Private toll roads are now seriously considered as an alternative to public (free-access) road infrastructure. Nevertheless, complete private provision without governmental control is only rarely considered. A main consideration against private roads would be that operators would be primarily interested in maximizing profits, which - given the market power they will have - will typically not lead to welfare-maximizing tolls and capacities. An important question is whether these discrepancies can be mitigated by a proper design of auctions for concessions of private roads. This paper therefore analyses capacity choice and toll setting by private investors in a competitive bidding framework organized by the government. We develop a two-link network simulation model with an untolled alternative to determine relative efficiency effects, and analyze rules for the government to organize the bidding process such that a more desired (welfare optimal) outcome is achieved. Our results show that, depending on the design of the auction, its outcomes may vary strongly, and may approach the maximum possible (second-best) welfare gains.