Consider a three-way battle between Alice, Bob, and Carol. Alice casts Confusion on Carol, and Carol fails her save. Bob casts Dominate Person on Carol, who again fails her save, and orders her to attack Alice.
What happens on Carol's turn?
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Based on the rules for overlapping enchantment effects, (emphasis added)
Sometimes magical effects that establish mental control render each other irrelevant, such as spells that remove the subject’s ability to act. Mental controls that don’t remove the recipient’s ability to act usually do not interfere with each other. If a creature is under the mental control of two or more creatures, it tends to obey each to the best of its ability, and to the extent of the control each effect allows. If the controlled creature receives conflicting orders simultaneously, the competing controllers must make opposed Charisma checks to determine which one the creature obeys.
... both spells are still effective. The affected creature, here Carol, will first roll for Confusion, then do what Bob wills on a 01-25.
Consider the spell descriptions.
This spell causes confusion in the targets, making them unable to determine their actions.
You can control the actions of any humanoid creature through a telepathic link that you establish with the subject’s mind.
Even though a confused creature is unable to determine what to do, they don't get to. Whoever has cast Dominate Person gets to choose that through the link.