My simulation gives two kinds of results - if I compute the mean number of losers divided by the mean of (winners+losers), I get 1/36 regardless of the (finite) number of rounds. The mean number of winners for 1-round, 2-round, 3-round and 4-round games are : 0.9718, 10.4356, 102.3750 and 996.1660 while the mean number of losers are 0.0282, 0.2896, 2.9552 and 27.7942.
But if I compute the mean in another way (calculating the ratio of losers to losers+winners, first and averaging that ratio), I get the ratio that increases with the limit on the number of rounds: 0.0282 for one-round, 0.0513 for two rounds, 0.0756 for three, and 0.0986 for four.