Optimal Online Bookmaking for Binary Games
This work addresses a specific problem in online betting for bookmakers, offering a novel solution but is incremental in the broader context of game theory or optimization.
The paper tackles the problem of maximizing a bookmaker's worst-case return in online binary betting by formalizing it as the Optimal Online Bookmaking game and providing an exact solution using a new technique called bi-balancing trees, which ensures equal house loss across decisive betting sequences.
In online betting, the bookmaker can update the payoffs it offers on a particular event many times before the event takes place, and the updated payoffs may depend on the bets accumulated thus far. We study the problem of bookmaking with the goal of maximizing the return in the worst-case, with respect to the gamblers' behavior and the event's outcome. We formalize this problem as the \emph{Optimal Online Bookmaking game}, and provide the exact solution for the binary case. To this end, we develop the optimal bookmaking strategy, which relies on a new technique called bi-balancing trees, that assures that the house loss is the same for all \emph{decisive} betting sequences, where the gambler bets all its money on a single outcome in each round.