ProbieTim So if I rolled 6, 5, 5, 4 and my opponent rolled 5, 4, 4, 3, I think I got 4 successes because 6 beats 5, 5 beats 4, 5, beats 4, and 4 beats 3. I think.
My understanding of the rules is that in this scenario
Tim: 6 5 5 4
Opponent: 5 4 4 3
Tim wins because the 6 is higher than 5. The other 3 dice are counted towards your "victories" or degree of success.
In this particular case Tim has beat the opponent on every die which is a Total Victory, so Tim crushed the opponent like a tetsubo smash.
For instance if Tim had rolled 6 2 2 2, Tim would still win, but without any victories so Tim just barely won.
The victories can carry over as bonus dice if it relevant to the next action.
If the examples above physical altercation
In scenario 1 Tim has 3 victories over his opponent so if the struggle continues then on the next dice roll Tim will get 3 bonus dice in his pool for his 3 victories.
On the 6 2 2 2 scenario, Tim has won, but will receive no bonus on the "second round" as it were.
There are good examples on the game's wikipedia
The exception to these is if the opponent only has 1 die in their pool. In that case the opponent only needs to roll equal to Tim's highest die which is treated a one victory in favor of the opponent.
One final note is the type of dice you use do matter, because the character with a dice pool disadvantage have a lower chance of success on larger dice.
So if you use d20s and Tim's Pool is 4 he rolls 20 10 5 2, however the opponent's pool is only 1 die he only has a 5% chance to match that 20.