I hadn't heard of this way of selling property before, but saw it referred to on a Rightmove advert today. I was a bit wary when I read the advert but having done some proper research, I'm entirely opposed to it.
For starters, it's not an auction. Despite the names for it (made up by whichever canny estate agent thought the idea up I assume), legally it is no different to a normal private treaty property sale.
Secondly, the buyer has to pay a massive fee up front which does not count towards the purchase price. This fee goes to the estate agent, not the seller. The seller doesn't pay anything to the agent so you can see why the idea might be appealing to them. Usual fee on this Modern Method seems to be 5%+VAT with a minimum of £5,000+VAT. Usual fee on a sale would be 1-2%+VAT. And the 5% is payable at the beginning, whether or not the sale completes, whereas the usual 1-2% is payable later, at completion, so you can see that the estate agent has an enormous incentive to push sellers into this.
For the seller it's not all rosy though; it doesn't give them any more certainty that the sale will proceed, since their agent is no longer bothered about pushing it through after receiving a fee up front. In fact, it's in the agent's interests for the deal to fall through and to seek another buyer to pay a further fee on the same property!
And a buyer will have to pay so much cash up front (5%+VAT=6%, plus I've seen reference to the 10% deposit being payable too - that's 16% of the full price in cash!), you have to wonder which buyers might be attracted to such a deal.
So, unless you're an estate agent and cavalier about potentially breaching your implied obligations to your client, I wholeheartedly steer you away from this whether you're a seller or a buyer. It has no certainty for either party and will get neither a better deal.