On tournament games and positive tournaments - de Armas, Roderico A. (Animo Repository, 2003)
This thesis deals with tournament games and positive tournaments. Given an n-node tournament T, a tournament game on T is as follows: Two players independently pick a node of T. If both picked the same node, the game is tied. Otherwise, the player whose node is at the tail of the arc connecting the two node wins. The optimal mixed strategy for a tournament 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 T can be uniquely partitioned into subtournaments P1, P2, ..., Pk, such that Pi beats Pj for all 1 less that or equal to i < : less than or equal to k. A formula for counting the number of nonisomorphic positive tournaments was derived. This formula was used to count the number of nonisomorphic positive tournaments for n = 1, 3, 5 and 7 and these were enumerated. All the theorems, corollaries and lemmas in this paper are results of the study conducted by David C. Fisher and Jennifer Ryan Tournament Games and Positive Tournaments published in the Journal of Graph Theory in 1995. The researcher provided the proofs of lemmas and corollaries. Proofs of theorems were simplified and illustrated.