We prove a separation between offline and online algorithms for finger-based tournament heaps undergoing key modifications. These heaps are implemented by binary trees with keys stored on leaves, and intermediate nodes tracking the min of their respective subtrees. They represent a natural starting point for studying self-adjusting heaps due to the need to access the root-to-leaf path upon modifications. We combine previous studies on the competitive ratios of unordered binary search trees by

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... redman WADS2011] and on order-by-next request by [Martínez-Roura TCS2000] and [Munro ESA2000] to show that for any number of fingers, tournament heaps cannot handle a sequence of modify-key operations with competitive ratio in o(√(logn)). Critical to this analysis is the characterization of the modifications that a heap can undergo upon an access. There are (Θ(n logn)) valid heaps on n keys, but only (Θ(n)) binary search trees. We parameterize the modification power through the well-studied concept of fingers: additional pointers the data structure can manipulate arbitrarily. Here we demonstrate that fingers can be significantly more powerful than servers moving on a static tree by showing that access to k fingers allow an offline algorithm to handle any access sequence with amortized cost O(log_k(n) + 2^^*n).