We formalize, under the name of games of addition, the strategic interaction between agents that can play non-simultaneously by adding payoff relevant actions to those that any other players or themselves have already taken previously, but may also agree unanimously to stop adding them and collect the payoffs associated with the truncated sequence of moves. Our formalization differs from that of extensive form games in that the order of the agents' moves is not predetermined but emerges endogenously when applying an adapted version of a solution concept proposed by Dutta, Jackson and Le Breton (2004). We provide results regarding the properties of solutions to games of addition, and we also compare their corresponding equilibria with those we would obtain if using extensive form games and subgame perfection as alternative tools of analysis.