Showing papers on "Tournament published in 1989"

Tournaments, Piece Rates, and the Shape of the Payoff Function

Abstract: In a recent article, Bull, Schotter, and Weigelt (1987) present an experimental comparison of behavior under piece rates and tournaments, drawing from the theoretical work of Lazear and Rosen (1981) and Nalebuff and Stiglitz (1983). Bull et al. successfully confirm systematic behavior consistent with each incentive scheme. In addition, they report greater effort variance under tournaments than under piece rates, implying that firms using tournaments may encounter costly and unexpected inventory buildups, shortages, and production bottlenecks. The authors attribute two-thirds of this additional tournament variance "to the fact that a tournament, unlike the piece rate, is a game and so requires strategic, as opposed to simply maximizing, behavior" (p. 3). They attribute the remaining one-third to increased "computational difficulty" involved in the tournament. We take issue with the methodology they use to make this attribution and argue that they overstate the variance due to computational difficulty. To analyze computational difficulty under the two schemes, Bull et al. contrast a piece rate experiment with a nongame tournament. In both experiments, subjects select a decision number (which proxies effort) between zero and 100. Under the nongame tournament, subjects know that the opponent consistently picks the decision number 37. Thus subjects face no strategic opposition and can theoretically compute expected earnings for each decision number and select the optimum. Nonetheless, nongame tournament subjects typically choose decision numbers farther away from those yielding optimal expected earnings. As shown in the first two lines of table 1 (taken from Bull et al., p. 16), the twelfth-round variance in the nongame tournament is 275.3, while the piece rate yields a variance of 33.66. Bull et al. assume that this difference reflects the difficulty of computing expected values for the nongame tournament. That is, since tour-

On the Numbers of Some Subtournaments of a Bipartite Tournament

Abstract: A bipartite tournament is obtained when the edges of a complete bipartite graph are oriented. Thus an m x n bipartite tournament T is a directed graph on disjoint vertex sets X = {x,, x2, ..., x,) and Y = {y,, y 2 , ..., y,}, where, for each pair (xi, y,), either xi dominates y, or y j dominates xi. Let the scores (or outdegrees) of xi and yj be denoted by ai and b,, respectively. We call A = [a , , a , , . , . , a,J and B = [b,, b 2 , . . . , bJ the score lists of T . The dominance matrix M of T is an rn x n matrix (mif) of 0s and Is, in which mfj = 1 if x i -+ y j , and mij = 0 if y , + xi. In [5 ] , M. G. Kendall and Babington Smith studied the maximum number of (directed) 3-cycles in ordinary tournaments (i.e., oriented complete graphs). In fact, there are only two different possible types of subtournaments of order 3 in an ordinary tournament. These are the 3-cycle (or the cyclic triple), and the transitive triple. The numbers of these in an ordinary tournament are determined by its score list. The smallest possible cycle in a bipartite tournament T is a 4-cycle. This, when viewed as a 2 x 2 subtournament, corresponds to the 2 x 2 submatrices of M , the [: ;jar[: 3 dominance matrix of T, of the type

Tournament Score Sequences

Abstract: By a tournament of order n > 0 we mean a loopless directed graph T = ( V, A ), where V = { 1,2, . . . , n ] such that for every pair of distinct vertices u, u f V, either uv E A or ZJU E A but not both. Thus, there are exactly ( z ) arcs in T. Apath of length k 1 2 0 from u to 21 is a sequence ( u , . . . , u ) of k distinct vertices such that ij E A for all vertices i and j in it, where j immediately succeeds i . If ( u , . . . , z j ) is a path of length k 1 2 2 and uu E A, then the sequence ( u , . . . , u, u ) is called a circuit of length k . The score of a vertex ZI is defined to be the outdegree of ZJ, that is, the number of arcs with initial vertex at U. This is denoted by the symbol S ( Z J ) . For convenience, we shaIl always assume that s(1) 5 s(2) r . . . 5 s (n ) . The sequence ( s ( i ) ) = (s(l), s(2), s(3), . . . , s ( n ) ) is called the score sequence of T. In 1953, Landau [4] proved that in every tournament T, there exists a vertex u such that for every vertex Z J different from u, there exists a path of length 1 or 2 from u to Z J . The following is an extension of this result.