Chapters 5 and 6 answer the questions.

We know that Lorenzo dose not like stay with "others", every person except his mother, father and grandma. In chapter 2, we know he tried his best to escape others. And in the chapter 4, he called Olivia as "half-sister". In chapter 6, the end of page 96, his said: "I don't know her." It shows he added his sister to the " Others List". This is the one of the reasons that he didn't allow his sister to stay.

"Ski week" is a awesome chance that he can stay alone. He had planed it for a while. In the basement with nobody, he can stay alone, play games, read book or let his imagination ran away ( Chapter 3 ). It's almost like his dream. But his sister just like a person can mess everything up. Nobody will allow someone to destroy their dream. It's another reason I found.

I think when he put his headphones on and switch on his play station, it's kind of be in his place. Because at that time, he can ignore everything. He will only fix on the games or music which can leave him alone ( Chapter 3, 5, 6 ). And it is way that he ignore his sister, too.

The first time he agree his sister to stay because his sister tried to wake all buildings up, it would let everybody including his mother know Lorenzo hided in the basement instead of taking the ski week ( Chapter 5 ). His mother would take him to the psychologist again to find out why Lorenzo has lied to her( It's an implied information from Chapter 6, the beginning of the page 97 ). The second time, he needed his sister to pretend as Alessia's mother to answer the phone which called by Lorenzo's mother. Or he will be discovered that he didn't go to the ski week. And if he has wanted his sister to help him, he must allowed his sister to stay ( Chapter 6 ).

At first he read about a letter that Olivia wrote to their father. She said she don't want his money anymore but care. He actually didn't understand, he thought Olivia got misunderstanding of their father ( Chapter 5 ). But this made him know more about his sister. And it help him understand his sister's pain from their parents.

And then, his sister explained him what is residual life estate means. His sister explained it just like she had been living like this for a long time (Actually she did, she found that when she was explaining, and then got shock. ). He didn't notice that but he thought this kind of life is horrible ( Chapter 6 ). When he understood his sister had lived like this, he began to feel sorry and wanted to help her.

His sister got illness, malaria, wasn't catching. When he found out this had gone really serious but nobody cared, he began to understand all the stuffs on Olivia. She was the person without love from her parents ( Chapter 6 ) . She's illness is painful, couldn't share and hurt not in body but nervous. So he began to feel sorry and help her.