When it is desired to compare t treatments in a paired-comparison experiment the simplest balanced design requires (t - 1) units of each treatment. Each replication, comprising jt(t - 1) comparisons, is analogous to a round robin tournament involving t players, with each comparison corresponding to a game between two contestants. This analogy was first pointed out by Kendall (1955). If we are primarily concerned with picking the best treatment it may be practicable to use a technique requiring less than (t - 1) units of each treatment per replication. The knock-out tournament (or cup-tie procedure) has been suggested for this purpose by Maurice (1958) and also by David (1959), Maurice compares the balanced design or round robin tournament with the cup-tie procedure for the case of four populations with means in the most unfavourable configuration (one population has a mean larger by 8 than the common mean of the other tlhree populations). She bases the comparison on the amount of replication (and hence the sampling cost) required to ensure detection of the difference 8 with probability at least P, and concludes that the cup-tie procedure is the more economical. In effecting comparisons with the balanced design she uses tables provided by Bechhofer (1954). David investigates some properties of knock-out and round robin tournaments and obtains an expression for the probability with which the best player wins a knock-out tournament, certain assumptions having been made about the strengths of the players. In this paper six tournament types are investigated for their effectiveness in selecting the best one of four players. The types, together with abbreviations used for convenience in