Broadening the Scope of Optimal Seeding Analysis in Knockout Tournaments

Optimal seeding in balanced knockout tournaments has only been studied in very limited settings, for example, maximizing predictive power for up to 8 players using only the relative ranking of the players (ordinal information). We dramatically broaden the scope of the analysis along several dimensions. First, we propose a heuristic algorithm that makes use of available cardinal information and show an improvement in the optimality of the solution. Second, we address tournaments with size up to 128 players. Since the large number of distinct seedings prohibits finding the optimal solution, we introduce an innovative, provably correct upper bound on the optimal value. More interestingly, our heuristic and upper bound achieve objective values that are close to each other. This shows the upper bound and the heuristic solution both approximate well the optimal values. Last but not least, we also investigate two novel objectives: the expected strength of the winner, and the revenue of the tournament. The analysis of both objectives shows that our solution is indeed robust and effective. Optimal Seeding in Knockout Tournaments: An Experimental Study Optimal seeding in balanced knockout tournaments has only been studied in very limited settings, for example, maximizing predictive power for up to 8 players using only the relative ranking of the players (ordinal information). We dramatically broaden the scope of the analysis along several dimensions. First, we introduce two additional novel objectives: the expected strength of the winner, and the revenue of the tournament. Second, we address tournaments with size up to 128 players. Since the large number of distinct seedings prohibits finding the optimal solution, we introduce innovative, provably correct upper bounds on the optimal value of each objective function. These upper bounds allow us to evaluate different ordinal solutions and show how ∗This work was supported in part by NSF grant IIS0205633-001 and in part by a BSF grant. †This is an extended version of the paper that appears in AAMAS 2010 Cite as: Broadening the Scope of Optimal Seeding Analysis in Knockout Tournaments, Thuc Vu,Yoav Shoham, Proc. of 9th Int. Conf. on Autonomous Agents and Multiagent Systems (AAMAS 2010), van der Hoek, Kaminka, Luck and Sen (eds.), May, 10–14, 2010, Toronto, Canada, pp. XXX-XXX. Copyright c © 2010, International Foundation for Autonomous Agents and Multiagent Systems (www.ifaamas.org). All rights reserved. some can in fact approximate well the optimal values. We also propose a heuristic algorithm that can further improve the optimality when there are cardinal data available. Ordinal Seedings in Knockout Tournaments: An Experimental Study The relative ranking of the players (ordinal information) has been often used in practice to seed players in a knockout tournament. The optimality of ordinal seedings has only been studied in very limited settings, for example, maximizing predictive power for up to 8 players. We dramatically broaden the scope of the analysis by addressing tournaments with size up to 128 players. Since the large number of distinct seedings prohibits finding the optimal solution, we introduce an innovative, provably correct upper bound on the optimal value. The upper bound allows us to evaluate different ordinal solutions and show how some can in fact approximate well the optimal values. We also investigate and show similar results for two additional novel objectives: the expected strength of the winner, and the revenue of the tournament. When there is cardinal information available, we propose a heuristic algorithm that can further improve the optimality. introducing two additional novel objectives (the expected strength of the winner, and the revenue of the tournament, Second, we address tournaments with size up to 128 players. Since the large number of distinct seedings prohibits finding the optimal solution, we introduce innovative, provably correct upper bounds on the optimal value of each objective function. These upper bounds allow us to evaluate different ordinal solutions and show how some can in fact approximate well the optimal values. We also propose a heuristic algorithm that can further improve the optimality when there are cardinal data available.