Can I ask the author of the research paper to use his program in my research? I am doing a research in a field of psychology and I would like to do almost the same research (just with different groups) as one other published research. I can make similar research but actually I want to compare my results with already existing ones. 

The thing is, that researchers made one simple application, more like a game. In their article they put a link for that game but if I want to use their game in my research I need to have theirs other application for recording each test session. I know for sure that I will ask them to use one part of application but if it's rational, I would like to ask them and use the whole game and the other application for the data collecting.  

I know that this is more individual question, but I would like to know what is the common practice in computer science. Do people usually/ rarely ask someones applications/ programs for they own research? Unfortunately, I don't have a time to collaborate with someone in computer science to make other similar game. And even if I had a time I wouldn't be able to exactly compare the results with that published research which I would like to do. Anyway, I don't know the researchers but I thought to send them an e-mail to ask them.
I am a researcher in the computer science field, and I'd say it's not unusual to ask authors of published papers for their code/applications. Sometimes, however, people might be reluctant to give you the software, because research code is often unstable and badly documented, and they might be ashamed (or worried that you will find bugs that could invalidate their previous publications!).

You should surely contact the authors, explain the situation, and stress that you are (of course) going to cite their paper. There's no harm in asking. In my experience, the worst-case scenarios are:
a) they will not reply;
b) they will reply, and ask you to be included among the authors of your future paper, in exchange for the software.

You could try to prepare for these two outcomes, while hoping for a positive answer :-)