In the problem of rational secret sharing, each player, who prefers to be the only one to learn the secret, acts rationally rather than merely honestly or maliciously in the traditional cryptographic settings. The core issue of rational secret sharing is to construct a mechanism so that all rational participants have incentive to provide their shares in the reconstruction phase. However, lack of enough incentive and how to achieve fairness in asynchronous model are always the hot potatoes for obtaining a deterministic protocol in this field. In this paper, we devote our attention to solving the above two difficulties in asynchronous model and succeed in obtaining a satisfactory result, 1/p-rational secret sharing protocol, in 2-out-of-2 case. Firstly, we apply the repeated interaction method to provide motivation so as to attract the players to participant in the collaboration and make it possible for a deterministic protocol. Secondly, inspired by 1/p-security of partial fairness, we propose a new concept, 1/p-advantage, to reduce the unavoidable probability of unfairness in asynchronous model to a satisfactory level. Besides, in order to perfect our protocol, we bring in the augmented perfect coin-tossing to determine the sending-order at the start of every round. Compared with previous ones, our protocol is applicable to the asynchronous model as well as the traditional synchronous model, which makes our protocol more practical. Furthermore, the new concept, 1/p-advantage, gives us a new perspective on rational secret sharing and makes our solution fresh.
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