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Why do all filled bids receive the same rate, which is the rate of the lowest filled bid, during the US Treasury auction process?

The explanation is given in NY FED - The Treasury Auction Process: Objectives, Structure, and Recent Adaptations. The Treasury first adopted the multiple-price format when it initiated bill auctions ...
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First, assume $N-1$ bidders' values $v_i$ are independently drawn from some interval, let's say $[0,1]$, according to the cdf $F$. Now fix some $x$ in this interval. What's the probability $P(m\le x)$ ...