# Tournaments, Fairness and the Prouhet-Thue-Morse Sequence

Palacios-Huerta, Ignacio, Economic Inquiry

I. INTRODUCTION

Mathematician Axel Thue discovered in Thue (1912) the Prouhet-Thue-Morse sequence while studying avoidable patterns in binary sequences of symbols, e.g., 0, 1. This sequence is defined by forming the bitwise negation of the beginning:

t = 0 1 1 0 1 0 0 1 1 0 0 1 0 1 1 0

where 1 is the bitwise negation of 0, 1 0 is the bitwise negation of 0 1, 1 0 0 1 is the bitwise negation of 0 1 1 0, and so on. Formally, the Prouhet-Thue-Morse sequence t = (t,,),,>o is defined recursively by [t.sub.0] = 0 and [t.sub.2n] = [t.sub.n], [t.sub.2n+1] = [t.sub.n]~ for all n > 0, where for u c {0, 1} we define u' = 1 - u.

This sequence t was already implicit in Prouhet (1851) and later rediscovered by Morse (1921) in connection with differential geometry. Worldwide interest in this sequence has developed during the past century as research has shown that it is ubiquitous in the scientific literature. In fact, this sequence occurs as a "natural" answer to various apparently unrelated questions, for instance, in combinatorics, in differential geometry, in number theory (e.g., the "Prouhet-Tarry-Escott" problem), in group theory (e.g., the Burnside problem), in real analysis (e.g., the Knopp function), in the physics literature on controlled disorder and quasi-crystals, in music, in chess, in fractals, and turtle graphics (e.g., the Koch snowflake), and in many others (see Allouche and Shallit 1999 for a survey).

Here we note that the Prouhet-Thue-Morse sequence can also be the natural answer to an important problem in economics: How should the order of a sequential tournament competition between two identical agents be determined to make it as fair as possible? That is, how should the order in a sequential tournament be determined to minimize any advantage that either competitor may have from both an ex ante perspective and an ex post perspective?

II. ANALYSIS

Tournament competitions are pervasive in real life and often characterize competitive situations such as competitions for promotion in internal labor markets in firms and organizations, patent races, political elections, student competitions in schools, penalty shoot-outs in soccer, chess matches, and many others (see Lazear and Rosen 1989; Prendergast 1999, and other references therein). There are numerous reasons why in a sequential tournament the order may give either a first mover advantage or a second mover advantage (see Dixit and Pindyck 1994; Cabral 2002, 2003, and other references therein). These include psychological reasons if, for instance, leading or lagging has an impact on the performance of the competing agents (Apesteguia and Palacios-Huerta 2010; Gonzalez-Diaz and Palacios-Huerta 2011). In other words, a given order that is fair from an ex ante perspective (e.g., randomly determined) need not be ex post fair if it gives any type of advantage (strategic or psychological) to either player.

Consider two players A and B that play against each other an even number of times. Typically a fair coin is used to randomly select the order of play, and the order follows a strict alternation of the chance to perform the task (e.g., in penalty shoot-outs in soccer, color of pieces in chess matches): A B A B A B A B A B A B .... The question we are interested in is the following: Is there a way to improve the ex post fairness of this sequential order?

Assume now that a coin flip selects A to perform his task first and B second in the first two rounds of a sequential competition. What should the order in the next two rounds be to make it ex post fair? If the order A B offers any kind of advantage to either player, then by reversing the order in the next two rounds we will tend to compensate for that advantage. The resulting sequence is: A B B A. And of course reversing is innocuous if no advantage existed in the first place. How about the next four rounds? The same principle applies: by reversing the order followed up to that point we will tend to compensate for any potential advantage that might have been given to either one of the players until then. …

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