$2$ Line Game

Consider the two$-$player game described below:

Figure i: The starting position of a simple game.

The starting position is given in Figure I. Player A moves first. The two players take turns moving. Each player must move her token to an open adjacent space in either direction. If the opponent occupies an adjacent space, then the player may jump over the opponent to the next open space if any. $($For example, if $\backslash ($ A $\backslash )$ is on $3$ and B is on $2,$ then A may move back to i$.)$ The game ends when one player reaches the opposite end of the board. If player A reaches space $4$ first, then the value of the game to $\backslash ($ A $\backslash )$ is $+1$ ; if player $\backslash ($ B $\backslash )$ reaches space $1$ first, then the value of the game to A is $-1.$

I. Draw the complete game tree, using the following conventions:

$-$ Write each state as $\backslash (\backslash$left$($s$\_\{$A$\},$ s$\_\{$B$\}\backslash$right$)\backslash ),$ where $\backslash ($ s$\_\{$A$\}\backslash )$ and $\backslash ($ s$\_\{$B$\}\backslash )$ denote the token locations.

$-$ Put each terminal state in a square box and write its game value in a circle.

$-$ Put loop states $($states that already appear on the path to the root$)$ in double square boxes. Since their value is unclear, annotate each with a $"?"$ in a circle.