Laws criminalizing perjury are not about being mistaken or less believable in your testimony. The crime is, very narrowly, stating something which you do not believe to be true, while under oath. If you make a statement that happens to be untrue but you believe it is true (you are mistaken), that is not perjury. In the US, moreover, you have to assert something literally untrue, not simply say something that could be interpreted as being untrue.
Rental arbitration hearings are nothing special, except that witnesses may or may not be sworn under the particular state's laws. If a witness is sworn and unambiguously asserts "Jones gave Smith $1,000 at that time", and another sworn witness asserts "Jones did not give Smith any money at that time", that might suggest that one witness testified as to something that they did not believe to be true. It might also show that they were standing in different places. Even if the circumstances show that the two witnesses observed the same facts, you would have to prove beyond a reasonable doubt that one party knew at the time that their statement was false. It is also possible that one witness made mis-remembered or misinterpreted what they saw. They might say or write something else (under oath, or elsewhere) that establishes that they knew better. An example is US v. DeZarn, which clearly established that the defendant could not have believed the statement that he testified to.
With no official record of what a party testified to, it may not possible to establish beyond a reasonable doubt that a party committed perjury, since it can be reasonable to doubt claims of what he actually testified to. The defendant's attorney may, however, have to create such a doubt. Some answers could be perjurous if the question and answer are short enough, for example "Did you cash the check?" "No", if witnesses to the hearing testify convincingly that this exchange took place.