Teaser 2720: Better tetra

From The Sunday Times, 9th November 2014 [link] [link]

I have four tetrahedral dice. Each has four identical triangular faces and on each face of each die is one of the numbers 1, 2, 3 or 4 (repeats being allowed on a die). No die uses the same four numbers and, for each die, the sum of its four numbers is ten.

I play a game with my friend. My friend chooses a die first and then I choose one of the remaining three dice. We each throw our chosen die and score the face-down number: sometimes it’s a draw, otherwise the higher number wins. I can choose my die so that I am always more likely to win.

So my friend changes the rules. We now throw our chosen die twice, add the two numbers, and the higher total wins.

Now he knows that there is one die he can choose that makes him more likely to win the game.

What are the four numbers on the winning dice?

The wording for this puzzle has been modified from the originally published version.