Tournament games and positive tournaments

Given a tournament T , the tournament game on T is as follows: Two players independently pick a node of T. 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. We show that the optimal mixed strategy for this game is unique and uses an odd number of nodes. A tournament is positive if the optimal strategy for its tournament game uses all of its nodes. The uniqueness of the optimal strategy then gives a new tournament decomposition: any tournament can be uniquely partitioned into positive subtournaments P 1 , P 2 , ⃛, P k , so P i “beats” P j for all 1 ≤ i > j ≤ k. We count the number of n node positive tournaments and list them for n ≤ 7.