Win-loss Sequences for Multiple Round Tournaments

Each of n teams numbered 1, 2, · · · , n play each of the other n − 1 teams exactly t times. Thus, each team plays t (n− 1) games, and the total number of games is tc2 = tn(n−1) 2 . Each game {a, b} produces a win for one team and a loss for the other team. Define ai, i = 1, 2, · · · , n, to be the win records for the n teams. That is, for each i = 1, 2, · · · , n, team i wins a total of ai games where 0 ≤ ai ≤ t (n− 1). Of course, n ∑ i=1 ai = tn(n−1) 2 . Suppose ai, i = 1, 2, · · · , n, are arbitrarily specified win records for the teams 1, 2, · · · , n subject only to the two conditions (1) 0 ≤ ai ≤ t (n− 1) and (2) n ∑ i=1 ai = tn(n−1) 2 . In this paper, we prove necessary and sufficient conditions that ai, i = 1, 2, · · · , n, must satisfy so that ai, i = 1, 2, · · · , n, is realizable. In [1] we solved this problem for the special case t = 1, and in this paper we solve the general case by reducing it to this special case t = 1. We have not come even remotely close to solving the general case by modifying the proof given in [1]. 1. Finding necessary conditions on ai, i = 1, 2, · · · , n. Suppose for 1 ≤ k ≤ n we choose any combination {n1, n2, · · · , nk} of k teams from the collection of n teams. Now these k teams play t ·C 2 = tk(k−1) 2 games among themselves. Therefore, the total number of wins among themselves for these k teams equals tk(k−1) 2 . Also, each of the k teams plays each of the n − k remaining teams t times for a total of tk (n− k) games. Therefore, (3′) is a necessary condition. (3′). For each 1 ≤ k ≤ n, any combination of k teams {n1, n2, · · · , nk}must satisfy k ∑ i=1 ani ≤ tk(k−1) 2 +tk (n− k) = tk 2 (2n− k − 1) . If we agree to write t (n− 1) ≥ a1 ≥ a2 ≥ · · · ≥ an ≥ 0, then the above necessary condition (3′) is equivalent to the following (3). (3). ∀k ∈ {1, 2, · · · , n} , k ∑ i=1 ai ≤ tk 2 (2n− k − 1) .