$12$ chess players took part in a tournament. Each played against each other exactly once. After the tournament every chess player did $12$ lists of names.
- On the first list, the player only wrote his own name.
- On the second list, they wrote their own names as well as all man they had won against.
- They proceeded to write lists: Every next list contains all names from the previous list and the new names that players from the previous list had won against.
It turned out that for all chess players, the $11$th and the $12$th list contained different amount of each name.
How many games ended draw in the tournament?
I have been thinking of the problem the hole day but I am clueless...