Essays on Tournament Design

Dmitry Ryvkin

Date of defense: June 2, 2006

Dissertation Committee:
Andreas Ortmann - CERGE-EI, chair
Dirk Engelmann - Royal Holloway, University of London
Kai Konrad - Free University of Berlin
Viatcheslav Vinogradov - CERGE-EI

Abstract:

 

A tournament is a principal-agent game where rewards are based on the relative performance of agents. Tournaments are observed in a variety of contexts, including the labor market, sporting events, research competitions, and rent seeking.

From the organizer’s perspective, a tournament is a mechanism that provides incentives and reveals the ranking of participating players. Such a mechanism can be designed in various ways, depending on the organizer’s objectives.

In this dissertation we discuss theoretically a relatively unexplored aspect of tournament design – the design of tournament formats. The latter are understood here as rules that specify how players are matched to compete, and how players’ relative performance is translated into their ranking. We develop a unified framework that addresses the problem of optimal design of tournament formats at an unprecedented level of generality.

In terms of organizers’ objectives, the focus of the dissertation is on the informational utility of tournaments. We leave the incentive provision problem aside and essentially view tournaments as estimators of the unobserved ranking of players. We characterize such estimators quantitatively by their predictive power – a measure of how reliably a tournament identifies better players. This view of tournaments is also relatively new in the literature, although organizers’ objectives stemming from it are important in many applications. In situations such as recruitment and promotion in firms or selection of public finance projects, a reasonable organizer’s objective would be maximization of the predictive power. In sports, where uncertainty and upsets increase the “entertainment value” of tournaments for spectators, organizers may tend to minimize the predictive power. Throughout the dissertation we mainly address the decision problem of an organizer who seeks to maximize the predictive power of a tournament.

The dissertation consists of a literature review and three essays. The first essay is co-authored by Andreas Ortmann, while the second and the third essays are singleauthored.

In the first essay, “Three prominent tournament formats: predictive power and costs”, we explore the predictive power of three benchmark tournament formats – contests, binary elimination tournaments, and round-robin tournaments. For each of the three formats we develop combinatorial techniques that give analytical expressions for predictive power. One interesting finding is that predictive power may depend nonmonotonically on the number of competitors and the overall noise level. We also show which of the three formats is preferred by an organizer whose objective is to select the best player most accurately in the presence of costs.

In the second essay, “The predictive power of composite noisy tournaments,” we propose a novel approach to construction of complex tournament formats out of elementary sub-tournaments (“building blocks”). The set of admissible building blocks is determined institutionally by players’ specific activities. Once the building blocks are defined, the proposed algorithm constructs all possible tournament formats satisfying certain natural constraints. As an illustration, we consider composite tournament formats built out of generalized contests of n players with m winners and explore their predictive power. It is shown that the predictive power is maximized by multi-stage formats that slowly eliminate weaker players. Such formats are the most costly ones, and the organizer will switch to other formats as costs increase.

In the third essay, “Some additional properties of tournament formats,” we discuss seeding in binary elimination tournaments, and marginal distributions of players’ scores in round-robin tournaments. Seeding can be applied as an additional control mechanism by an organizer who has some a priori knowledge about players’ ranking. We discuss how the predictive power of binary elimination tournaments is influenced by seeding in various regimes, and show that in some situations an optimal seeding scheme that maximizes the predictive power can be identified. Marginal distributions of scores in round-robin tournaments show what is the probability for a particular player to get a certain number of points. Our closed-form expression for this probability constitutes an interesting mathematical result.