ions of the good by consumers (b1, , b2 b1). This is represented in Figure 3.9. Clearly, only K consumers can obtain a unit of the good, SO some form of rationing is necessary. Because the firm does not know the realizations of consumer valuations, the aggregate demand function is not known to the firm. Many different allocation mechanisms are available to solve the problem of selling multiple units of scarce capacity. A well-known mechanism is the Vickrey auction (1961). Each buyer submits a bid p1. The winning bidders must pay the highest losing bid. Suppose that the consumer bids are in increasing order Then the highest losing bid is p1-K. The buyer's (weakly) dominant strategy is bid his true valuation, pi*=bi. To see why, note that a buyer will never bid above his valuation. If bidding above the valuation resulted in obtaining a unit of the service that the buyer would not have received when bidding at his valuation, the buyer necessarily would pay more than his valuation for the unit, resulting in a loss of consumer surplus. The buyer will never bid below his valuation, for to do SO would potentially result in the loss Networks in Telecommunications of a unit that could have been obtained at a price below the consumer's valuation. This would mean forgoing potential consumer surplus. Accordingly, the market clears at the highest losing bid, pl-K*=bl-K., = This is the appropriate market-clearing price, SO that the Vickrey auction results in an efficient allocation of the network's capacity. The K highest bidders obtain units of capacity, and because these are the buyers with the highest valuations, it follows that capacity is allocated to the highest-value consumers. The equilibrium price is shown in Figure 3.9. The K consumers with values bi>bi-k each obtain a unit of capacity and pay bI-K. 2. Reliability Pricing of Capacity Reliability of service is represented by Q, which is the likelihood that a consumer receives a unit of service. The network firm offers consumers a nonlinear payment schedule based on re