first price auction the equilibrium bid of a buyer with valuation is . That is, the payment of the winner in the sealed first-price auction is equal to the expected revenue in the sealed second-price auction. Proof of revenue equivalence Suppose that buyer 2 bids according to the strategy , where is the buyer's bid for a valuation . We need to show that buyer 1's best response is to use the same strategy. Note first that if buyer 2 uses the strategy , then buyer 2's maximum bid is and so buyer 1 wins with probability 1 with any bid of 1/2 or more. Consider then a bid on the interval . Let buyer 2's value be . Then buyer 1 wins if , that is, if . Under Vickrey's assumption of uniformly distributed values, the win probability is . Buyer 1's expected payoff is therefore Note that takes on its maximum at . Use in network routing In network routing, VCG mechanisms are a family of payment schemes based on the added value concept. The basic idea of a VCG mechanism in network routing is to pay the owner of each link or node (depending on the network model) that is part of the solution, its declared cost plus its added value. In many routing problems, this mechanism is not only strategyproof, but also the minimum among all strategyproof mechanisms. In the case of network flows, unicast or multicast, a minimum cost flow (MCF) in graph G is calculated based on the declared costs dk of each of the links and payment is calculated as follows: Each link (or node) in the MCF is paid , where MCF(G) indicates the cost of the minimum cost flow in graph G and G − ek indicates graph G without the link ek. Links not in the MCF are paid nothing. This routing problem is one of the cases for which VCG is strategyproof and minimum. In 2004, it was shown that the expected VCG overpayment of an Erdős–Rényi random graph with n nodes and edge probability p, approaches as n, approaches , for . Prior to this result, it was known that VCG overpayment in G(n, p) is and with high probability given Generalizations The most obvious generalization to multiple or divisible goods is to have all winning bidders pay the amount of the highest non-winning bid. This is known