This paper investigates regulation by auctions of private supply of congestible infrastructures in two networks settings: 1) two serial facilities, where the consumer has to use both in order to consume; and 2) two parallel facilities that are imperfect substitutes. There are four market structures: a monopoly and 3 duopolies that differ in how firms interact. The effects of an auction depend on what the bidders compete. With a bid auction, the bidders compete on how much money they transfer to the government. This auction leads to the same outcome as the unregulated game (for a given market structure), since this gives the maximum profit to transfer. An auction on the capacity of a facility leads to an even lower welfare than no regulation, because firms set very high capacities and usage fees. Conversely, an auction on generalised price or number of users leads to the first-best outcome. Moreover, these two auctions are robust: they attain the first-best regardless of whether the facilities are auctioned off to a single firm or to two firms, and for all market and network structures. On the contrary, the performances (relative to the first-best) of the bid and capacity auctions strongly depend on these considerations.