I've decided I've been lurking too much recently and not actually posting anything, so I wanted to get an idea I've been toying with out there. I've been thinking about a high-fantasy game that runs with a very simple resolution mechanic, as most of the gameplay would focus on exploration and character interaction, rather than combat. I'm thinking about maybe some kind of hidden bidding system, where players have (in this case I was thinking cards) that they can bid face down against their opponents, with highest total winning. I think there would have to be some kind of system governing how many cards a player can bid, and what kind of bonuses they get to them.

My first thought goes something like this. With a normal playing card deck, black cards could be used for combat, while red cards would govern other skill checks and opposed conflicts. A player involved in a conflict would place a number of cards face down, reducing the number in their 'hand'. The GM would bid a number against them based on the enemy they were fighting. Each side would have a chance to look at the number of cards the opposing side had bid, and fold if they wanted, reducing the damage or negative consequences for their failure, and keeping half of their bid cards. If neither side folded, both players would reveal their cards, and the highest total would win the conflict. Any used cards would be discarded, so the characters would get more and more exhausted as their hands were whittled down by sucessive conflicts.A character could take a few moments to rest and recover, and draw up some additional cards.

I'm new to the idea of diceless systems, but does this idea sound all right?

It sounds ok, but the secret bid part makes me feel cautious (that the only word i can put to this). It kind of puts me in mind of the card game WAR.
There was a card based diceless game put out 9-10 years ago by TSR i think for dragonlance called saga system. Here they designed cards with suites that matched the character attributes and were numbered 1 through 10. Whenever you used a card that matched the suite of the attribute that you used it for it was called a trump and allowed said player to flip over the top card of the deck and add its result to the total. so in theory the player could play a low card and end up with a high result if they trumped. However, the cards were not only the representation of resolution, but were also used to establish the characters adventuring status/reputation, and their hit points. The more adventures the hero had and the more well known they were the more cards they would gain in their total hand. The more cards you had in your hand the higher your possible hit points. What i mean by that is that whenever you took damage you had to discard a number of cards that equaled to the amount of damage that the character took. However, unlike discarding for action where you recovered you hand at the end of the round, when you discard for damage it shortened your hand by the number of cards you had to discard to cover the damage.

So if you are going to create a game that works with cards as part of the resolution system you should think about how they can be used through out the game.

There must be something in the air, because I've played in two homebrew games just in the last month that use cards for conflict resolution.

The one I remember best did a nice job of simulating swordplay, and worked as follows:

• each player has a max hand size determined by weapon speed & skill with that weapon
• players begin the fight by drawing cards up to 1/2 max hand size
• on your move you can

• attack (laying down a card); score a wound if the card is = or lower than your skill rank. Since the highest skill rank was 10, all face cards were autofails.
• force your opponent to set two cards from his hand face-up on the table
• improve your hand by drawing two cards, up to your max hand size

• in response to an opponent's successful attack you can attempt to parry

• play a card from your hand OR from the top of the deck
• a lower card successfully parries
• a lower card in the same suit parries and ripostes (scoring a hit unless your opponent can parry or riposte)

• alternatively, in response to an opponent's successful attack you can dodge by discarding any two cards from your hand

The game also featured simple a card-based resolution for situations other hand melee combat. As noted earlier, skills had ranks ranging between 1 and 10, and to score a success you needed to draw and immediately play a card = or < your skill rank. To add texture, skill challenges were classed as "easy," "normal," and "hard." An easy challenge required 1 success on a draw of 2 cards; a normal challenge required 1 success on a draw of 1 card, and a "hard" challenge required 2 successes on a draw of 2 cards.

The system was well balanced but placed the spotlight on melee at the expense of other skill areas. (A common fault in gaming, all the way back to D&D). We agreed afterward that a better system would employ the melee system for important skill challenges and the quick system for minor skill challenges.