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...

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...

Given a tournament with an acyclic tournament as a feedback arc set we give necessary and sufficient conditions for this feedback arc set to have minimum size.

This paper analyzes the performance of a genetic algorithm that utilizes tournament selection, one-point crossover, and a reordering operator. A model is proposed to describe the combined effect of the reordering operator and tournament selection, and the numerical solutions are presented as well. Pairwise, s-ary, and probabilistic tournament selection are all included in the proposed model. It...

We describe the participation of our program to the 2007 General Game Playing Tournament. It finished at the third place of the qualifying tournament. After an informal description of General Game Playing and of its Game Description Language, we present in details the structure of our player as it participated to the qualifying phase of the tournament. We then present the context of this phase ...