@amiramnoam - Total set of possible combinations of month and dates are :
possible data set:
{J: 14, 15, 18;
F: 13, 15, 16, 17;
M: 13, 15;
D: 13, 14, 16; ) (Thanks to @laurakodi for this summary. I had writting the same summary on a piece of paper but when I started typing my answer, I saw this summary. I thought - perhaps the answer is already solved by someone so I just took the summary and have worked out answer on my own. No issues about whether i get prize or not - It was fun!! Thanks for the brain teaser ) Anyway - here is my logic
Let us make an assumption that Joseph knows month and Roi knows date. For Joseph to say for sure that Roi does not know birthdate for sure, Joseph must know that the month he knows does not have ANY unique date (meaning all the dates in the month he knows have to be present at least in one more month). So, the months Jan and Feb which have 17th and 18th as unique dates are eliminated. Now, Roi, hearing Josephs statement, uses same logic and eliminates Jan and Feb. Then he says NOW he knows for sure. That means the date he has can not be duplicated in both March and December otherwise he would not know the month for sure. So, the dates 13 is eliminated from both December and March. Now, it can only be March 15, December 14 or December 16.
Joseph uses this logic based on what Roi says and then mentions he knows date too so, he is seeing only one date left in the Month he knows. Since this is only in March, the date is March 15.

Hope that logic is correct.

Regards,

@vm2904