You do not have to offer half or any money as you completed your side of the transaction,  that is,  you had tracking which showed delivery to the buyer's home,  where it was left,  is out of your control.

Ask your buyer to open a case for item not received.

If this was a Simple Delivery transaction,  eBay will fully refund the buyer,  and you get to keep the payout.

If it wasn't you enter the tracking number into the case details,  showing delivery you will not need to refund and the case will close.  

@dex_sin