Many problems in multiagent decision making can be addressed using tournament solutions, i.e., functions that associate with each complete and asymmetric relation on a set of alternatives a non-empty subset of the alternatives. For a given tournament solution S, Schwartz calls a set of alternatives S-retentive if it satisfies a natural stability criterion with respect to S. He then recursively ...

In this study a new class of tournament designs is introduced. In particular, each game of the tournament involves several (two or more) teams competing against one another. The tournament is also required to satisfy certain balance conditions that are imposed on each pair of players. These balance conditions are related to both the total number of players on each team and the number of teams i...

A tournament can be viewed as a majority preference relation without ties on a set of alternatives. In this way, voting rules based on majority comparisons are equivalent to methods of choosing from a tournament. We consider the size of several of these tournament solutions in tournaments with a large but finite number of alternatives. Our main result is that with probability approaching one, t...

Let H be a tournament, and let ≥ 0 be a real number. We call an “Erdős-Hajnal coefficient” for H if there exists c > 0 such that in every tournament G with |V (G)| > 1 not containing H as a subtournament, there is a transitive subset of cardinality at least c|V (G)| . The Erdős-Hajnal conjecture asserts, in one form, that every tournament H has a positive Erdős-Hajnal coefficient. This remains ...

We give a short constructive proof of a theorem of Fisher: every tournament contains a vertex whose second outneighborhood is as large as its ®rst outneighborhood. Moreover, we exhibit two such vertices provided that the tournament has no dominated vertex. The proof makes use of median orders. A second application of median orders is that every tournament of order 2nÿ 2 contains every arboresce...

Bracketology, the art of successfully picking all the winners in the National Collegiate Athletic Association’s (NCAA) annual men’s Division I college basketball championship tournament, has become a favorite national activity. In spite of the challenges and uncertainty faced in this endeavor, patterns exist in how the seeds appear in each round, particularly the later rounds. This paper statis...

A king u in a tournament is a player who beats any other player v directly or indirectly. That is, either u → v (u beats v) or there exists a third player w such that u→ w and w → v. A sorted sequence of kings in a tournament of n players is a sequence of players, S = (u1, u2, . . . , un), such that ui → ui+1 and ui in a king in sub-tournament Tui = {ui ,ui+1, . . . , un} for i = 1,2, . . . , n...

We study biased orientation games, in which the board is the complete graph Kn, and OMaker (oriented maker) and OBreaker (oriented breaker) take turns in directing previously undirected edges of Kn. At the end of the game, the obtained graph is a tournament. OMaker wins if the tournament has some property P and OBreaker wins otherwise. We provide bounds on the bias that is required for OMaker’s...

We examine the size s(n) of a smallest tournament having the arcs of an acyclic tournament on n vertices as a minimum feedback arc set. Using an integer linear programming formulation we obtain lower bounds s(n) ≥ 3n − 2 − blog2 nc or s(n) ≥ 3n − 1 − blog2 nc, depending on the binary expansion of n. When n = 2k − 2t we show that the bounds are tight with s(n) = 3n−2−blog2 nc. One view of this p...

Tournament solutions, i.e., functions that associate with each complete and asymmetric relation on a set of alternatives a non-empty subset of the alternatives, play an important role within social choice theory and the mathematical social sciences at large. Laffond et al. have shown that various tournament solutions satisfy composition-consistency, a structural invariance property based on the...