Quadruple systems containing AG(3,2)
A quadruple system of order v, denoted QS(v) is an ordered pair (X, Q) where X is a set of cardinality v and Q is a set of 4-subsets of X called blocks, with the property that every 3-subset of X is contained in a unique block. The points and planes of the affine geometry AG(3, 2) form a QS(8). We prove that a QS(v) containing a proper subsystem isomorphic to AG(3, 2) exists if and only if v>=16 and v=2 or 4 (mod 6).
Abstract We study the learning behavior of a population of buyers and a population of sellers whose members are repeatedly randomly matched to engage in a sealed bid double auction. The agents are assumed to be boundedly rational and choose their strategies by imitating successful behavior and adding innovations triggered by random errors or communication with other agents. This process is modelled by a two-population genetic algorithm. A general characterization of the equilibria in mixed population distributions is given and it is shown analytically that only one price equilibria are attractive for the GA dynamics. Simulation results confirm these findings and imply that in cases with random initialization with high probability the gain of trade is equally split between buyers and sellers.