Mathieu Thuillier: The ballot theorems

Abstract:
The ballot problem was first enunciated by the French mathematician Bertrand in 1887. It solves the following question: in case of an election between two opposing candidates, what is the probability that the winner has been ahead throughout the whole count? Bertrand solved immediately the problem and concluded that the probability was equal to (alpha+beta)/(alpha-beta).

This single result led to a flurry of research, aiming at generalizing that result. In this paper, we will focus on the original ballot theorem and its first generalization, that is the probability for a candidate to have throughout the count k times as many votes as the losing candidate. Beyond these results and the proofs attached to them, we will explore the link between ballot problems and the Catalan numbers, as well as some of the direct consequences of
the results presented, especially in the theory of random walks. We will also investigate one direct application of the theorems in the eld of electronics.

In a final section, we will focus on the perspectives for researchers, as they attempt to generalize the ballot theorems, eliminating the restrictions, among which the fact that the original ballot problem deals only with integers.

Supervisor: Yishao Zhou