Karma, Atributes in the bargain

Karma, in the bargain:
If writing a game that uses a karma system, AKA not rolling dice but using some non random way of determining the outcome of things, one of the great challenges is having situation not be too cut and dry.
The Amber dice less role playing system does this by creating a situation where no one really knows how they stack up as compared to the rest of the group. You might know, “I am stronger than Bill, but Hank is the strongest of us all, however Kelly is a far better swords woman than any of us and very dangerous with a blade.”  The game sets that up in character generation with a bidding system, and carries through the game. In Amber it’s very thematic, replicating quite well the relationships all of the princess and princesses of Amber have with each other in the books.
That’s one way to do it. And it works if the players are using numerical values to measure their characters abilities and the GM can say for sure “Yes your character is physically stronger than his character.” Furthermore in Amber if two characters get into a contest the character with the higher relative attribute wins, every time.  In the context of the setting the system makes sense, works and is handled really well in game, allowing you to test your opponent’s abilities and manipulate situations to their advantage, just like the books. For my tastes though it might be too cut and dry as a resolution system.
I am not sure that I want to go that exact route. In that I want characters to be more a set of skills rather than a set of attributes. 
Here is what I’m thinking, in the king Arthur stories, Arthur was a night, a king, a leader,  has a group of nights he works with, a wife, a wizard buddy named Merlin, holds x-Calibur, and so on. I have never read where anyone tacks on “and he could bench 230 lbs twelve times.”
I am not sure if raw data about a character is that important to character? Are attributes like strength and Dexterity just holding over from a war gamming past, like the RPG equivalent of a vestigial tail?
My thought would be that as players narrate their action if they can call on their skills they can reduce the difficulty of an action, until if they describe enough detail they could eliminate a challenge without ever having to “roll.”
I use the term roll as a generalization I was thinking more of a bidding system. The gm would secretly set the difficulty and the players have a number of chips at their disposal, as the player describes what he is doing the gm reduces the difficulty. If there is any difficulty left at the end of the player’s narration, the player would bid blindly against the GM difficulty. If the player’s bid is higher the action would be successful, if the gm’s difficulty is higher the action would be a failure.
That would be the basic idea. The result of every action would be based on narration first, then resource management (the bid), and finally luck (if you consider the gm having set a low difficulty to start with luck)
I am hoping that by moving the emphasis away from having “high stats” to statistically improve the odds of performing an action “Ie hitting the orc.” Putting that emphasis now onto the idea that this character can do stuff and the player has to say how he is doing stuff. I may be able to move character effectiveness into the realm of the story and away from the realm of numbers.

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