The minimax bookie: the two-horse case

A bookmaker takes bets on a two-horse race, attempting to minimise expected loss over all possible outcomes of the race. Profits are controlled by manipulation of customers' betting behaviour, which is assumed to be determined uniquely by the price quoted for each horse. We consider different strategies for choosing these prices as bets accumulate, and examine the penalty incurred by the use of a strategy other than the optimal one, in the general case where the distribution of the customers' betting probabilities is unknown.