by Jess Behrens

*© 2005-2019 Jess Behrens, All Rights Reserved*

Last Year, I made several posts about __traditional, economic game theory__ in which I used it to analyze the tournament a round at a time. Much of what I said in those posts remains true. However, as I covered in __Lesson #1__ of what I learned from the 2019 Tournament, many of the conclusions I drew were based on a network methodology that was far too __specific__. This logical error extended to the way in which I defined the Evolutionary Game Theory (EGT) species, all of which is covered & corrected in __Lesson #2__. Because these EGT archtype's were the strategies available to Player 1 & Player 2 in my Nash equilibrium analysis, the Nash strategy results I posted last year are not valid. Despite this, the relationship between Nash game theory & EGT species' persists when I apply the same methodology to the newly constructed network weighting, which I will now describe.

In order to assert that Nash theory works in the Men's NCAA Tournament by round, I need to be more specific about the definition of the game. The game I'm examining involves two players with perfect information and is __non-cooperative, asymmetric, non zero-sum, & simultaneous in nature__. Player 1 is always the better (high) seeded team & player 2 is always the more poorly (low) seeded team. There are 4 strategies from which each player can choose, and they correspond to the 4 EGT strategies I've described throughout this blog: Hawk, Owl, Dove-Owl, & Dove.

These rules imply that the game can only apply to the first four rounds of the tournament (up through the Elite 8), when each game is guaranteed to involve two teams who do not have the same seed. Also suggested is that patterns in team psychology, as effected by seeding, must play a role in the outcome.

I will present results from three of the first four rounds cumulatively & for each round separately: the second round, the sweet 16, & the elite 8. I've left out the first round. This is because the Win & Loss Networks used to assign teams to an EGT archetype, & thus their Player 1 & Player 2 strategy, are based on a teams propensity to win or lose in the first round.

As I covered in __Lesson #1__, all years from 2005-2019 are included in team strength designations, but only 2005-2017 are used to develop the first round sensitivity & specificity values that weight the network; 2018 & 2019 are not included in any of the weighting and represent a true test population across all rounds. Thus, the second round represents the first opportunity for EGT species to truly be a factor that isn't purely 'fitted' to the results it's trying to predict for all 15 years (2005-2019).

Second Round

Figure 1 shows the results of each second round game for the 2005-2019 tournaments as if they followed the game theoretic rules I described in the preceding section: a perfect information, two player, non-cooperative, asymmetric, zero sum, & simultaneous game.

The two matrices in Figure 1 (& Figures 2-4) represent the same data. The top matrix shows the total number of winners over the years 2005-2019 by the combination of EGT strategies participating in a game. The bottom shows the proportion that those wins represent of the total for that combination of seeds & strategy. The left cell in each strategy grouping corresponds to the number of times Player 1 wins & vice versa for the right cell/Player 2. Thus, in the top matrix of Figure 1, a higher seeded Hawk played a lower seeded Hawk 13 times, with the higher seeded Hawk winning 9 games & the lower seeded Hawk winning 4.

Green font is used for player 1, the high seed, and blue is used for player 2, the low seed. The mid-shade gray cells are represent dominated strategies for the high seed & the dark gray dominated strategies for the low seed.

Of the two squares, the lower is used to determine which strategy should be selected by each player because it represents the real probability of winning if the players *could* choose which EGT strategy they wanted to put into a second round game. Based on all of that, the second round has a pure strategy solution: each player should select an Owl to play in the second round.

Given that the Owls are the primary beneficiaries of the Network, this result makes sense. It indicates that the benefits of the networks extend past the first game.

However, it also begs the question: are these 'players' doing what Game Theory says they should? Of course, this is a difficult question to answer, because the teams match up based on tournament seeding & so it isn't possible for Player 1 to actually 'choose' to be anything other than what it's season statistics, as measured in the networks, dictate. Even so, it's worth looking at the actual percentage of 'plays' by each player to see if they approximate what Game Theory suggests would be optimal.

This technique of comparing the actual percentage of strategies played to the Nash Mixed Strategy percentages has been used in the economic literature. Ignacio Palacios-Huerta published a study of FIFA penalty kicks where he showed that the goalie & kicker's strategy choices conform to what the Nash Mixed Strategy predicts would be optimal. The citation is below, if you'd like to read it:

Ignacio Palacios-Huerta (2003) "Professionals Play Minimax" *Review of Economic Studies*, Volume 70, pp 395-415.

If the percentage of strategy 'plays' for each player does conform to Game Theoretic expectations, then it would suggest a form of 'strategizing' that isn't currently believed to exist. However, it would be especially difficult for these numbers to ever perfectly conform to the predicted Nash Mixed Strategy equilibrium for two reasons:

The network is limited. Teams are broken into their EGT strategies based on their network specific location relative to one another as derived from the transformation of a season's worth of data. In other words, they have to out-compete one another for a limited number of spots within the network. They earn their 'spot'. Thus, it will never be possible for all 'plays' (100%) to be 'Owl', as is suggested as optimal in the 2nd Round, for example.

Seeding leads to sub-optimal strategies competing in later rounds. These sub-optimal 'plays' have to be incorporated into the Game Theory matrix.

Table 1 shows the number of times each 'player selected' Hawk, Owl, Dove-owl, or Dove in the second round.

Going back to Figure 1, the pure strategy predicts that both 'players' should always select to be an Owl in the second round. While these results are not anywhere near 100% Owl, that strategy does dominate the others, especially in the case of the Player 1, the Higher Seed. I don't know at what point one can say a given percentage conforms with a 100% expectation, but certainly Player 1 utilizing 'Owl' nearly 75% of the time suggests a nearly dominant reliance on that strategy.

Sweet 16

As the tournament moves into the second weekend & the Sweet 16, the relative strength of the the EGT strategies begins to shift. A new designation shows up as well, one where I'm not sure whether or not the strategy is dominated (light gray). If the game has perfect information, the Dove-Owl strategy, which is only available to Player 1 in Figure 2, would seem to be a wash, netting either a complete win or total loss. Over time, the value to player 1 & 2 would be 0.5, a value that is either smaller or equal to the Hawk & Owl strategy for Player 1. This would seem to make it a dominated strategy.

Nonetheless, I __checked an expert__ to see of the mixed strategy equilibrium shown in Figure 2 applies. The result of his web based analysis indicated that, in a general game, a mixed strategy equilibrium applies. In a zero sum game, however, Player 1 has a pure strategy and should always play a Dove-Owl.

What's fascinating to me about this is that the only higher seeded Dove-Owl to win a game in the Sweet 16 is (wait for it........wait for it......wait.....for.......it......) the Zion Williamson & RJ Barrett led Duke Blue Devils from this last season. And they, like the 2010 Kansas Jayhawks who were in a similar situation, were in a one point, last second game against a lower seed in the second round. Duke won by a point. Kansas lost by a point. Perhaps having multiple first round draft picks on a team can compensate for poor position within the tournament networks, which both 2010 Kansas & 2019 Duke (2017 Villanova, too) experienced? Buttttttt, it is rare.

No matter how you look at it, Figure 2 definitely shows that the Hawks are beginning to assert themselves in the Sweet 16. I believe this stems from the fact that those Hawks who won two games now have had a few days off to prepare for this game.

Again, though, do the seeds, as 'players' in a game theoretic sense, conform to the mixed strategy equilibrium that is predicted in Figure 2? As Table 2 shows, amazingly, the answer is both yes & no.

From Table 2, we see that the higher seed (Player 1) is still very much dominated by the Owls, who are (as I have pointed out numerous times) the main beneficiaries of the networks. Player 2, however, plays almost exactly what Figure 2 suggests it should to keep Player 1 guessing. Player 2 is off by about 1% for Hawks & 3% for Owls. Could it be this represents a sort of over-confidence on the part of the higher seeds, which makes them vulnerable to Player 2, the lower seed? Could it be that, given their status as a lower seed, Player 2 teams are forced to be more strategic?

Elite 8

There are no lingering questions in Figure 3. The Elite 8 is a Hawk dominated round. The benefits of the network have now worn off & the Owls are just trying to survive. Of course, given that they have won 9 of the 15 tournaments, one of them usually does.

When we look at the number & percentage of times that each strategy was 'played' by each player in the Elite 8, which is shown in Table 3, we can see that both are still very overwhelmingly dominated by the Owl strategy.

As was the case in the other rounds, Owl dominance makes some sense given the number of times Owls have won the tournament over the 15 years included in this analysis. However, it is clear from the actual results that those teams fortunate enough to be Hawks experience a significant advantage in the Elite 8. If it were possible for a team to suddenly change itself, that team would have a tremendous advantage.

All Games, Rounds 2 - Elite 8

Finally, if we look at all games between Rounds 2 & the Elite 8, we end up with a mixed strategy equilibrium. Of course, this makes sense given the fact that a mixed strategy occurs in the Sweet 16 with dominance switching from Owls (2nd Round) to Hawks (Elite 8) over the three rounds.

*The Tournament 'wants' a Hawk to play an Owl in the Championship*

Figure 5. Hawks vs. Dove-Owls, Lotka-Volterra Population Equations

Figure 5 shows the Lotka-Volterra carrying capacity for Hawks as a function of Dove-Owls & vice versa. As I describe in __Lesson #3__, the lines in Figure 5 are isoclines which represent the high end of the 95% confidence interval for the total number of each species across 20,000 Evolutionary Game Theory simulations. The slope of each line is determined by the competitive impact the two species have on one another. In Figure 5, the Hawk line is dominating the Dove-Owl line. A community that included only these two species would tend to favor the Hawks.

The points in Figure 5 represent the average number of Hawks & Dove-Owls per simulation across the ensemble of 20,000. If you look closely at these points (Figure 5 is interactive), you'll see 3 points to the left of both lines, three to the right of both line, & 9 in between the two lines. The three on the left are 2005, 2007, & 2012; on the right are 2017, 2010, & 2019.

In 4 of those 6 years, the two teams playing in the championship were of the same species:

In 2005 & 2012, Illinois, North Carolina, Kentucky, & Kansas were all Owls.

In 2010 & 2019, Duke, Butler, Texas Tech, & Virginia were all Hawks.

In all of the other 11 years, a Hawk played an Owl in the Championship, which has a Poisson p < 0.000001 based on 20,000 Tournament simulations. If you read through both __Lesson #3__ & __Lesson #3, cont'd.__, you'll see that the only species the Hawks dominate are the Dove-Owls.

__Land the plane:__ When you combine the Nash equilibrium results with Figure 5, it suggests that:

That the Dove-Owls are Hawks who were off just a smidge & were unable to benefit much from the networks (Figure 6 shows a 3d 'picture' of all 4 species).

The Nash strategic advantage Hawks' experience after Round 2 derives in part from their dominance of the Dove-Owls.

Evolutionary pressures due to the over-population readily apparent in most of the

__Lotka-Volterra plots__coupled with these Game Theory (Nash Mixed Strategy Equilibrium) results & the__evolutionary game theory clustermap plots__are responsible for the highly significant total of 11 tournament finals involving a Hawk & an Owl (p<0.000001).The fact that virtually all of the

__Lotka-Volterra plots__paint the tournament to be an overpopulated ecological system dictates that the optimal Game Theory strategy is virtually impossible to match. Of course this makes the fact that the lower seed in the Sweet 16 does match what Game Theory predicts especially significant. It also suggests that the change in strategy & seed strength during the Sweet 16 is particularly important to the overall tournament results.

Figure 6. 3d Plot of Evolutionary Game Theory Species

So, that's it. Hope you enjoyed the presentation. And kudos to Zion Williamson & RJ Barrett for doing what no other team in their position has done in the past 15 years. Let that sink in when you want to say that you're disappointed with how Duke's season ended.

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