As mid march rolls in and the Washington, DC area sees its first real snow of the year (wait, what?!), office workers across country begin spending their precious time furiously filling out brackets. This year I decided to go all in and join a $5 NONREFUNDABLE bracket pool.

After committing to spending such a large fee, I began reading the rules (which is obviously the rational order of doing things). Traditionally, March Madness Brackets are scored to reward to players for making the correct pick with a bonus for making correct picks in later rounds. For every first round winner you pick correctly you get one point, for every second round winner you pick correctly you get two points, and so on through the sixth round (aka the National Championship game).

However, this scoring system was different. In order to reward upsets, my league’s scoring system assigns points based on making picks that other contestants didn’t make. In my league, if you make a correct prediction your point total equals the round times the number of people who MISSED that pick.

Points = Round # x Number of People who Picked Another Team

For example, let’s say I pick 15th seeded Troy to beat 2nd seeded Duke in their first round game and 99 out of the 100 other contestants pick Duke. If Troy wins, I get 1 x 99 points. If Duke wins I get zero points, but the 99 people who picked Duke 1 x 2 points. Two points because two of us (another contestant and I) did not pick Duke.

This changes the strategy of a rational March Madness consumer (as all March Madness contestants of course are). In a traditional March Madness pool, you want to pick the team with the highest chance of winning to maximize your expected point value. For example, if 6th seeded Maryland has 51 percent chance of winning their first round game against 11th seeded Xavier, you should pick Maryland because the expected payoff is 1 x .51 versus 1 x .49 for Xavier. However, under my league’s scoring system the percentage of people not making that pick is also included in the expected payoff calculation. If 58 percent of people choose Maryland than the expected payoff of choosing Maryland is 1 x .51 x .42 = .21 versus 1 x .49 x .58 = .28 for Xavier.

Game Theory

That’s all well and good, it seems all you need to do is to find undervalued underdogs and pick them. However, things are not so simple.

See, assuming that all of my fellow Bracket contestants have perfect information and strict rationality, we all are aware of this market inefficiency. Everyone knows that we should pick Xavier in the above scenario, so as more and more people pick Xavier their expected payoff goes down and Maryland’s goes up.[1]

It turns out we are playing a one round simultaneous move game, in game theory parlance. That means all contestants reveal their picks simultaneously and no further decisions can be made. To solve this game, i.e. figure what strategy to play, we look for Nash Equilibriums. A Nash Equilibrium is a situation where no single player has an incentive to change their strategy.

What is the Nash Equilibrium here? Each contestant should select teams proportional to the likelihood of that team winning. In our example Maryland has a 51 percent chance of beating Xavier, so a contestant should choose Maryland with a 51 percent likelihood. If everyone uses this strategy, on average 51 percent of people will choose Maryland and 49 percent will choose Xavier. That means the expected payoff of choosing Maryland is 1 x .49 x .51 = .25 and the expected payoff of choosing Xavier is 1 x .49 x .51 = .25. Everyone is indifferent between the two teams, and therefore has no incentive to switch picks.

This concept can easily be extended to the rest of the bracket and through every round. Game solved.[2]

Behavioral Economics

But if there is one thing you learn from how Washington, DC deals with snow it’s that people are not perfectly rational.

This is where behavioral economics comes in. Behavioral economics helps to rectify the mythical rational consumer with how people behave in the real world.

To get a sense of how (ir)rational my fellow March Madness contestants are, I downloaded ESPN’s data on users’ March Madness picks. Ideally, I would like the data from the other people in my pool but that is not available to me. The ESPN data has a very large sample size, but ESPN uses the traditional scoring system, so individuals are incentivized to pick favorites. Additionally, some number of these brackets are probably not legitimate attempts. Still, it’s the only contestant picking data I have.

The ESPN data can give us insight into how contestants behave irrationally. People should always pick the favorite given the ESPN scoring system, but that is not the case. For example, a strong favorite like Baylor was chosen by 89 percent of individuals to win in the first round even though FiveThirtyEight gives them 90 percent chances of winning in the first round each. [3] Actually the user behavior is not too far from the Nash Equilibrium of my league!

Given that my league gives even more incentive to pick upsets than ESPN, we can assume that distribution of picks in my league will be similar to the Nash Equilibrium but skewed more heavily towards upsets. In my results below, I did not guess the degree of this skew was and used the raw ESPN data.

Greedy Algorithm

How do I make my picks? There are 9.2 quintillion possible brackets[4] so calculating the expected score of each bracket is not going to happen. Instead, I created a greedy algorithm.[5]

My greedy algorithm iterates through all 32 picks and makes the best pick available each time. A “pick” is choosing a specific team to make it to a specific round. I only have to make 32 picks because once I have chosen 32 teams, the remaining the 32 teams are the de facto first round losers.

The first iteration of the algorithm calculates the expected payoff of choosing each team to win the national championship game. It looks at each team and sums the expected payoff of choosing that team to win their first round game, their second round, and so on through the national championship game.[6] The team with highest expected payoff is chosen to win the national championship game.

Each subsequent iteration continues similarly but with the constraint that certain teams will only be able to advance so far before running into a previously picked team. In other words, I can’t choose everyone to win the national championship.

Results

This model produces a fun and perhaps unconventional bracket. My Final Four teams are Virginia, Louisville, Gonzaga, and Kentucky. A five, two, two, and one seed. Three number one seeds are knocked out in the second round. Each of these number one seeds (Villanova, Kansas, and UNC) have ESPN second round user pick rates (88 percent, 87 percent, and 92 percent) above their FiveThirtyEight second round win probabilities (80 percent, 81 percent, and 80 percent). They are being overvalued for my league’s scoring system.

St. Mary’s and Southern Methodist are the two surprise elite eight teams. FiveThirtyEight rates them as the strongest six and seven seeds respectively. My final bracket calculates their expected payoffs as fourth and sixth largest.

And my champion? Gonzaga.

FiveThirtyEight gives Gonzaga a 14 percent chance to win the tournament but only seven percent of ESPN users picked them. My model gives them a whopping 20 percent better expected payoff as champions than choosing Villanova, the second highest.

Notes and Footnotes

All data is current as of the evening of Monday March 13, 2017.

[1] If the percentage of people picking Maryland drops to 50 percent (from 58 percent) and the corresponding percentage of people picking Xavier increases to 50 percent (from 42 percent) that expected payoffs adjust. Maryland’s expected payoff increases to 1 x .5 x .51 = .26 (from .21), while Xavier’s expected payoff decreases to 1 x .5 .49 = .25 (from .28). All of a sudden Maryland is looking like a good choice!

[2] I am sorry if I offended any game theory professors with my back of the envelope game theory calculations.

[3] Remember, for an individual contestant the expected payoff of choosing Baylor to win the in the first round is 1 x .9 = .9, while choosing their 14th seeded opponent New Mexico State is 1 x .1 = .1.

[4] Not including the play-in games.

[5] A greedy algorithm is an algorithm that makes locally optimal choices in order to reach a final decision. The advantage is that they are fast (in my case about .5 quintillion times faster than the brute force approach) and often lead to good enough solutions. The disadvantage is that they may not lead to good enough solutions. This happens if a decision made early on in the algorithm has significant negative consequences down the road.

[6] Remember the expected payoff of choosing a team to win any given game is the round times the probability of winning times percentage of people who did not choose that team.