  We study two-player games with counters, where the objective of the first
player is that the counter values remain bounded. We investigate the existence
of a trade-off between the size of the memory and the bound achieved on the
counters, which has been conjectured by Colcombet and Loeding.
  We show that unfortunately this conjecture does not hold: there is no
trade-off between bounds and memory, even for finite arenas. On the positive
side, we prove the existence of a trade-off for the special case of thin tree
arenas. This allows to extend the theory of regular cost functions over thin
trees, and obtain as a corollary the decidability of cost monadic second-order
logic over thin trees.
