In normal dice play expected value doesn't come into play. But if we were to do a long term sum of results (like amalcon proposed) then it expected value would come into play, and would determine the winner.
This illustrates the difference between premature rounding (normal dice play) and non premature rounding (long term summing of results).
But a long term sum of results is a completely different game. It doesn't matter what's on the face of the dice at all; all that matters is the average value per roll. It's not dice anymore! And you've defined the winning condition completely differently!
I also don't understand how normal dice play counts as "premature rounding". It's just how playing dice works -- you compare the numbers versus each other.
"But a long term sum of results is a completely different game."
That's exactly the point! A naive intuition about dice is that they have a single probabilistic long-term score. E.g. 6-sided die is 3.5. So if one dice beats another, it should be transitive (the intuition goes). It's the summation (implied by averaging over time) that leads to the naive intuition.
Maybe this description of the two games might help clarify the link.
Game 1: Player 1 chooses a dice, then player 2 chooses a dice. They both roll numbers, say a and b. Then player 2 gives player 1 (a - b) dollars.
Game 2: Player 1 chooses a dice, then player 2 chooses a dice. They both roll numbers, say a and b. Then player 2 gives player 1 (a - b > 0 ? 1 : -1) dollars.
You can kind of squint and see that game 2 is the same as game 1 just with (a - b) "rounded" to either 1 or -1.
This illustrates the difference between premature rounding (normal dice play) and non premature rounding (long term summing of results).