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Phylogenetic methods typically rely on an appropriate model of how dataevolved in order to infer an accurate phylogenetic tree. For molecular data,standard statistical methods have provided an effective strategy for extractingphylogenetic information from aligned sequence data when each site (character)is subject to a common process. However, for other types of data (e.g.morphological data), characters can be too ambiguous, homoplastic or saturatedto develop models that are effective at capturing the underlying process ofchange. To address this, we examine the properties of a classic but neglectedmethod for inferring splits in an underlying tree, namely, maximumcompatibility. By adopting a simple and extreme model in which each charactereither fits perfectly on some tree, or is entirely random (but it is not knownwhich class any character belongs to) we are able to derive exact and explicitformulae regarding the performance of maximum compatibility. We show that thismethod is able to identify a set of non-trivial homoplasy-free characters, whenthe number $n$ of taxa is large, even when the number of random characters islarge. By contrast, we show that a method that makes more uniform use of allthe data --- maximum parsimony --- can provably estimate trees in which {\emnone} of the original homoplasy-free characters support splits.

Circumstances in which parsimony but not compatibility will be provably misleading