An employee (complainant) claims to have sensitive hearing has complained about a co-worker (defendant) that his radio is too loud. The defendant says it helps him concentrate and therefore, more productive. The complainant disagrees arguing that the co-worker turns it up louder on purpose, to which the defendant disagrees stating that the complainant talks loud and he can barely hear his music. They are in an area where earbuds or headphones are not allowed. What should the manager do? Remove the radio? Relocate the complainant to another area away from the radio? Tell the two co-workers to sort it out themselves? What would you do?
I sympathize. Cube farms are incubators for clashes in environmental preferences -- sound, light, temperature, and others.
I'm going to assume that if there were a volume at which the one could enjoy it while the other wouldn't be bothered, then they'd already be doing that. This answer addresses the case where they can't agree.
Unless listening to the radio is part of the job or something to be expected in that particular workplace (like, maybe you work at a radio station), the radio-using coworker should defer to the coworker who is bothered by it. (Presumably it hurts his concentration, just as the other says it helps his.) If you allow him to say that it helps him concentrate so the other person should just suck it up, then you open the door for other things that help people concentrate. I know people who extol the virtues of burning incense, for example, but I don't think you want them to do so in your workplace.
If you can separate the two without causing other problems, it's worth doing so -- avoiding easily-avoided friction is a win for everybody. If doing so would give one of them a benefit, though (like a better workplace), you should factor in concerns about fairness. I had a coworker once who admitted that he was deliberately loud and annoying because he hoped this would get him out of the cube farm and into a private office. Don't reward that sort of thing.
From your description it sounds like this problem has already escalated; the one is turning his radio up louder to annoy the other, who is responding by talking louder. If that is going on then they both need to grow up, and that's best accomplished by their manager(s) sitting each of them down privately and talking to them about professionalism.
Since you are a new manager, this is how I would proceed. First I would determine for myself if the music is too loud and/or if the other person talks loudly. You need to know if one or both or neither complaints are valid. Go spend some time casually in their work space and make note of how it seems to be to you Don't be obviously looking at this. Go by for other reasons, but pay attention when you are there. Further, observe how these people work with each other. There is a strong likelihood that neither the loud talking or the radio is the real issue. These kinds of things tend to come up when two people don't like or respect each other. So watch their interactions for a couple of days.
Next evaluate which of the two people is your better employee. It is likely that you may have to take an action that makes one of these two unhappy; know in advance which person is the one you most want to support especially if the both complaints have some validity. This does not mean the better employee gets a free pass on his behavior if he is in the wrong. It is one factor of many that you can consider. It really only comes into play when someone is significantly better, the job is critical, and especially if the position is hard to fill at that level of performance.
Next discuss the whole issue with HR. As a new manager, you may not be used to consulting with HR about personnel issues. However, they can help you a good bit in knowing what the options are and what process you need to follow. Don't start out telling them the actual issue and people involved. First talk to them abuot what the copmany policies for conflict resolution are. Once you know how much freedom you have to decide what happens, then you have a better idea of what you can do.
I am suspecting you will find that both people are contributing to the problem. You will have to watch whatever solution you have to make sure these two don't escalate into new issues espcially if one person feels he has won over the other.
Of the options avaiable, if it is possible to move these people so they don't interfere with each other, that is probably the best move. It is the least likely to generate further complaints. Neither person will feel as if he is the winner as long as one does not end up with a significantly better location than the other. It will seem the most fair solution to most people.
But it may not be possible. I don't know how closely these two people need to work. Or if they both need to work with the same equipment.
If you have to come down on one side or the other, consider other people who work in the same area. When I worked in an office where the radio was played, majority ruled on whether it was played and what station it was on. Taking this route makes it a group decsion. Of course this doesn't help if the group consists of only these two people.
Whichever side you come down is bound to make someone unhappy. Consider in your solution if you can find some other thing to give the person who loses this discussion as a sweetener and to show that you are not just favoring one person.
Here's a compromise you might suggest. Personally, I like to listen to music, but I also need to have an ear out in case someone calls my name. So I put in just one earbud (in the ear that is turned away from my coworkers). Suggest that he find one of those really small ones which is essentially just a long wire with a tiny earbud; the worker should be able to hide the fact that he is wearing it from customers, and it will be inaudible to the coworker.