Double Rock Paper Scissors

Alice and Bob play a game of Extra Tricky Double Rock Paper Scissors (ETDRPS).  It's like Rock Paper Scissors, but with a few extra rules.

1. One game of ETDRPS consists of two simultaneous Rock Paper Scissors games.  One game is played between Alice's right hand and Bob's left hand.  The other game is played between Alice's left hand and Bob's right hand.  These two games are won/lost/tied separately.
   
   
2. Bob wins ETDRPS only if he wins more Rock Paper Scissors games than Alice does.  If each player wins one game, or if both games tie, then Alice wins ETDRPS.
    
3. Alice may not use paper.

What strategies should each player use?  What is the probability of each player winning?

This puzzle was "borrowed" from the Caltech Harvey Mudd Competition, which is supposed to be high school level.  If you ask me, most of them are exceedingly difficult, but this one isn't too bad.

See the solution