A Garden of Hackenbush Stalks

In Blue-Red Hackenbush each stalk is a vertical chain of edges rooted in the ground, each edge coloured blue or red. Blue (Left) chops a blue edge on their turn; Red (Right) chops a red edge. Chopping an edge removes it and everything above it. A player with no legal move loses. Three stalks stand in a row; reading each from the ground upward, they are: blue-red-blue; red-blue-red-red; blue-blue-red. The value of a position is the number v such that, after adding a stalk worth -v to it, the second player wins however play begins; a lone blue edge has value 1, a lone red edge has value -1, and positive values favour Blue. Give the value of the whole position as a reduced fraction.

• graph theory

Source: Hackenbush values are from Berlekamp, Conway and Guy, Winning Ways. Statement written for AxiomIQ.