The properties in GP's post are not likely to be sold at auction - very few homes are sold that way in the US.

The 'bidding war' referred to is a standard listing/sale that receives multiple offers. If the interest is high enough, the seller may counter all of the offers and reveal the highest price, forcing the bidding.

Even without revealing the highest offer, the offers themselves constitute a silent auction, which is where a lot of the information asymmetry comes from. If I could speak to each of the other people and tell them I had X amount to spend we could all agree to walk away or make offers without going through a bunch of hassle and (for 32 of 33 offerers) heartbreak. The process really highlights to me how broken certain types of 'market' can be when there's a monopoly of information.

What you are talking is second price auction.

Winner gets the house, but pays the price of the runner-up. I wish this existed.

Why would a seller agree to this type of auction?

To get people to be more aggressive with their bids.

toomuchtodo: I am not sure if i would call it "true" value. The true value is what you'd get if you were to sell it again. Maybe at that same day, which sounds to me like the runner-up bid.

Why would the seller go along with the idea that the buyers would negotiate amongst themselves to choose a price?

They wouldn't, because it's not in their interest.