The present study concerns itself with the analysis of cooperative games in which players are required to join the game in a sequential manner. In such games, the value generated by those players who have joined at any given point must be irrevocably divided among the other players. Two criteria are proposed for the value division mechanism: firstly, that players should have incentives to join as early as possible, and secondly, that the division should be considered fair. In order to satisfy this requirement, it is necessary that each player’s expected share in the mechanism should be equal to her Shapley value, assuming that the players’ arrival order is uniformly random. The primary technical contribution of this study is a comprehensive characterisation of 0-1 value games for which suitable mechanisms are available. We demonstrate that the Rewarding First Critical Player (RFC) mechanism is complete, in that a 0-1 value function admits a mechanism with the aforementioned properties if and only if RFC satisfies them. Furthermore, we provide an analytical characterisation of all such value functions.