Background:
This conflict resolution system for HeroQuest borrows directly from the Pendragon RPG, which may be easier for players with experience with that game to learn. I originally introduced this house rule because some of my players were irritated by being regularly defeated by lesser opponents when both are successful. For example, a character with a skill of 5 will by definition always have a roll of 5 or less when he succeeds. A character with an ability rating of 20 will loose more than 3/4 of the time when both characters succeed. The Pendragon Way solves this problem by always rewarding the higher successful roll.

Overview:
In this system, the highest successful roll in a contest always wins. If both characters have the same number of masteries, then the masteries cancel out and the ability ratings are compared as usual. If a contestant has an advantage of one or more masteries, then the player with the advantage may 'bump' the result once for each extra mastery.

How does it work? 
Roll 1D20 and compare to your ability rating.
If you roll lower than your ability rating then you succeed.
If you roll higher than your ability rating then you fail.
If you roll your rating exactly then you have a critical success.
An unmodified roll of 20 is a fumble unless the caracter has a 19 ability rating, in which case it is a failure.
An unmodified roll of 20 is a critical if the character's ability rating is 20.
Each mastery advantage "bumps" the result.

Examples: 
A character with a 6w1 ability rolls a 10. This is a failure which bumps to a success.
A character with a 6w1 ability rolls a 5. This is a success which bumps to a critical.
A character with a 6w1 ability rolls a 6. This is a critical result, which cannot be bumped further.*
A character with a 6w2 ability rolls a 10. This is a failure which bumps twice to a critical.

* The excess bump may be used to "bump down" the opponent's result.