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# Hawk, Doves, & Owls: Defining the Math for Evolutionary Game Theory Simulations - Chp. 4

by Jess Behrens

In order to show that Evolutionary Game Theory (EGT) works for the purposes I've outlined here, it is necessary that the math used in developing the Hawk/Dove/Owl game provides insights into the tournament's workings & results. EGT dates to the 1950s, when ecologist Maynard Smith was asking the question: if evolution is driven by the 'survival of the fittest', why are behavioral traits such as sharing and altruism found so frequently in nature? Smith corresponded frequently with John Nash, who won the Nobel Prize in Economics along with John Harsanyi & Reinhard Selten, for his work on Game Theory & mixed methods equilibrium.

Smith's work on the Hawk/Dove game showed that the strategy of sharing resources, as doves do with each other, can outperform the hawk strategy of direct conflict when the overall population & energetic value of the available food favors them. In other words, when the food being fought over doesn't provide enough energy to compensate for the extra energy Hawks expend using their strategy or when there are so many hawks in a population that repeated conflicts lead to injury, they are at an energetic disadvantage to their more easy going competitor's, the Doves. Thus, while an individual dove may never receive more energy from an interaction with a dove as an individual Hawk does, it is still possible for the doves to out compete the Hawks at a population level (i.e. collectively have more energy than the Hawks). Applying this idea to the NCAA Men's basketball tournament requires use of computer simulations, which I will describe below. If done correctly, the basketball team populations I've defined in previous posts should conform to the expectations developed in the EGT literature. It should also provide insights into the success/failure of individual teams within those populations.

Figure 1 defines the important terms for 3 of the populations I defined in this post. The Dove-Owl is a hybrid that I identified within my data set & includes teams that are typically strong enough to win at least one game, and maybe two, but struggle to make the Elite 8. While the version of this game I use in my simulations isn't exactly what you see in Figure 1, the Dove-Owl uses the same terms as the other species.

Figure 1. Hawks, Doves, & Owls - Evolutionary Game Theory Math

The Hawk/Dove/Owl game is relatively straight forward. There are only two terms, V & C, that are included in the math. V is defined as the benefit received for any interaction among the competing species. In nature, this value represents the energetic value of the food or shared resource being competed over by members of the population. C is the additional cost to the Hawks associated with their specific behavioral strategy.

Equations in the squares found in Figure 1 represent the average value to each species that comes from a competition between them. So, on average, or over a relatively long period of time, doves receive half (V/2) the energy associated with a competition between two doves and a quarter (V/4) of the energy in competitions with Owls, etc. As Figure 1 shows, the Owl is at an energetic advantage to both the Doves & the Hawks. In adding the Dove-Owl, I've simply modified Figure 1 such that all Dove-Owls behave like Doves (0 or V/4) when competing with Hawks and Owls. They receive the same benefit as Owls in interacting with Doves (3V/4) & share like Doves/Owls (V/2) when competing with another Dove-Owl.

Normally, Evolutionary Game Theoretic simulations run with the same opportunity for an interaction to occur among members of a given species over a specified time period. The NCAA tournament is a bit different, having 6 (or 7, depending on the year) rounds that involve progressively fewer teams. To make my simulation closer to the reality of the tournament, I limit the length of a given iteration to 6 rounds. I also limit the number of possible interactions in each round to those found in the tournament. Because the tournament expanded from 65 to 68 teams in 2011, the round by round breakdown in my simulations also changed. Thus,

• From 2005 - 2010, the round by round competition breakdown is: 32, 16, 8, 4, 2, 1. In these years the 'Type' of team that won the single 16 vs. 16 play in game is included as the the 64th team and one team is dropped.

• From 2011 & On, the round by round competition breakdown is: 34, 16, 8, 4, 2, 1. All teams in the play-in games are included.

Interactions in the simulations I will present over the next series of posts follow the Evolutionary Game Theoretic rule of occurring at equilibrium & where individual variation within each species is not allowed. This is just another way of saying that when a Hawk faces a Hawk, the math you see in the equation for that square (the upper left) determines how much energy the Hawk population total receives, etc.

Furthermore, it means that each interaction is determined at random using the percentage of population for each species type within the given tournament year. Said another way, tournament seeding is not a factor. Doves don't often play other doves in the first round (the seeding committee normally gets it right). However, in this simulation we're more interested in all the potential energetic interactions for a given tournament year independent of seeding. So, if two doves are selected at random to play in the first round, the entire dove population gets V energy (V/2 for one dove & V/2 for the other dove).

Another major difference betweeen the NCAA tournament structure & the normal EGT simulation rules is the fact that individuals within a given population don't usually 'die' (or stop playing when they lose). I modified my rules to better match the NCAA Tournament by hypothesizing that the Doves & Dove-Owls are competing at near starvation levels. If a Dove doesn't receive at least half of the benefit from a given competition, which effectively means any interaction where they aren't matched against another Dove, the total number of Doves within the population is reduced by one and the percentage for each population type is recalculated. The same rule applies to Dove-Owls in rounds 2-6.

What this means is that the ability of the Doves & Dove-Owls to out compete the hawks through sharing is diminished with each successive round by limiting the number of interactions and reducing the number of competitors. One could argue that these rules are the method by which the NCAA ensures that a 'good' team wins the national championship.

The last piece of information I need to include that is vital to understanding the results I will present in future posts is how I determined the values for V & C. To decide this, I used the data provided by the tournament itself. As such, V increases linearly with each round in the tournament. So, V = 1 in the first round & 6 in the sixth round. The C term was a bit more difficult to determine. For that, I turned to the Poisson distribution & compared the relative success and failure of Hawks & Owls in each round. Table 1 shows the results of that analysis.

Table 1. Wins by 'Species' in Tournament Games between Hawks & Owls by Round.

As I said earlier, the committee typically does a good job of making sure that the doves & dove-owls are primarily located in the lower seeds. However, as you can see, games between Hawks & Owls do occur in the first round, 12 times to be specific. Note: Definitions of Hawk & Owl were made prior to the creation of Table 1. There is no working backwards here. Cells highlighted in green mean that the the Hawk team won significantly fewer games than expected. Likewise, red means that they won significantly more than expected. As Table 1 clearly shows, Hawks are at a disadvantage in the first round and at an advantage in the fourth round (Elite 8). Rounds 2, 3, 5, & 6 showed no statistical difference from expectation.

Thus, C = 0 in rounds 2, 3, 5, & 6, making the Owls & Hawks energetically equivalent in these rounds. Round 1 is the primary round where Owls experience a real advantage, so C = 1 in that round. Since Hawks experience a real advantage in the fourth round (Elite 8), they pick that cost back up. Thus, for the fourth round, I set C = -1.

Each of the 14 years of the tournament data to which I have access was simulated 1000 times. I included 2018 even though the results of this years tournament were not used in constructing the networks. Table 2 shows results & Poisson significance for each of the four species when all 14 years are grouped together. Cells highlighted in red identify significantly more wins than expected and green highlight significantly fewer than expected wins. Just to be clear, Table 2 shows only results from the actual tournament data. It does not include any simulation results and I place it here to show what this project is trying to explain: The relative success of the Hawk type & whether or not that advantage is affected by individual variation in the population structure in a given tournament year.

Table 2. Poisson Significance of Tournament Results by Round & Species Type

Future posts will begin to tear these years apart and compare the results relative to the what the simulation's suggest may bet the role of tournament population structure.

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