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2007 NCAA March Madness Tournament: A game theory approach to finding an optimal strategy for office pools

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  1. Introduction
  2. Base scoring system
  3. Determination of the expected value
  4. Adjustment for more than two players procedure
  5. Determination of the Payoff Matrix
  6. Finding an optimal strategy
  7. The advancement probabilities
  8. Conclusions and interpretation of the study
  9. Appendixes
  10. Bibliography

The focus of this paper is to use aspects of game theory to form a strategy for maximizing a player's chances of winning a NCAA Tournament Bracket from the sweet sixteen rounds on. For each match-up, a strategy for selecting a team to advance will be determined by analyzing a payoff matrix representing a two person, non-zero sum game. Overall bracket completion strategy will be the found by using these strategies to advance teams through each round. Both pure and mixed strategy techniques are used in the determination of the overall completion method. Mixed strategies will incorporate the use of both maximin and minimax techniques.
There are three objectives of this paper: 1) To gain an understanding of some common applications of game theory, 2) To gain experience using game theory to solve problems, 3) To analyze how a mixed strategy developed using game theory will fare against brackets submitted for the 2007 tournament.Thanks to extensive research performed by statisticians, mathematicians, and sports fans alike, published on the internet are each team's probabilities stating the chances a certain team has to advance into the next round. These probabilities will be known hereafter as the advancement probability. Using different aspects of game theory, these probabilities will be used to form a strategy which will give the player, hereafter known as player A, the maximum potential to take first place in a March Madness pool. Each match-up will be determined separately, and is likely to use both pure and mixed techniques. A mixed strategy is on which gives the player the probabilities with which each choice should be made, resulting in the maximum expected value. Mixed strategies will be found using maximin. The definition of a choice here is the decision a player has to make regarding which team should be picked to advance to the next round. The mixed strategy found will be used to fill out experimental brackets. A strategy change will take place to accommodate for the change from a two-player to an n-player game through the use of betting odds.

[...] In reality, there really is no way to play a mixed strategy when filling out one single bracket. In this case, the team with the highest expected value should always be selected to continue. When filling out multiple brackets, a variation of a contrarian strategy should be used. A contrarian strategy is where the champion team is determined before filling out a bracket, and the bracket is filled out based on that. This team is specifically not the team pegged as most likely to win. [...]

[...] These situations include games that involve more than one player. Because this is a parimutuel bracket, the number, of people making the same choice, will affect the payoff. Since all choices are expected to be made simultaneously (i.e. no player has any knowledge of the decisions the other players will make before choosing themselves), it is impossible to know what n is beforehand. For analysis purposes then, the pre-assigned probability of a team advancing can be used to determine an approximation of . [...]

[...] While the aspects of game theory provide a good strategy for player A to use, the statistics used will inevitably lead to a misleading optimal strategy for the methods used. Furthermore, for non-zero sum games, while maximin and minimax methods may provide a good estimate, they do not necessarily provide the optimal solution. In conclusion, while the strategies determined here are not optimal, they certainly are not a bad approximation. When playing multiple entries, the data provided here can give some insight on whether or not to advance a certain team more than once. [...]

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