March brings March Madness, the annual conclusion to the U.S. men's college basketball season with two single elimination basketball tournaments showcasing the best college teams in the country. Almost as mad is the plethora of office pools across the country where the object is to pick a priori as many game winners as possible in the tournament. More generally, the object in an office pool is to maximize total pool points, where different points are awarded for different correct winning predictions. We consider the structure of single elimination tournaments, and show how to efficiently calculate the mean and the variance of the number of correctly predicted wins (or more generally the total points earned in an office pool) for a given slate of predicted winners. We apply these results to both random and Markov tournaments. We then show how to determine optimal office pool predictions that maximize the expected number of points earned in the pool. Considering various Markov probability models for predicting game winners based on regular season performance, professional sports rankings, and Las Vegas betting odds, we compare our predictions with what actually happened in past NCAA and NIT tournaments. These models perform similarly, achieving overall prediction accuracies of about 58%, but do not surpass the simple strategy of picking the seeds when the goal is to pick as many game winners as possible. For a more sophisticated point structure, however, our models do outperform the strategy of picking the seeds.

March Madness and the Office Pool

Fixing balanced knockout and double elimination tournaments

A double-elimination tournament is a competition where no participant is eliminated until they have lost two matches. It is structured as two single-elimination tournaments: the winner bracket and the loser bracket. Players who lose once in the winner bracket are mapped to positions in the loser bracket, according to a mapping called the link function. Surprisingly, although the same structure of the winner and loser brackets is used universally, there is no standard definition of the link function. By investigating several design goals, we show that the functions used in practice are not optimal. We propose a similar function that is optimal with respect to our design goals. In order to demonstrate some of the possible research questions about double-elimination tournaments, we address the manipulability of the outcome of a double-elimination tournament. We show that they are vulnerable to manipulation by a coalition of players who can improve their chance of winning by throwing matches, a phenomenon recently observed in Olympic Badminton. We also discuss the computational complexity of manipulation by a tournament organizer (agenda control) in two settings: by changing the player seeding in the winner bracket, or by picking the mapping of losers to the loser bracket. We provide algorithms, hardness proofs, and we formulate open problems for future research.

The structure, efficacy, and manipulation of double-elimination tournaments

We consider the problem of using simulation to efficiently estimate the win probabilities for participants in a general random knockout tournament. Both of our proposed estimators, one based on the notion of “observed survivals” and the other based on conditional expectation and post-stratification, are highly effective in terms of variance reduction when compared to the raw simulation estimator. For the special case of a classical 2n -player random knockout tournament, where each survivor of the previous round plays in the current round, a second conditional expectation based estimator is introduced. At the end, we compare our proposed simulation estimators based on a numerical example and in terms of both variance reduction and the time to complete the simulation experiment. Based on our empirical study, the method of “observed survivals” is the most efficient method.

http://www.jise.ir/article_3970_02d5bdcdb9c2ffe9b8b030153c8cf883.pdf

Efficient Simulation of a Random Knockout Tournament

The selection of an appropriate competition format is critical for both the success and credibility of any competition, both real and simulated. In this paper, the automated parallelism offered by the RoboCupSoccer 2D simulation league is leveraged to conduct a 28,000 game round-robin between the top 8 teams from RoboCup 2012 and 2013. A proposed new competition format is found to reduce variation from the resultant statistically significant team performance rankings by 75 % and 67 %, when compared to the actual competition results from RoboCup 2012 and 2013 respectively. These results are statistically validated by generating 10,000 random tournaments for each of the three considered formats and comparing the respective distributions of ranking discrepancy.

/pdf/simulation-leagues-analysis-of-competition-formats-bl5kpcc5mu.pdf

Simulation Leagues: Analysis of Competition Formats

Tournaments are popularly used in sporting events to select a champion. They are also used in experiments where paired comparison procedures are needed. Knockout tournaments are most useful when the number of treatments is too large to use the more well-known round-robin tournament. The mathematical and statistical literature does not address questions such as how one should seed teams in a tournament or which tournament structures are appropriate. For double-elimination (DE) tournaments even basic questions such as how many structures there are remain unanswered. This article addresses some fundamental questions concerning DE tournaments, including the number of DE tournaments and the probabilities of teams winning a DE tournament. Edwards gives many results about single-elimination (SE) tournaments, such as the probability of winning an SE tournament and a notation for labeling and counting them. In this article I develop similar results for the DE tournaments. The results given apply to an arb...

Double-Elimination Tournaments: Counting and Calculating

Gender is a critical social factor influencing how children view the world from very early childhood. Additionally, during the early elementary years, parents can have a significant influence on their child’s behaviors and dispositions in fields such as science. This study examined the influence of parent gender and child gender on 2nd- and 4th-grade children’s strategies for solving science problems with their parents, as well as their attitudes about science. The behaviors of 13 parent-child dyads as they solved hands-on science problems together in an informal setting were examined. A child interview and a parent questionnaire were used to elicit information about their attitudes toward life science compared to physical science. Quantitative and qualitative analyses of the parent-child interactions revealed differences in the amount of encouragement offered to boys versus girls. Additionally, there were differences in questioning behaviors by parents as a function of parent gender and child gen...

Christopher T. Edwards

Papers

Double-Elimination Tournaments: Counting and Calculating

Gender Influences on Parent-Child Science Problem-Solving Behaviors