Problem $1$

Consider the two$-$player game described in the figure below.

The figure illustrates the starting position of a simple game. Player A moves first. The two players take turns moving, and each player must move his token to an open adjacent space in either direction. If the opponent occupies an adjacent space, then a player may jump over the opponent to the next open space if any. $($For example, if $A$ is on $3$ and $B$ is on $2,$ then A may move back to $1.)$ The game ends when one player reaches the opposite end of the board. If player A reaches space $5$ first, then the value of the game to $A$ is $+1$ ; if player $B$ reaches space $1$ first, then the value of the game to $A$ is $-1.$

Draw the complete game tree $($see Chapter $5$ slide pages $10,11,$ and $15$ for game tree$)$ as shown in the figure below. For each terminal state $($leaf node$),$ write its game value $($e$.$g$.$ the utility value in Chapter $5$ slide p$\mathrm{.}10).$ For each loop state $($a state that already appears on any path to the root$),$ write a question mark $("?")$ beside the state, and stop expanding from the state.

Page $1$

Redraw your game tree as shown in the figure below. $($Do not include loop states.$)$ Run minimax algorithm as in Chapter $5$ slide p$\mathrm{.}23.$ You must write all minimax values for all nodes.

Does your minimax algorithm ran on Problem $1\mathrm{.}2$ give an optimal decision?