Is it permissible to take a sum of money in exchange for withdrawing from an auction?

What is termed "reconciliation" in the question is known as "annulment" (iqalah) among scholars. They have differed on whether it is a فسخ (dissolution) or a بيع (sale). If we consider it a sale, as is the view of the Maliki school, then paying the amount is permissible, because it is akin to the seller buying back the sold item at a lower price than what you originally paid him for it. Here, the annulment differs from the forbidden bay' al-'inah (a type of buy-back sale). As for the amount paid by a bidder in some auctions, it is an earnest money sale (bay' al-'urboun), which some scholars have permitted.