So i have this question:
and this in the answer on the answer sheet:
By backward induction, we know that I prefers g to h since 3>2 and likewise I prefers l to i.
Now im a bit stuck in order to find p. I know that on the right hand side I's utility is the expected value that yields 4 with probability p and yield 1 with probability 1-p.
Hence the expected utility for I is 4p+(1-p)=3p+1. Then if i set this equal to the left hand side value of I , which is three:
3p+1=3, i get than p=2/3.
From here I am lost. How can i get the values of the answer sheet? Please explain in the simplest way possible.
You are almost there. What you have found is that for $p=2/3$ it does not matter whether $I$ chooses $a$ or $b$. But what happens if $p$ gets larger than $2/3$? In that case $3p+1>3$ and thus $I$ should choose $b$. On the other hand if $p < 2/3$ then $3p+1<3$ and thus $I$ should choose $a$ in the first move.
Now when $I$ chooses $b$ in the first move then the expected outcome for $II$ will be $2p+3(1-p)=3-p$. If $I$ chooses $a$ in the first move then the expected outcome for $II$ is $1$ because $II$ will choose $c$ over $d$.
Hence the you arrive at the solution.