Rational secret sharing aims to design the realistic secret sharing scheme and to guarantee its fairness by introducing selfish players. However, due to the requirement that the dealer knows all kinds of players payoffs accurately, the stability of the reconstruction game is not taken into account. It might cause that, during the execution of the existing rational secret sharing schemes, the players who deviate from the prescribed protocol obtain the secret, whereas the others who follow the prescribed protocol faithfully cannot. To prevent this unfair solution, combined with the incentive compatibility principle of mechanism design, this paper makes use of the method that the number of reconstruction rounds is randomly chosen by the dealer, and devises the reconstruction order adjustment mechanism to effectively restrict the selfishness of rational players. Then, a rational secret sharing scheme with unknown reconstruction rounds is proposed. The analysis shows that the proposed scheme can realize the subgame perfect equilibrium of the reconstruction game and ensure its stability. That is, the fairness of the proposed scheme is achieved. In addition, comparied with several typical schemes in communication type, the number of reconstruction rounds and additional assumption, the results illustrate that the presented scheme is not only fair, but also has better practicality.