Even though we don't really think this is a good idea quite a lot of customers asked for such an option so here is an example how this can be achieved with a rule. We still however recommend using the To solve functionality and setting it up with the "Prioritize last agent" algorithm. To do so, navigate to Configuration --> System --> General --> Scroll to the "To solve" section
If you are adamant on automatically assigning the tickets to agents on replies then go to Configuration->Automation->Rules of your LiveAgent admin panel and create a rule there like this one:
Important part on that rule is the "Action initiator" part which assigns the ticket to initiator of the action/rule so to the agent/admin who added the offline message/reply in ticket.
Also make sure that you select appropriate message group types, otherwise the rule will be applied to every action, including adding tags, resolving tickets without response, etc.
Remember that this is just an example and you can customize the rule with additional conditions or actions the way you want it.