Incentivizing for early arrival (I4EA) mechanisms are designed to work in online cooperative game scenarios to distribute value in a way that refrain people from delaying their joining. While Reward First Critical player (RFC) mechanism contributes significantly to I4EA mechanisms, there’s still major limitation and problems within RFC and Greedy Monotone (GM) decomposition. In this paper, we introduce two decomposition methods utilizing polyhedral theories to address and solve existing problems and extend RFC to a wider applicable range. First, a monotonicity-preserving decomposition breaks monotonic games into simple monotonic components without introducing artificial excess value, thereby overcoming the limitations of GM decomposition that GM may not produce optimal solution. Second, a top-down linear decomposition expresses any game satisfying given linear constraints as a non-negative combination of simple basis games with the same constraints. It reduces incentive verification to a finite set of canonical instances, enabling verifying if a type of game incentivizes for early arrival under RFC. Both methods produce valid RFC decompositions in polynomial time, characterize their correctness and computational complexity, and show their applicability to online cooperative scenarios.