We study the complexity of solving decision trees with a Skew-Symmetric Bilinear (SSB) utility function. The SSB model is an extension of Expected Utility (EU) with enhanced descriptive possibilities. Unlike EU, the opti-mality principle does not hold for SSB, which makes its optimization trickier. We show that determining an SSB optimal plan is NP-hard if one only considers deterministic plans while it is polynomial time if one allows randomized plans. With the Weighted EU model (a special case of SSB), the problem becomes polynomial in both settings. Our numerical tests show the operationality of the methods.