For the multi-attribute group decision-making problems, where the information of the attribute weights and the expert weights is completely unknown and the preference information is in the form of hesitant 2-tuple linguistic, a multi-attribute group decision-making method based on the prospect theory and the grey relation analysis is proposed. Firstly, the weights of the experts are determined by the matrix vec operator and the grey relation analysis, and the weights of attributes are calculated by the maximizing deviation method. Subsequently, a comparison method of the hesitant 2-tuple linguistic elements is given, and the positive and negative ideal solutions based on that are determined and used as the decision reference point. Then the hesitant 2-tuple linguistic prospect value function according to the prospect theory and the grey relational coefficient is acquired, and then the ratio of the gains to losses of the alternatives is obtained, and the alternatives are ranked accordingly. Finally, the proposed method is applied to a numerical example of investment decision, and the results show the rationality and effectiveness of the method.