 pay. Let FI - -k(b) be the probability distribution for the I - K order statistic.7 The probability that the I - Korder statistic is less than or 7 The cumulative probability distribution of the (I - K)th order statistic is F1-k(b) = -k()f(b)i(1-F(b))-ifor all b in the unit interval. Pricing Network Services equal to b is equal to the probability that the number of consumers with willingness to pay bi less than or equal to b is greater than or equal to I - K. Thus, So the likelihood of a type-b consumer winning the Vickrey auction equals the reliability level chosen by a type-b consumer with reliability pricing. In a Vickrey auction, only winners make payments for service, and these pay- ments are necessarily made after demand is observed by the firm. With reliability pricing, all consumers make payments, and payments are made before demand is observed. Despite this difference, the expected payment of a type-b consumer in the Vickrey auction equals the payment for reliability of a type-b consumer. In the Vickrey auction, all winning consumers pay the bid of the highest-type losing consumer, = The probability distribution of the willingness to pay of the highest-type losing consumer is the order statistic for the (I - K)th highest type, because capacity equals K and there are I consumers. Thus, the expected payment in the Vickrey auction of a type-bi consumer is the expectation of the (I - K)-order statistic, contingent on the order statistic being less than that con- sumer's willingness to pay: Expected payment It follows that the expected payment in the Vickrey auction equals the actual payment with reliability pricing,8 This equivalence holds because Q*(b))=F1_k(b). 3. Priority Pricing of Capacity Another approach to pricing network capacity is priority pricing. 9 This widely used system assigns access priorities to consumers, with higher prices charged for higher priorities. Postal systems assign priorities for express mail deliveries as a means 8 Apply integration by parts to the expected payment in the Vick