Hmm. How would you even go about an Amber-style ability auction in D&D...
First off, do you only auction ability scores? In Amber, your auction currency needs to last if you want personal connections, special powers (feats?) and objects too.
Then, how do you define the concept of "rank"? You can't just say that 9 currency points gives you a Strength of 15, and then allow others to spend 7 points to buy up to Strength 14 - that would make the characters functionally indistinguishable.
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Your bid for each ability determines your starting score (per the usual costs, PHB page 13), except that the highest bid is given one point from each non-winning bid.
Other things to bid on would include "an acquaintance", "a patron", "background feat", "starting feat", "random hereditary magic item", and "hand-picked magic item". Only the winning bid gains each of these, the others' points are wasted.
Example: Bob, Sue, Matt and Claire set out to create characters. Bob bid 6 points on Charisma and win that bid. This nets him one point from each of the others and a 15 Charisma (what 6+3=9 points gets you). Sue bid 5 points, and would normally get a 13 Charisma, but since she was outbid she loses one point and ends up getting a 12 Charisma for her points. This ensures the winner gets to be somewhat special. Matt and Claire bid 3 and 1 points, respectively. This gives them scores of 10 and 8, respectively.
You would need to define 10+ points as getting you a 16, as well as -1 point as getting you a 7.
You would also need to resolve draws somehow; probably easiest to give each character ten times as many character points (270 instead of 27), making draws much more rare. (This post doesn't incorporate this idea.)
An acquaintance = will help you in minor and discreet ways, will not accompany you on adventures or risk life and limb for you.
A patron = will actively support you and look out for you, is a source of quests, will accompany you once on an adventure (CR 5 NPC) but if killed is lost (even if resurrected).
A background feat = you start with a bonus feat you select from the following limited list: Dungeon Delver, Healer, Keen Mind, Linguist, Magic Initiate, Martial Adept, Ritual Caster, Skilled, or Tavern Brawler.
Starting feat = you start with a bonus feat, just like variant human. You need to bid at least 5 points or your winning bid gains you a background feat instead.
Random hereditary magic item = Roll once on the Magic Item Table F, DMG page 146. This item once belonged to your family, mentor or organization but has now been given to you. It need not be directly useful or even usable by your character. Apply at least one Special Feature (creator or intended user, detail from its history, minor property, or quirk) to your heirloom item.
Hand-picked magic item = Choose one item from Magic Item Table F, DMG page 146. You start the game in possession of this item. Make up its history and special feature(s) as you please. You need to bid at least 5 points or your winning bid gains you a random item instead.