Two friends say Bob and Sue play a game where Bob selects one of three
boxes containing three numbered ping-pong balls: Box A (2,6,7); Box
B(1,5,9); Box C(3,4,8).Then Sue chooses one of the remaining two boxes.
Each randomly select one ball from their boxes and the higher number
wins the game. If picking a ball has equal probability, who has a
higher chance of winning, and why?

p.s. Sorry to ansel001, I went to adjust the question but deleted it
instead. After revising, Box B contains balls numbered (1,5,9), was
(1,5,6). This gives an equal chance of winning from each box, that
being 4. I believe the key to the problem could be to consider the
Monty Hall effect, but I may be wrong. Sorry again, for the confusion.

Box A vs B:
     B(1) B(5) B(9)
A(2)  W    L    L
A(6)  W    W    L
A(7)  W    W    L
p(a>b)=5/9

Box A vs C:
     C(3) C(4) C(8)
A(2)  L    L    L
A(6)  W    W    L
A(7)  W    W    L
p(a>c)=4/9

Box B vs C:
     C(3) C(4) C(8)
B(1)  L    L    L
B(5)  W    W    L
B(9)  W    W    W
p(b>c)=5/9

If Bob chooses A then Sue will pick C and win 5/9 times
If Bob chooses B then Sue will pick A and win 5/9 times
If Bob chooses C then Sue will pick B and win 5/9 times

Thus both playing smart the expected value of Sue winning is 1/3*3(5/9)=5/9

The "Monty Hall effect" would come into play if the boxes were (1,1,1)
(1,1,1) and (9,9,9) and Bob let Sue pick a closed box, then opened one
of the remaining boxes to reveal (1,1,1) and gave her the chance to
change, which she would then take.

wobbleone,

Yes, your revision is more like what I was expecting the first time. 
Mathisfun is correct.  The second person has a 5/9ths chance of
winning no matter what the first person picks.  This does not violate
the law of transitivity however.  Each box has an average expected
value of 5.  Note that when A beats be with probability 5/9, the
average margin of winning is 3 and the average margin of losing is
-3.75.  The overall margin of winning or losing is zero as would be
expected, since they both have the same average expected value.