Break Accuracy: Optimizing the Sorting Power of Preliminary Rounds Using Tapered Points

by R. Eric Barnes, Paul Kehle, Nick McKenny and Chuan-Zheng Lee • HWS

Ideally, preliminary rounds at tournaments sort teams so that the best teams break into elimination rounds. At the World Championships of debate, the scoring system during the nine preliminary rounds does a poor job of sorting teams accurately. Adding additional rounds would increase the accuracy and fairness, but this is impractical. Using mathematical models and computer simulations of tournaments, we show that using a slightly different scoring system over the nine preliminary rounds would improve the accuracy of the break even more than would doubling the number of preliminary rounds to 18. Other implications and insights into tabulation and sorting accuracy are also discussed.

This article is dedicated to the memory of Steve Penner. We discussed these ideas well before any of the computer simulations started.


Debate tournaments are composed of preliminary debate rounds, in which all teams compete, followed by elimination rounds, in which a subset of teams who performed best in the preliminary rounds compete in a single-elimination format.  The number of teams that advance (or “break”) into elimination rounds is determined by the overall number of teams at the tournament.  The primary purpose of preliminary rounds is to identify which teams deserve to break into the elimination rounds.  Although there are many practical complications, this can be seen essentially as a mathematical sorting problem.  The goal of this paper is to study how the scoring of the preliminary rounds can be adjusted to optimize this sorting process and maximize the fairness of tournaments.  That is, how can we ensure that the teams that perform best are most likely to break?

Our focus will be on the World Universities Debating Championships (WUDC or “Worlds”), which are held in the British Parliamentary (BP) style of debate, with four teams in each debate, and two debaters on each team.  Judging is done by a panel of judges who strive for consensus, voting when necessary, and ultimately deliver a single decision.  The scoring used to rank teams in a BP tournament traditionally awards 3 points to the first-place team, 2 points to second-place, 1 to third and 0 to fourth.  This creates some unique dynamics and some opportunities for improvement.

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About the Authors

R. Eric Barnes is an associate professor of philosophy at Hobart and William Smith Colleges (HWS) in Geneva, NY, where he is also the director of debate.  His research is about moral theory, applied ethics and competitive debate.  In 2007, he invented the Round Robin format for British Parliamentary debate, and the HWS Round Robin has served as a laboratory for debate research as well as a venue for high-quality debating.  Prof. Barnes has published articles about (and against) judging theory, about expanding the break at the WUDC (in the run up to that decision), and about the reliability of judging calls in BP.  He is committed to improving the activity of competitive debate by promoting both better arguments and better ways to run tournaments.

Paul Kehle is a professor of mathematics education at Hobart and William Smith Colleges (HWS) in Geneva, NY.  He conducts research in computational discrete mathematics, mathematical modeling, and mathematical cognition (by humans).  Through his teaching of pre-service teachers and professional development work with teachers, he promotes the study and teaching of mathematical modeling in grades K through 12; and he develops curriculum to support the teaching and learning of computational thinking and mathematical modeling.  His only interest in debate is understanding the mathematics and fairness of debate tournaments.

Hugh N. McKenny is a senior at Hobart and William Smith Colleges (HWS) in Geneva, NY.  He is completing a triple major in Mathematics, Cognitive Science, and Education while also participating in a Teacher Education Program to become a certified teacher of mathematics in New York State.  While his interests range well beyond his three majors, he particularly enjoys exploring problems in discrete mathematics and working to understand the nuances of mathematical problem solving.  

Chuan-Zheng Lee received his BE (Hons) degree from the University of Auckland and is studying towards a PhD in electrical engineering at Stanford University, doing research on the theory of machine learning.  He has served on the tab teams of three WUDCs, four Australs, three Yale IVs, USUDC, Hart House IV and HWS IV, and he is on the development team of Tabbycat, a popular tabulation platform used around the world.  He has also judged semifinals at WUDC, Australs, NAUDC, Yale IV and Hart House IV.  He is interested in supporting the global debating community through improving software and encouraging the sharing of knowledge among tab directors.