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We study two-player games with counters, where the objective of the firstplayer is that the counter values remain bounded. We investigate the existenceof a trade-off between the size of the memory and the bound achieved on thecounters, which has been conjectured by Colcombet and Loeding.  We show that unfortunately this conjecture does not hold: there is notrade-off between bounds and memory, even for finite arenas. On the positiveside, we prove the existence of a trade-off for the special case of thin treearenas. This allows to extend the theory of regular cost functions over thintrees, and obtain as a corollary the decidability of cost monadic second-orderlogic over thin trees.

Trading Bounds for Memory in Games with Counters