Ballot theorems, old and new

" There is a big difference between a fair game and a game it's wise to play. "-Bertrand (1887b). 1 A brief history of ballot theorems 1.1 Discrete time ballot theorems We begin by sketching the development of the classical ballot theorem as it first appeared in the Comptes Rendus de l'Academie des Sciences. The statement that is fairly called the first Ballot Theorem was due to Bertrand: Theorem 1 (Bertrand (1887c)). We suppose that two candidates have been submitted to a vote in which the number of voters is µ. Candidate A obtains n votes and is elected; candidate B obtains m = µ − n votes. We ask for the probability that during the counting of the votes, the number of votes for A is at all times greater than the number of votes for B. This probability is (2n − µ)/µ = (n − m)/(n + m). Bertrand's " proof " of this theorem consists only of the observation that if P n,m counts the number of " favourable " voting records in which A obtains n votes, B obtains m votes and A always leads during counting of the votes, then the two terms on the right-hand side corresponding to whether the last vote counted is for candidate B or candidate A, respectively. This " proof " can be easily formalized as