@Michaela G. I understand your personal feeling on the matter, but I keep my feeling out or my business decisions. I was aware that there were no deposits taken as was evident in escrow. I believe it wouldn't be wise for anyone to allow tenants to stay in their property without a rental deposit.
My main concern is a pet deposit. To you point on changing terms, if a previous owner made many errors in their management practices it is my responsibility to correct these errors. The tenant can get as upset as they'd like. They have the option to move or stay. I'd much rather have a tenant that agrees with my terms, which are very fair by the way, then to risk serious financial strain in order to keep a flawed program in place for a tenants comfort level.
Maybe it's because I am new to things but I can't see myself ever letting a tenant dictate terms to me through the fear of vacancy. I believe my market is stron enough that I'll find another renter fairly quickly. I wouldn't deliberately push a good tenant away but my terms are my terms. The terms the tenant has were with the last owner. When the home was sold I have the right to change them. Their lease was on a week to week basis. It is renewed each consecutive week as long as both parties agree.