Method 1 (preferred by many folks):
A. Set up the auction with starting price at the amount agreed, and notify your potential bidder.
B. Cancel all bids other than that of the person to whom you promised the item, and end the auction early after your transaction partner bids.
Advantage: No extra eBay fees
Disadvantage: You might have a lot of work cancelling bids.
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Method 2:
A. Set up auction as a Buy It Now (BIN), with a reserve at or just below BIN (Remember, your Reserve Fee will not be refunded)
B. Email prospective buyer immediately after listing the auction, giving buyer the auction number, and informing them the auction will be "limited" for only a short time.
C. The buyer then has a period of time in which the auction is not indexed, and can be bought without bids from competitors.
D. If the prospective buyer is a no-show during the "waiting for index time", the auction is soon visible to all eBayers, and can be purchased by anybody, including the prospective buyer.
Advantage: This gives the seller a chance to sell the item, even if the prospective buyer does not take advantage of the offer.
Disadvantages: The buyer might not see the auction in time, and somebody else might win. Your reserve fee is not refunded, even if the bid does or does not meet the reserve.
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Method 3:
Set up a fixed price sale with a VERY high price, and with a "best offer". Most folks will not bid because of the high price. However, if your prospective buyer makes a Best Offer at the price you previously agreed, you can accept that offer and complete the transaction.
This method is somewhat more convenient, in that you won't have to cancel any bids.
Disadvantage: If the prospective buyer is a no-show, you can lose the listing fee.
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Method 4:
Set up your auction, with the title and/or description/ body containing a phrase similar to "This is an auction specifically for [insert ID here]".
Advantage: No extra eBay fees
Disadvantages:
A. Strange folks may bid anyway, requiring you to cancel bids. If somebody else wins, you could file an "Unwelcome Bidder" complaint, but that process takes time.
B. If you do not include a description of the item(s), both you and the buyer will have an interesting time if a Significantly Not As Described (SNAD) issue arises.
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Let us know which works for you.
A great place for quick answers; no waiting, no voting,
"Do It Yourself": eBay Customer Support .