I was researching a bit on this topic, and it became clear during the research that having a general model to foster abstract reasoning for the problem was fundamental.
So I will write about two things and related discussion:
- The "Private rooms & digital arbiter" thinking model
- an application for this model: The  "Hardcore Pawn or Multiplayer protocol"


The "Private rooms & digital arbiter" thinking model

This model comes with some rules:
1. The arbiter has the offer statement, called Agreement, and the control of the value of BTC that is going to be exchanged for the FIAT money
2. Nobody has access to the information of what happens inside the private rooms except Alice inside her room and Bob inside his room.
3. Alice and bob can exchange value inside the rooms through an appropriate conduit or communication mean that connects the rooms.
4. Alice and Bob enter their rooms and they perform some or no operation
5. When they come out of the rooms to talk to the arbiter, one in several scenarios can happen


-table 2

6. The Arbiter is able to automatically decide in each scenario what to do with the funds from Alice that it holds, making sure that the agreement between Alice and Bob is respected. Ideally this shall happen with digital and automated information.

I am aware that the digital perfect arbiter concept is not possible or concrete in many cases. This is a conceptual model to allow reasoning on the matter. In fact one of the reasoning could be done on how to implement an arbiter in different scenarios. Interesting alternatives could be based on P2P models that interact with blockchain smart contracts, similar for example to the solution from Peer3, but focused for Bitcoin:
Blogpost: https://medium.com/@peer3_to/building-a-peer-to-peer-internet-secured-by-web3-f2ef870922d9
Github: https://github.com/peer3to/state-channels-plus
The research was done trying to understand how the arbiter can perform its job using just the information that Alice and Bob have, and how the model can be improved to best represent the general problem.
It's also possible that it can be demonstrated that the arbiter cannot function automatically or cannot function at all in this scenario. It would be nice to read such a demonstration.

Most services are relying on receipt or other proofs from banks or FIAT payment processors to be able to determine how to take an arbiter decision and solve any controversy.
My opinion is that this interaction system can be improved with the goal of automatic controversy resolution and with a reduced cost of the operation and the spread. The following is the attempt of proofing that with some different compromise a protocol that respects these conditions is possible.

Notably the information of what happened into the rooms is property of Alice and Bob, and always exits the rooms with them. However a mean for them to proof the truth automatically to the Arbiter or generally to allow the Arbiter to take a final decision that respects the contract has not been found.

The  "Hardcore Pawn" or Multiplayer protocol
To be able to take a decision in every case listed in Table 2 the arbiter sets new rules for a multiplayer game protocol.
1. The players have to agree on a multiplicator protection factor called Mp of a minimum of 2.5
2. if Bob wants to buy 0.1 BTC, he has to give to the automated Arbiter 0.1 * Mp BTC
3. If Alice wants to sell 0.1 BTC she has to give to the automated Arbiter 0.1 BTC plus 0.1 * Mp BTC
4. if Alice communicates i have received to the automated arbiter, the arbiter
-> a. Sends a payment to Bob for 0.1 BTC plus 0.1 * Mp BTC
-> b. Sends a payment to Alice for 0.1 * Mp BTC
-> c. In all other cases the Arbiter guards the funds

Possible optimizations
1. Alice sets a date of expiration for the offer, but this does not count if the offer is accepted before it expires
2. Tie the mp minimum to a fluctuation factor calculated on the recent historic price of BTC, rising the mp when the market is more volatile.
3. The arbiter requires a poll, or a keep alive signal signed from Bob and Alice, to release the funds if one of the players goes missing

This approach tries to use incentive to split the stakes on top of the actual transaction and take away some complexity from the actual proof and the automation of the parallel transaction proof. Maybe it could work for small p2p transactions.

so some questions come to my mind:
- was the use of automated Zero Knowledge ever attempted for P2P FIAT -> Bitcoin payments?
- do you think the model is helpful in exploring and discussing this problem?
- how would you improve the model?
- considering that banked citizens are in theory possessors of their digital FIAT money and of the privacy of their transactions, do you think it would be their right to be able to automatically communicate their transactions through an API of their bank provider?

Hopefully this will be a fun discussion

PS extended article on medium: https://medium.com/p/ec89cb77dbea/edit