A short proof of convexity of step-out–step-in sequencing games

The Step out–Step in sequencing game is a particular example of a game from the sequencing game framework of Curiel, Perderzoli, and Tijs, where coalitions of players in a queue may reorder themselves to improve the their overall cost, under some restrictions. Musegaas, Borm and Quant proved, in two papers, that a simple greedy algorithm correctly computes the valuation of a coalition, and that the game is convex. These proofs entail rather involved case analyses; in this note, we give short proofs of both results.