Rational choice when preferences are not required to be transitive and complete has been discussed in the literature for years. In this article transitivity and completeness of the preference relation is also not assumed. It will be shown that nevertheless the existence of a competitive equilibrium can be proven when those properties are replaced by a domination property which allows that there could be cicles among those alternatives which are of less importance for the individual and which he

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... or she would never choose if better ones are available. Moreover, one can show that the compensated demand function is continuous under very weak conditions, and because of this, Shephard's lemma follows without assuming transitivity and completeness of the underlying preferences.