how do I know for sure if a free agent will sign with me? Is there any chance that he would choose the other ( lower ) offer?

During early free agency the competing offers are masked (they don't show you the division of bonus vs base) - but generally the player will choose the largest bonus available, with a smaller amount linked to the base salary. (The next version will also favor shorter contracts with larger bonuses, but that is not the case now) So you won't be able to know for -sure- that you have the best offer (and of course someone else may make a better offer at the same time that you up your offer). The player's choice is purely financial, though.