Sequential auctions for identical items with unit-demand, private-value buyers are common and often occur periodically without end, as new bidders replace departing ones. We model bidder uncertainty by introducing a probability that a bidder must exit the auction in each period. Treating the sequential auction as a Markov process, we demonstrate the existence of a unique steady state. In the absence of uncertainty, the steady state resembles a posted-price mechanism: bidders with values above a threshold almost surely win items by repeatedly bidding the threshold price, while those below the threshold almost surely do not. The equilibrium price corresponds to the threshold value that balances supply (bidders with values above the threshold) and demand (auction winners). When uncertainty is introduced, the threshold value persists but becomes less precise, growing "fuzzier" as uncertainty increases. This uncertainty benefits low-value bidders, those below the threshold, by giving them a significant chance of winning. Surprisingly, high-value bidders also benefit from uncertainty, up to a certain value limit, as it lowers equilibrium bids and increases their expected utility. On the other hand, this bidder uncertainty often reduces the auctioneer's utility.

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