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 . [...]
[...] As stated before, the ability to use either or adds a lot of complexity to the determination of an optimal strategy, so will be used in all calculations. Maximizing the expected value of player results in the equation for (see Appendix A for the systematic deduction): A similar process is used to develop a strategy to minimize the expected value of B's payoffs. (This is shown step-by-step in Appendix B). An expression for using the minimax method is given as follows: Once the probability that A should select row A1 has been determined, to determine , all that is left is to plug into equation to get the expected value of A's payoffs. [...]
[...] Finding an Optimal Strategy Player A's strategy to maximize its payoff can be split into two categories, pure or mixed. By playing a pure strategy, A would select an entry in this matrix 100% of the time. This is best applied when an equilibrium point is present. An equilibrium point for A is an entry in the matrix where A's payoff is the minimum in its row, and the maximum in its column. Here it can be said that A's payoff cannot decrease regardless of what B does, and vice-versa for B. [...]
[...] In addition, intended for this paper was an examination of how the experimental bracket did against a number of submissions from peers using this year's tournament results. This idea was put to rest since the tournament is well over, and the data would not be worth collecting. The methods used to estimate , and here are simplifications. When determining the number of players who potentially select the same team, statistics should be gathered and used to determine a value for . [...]
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