You can model "wins in the tournament" as a continuous variable. You will have to apply a post-processing step to your output where you round the final number to a meaningful integer number of wins.
This method is restricted to regressions.
Another possibility is use a multinomial logistic regression. This will ignore the ordering of the # of wins and treat each possible outcome of # of wins as a distinct class. In this case, you would have 7 classes. Again, the multinomial logistic regression assumes the outcome variable is nominal, not ordinal i.e 0 wins, 1 win, 2 wins, aren't seen as numerically ordered but as distinct classes just as red hair" and "blonde hair" are seen as distinct classes with no notion of "greater than" or "less than" between the classes. Modeling an ordinal variable as a nominal variable may reduce the accuracy of the model
The paper linked in the hyperlinks above can give you more guidance on more advanced methods. In addition, you can look at this paper here