Maybe we have competing cities. In an anarchist society, there is no reason why there could not be a city that is centrally planned and residents enter into voluntary contracts to pay monthly fees to a central planner to plan these kinds of things. As long as the planner does not use some kind of force to gather fees (practically taxes) on people who have not entered into these voluntary contracts it would be perfectly compatible.
That being said, in a non planned city, you might not have competing roads, but various forms of transportation would compete with each other. You might have subway systems competing with the roads competing with dedicated bus routes competing with various raised monorail systems competing with things we might not have ever thought of.