What is the ruling on not informing a competitor about the prices you are dealing with, because he sets the same price that you set?

You are not obligated to tell your competitor your selling prices, but it is not permissible for you to lie and tell him unreal prices, because lying is Islamically forbidden. If you need to avoid telling him the truth, you can use tawriyah (double entendre) and ma'areedh (allusive speech), which means uttering a word that has two meanings, intending the far, less obvious meaning. This is permissible if there is a compelling legitimate interest that necessitates it, or an unavoidable need that cannot be met otherwise. However, if there is no such need, it is disliked, and it becomes forbidden if it leads to taking what is false or denying what is due.