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We do live in 3 dimensions. You could build a road next to another road, above another road, below another road, or on a completely different path. ...
This system would produces extensive waste. Rather than being efficient, it would be quite inefficient. Therefore, the system would never work that way. The providers of the roads would lack the economy of scale in both production and funding that the public sector. To maintain profitability, companies would need to take measures to severely limit competition and the need to still generate a return to investors would probably lead companies to provide the least acceptable quality while charging as much as possible. Without some government control or a pool of captive customers, no private roads of any size would be built due to the massive capital costs and the uncertain return.
However the 'owners' of the roads would want to encourage as much driving as possible.