As Argentina's Torneo Inicial begins its last three rounds, let's try to compute the probabilities of championship for each team. Our tools will be Monte Carlo and sandwiches.
The core modeling issue is, of course, trying to estimate the odds of team A defeating team B, given their recent history in the tournament. Because of the tournament format, teams only face each other one per tournament, and, because of the recent instability of teams and performance, generally speaking, performances in past tournaments won't be very good guides (this is something that would be interesting to look at in more detail). We'll use the following
oversimplifications intuitions to make it possible to compute quantitative probabilities:
- The probability of a tie between two teams is a constant that doesn't depend on the teams.
- If team A played and didn't lose against team X, and team X played and didn't lose against team B, this makes it more likely than team A won't lose against team B (e.g., a "sandwich model").
Guided by these two observations, we'll take the results of the games in which a team played against both A and B as samples from a Bernoulli process with unknown parameter, and use this to estimate the probability of any previously unobserved game.
Having a way to simulate a given match that hasn't been played yet, we'll calculate the probability of any given team wining the championship by simulating the rest of the championship a million times, and observing in how many of these simulations each team wins the tournament.
Clearly our model is overly rigid — it doesn't feel at all realistic to say that those two teams are the only with any change of winning the champsionship. On the other hand, the balance of probabilities between both teams seems more or less in agreement with the expectations of observers. Given that the model we used is very naive, and only uses information from the current tournament, I'm quite happy with the results.