Probabilities within optimal strategies for tournament games

Let T be a tournament. The tournament game on T is: Two players independently pick a node. If both pick the same node, the game is tied. Otherwise, the player whose node is at the tail of the arc connecting the two nodes wins. Fisher and Ryan showed that for any tournament T, the tournament game on T has a unique optimal strategy. If one node beats all others, the optimal strategy always picks that node. Otherwise, we show the probability that a node is picked in the optimal strategy is at most 13. We also find bounds on the minimum nonzero probability of a node in the optimal strategy.