Elo and Draws.
I still had some nagging thoughts after my recent examination of the distribution of Elo. In that blog post, I recognized that a higher probability of a draw would lead to tighter standard error around the true 'ability' of a player, as estimated by an Elo ranking. Without any data, I punted on what that probability should be. So I decided to look at some real data.
I started working in a risk role about a year ago. Compared to my previous gig, there is a much greater focus on discrete event modeling than on continuous outcomes. Logistic regression and survival analysis are the tools of the trade. However, financial risk modeling is more complex than the textbook presentation of these methods. As is chess. A loan holder might go bankrupt, stop paying, die, etc. Similarly, a chess player might win, lose or draw.
There are two main ways of approaching multiple outcome discrete models that leverage the simpler binary models: the competing hazards view, and the sequential hazards view. Briefly, risk under competing hazards would be like traversing the Fire Swamp: at any time, the spurting flames, the lightning sand or the rodents of unusual size might harm you. The risks all come at you at once. An example of a sequential hazard is undergoing surgery: you might die in surgery, and if you survive you might incur an infection and die of complications; the risks present themselves conditional on surviving other risks. (Both of these views are mostly just conveniences, and real risks are never so neatly defined.)
Returning to chess, I will consider sequential hazards. Assume two players, and let the difference in true abilities between them be denoted \(\Delta a\). As with Elo, we want the difference in abilities is such that the odds that the …read more