According to the Islamic Shari`ah, the basic principle for transactions is permissibility unless there is a specific injunction indicating otherwise. Only when the general Islamic parameters that secure justice and fairness are infringed upon, a particular transaction turns to be impermissible.

Dr.MonzerKahf, Scholar in Islamic Economics & Financial Expert, states the following: Auctioning off items is permissible in Islam. The Prophet (peace and blessings be upon him) did that in the famous authentic Hadith of the man who came begging and the Prophet (peace and blessings be upon him) asked him to bring whatever he had at home. The man brought a little mat and pitcher. The Prophet (peace and blessings be upon him) auctioned them off and sold them to the highest bidder and instructed that man to buy food for his family and an axe to get firewood from the mountain and sell it as a way to earn a living. He (the Prophet, peace and blessings be upon him) said: This is better than begging.
Moreover, auction is permissible for personal matters and for charitable purposes too. It is also permissible to bid higher prices than the bids made by others in auction, as an exception from the general rule that prohibits bidding against others, as long as it is announced as an auction sale, not a private negotiation or musawamah sale.

Shedding more light on the issue, we’d like to cite for you the following fatwa issued by the prominent Saudi scholar and lecturer Sheikh M.SalehAl-Munajid, who adds:

Islam permits selling by auctions and does not forbid it, according to the most correct and well-known opinion of the scholars. This is based on the following evidences:
1. Jabir said: A man had decided that a slave of his would be manumitted after his death, but later on he was in need of money, so the Prophet (peace and blessings be upon him) took the slave and said, “Who will buy this slave from me?”Nu’aymibnAbd-Allah bought him, and he (the Prophet (peace and blessings be upon him) handed the slave over to him. (Narrated by al-Bukhari, 2034; Muslim, 997). Al-Bukhari included this hadeethin a chapter titled Baabbay’ al-Muzaayadah(Chapter: selling by auction).”
Ibn Hajar said:Ibn Battal replied that the words of the Prophet (peace and blessings be upon him) in this Hadith, “Who will buy this slave from me?” indicate that he was offering him to the highest bidder so that the needs of the bankrupt man for whom he was selling him could be met. (Fathal-Baari, 4/354).
2.`Ataa’ said: I met some people who saw nothing wrong with selling booty to the highest bidder. (Narrated by al-Bukhariin Kitaabal-Buyoo(the book of sales), Baabbay’ al-Muzaayadah(Chapter: selling by auction)).
Rationalevidence: In an auction, the vendor offers his goods for sale, and the purchaser offers to buy them for a certain price. If the vendor does not accept that price, that is the end of the matter and there is no transaction. Then he will say, “Who will offer more?” A second purchaser can then offer a higher price, and so on. In this case, each offer is a separate and independent deal, and there is nothing wrong with that.