Blotto with an Edge

Alice has 5 indivisible soldiers and Bob has 4. Each secretly distributes all of his or her soldiers over three battlefields (any non-negative integers summing to the total, zero allowed). On each battlefield, the side with more soldiers wins it and the other loses it; equal numbers is a draw. Alice's payoff is (battlefields she wins) − (battlefields she loses), and Bob's is the negative of that. With both sides playing optimally (mixed strategies allowed), what is the value of the game to Alice? Give a reduced fraction.

• distributions
• graph theory
• permutations

Source: Colonel Blotto game (Borel, 1921); this 5-versus-4 instance written for AxiomIQ.