The unit normal loss integral (UNLI) frequently emerges as a solution for the computation of various Value of Information (VoI) metrics. However, one limitation of the UNLI has been that its closed-form solution is available for only one dimension, and thus can be used for comparisons involving only two strategies (where it is applied to the scalar incremental net benefit). We derived a closed-form solution for the two-dimensional UNLI by the integration by parts technique, enabling closed-form VoI calculations for three strategies. We verified the accuracy of this method via simulation studies. A case study based on a three-arm clinical trial was used as an example. VoI methods based on the closed-form solutions for the UNLI can now be extended to three-decision comparisons (e.g., VoI analysis of a risk prediction model), taking a fraction of a second to compute and not being subject to Monte Carlo error. An R implementation of this method is provided as part of the predtools package https://github.com/resplab/predtools/.