dice game, who will beat who??

So there's this game me and a friend have been playing.

One player has 3 six-sided dice, the other has 2 six-sided dice.
Both players roll their dice, once only - no rerolls to get better numbers!
The 2 highest dice win a point for their player. If there is a tie, player 2 wins.

EG players 1 rolls (5, 2, 1) and player 2 rolls (2, 1) - each player gets 1 point.

Now the question - which player is going to win overall??
Player 1 has more dice, but player 2 has better odds of winning with the dice he has...

Some sort of statistics or probability table on this would be awesome. It's a few years since I did anything like this so my calculation-fu is a bit weak :( :(

I'd rather be Player #2, the player rolling just two dice.

Based upon a few computer simulation of this game, of 100 million games each...

...the player rolling three dice wins this game approximately 22.586% of the time.
...the player rolling two dice wins this game approximately 32.566% of the time.
...ties occur approximately 44.847% of the time.

As expected, each 100 million run simulation saw nearly identical percentages.

The above assumes I understand the rules of the game correctly. (Another example of two would have helped me. Sorry.)

Is this correct:

Red: 6-5-2 Blue: 6-1 Blue gets one point.
(The two highest dice are the sixes so the 5 that Red rolled never comes into play.)

Red: 5-1-1 Blue: 5-4 Blue gets one point.
(The two highest dice are the fives so the 4 that Blue rolled never comes into play.)

Red: 3-3-2 Blue: 3-2 Each player gets one point.
(The two highest dice are the 3s. Blue gets a point (tiebreak on Red's first 3) but
Red also gets a point for having rolled an "uncontested" second 3.)

Red: 3-3-2 Blue: 4-3 Blue gets two points.

Red: 5-4-3 Blue: 3-3 Red gets two points.

Red: 6-2-1 Blue: 6-6 Blue gets two points.



Here's the first 30 games from my program:

Red: 5-3-1 Blue: 5-2 Red = 0 Blue = 1
Red: 4-2-1 Blue: 5-2 Red = 1 Blue = 1
Red: 4-3-2 Blue: 5-2 Red = 1 Blue = 1
Red: 6-4-4 Blue: 6-1 Red = 0 Blue = 1
Red: 4-2-1 Blue: 4-3 Red = 0 Blue = 1
Red: 6-2-1 Blue: 5-1 Red = 1 Blue = 1
Red: 6-2-1 Blue: 2-2 Red = 1 Blue = 1
Red: 4-2-2 Blue: 5-4 Red = 0 Blue = 2
Red: 5-4-4 Blue: 6-4 Red = 1 Blue = 1
Red: 4-2-1 Blue: 6-5 Red = 0 Blue = 2
Red: 3-3-2 Blue: 4-3 Red = 0 Blue = 2
Red: 5-1-1 Blue: 6-3 Red = 1 Blue = 1
Red: 6-4-1 Blue: 5-4 Red = 1 Blue = 1
Red: 6-3-2 Blue: 4-2 Red = 1 Blue = 1
Red: 2-2-1 Blue: 5-5 Red = 0 Blue = 2
Red: 5-5-5 Blue: 5-3 Red = 1 Blue = 1
Red: 5-1-1 Blue: 6-1 Red = 1 Blue = 1
Red: 6-5-4 Blue: 2-1 Red = 2 Blue = 0
Red: 2-2-1 Blue: 5-4 Red = 0 Blue = 2
Red: 6-5-1 Blue: 4-2 Red = 2 Blue = 0
Red: 6-5-1 Blue: 6-5 Red = 0 Blue = 1
Red: 5-2-2 Blue: 5-3 Red = 0 Blue = 1
Red: 6-5-3 Blue: 6-5 Red = 0 Blue = 1
Red: 4-4-1 Blue: 4-3 Red = 1 Blue = 1
Red: 5-4-2 Blue: 2-1 Red = 2 Blue = 0
Red: 4-3-1 Blue: 6-1 Red = 1 Blue = 1
Red: 5-3-2 Blue: 2-1 Red = 2 Blue = 0
Red: 6-5-3 Blue: 2-1 Red = 2 Blue = 0
Red: 3-3-1 Blue: 5-3 Red = 0 Blue = 2
Red: 3-1-1 Blue: 3-3 Red = 0 Blue = 2

Sorry! Obviously my example wasn't as clear as it should have been...
I'll take your example games and write the scores I would have given.

'Red: 6-5-2 Blue: 6-1 Blue gets one point.'
Incorrect, each player would get 1 point.

'Red: 5-1-1 Blue: 5-4 Blue gets one point.'
Incorrect, blue would have received 2 points.

'Red: 3-3-2 Blue: 3-2 Each player gets one point.'
Correct.

'Red: 3-3-2 Blue: 4-3 Blue gets two points.'
Correct.

'Red: 5-4-3 Blue: 3-3 Red gets two points.'
Correct.

'Red: 6-2-1 Blue: 6-6 Blue gets two points.'
Correct.

Thanks heaps for this!!

If Blue wins ties, then I think the game hugely favors the 2 dice player.