Host offered a refund and now refuses to pay the amount offered

[Marionne2]

I'm currently engaged in a battle with Airbnb over a recent stay.  My family and I arrived at the rental property and found some issues including a significant issue with the A/C.  I contacted the host via Airbnb messages, and after offering to bring an AC the next day, he messaged "I am sorry if you’re not satisfied. You can cancel reservation today.  If you check out I will refund you. "  Please note that I did not ask for a refund - although I was happy to see the offer, the host offered the refund without my asking.  Later in the message chain he sent "Yes I will refund you minus cleaning fee .  But Airbnb fee is I am not sure how it is processed I won’t charge Airbnb fee."  Given that I wasn't feeling confident that the issues would be able to be quickly resolved, I took them up on the offer to get a refund and we departed.  The next day I received a message from the Host that their refund policy was restricted and I would only receive 50% of the nightly amount.  There was nothing in their message which indicated anything about only refunding 50%, the message was very clear that they were offering a refund less cleaning costs.  I waited for the Refund Request to expire, and contacted Airbnb.  Their response was literally that "there's little that we can do if the Host has changed his mind regarding the refund."  So according to Airbnb, a Host can lie to get someone to check out, then refuse to pay them what was indicated.  I've been battling Airbnb about this for almost 2 weeks now and am trying to determine what my other options are; this is outright fraud.   Airbnb closed my case and I've called to get it reopened.  Airbnb is refusing to take responsibility for their role in the reservation and refund process, and is allowing Guests to be cheated.  If they are going to allow Hosts to make statements about refunds, then they need to hold the Hosts to what was offered.

[Emiel1]

@Marionne2 

If you cancel a reservation, the attached cancellation policy will apply and automatically calculate the refund (if any). I host can decide to give some additional refund, but is not obliged to do so.

In your case the reservation should not have been cancelled, but shortened by the host,with amending of the calculated price, so the unused nights will be refunded, including Airbnb service fees, in only one operation (to be accepted by the guest first before it is executed).

Airbnb can do not much for you, as they can not force the host to provide an additional refund.

[Marionne2]

Interestingly, after offering me a refund less the cleaning fee, the Host told me that I had to be the one to cancel the reservation, and that they did not know any other way to initiate the process. 

Based on conversations I have seen from Hosts, it sounds like Airbnb does have some tools at their disposal to issue refunds and withhold them against future payments.  Given that the Host's offer of a refund is captured in Airbnb messages, you would think that Airbnb can/would make this right.