Dice problem!?

Question

In a game of dice where your opponent has one die and you have another, the winner is the higher number (you each get to roll once).

There are 4 dice...
Die A: four 4's and two 0's
Die B: six 3's
Die C: four 2's and two 6's
Die D: three 5's and three 1's

If the opponent gets to pick the first one (unknown), which should you pick to win?
Why?

Answer 1

The answer is different depending on how the game is played. If there are only four dice, I can tell which one he has & will pick accordingly. But it is unclear. Most of my answer here is based on the idea that I have all four dice & so does her. He picks one of his & I pick one of mine -- so there is the possibility that we have the same one.

Choose Die A if he picks randomly. Pick D otherwise. If I know which one he picked, then I choose D if he has A, A if he has B, B if he has C and C if he has D.

It is:

50% against another A
67% against a B
56% against a C
33% against a D

Die B is:
33% against an A
50% against another B
67% against a C
50% against a D

Die C is:
44% against an A
33% against a B
50% against another C
67% against another D

Die D is:
67% against an A
50% against a B
33% against a C
50% against another D

Die A is better against two dice, even with one and worse with one
B and C are even with two, better against one and worse against the other.
C is worse against two other dice, even with one and better than one.

If my opponent is picking randomly, then I would certainly pick A. If he is not picking randomly, then I might want to choose D -- because if he picks the best one, I want to pick the die that beats the best one. He is unlikely to pick the one die that is best against D -- because it is not best against anything else. To control for this, I might choose a mixed strategy where I sometimes pick A and sometimes pick D.

Interesting game theory problem.

Answer 2

die c- it has the highest average value per roll