  We consider models of growing multi-level systems wherein the growth process
is driven by rules of tournament selection. A system can be conceived as an
evolving tree with a new node being attached to a contestant node at the best
hierarchy level (a level nearest to the tree root). The proposed evolution
reflects limited information on system properties available to new nodes. It
can also be expressed in terms of population dynamics. Two models are
considered: a constant tournament (CT) model wherein the number of tournament
participants is constant throughout system evolution, and a proportional
tournament (PT) model where this number increases proportionally to the growing
size of the system itself. The results of analytical calculations based on a
rate equation fit well to numerical simulations for both models. In the CT
model all hierarchy levels emerge but the birth time of a consecutive hierarchy
level increases exponentially or faster for each new level. The number of nodes
at the first hierarchy level grows logarithmically in time, while the size of
the last, "worst" hierarchy level oscillates quasi log-periodically. In the PT
model the occupations of the first two hierarchy levels increase linearly but
worse hierarchy levels either do not emerge at all or appear only by chance in
early stage of system evolution to further stop growing at all. The results
allow to conclude that information available to each new node in tournament
dynamics restrains the emergence of new hierarchy levels and that it is the
absolute amount of information, not relative, which governs such behavior.
