I am the primary renter with three sublet agreements for each of the 3 rooms of a home in Texas that I am currently renting out. I want to evict one of the tenants because of complaints from the other two tenants, but the tenant is requesting arbitration. How does this work?
As the primary renter who is in turn renting to subtenants, you are the subtenants' landlord and have the ability to terminate and evict for disruptive behavior if necessary. You'd need to follow Texas procedure to terminate the tenancy for misbehavior, and then file for eviction if necessary. We'll assume for the purposes of your question that the tenant's behavior is so disruptive that it would support your decision to terminate.
When your tenant requested arbitration, he was probably thinking of ways that he could answer your claims in a setting other than in a court case for eviction. These cases are expensive to defend and, as important, a loss for a tenant means an eviction lawsuit in his rental history, which may make it hard to obtain rentals in the future. Your tenant may want to argue his case in a setting where, if he loses, it won't be "official."
Most tenants who are thinking along these lines ask for mediation, not arbitration. There's a huge difference. With mediation, a neutral mediator helps the parties come to a mutually agreed upon resolution. The mediator makes no decisions, and can't enforce the decision (although the parties can decide that they will make their conclusions enforceable in court). By contrast, an arbitrator is like a judge, who makes a decision for one party or the other after hearing the evidence. Arbitrators' decisions are always enforceable in court.
Mediation is more informal and much less costly than arbitration. Mediation is the way to go if you and the tenant are willing to try to work out a resolution of the noise issues. Arbitration isn't appropriate in this setting—by the time you finish with the arbitrator (and pay for it), you may as well have gone to court, where evictions proceed very quickly.
You can get information on mediation in Texas, which has a state-mandated mediation system for disputes just like yours. (Texas Civil Practice & Remedies Code, § § 154.001 and following).