Who has the edge, and how would you play this...?

[RocketManJames]

Here is a dumb game. I am confused by who has the edge and how you would play this.

Person A has 2 marbles. He secretly puts both marbles in his left pocket, both marbles in his right pocket, or he puts one in each. Person B pays $1 (or X) to guess if it is left, right, or both.

Payout:
Choosing Left or Right correctly: +$1.50 (or 1.5X)
Choosing a Tie correctly: +$6 (or 6X)

Now, given this, how should Player A play this? What percentage of the time should he split the marbles? He probably doesn't want to do it too often, as he has to pay a lot more if Player A chooses the split. But, if he never does, Player B will soon figure that out, and never choose "both" thereby winning easily since he's getting paid 3:2 on an even bet (if Player A never splits the marbles).

So, *without any a priori knowledge* of the probabilities that Player B will choose Left, Right or Split... How should player A play this? What about for player B? How do (or should) their respective strategies change as the game goes along?

Excuse me if this is a well-known type of problem that many people have solved over and over on this Forum. I was just thinking about this today and I can't really figure it out.

-RMJ