Rings are quite common in both road traffic networks as well as in telecommunication networks. In the road traffic context, we often find rings surrounding towns and cities. Traffic over these rings is either bidirectional or we may find two rings that surround the town carrying traffic in opposite directions (clockwise and anti-clockwise). Telecommunication networks based on rings have been often used both as local area networks as well as in metropolitan area networks and here too we find both bidirectional networks as well as networks consisting of two rings carrying traffic in opposite directions. Each decision maker (e.g. the drivers, in the case of road traffic, and perhaps Internet access providers, in the case of telecommunication) is faced with a simple routing decision: whether to go clockwise or anti-clockwise. Assuming a simple source-destination demand matrix, we analyze this problem as a non-cooperative game and derive several interesting characteristics of the equilibria.