Two classes of one machine sequencing situations are considered in which each job corresponds to exactly one player but a player may have more than one job to be processed, so called RP(repeated player) sequencing situations.In max-RP sequencing situations it is assumed that each player's cost function is linear with respect to the maximum completion time of his jobs, whereas in min-RP sequencing situations the cost functions are linear with respect to the minimum completion times.For both classes, following explicit procedures to go from the initial processing order to an optimal order for the coalition of all players, equal gain splitting rules are defined.It is shown that these rules lead to core elements of the associated RP sequencing games.Moreover, it is seen that min-RP sequencing games are convex.