We analyse river sharing games in which a set of agents located along a river shares the available water. Using coalition theory, we find that the potential benefits of water trade may not be sufficient to make all agents in the river cooperate and acknowledge property rights as a prerequisite for trade. Specifically, a complete market for river water may not emerge if there are four or more agents along the river. Instead, a partial market may emerge where a subset of agents trades river water, with the possibility that other agents take some of the river water that passes their territory.