by Jess Behrens

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

Up to this point, I've worked faithfully under the impression that the tournament is essentially a random walk. The results of my analysis have caused me to start to doubt that. **In fact, I no longer think that the tournament is the result of statistical processes (i.e. trend, with uncertainty). I believe that the evidence shows that the champion, and perhaps the runner-up, can be explained by taking a mechanistic view of the tournament. This mechanism seems to be related to the major first round upsets as well.**

Of course, everything I will show in this post was derived from my NCAA database & index methodology. Those 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 1-65/1-68 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 10,000 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.

What I will cover today is an update to this method based on __Nash's__ __Game Theory__ & __Mixed Strategies__ work. Game theory runs through all of this work, from its basis in Evolutionary Game Theory to the use of the Key Player metric in delineating the 4 different species types. **In this post, I will use traditional 2 player game theory 'squares' to improve the evolutionary game theory simulations and to illustrate how the Dove-Owls play a disproportionately important role in identifying both who wins the championship and which highly seeded teams will lose in the first round.**

Game Theory & the NCAA Tournament

For game theory to work as a tool in this analysis, and for a mixed strategies Nash equilibrium to emerge, several things must be true.

The two players & the available strategies to those players must be consistent.

Each strategy must be available to both players.

Those players must be able to choose the strategy they want to play. Said another way, the players must be 'rational actors' and, thus, able to learn from playing the game.

All three of these requirements can be fulfilled in the second & third (Sweet 16) rounds of the 6 total rounds of the tournament.....with one caveat that is the central principle of this post. As for why only these two rounds work: the first round is Owl dominant & the third round is Hawk dominant, __both of which were factors in determining the cost term used during the simulations.__

As far as point #1 is concerned, Player 1 will always be the higher seeded team & Player 2 will be the lower seeded team. Since the second and third rounds are still within the regions, there will never be a case where the two teams playing one another are the same seed. Since there are 4 species in the tournament, point 2 necessitates that each of these be considered as strategies available to the 2 players. In the second & third round, but not the first, the Dove & Dove-Owl strategies are completely dominated by the Hawk & Owl strategies. **Thus, the second & third rounds can be simplified into a 2 x 2 game involving the Hawks & the Owls. As a 2 x 2 game, as long as Point 3 is fulfilled, it is possible that a mixed strategies Nash equilibrium will exist.**

Point 3, according to traditional economic theory, is not fulfilled in this analysis because these strategies are based on the idea that team behavior is an aggregate function. They do not have a single decision maker. Basketball games involve thousands of individual decisions, according to this train of thought. There are several reasons __I've already enumerated __I disagree with this belief. Teams are 'rational' actors, even though that rationality comes from their cumulative win/loss experience over the course of the season.

You can see it in how the Hawk strategy is at a distinct disadvantage in the first round and a distinct advantage in the Elite 8. However, the evolutionary game theory based method used here does explicitly state that they don't have complete latitude in who they are because their 'intelligence' is the result of the sum of their experiences.

**But one could easily argue that this is the same thing that happens with individuals playing Nash games: their tendencies are the results of their conditioning. And the result of the game, then, is a function of the fact that both players are the result of their conditioning and the rules of the game. This much is no different from how this method considers each team to be an individual independent of the players on those teams. ** We're just formalizing the 'conditioning' to be the standardized transformation of that teams season statistics.

**From my perspective, if one were to consider all Nash games, and the players in them, to be the result of their experiences (assuming you could quantify the 'season statistics' for all of these individual players), the result would look no different from what I'm about to present. In fact, I think that there is an error in Nash's thinking that the methods presented here can correct: People, playing 'games', have tendencies and they do not, ever, have full latitude to choose to overcome those tendencies. Certainly, they can learn. But, that requires additional experience, which becomes a response to them winning or losing based on those tendencies and feeds into the 'season statistics'. **They don't know the entire game, or their choices, immediately or ever. If Nash theory requires truly rational participants who have full knowledge of the game, then it limits itself to a very small subset of actual games.

**Sorry, if the above criteria have to be true for game theory to apply, every single generalization of Nash theory to the fields of international relations, sociology, etc. are all incomplete &, technically, wrong. The distinction that says Nash theory can not be applied to groups is based on a desperate need to cling to the idea of it being one brain that selects a distinct strategy and nothing more than that, even if/when the results are useful. It becomes an argument about which principles of game theory one is willing to violate, not something more profound because individuals have the same tendencies and conditioning as groups. **By the way, if you're looking for singularities in a basketball game, all of the players are working toward one goal and their actions are all intended to bring about a win. Having played basketball since I was a kid, there is definitely a group mind. Furthermore, if you really want a singularity, then the coaches brain and impact on the game is singularly unique. But even he isn't a completely rational or a real player in the game.

**Finally, if you consider how the tournament works, specifically that it is based on seeding, then the selection committee effectively chooses the potential strategies in the second & third round. It's their democratically chosen seeding that effects the presence or absence of a potential Nash mixed strategy, even if they do it without knowing the species types of the teams involved. What's more, while the seeding criteria may not change, the committee itself learns from what happens year over year, even if they don't realize that the teams can be generally categorized into 4 different species types. It's just like any other misunderstood Nash game you can find in the economics literature, where the game has been 'defined' by the researchers without fully understanding or enumerating all the possible strategies.**

Nash theory & the 2nd Round of the NCAA Tournament

Having said all of that, for the second round to be a Nash game, the Dove-Owls must be counted as Owls. It is as if that strategy 'surges', or reaches its greatest strength, in the second round. They change from what is a very clear, weaker role in the first round. And while some of them play in the Sweet 16, none of them have ever won a Sweet 16 game, even though that is a possible outcome. Figure 1 shows the 2 x 2 Nash 'games' for the second & third rounds of many different 'groupings' of tournament years. I will spend the remainder of this very long blog post discussing the implications of these Nash squares as well as how I arrived at the different groupings of years. **While there are multiple Nash mixed strategy equilibrium's in the 2nd Round, there are only partial mixed strategies in the third round. This shows that the progression of the tournament from more of an open system to one where either the Owls or the Hawks dominate.**

Figure 1. Nash Equilibrium 2x2 Squares, Second & Third Rounds, by Tournament Year Group

In __Chapter 16__, I described a clustermap, derived using the best fit regression slope between Doves & Dove-Owls across all possible pairs of 14 tournament years, which is presented again in Figure 2. Several of the labels in the 'Group' row correspond to the bracket identifiers in Figure 2. Before describing them, however, it's imperative that I describe the important elements in Figure 1 and how they pertain to determining the Nash strategies for each group.

For those new to Nash theory, Player 1 is on the left & Player 2 along the top in Figure 1. Each grouping of 4 cells, in the middle, with decimals in them, represent one 'game' based on a grouping of years, which are labeled by 'Group' in the top row. The decimals themselves represent the proportion of times the applicable strategy won a match-up between the two strategies in that game. The left percentage always applies to Player 1, the higher seed, & the second (after the comma) applies to Player 2, the lower seed. Thus, the upper left square always references a game where a Hawk played a Hawk; & the lower right where two Owls played. The lower left will always reference a game between a higher seeded Owl and a lower seeded Hawk. Conversely, the upper right square will reference a game where a higher seeded Hawk played a lower seeded Owl.

Figure 2. Seaborn Clustermap, Doves vs. Dove-Owls, Regression Slope

Directly below the Player 1 line, you'll see two squares P1% & P2%. These rows contain the result of a Nash mixed strategies game theoretic analysis for Player 1 (p & 1-p) & Player 2 (q & 1-q). You'll see that p & q are always below the Hawk column; 1-p & 1-q are always below the Owl column. This is because p & q contain the fraction of time Nash theory indicates that the applicable player should play Hawk in a game for the applicable grouping of years; while 1-p & 1-q contain the fraction of time Nash theory indicates that the applicable player should play Owl in a game for the applicable grouping of years. These two decimals, 'p, 1-p' & 'q, 1-q' will always, separately, add up to 1.

**Groupings with a decimal for all four (p, 1-p & q, 1-q) are mixed strategy Nash equilibrium's. Years where 2 are decimals (p, 1-p OR q, 1-q) is a partial mixed strategy Nash equilibrium. Finally, years where all four are either 1 or 0 represent dominant strategies. A Nash mixed strategy, or a partial mixed strategy for p, implies that Player 1, the higher seeded is not dominant.**

If you look at the 'All Tournament Years' Group in Figure 1, you'll see that both p & q equal 1 in the 2nd round. This means that, when examined across the entire data set, the 2nd round is purely hawk dominant. If you recall from __Chapter 16__, the left 7 years along the bottom of Figure 2 heavily favor Owls as 6 out of the 7 years have an Owl champion. Likewise, the right seven heavily favor Hawks, with 5 of the 7 years having a Hawk champion. Something interesting happens when one examines these two halves as separate Nash games.

Figure 2 has 3 bracket labels: A, B, & Hawk Dominant. A & B combine to include all 7 Owl dominant tournament years. This is why the 2nd & 3rd Groups in Figure 1 are 'Owl Dominant, Cluster A & B' & 'Hawk Dominant, C & D'. C & D aren't identified yet, but I will describe them later. However, the Nash equilibrium values for 'Owl Dominant' & 'Hawk Dominant' apply to these two groupings. As you can see in Figure 1, the Owl Dominant years are have a partial mixed strategy Nash equilibrium, with Player 1 being purely Hawk dominant & Player 2 showing a mixed strategy.

**The Hawk Dominant years, however, are a mixed strategy Nash equilibrium, with both Player 1 & Player 2 benefiting from there own combination of the Hawk & Owl strategies. The presence of this mixed strategy presents an interesting dilemma, because it suggests that in Hawk dominant years, neither strategy is immediately dominant after the first round. If neither strategy is immediately dominant, and a Nash mixed strategy is present as long as the Dove-Owls are counted as Owls, then perhaps the Dove-Owls improve after the first round and play like Owls in the second round? **

Evolutionary Game Theory Simulation Results of the Dove-Owl Surge

To test this, I altered the Evolutionary Game Theory simulations so that the Dove-Owls in all years play like they are Owls. They remain Dove-Owls and add to the Dove-Owl fitness, but the outcome of all possible interactions is determined as if they are Owls in the second round (__Chapter 4__). In all other rounds, the Dove-Owls behave normally. This is because the second round is the only one where there is a Nash mixed strategy equilibrium.

To see the effect of this energetic change, consider Figures 3 & 4. Figure 3 is the Hawk/Owl competition plots for Owl & Hawk dominant years, displayed separately, *without* the Dove-Owl surge, while Figure 4

Figure 3. Hawk vs. Owls Fitness Regression Plot, without Dove-Owl Surge

Figure 4. Hawk vs. Owls Fitness Regression Plot, with Dove-Owl Surge

is the same *with* the Dove-Owl surge. The effect is immediately apparent: allowing the Dove-Owls to act like Owls in the second round has a dramatic impact on the linear nature of the fitness relationship between Hawks & Owls. The Pearson drops from -0.83 to -0.62 in Owl dominant years and from -0.69 to -0.4 in Hawk dominant years. If you recall from __Chapter 6__, the Owl dominant years tend to be much more linear than Hawk dominant years, which holds up during the surge as well. However, allowing the Dove-Owls to function as Owls in the second round definitely challenges that.

The effect of this 'surge' is less apparent in the Dove vs. Dove-Owl cluster map, although it is definitely there. Figure 5 shows that cluster map, which methodologically is no different than Figure 2 aside from

Figure 5. Seaborn Clustermap, Doves vs. Dove-Owls, Regression Slope with Dove-Owl Surge

the change in the Dove-Owls during the second round. Comparing Figures 2 & 5 show that the only substantial difference is that 2011 and 2014 are pushed to the far right of the cluster map. Also, the red dots, which represent a regression coefficient of one (2007 regressed with 2007, 2008 with 2008, etc. will always be a perfect 1.0), proceed from upper left to lower right in a much more linear manner. Thus, the order within the groups has changed, but the only group to have changed is 2011 & 2014. The left 7 years include the same Owl dominant years & the right 7 years include the same Hawk dominant years. It's just that now 2011 & 2014, the two Owl champions that occur within the Hawk dominant years, have been separated out into their own cluster at the far right of the Hawk dominant group (C & D). I've updated the cluster groupings with C & D, which of course correspond to the Nash equilibrium 'Groups' C & D in Figure 1.

If you look at Figure 1 Groups A, B, C, & D, you'll see that Group A & Group C both have a Nash mixed strategy equilibrium. Only Group B is Hawk dominant (p=1 & q=1), which is the default status for all the tournaments grouped together. Finally, newly minted Group D is the opposite, and is Owl dominant (1-p = 1 & 1-q = 1). **Including the Dove-Owl surge in the simulations has not only ironed out the regression slopes into a linear progression, it suggests that it may be responsible for the unique results that occurred in these two tournaments. It seems to say that this surge 'broke' the linear nature of the tournament.**

But did it? What do the linear competition plots suggest? Figure 6 shows the updated competition plots displayed in Figures 3 & 4, but for each of the 4 clusters identified in Figure 5. As expected, the

Figure 6. Hawk vs. Owls Fitness Regression Plot, with Dove-Owl Surge, Clusters A to D

linear relationship between Hawks & Owls decreases as you move through the clusters from left to right (A to D), just as the linear nature decreased from Owl dominant years (left) to Hawk dominant years (right) in Figure 2. In general, Figure 6 demonstrates that as one moves from left to right in Figure 5, the tournaments are less able to deal with the surge in strength associated with the Dove-Owls. And, as expected, the Pearson drops to -0.24 in 2011 & 2014, which means the driving relationship within the Hawk/Owl/Dove game is either extremely challenged, or completely broken.

Cluster Designation, Nash Mixed Strategy Status, & the Tournament Champion

But how does one make use of this information? What, if any, relationship exists between Figures 5 & 6 and results in the tournament? Well, it just so happens that there seems to be a 1 to 1 relationship, with minor exceptions, between where the Tournament Champion falls within the tournament vector

Figure 7. Championship Queries & Dove vs. Dove Owl Cluster, with Dove-Owl Surge

and the cluster designation displayed in Figure 5. In fact, the champions in these different years fall in very tightly defined ranges. Figure 7 shows output from these tightly defined queries. **Again, I can not emphasize enough that the teams mentioned in Figure 7, and the plots shown in Figures 3, 4, & 6, are either Hawks or Owls BUT their cluster designation's are based on a cluster map that considers the slope of a regression that looks at Dove-Owls & Doves.**

Interestingly, the most common champion, which includes 7 of the 14 champions in this data set, occur during the only years where there is a Nash mixed strategies equilibrium in the second round (Cluster A & C). Also of note is the fact that all 5 years where 2 or more teams seeded 1-3 either lost or were taken to overtime (2010, 2012, 2013, 2015, & 2016) occur in these years as well as the biggest upset in tournament history (#16 UMBC over #1 Virginia). **It is, perhaps, reasonable to hypothesize that these upsets are a result of the competitiveness of these years & that they feed into the Nash mixed strategy equilibrium that occurs in the second round.**

The Cluster A & C query includes teams that fall on Index 44, Rank 3 & 4 as well as Index 45, Rank 2 to 13. There is one obvious exception, Oklahoma State in 2005. But, we'll get to that later because 2005 was a unique year. Illinois, the tournament runner up, is also in that query, as are both Villanova & Kansas from this last tournament. Of course, those two teams met in the Final Four, with Villanova devastating my Jayhawks. All of this fits with the pattern seen in Figure 5 since 2018 falls in the Owl dominant years & Villanova was an Owl while Kansas was a Hawk.

The next query, Cluster B, falls in a different range. As was pointed out in __Chapter 16__, Index 44 was written as a density function. The higher the rank, the denser the team. Cluster B, then, includes teams farther down Index 44, running from Rank 7 to Rank 10, who also fall between Ranks 41 & 60 in Index 20. Cluster B includes both North Carolina & Gonzaga from 2017, by the way. In that year, Carolina was an Owl & Gonzaga a Hawk, which conforms to the relative location of 2017 in Figure 5. Of course, Cluster B does not have a Nash mixed strategy equilibrium, with the Higher seed being Hawk dominant (p = 1) and the lower seed being Owl dominant (1 - q = 1).

**Finally, Cluster D is maybe the most interesting. Index 35 & Index 47 are similar, and both have a strong relationship to the mean/median of the entire tournament vector. In these two years, 2011 & 2014, both champion & the runner up fall very close to the median of the vector, which is interesting given the Cluster D competition plot seen in Figure 6. It's as if the Dove-Owl surge has broken the way the tournament normally works and, as a result, the system reverts to the average/median.**

If you look closely again at Figure 7, you'll see that the 2005 Tar Heels are not included in the queries. I did that intentionally, as the only years displayed in Figure 7 correspond to the years identified in the clusters shown in Figure 5. I've updated the Cluster B query to include 2005 & have extended Index 20

Figure 7a. Cluster B Tournament Champions, Including 2005

one rank (Ranks 40-60 rather than 41-60 as shown in Figure 7). As you can see, Carolina shows up....but so do both Gonzaga & Arizona (as well as the Texas team that, like Arizona, lost in the Elite 8 back in 2006). What's important to the narrative here is that there is still some uncertainty in what is presented here. Gonzaga fell in a lot of ranks associated with losing in the first round that both Carolina & Arizona did not. Thus, when applying these clusters to the process of selecting a champion, it is still imperative to consider a teams entire profile.

Also, consider the other exception in Figure 7: Oklahoma State. If you look back, you'll see that Arizona played Oklahoma State in the Sweet 16 and the Wildcats by 1 point. Arizona then took Illinois to overtime in the Elite 8 and lost by a point. So, the exceptions in Figure 7 & 7a, aside from Gonzaga, follow the general chart of relative strength established by the overall method. They also make 2005 a truly unique tournament.

The Dove-Owl Surge & The Fitness Remainder (OT-HT+DT)

**When one re-considers the fitness relationships among the 4 species types, with special focus on the Dove-Owls within the evolutionary game theory simulation, a new & useful concept emerges: the remainder. **Because Dove-Owls, aside from when they surge in the 2nd round, behave like Doves anyone other than a Dove, they are heavily effected by the energetic status of both the Owls & Doves. **All three of these strategies, Owls/Doves/Dove-Owls, share. So, energy gained in one round is available to the others in subsequent rounds. The only strategy that does not share is the Hawk.**

Thus, if one subtracts the Hawk fitness total (HT) from the Owl fitness total (OT) and adds in the Dove fitness total (DT) for every one of the 10,000 simulation iterations, they have an estimate of the total amount of remaining energy available to the 'sharing' strategies if the simulation were allowed to continue beyond 6 rounds. It represents the remaining fitness total that Dove-Owls would 'grow into'.

The relationship between this hypothetical sum and the Dove-Owls should be positive in years where the Dove-Owl surge forces a Nash mixed strategy equilibrium (Clusters A, C, & D) and negative in the

Figure 8. Lotka-Volterra Interspecific Competition, Remainder (OT-HT+DT) vs. Dove-Owls, by Cluster

other years (Cluster B). As Figure 8, which is a Lotka-Volterra plot displaying this hypothetical total as if it were another species, shows, that is indeed the case. Cluster B is the only negative slope. Cluster A & C display a reasonable upward slope, while Cluster D is exponentially sharper. Of course, this helps confirm the previous assertion that the Dove-Owl surge essentially 'broke' 2011 & 2014.

Hawk vs. Owl Competition, An Upadated Lotka-Volterra Plot for All Tournament Years

__Chapter 13__ includes a Lotka-Volterra Plot, with description, for Hawk vs. Owl fitness that I then updated in __Chapter 16__. The plot in Chapter 13 is based on the original simulation where the __cost term__ was equal to 1.0 in the first round. Chapter 16's updated the cost term to 1.5 in the first round. This version,

Figure 9. Lotka-Volterra Interspecific Competition, Hawks vs. Owls, All Tournament Years

Figure 9, includes both the Dove-Owl surge & sets cost = 1.5 in the first round. If you compare Figure 9 to the versions in Chapter 13 & 16, you'll see that the same basic form applies: an unstable equilibrium between Hawks & Owls. The 'mouths' of the two lines, the upper left (Y-Axis intercept) & the lower right (X-Axis intercept) continue to spread. This is expected given the increased uncertainty associated with a larger cost term and the surge of strength in the Dove-Owl population.

If you use the same description of relative strength that was outlined in Chapter 13, this version is 'correct' in 10 of 14 years. Only 2010, 2012, 2017, & 2018 are incorrect. The area between the lines to the upper left should favor Owls and does 4 out 5 times (2012 is the exception). Conversely, the area between the lines to the lower right should favor Hawks and is only right once out of 3 times (2017 & 2018 are exceptions). Finally, the ares above and below the lines should favor the Hawks, and does so 4 out of 5 times (2010 is the only exception).

**While the cluster maps provide more accurate champion predictions, I include these Lotka-Volterra plots to illustrate that, in general, this method conforms to what would be expected if these were really bird populations living in the wild. The actual tournament champion predictions would be better made with the cluster maps. However, it is good to see another body of literature that is related to this method generally confirm the expected outcomes.**

In conclusion, I think that the evidence presented here makes a compelling case for the tournament champion being the result a more mechanistic than statistical process. These results also suggest that traditional game theory can be used to describe the Tournament if one considers the interplay between the teams evolutionary game theoretic species, a teams seed, & the relative competitive nature of the Doves & Dove-Owls in a given tournament year. To summarize:

Transformed, standardized season statistics determine a teams location in the vector

Social Network Analysis and Key Player designation determine species designation

Relative proportion of species determine Fitness Totals for each species, and subsequently the degree of competition among species

Cluster of Dove & Dove-Owl species competition strength determine the effect of the Dove-Owl surge on which species type will win the tournament & the presence (or absence) of a Nash mixed strategy Hawk vs. Owl equilibrium in the second round.

All of this will effectively 'point' to where the tournament champion is located within the vector

Congratulations on making it through a very long and dense blog post. You get a star for the day.

__<--Chapter 16__ Chapter 18-->

#basketball #NCAATournament #NCAA #LotkaVolterra #PredatorPrey #Seaborn #Matplotlib #MensCollegeBasketball #NetworkAnalysis #EvolutionaryGameTheory #MonteCarloSimulations #KeyPlayer #MixedStrategiesEquilibrium #NashEquilibrium