded 7-10 and located in the top group are far more likely to lose than would be possible given a random distribution (poisson p <0.001). Thus, even among the exceptions, there are patterns, which bodes well for using a conditional forest to find those exceptions. Finally, Texas Tech, who I mentioned in Lesson #1 was a victim of over weighting specificity, is now firmly in Group 1. And to reiterate, that is the result of removing queries, not adding new observations (queries).
Table 2 shows the 3rd & weakest group of teams in the tournament. I've included it here to illustrate that it also follows the general principle outlined above - the teams who win out of this group for the most part play other teams within the same group. For example, West Virginia in 2018 beat two other teams in group 3.
Table 2. Group 3, 2018 & 2019 NCAA Tournament
In fact, of the 544 total teams in this group (2005-2019), only 7 have made it to the sweet 16. Of those 7, only 2006 West Virginia & 2019 Duke beat a team that was outside of group 3. West Virginia beat Northwestern State in the second round in 2006, who was (surprise, surprise) in group 1. As we all know, 2019 Duke beat UCF.............by a point. On the last play of the game. If you want to give that team credit for anything, it's for SURVIVING. They were not positioned well at all. Rather than expressing disappointment that they didn't win the National Championship, give RJ & Zion an award for actually winning 3 games. Frankly, if it were any other team without the presence of 2 NBA Plug & Play All Americans & potential future Hall of Famers, they could have lost to North Dakota State. You read that correctly. And, for the record, I'm not hating on Duke here. I'm saying that their exceptional talent & Hall of Fame coach won them 2, maybe 3, games that another team wouldn't have won.