After the group's initial visit to Treno, return to the King's Mansion & the auction. Some Key Items & some extremely useful equipment are auctioned off here, & this auction is the key to receiving a very powerful weapon!  Upon entering, the auctioneer lists off the items up for bid. To purchase an item, approach the stage until you see a "?" over Zidane's head. Nobles in the audience will bid, and they have a lot of Gil to throw away. Whenever you want to bid, press the X button & a menu appears, which  automatically inserts a bid 100 Gil higher than the last, but you can raise or lower your bid as you see fit.
Items Highest Bid
Mini Cid 3400
Dark Matter 15400
Fairy Earrings 9500
Promist Ring 23500
Thieves Gloves 35600
Magician Robe 8600
Doga's Artifact 8800
Une's Mirror 13800
Rat Tail 13700
Feather Boots 18600
Ribbon 90000
Madain's Ring 10200
Reflect Ring 14400
Griffin's Heart 7400
Pearl Rouge 24500
Magical Fingertip ???
Rare and Key Items that you can bid on at the auction include the following:
Mini-Cid, Griffin's Heart, Doga's Artifact, Une's Mirror, the Magical Fingertip, and the Rat Tail. These items have some strange uses & can be resold...
The trick to the auction is to avoid getting into a bidding war. If you attempt to outbid the nobles at every turn, they will toss out fake bids just because they know you'll outbid them. The smartest thing to do is let the nobles bid amongst themselves for a while. Then, when it seems like the auctioneer is ready to sell the item, throw in a last-minute bid just slightly higher than the last. This should get you the item you want at a reasonable price every time. If you allow the nobles to jack up the prices, then you lose out on profiting from the resale value. Check the resale value of these items in the next section before paying too much for an item. If you leave the auction before a Key Item is sold, then it will be available the next time you enter the auction.
Street Value of Auction Items:
After purchasing certain Key Items from the lavish Treno Auction, talk to people in the area around the Café Carta. Some nobles will see that you have items they desire, and will offer Gil for them. But don't settle for the first offer! Turn down the first amount to get them to increase their offer.
Two notes here: You must sell the Griffin's Heart to the Adventurer before that person becomes interested in the Rat Tail. Also, the Scholar in Treno's Synthesis Shop will not top his offer of 10000 Gil.
Expected Resale Prices in Treno Streets: Item  Noble  1st offer  2nd offer
Une's Mirror  Nobleman by Café  12000 Gil  15000 Gil
Doga's Artifact  Scholar in Synth shop  10000 Gil  N/A
Griffin's Heart  Adventurer by Café  8000 Gil  10000 Gil
Rat Tail  Adventurer by Cafe  20000 Gil  25000 Gil
Important items :
Dark matter :
Can be used in the battle (do 9999 damage to enemies),and can be used to learn Summon Odin.
Doga's Artifact & Une's Mirror :
Go to the Black Mage Village Inn & check the gramophone. From there, certain arranged version of FF3 music will be played. This event is only available after Terra is destroyed.
Magical Fingertip:
Give it to the old man in Daguerrero Library, he will reward you with a weapon sword for Steiner,Excalibur. Notice that Magical Fingertip will not appear until you have bought Une's Mirror, Griffon's Heart, Rat Tail, Doga's Artifact and sold them to people in Treno.
Auction House