The bill that arrives after somebody leaves
Utilities bill for a period that has already happened, so the final one lands three to six weeks after somebody moves out and covers weeks they were living there. It is genuinely partly theirs. The reason it so often goes unpaid is not bad faith — it is that by the time it arrives, everybody has mentally closed the account.
Hold something back, or agree an estimate
The two arrangements that work are both decided before the person goes. Either the household holds a small agreed amount from whatever is owed to them until the final bills arrive, or everybody agrees an estimate on the day and settles at that figure whatever the bill turns out to be.
The estimate is less accurate and much better in practice, because it closes the matter while both parties still care about closing it.
Take a meter reading on the day they leave
It converts an argument about fairness into arithmetic. Without a reading, the split of a bill spanning a move-out date is somebody's opinion about how much electricity a person uses, which is not a conversation with an answer.
Set a date for settling, not a promise
"We'll sort it when the bill comes" is how this ends up unresolved. "We'll settle up by the end of October" is a date somebody can be reminded about without it being an accusation.
The same applies in reverse: if you are the one leaving, ask what the arrangement is rather than waiting to be chased. It is a far easier message to send than to receive.
Once you have agreed the split
Something has to apply it every month.
Pairly holds the rent, the bills and who pays what, splits them evenly or by income, and keeps a running total of who has covered more. Both of you see the same month.