Can one atom in this type of reaction win the "tug of war"? What might happen if it did?

Answer: YES

Explanation:

POLAR BONDS will form.

A polar bond is a covalent bond between two atoms where the electrons forming the bond are unequally distributed. This causes the molecule to have a slight electrical dipole moment where one end is slightly positive and the other is slightly negative.

I'm assuming that when you mean "tug of war," you're referring to the pulling of electrons from both atoms in a chemical reaction.

Essentially, yes. If one atom is more strong (the real term is "more electronegative") than the other, it will be able to successfully pull the electron pair entirely from the other atom.

When this happens, one atom will become positive, and the other negative. Because of these opposite charges, the two atoms will attract towards each other, thus forming what is known as an "ionic bond."

-T. B.