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Current instrumental variable methodology focuses mainly on estimating causal effects for a dichotomous or an ordinal treatment variable. Situations with more than two unordered treatments are less explored. The challenge is that assumptions needed to derive point-estimators become increasingly stronger with the number of relevant treatment alternatives. In this article, we aim at deriving causal point-estimators for head-to-head comparisons of the effect of multiple relevant treatments or interventions. We will achieve this with a set of plausible and well-defined rationality assumptions while only considering ordinal instruments. We demonstrate that our methodology provides asymptotically unbiased estimators in the presence of unobserved confounding effects in a simulation study. We then apply the method to compare the effectiveness of five anti-inflammatory drugs in the treatment of rheumatoid arthritis. For this, we use a clinical data set from an observational study in Norway, where price is the primary determinant of the preferred drug and can therefore be considered as an instrument. The developed methodology provides an important addition to the toolbox for causal inference when comparing more than two interventions influenced by an instrumental variable., Competing Interests: Declaration of conflicting interests The author(s) declared the following potential conflicts of interest with respect to the research, authorship, and/or publication of this article: ICO developed the concept and secured the funding, AK developed the methodology and did all analyses. AK and ICO wrote and reviewed the manuscript.