There is no guarantee of market efficiency for single sellers, and while you aren't being explicit about it, I suspect that your belief that people are rational about selling their vehicles rests on a view that this markets efficiently determine price.
Forgive me for inferring too much from your statement if I am, but this is exactly the kind of situation in which the zero arbitrage principle doesn't apply. if I make a bunch of offers to buy a car for less than it's worth, then all I need to do is find one person who is selling that car and not shopping around. There's no reason to expect any market forces to govern that transaction at all.
There is no reason to believe that selling your car for whatever the first person you find offers is likely to give you a fair price.
I'm basically speaking from personal, anecdotal experience selling (and buying) a few low-end cars via Craigslist and equivalents.
I have other things I'd rather do with my time than either handle a bunch of prospective buyers inspecting, test driving, and low-balling my car (and dealing with potential fraud), or myself inspecting, test-driving, and low-balling other people's cars, even though that may be a way to save/earn some money.
From an economics point of view, the transaction costs are significant (even if not dollar-denominated), so the arbitrage opportunity exists for someone who can minimize or simply disregard those costs.
just because associated costs exist doesn't mean that they are reflected in the offered price. In order for that to be the case you would have to make a zero arbitrage argument, which doesn't apply to markets where sellers simply sell to the first buyer offer.
I'm not sure why you keep bringing up selling to the first offer and zero arbitrage.
goatherders and I both described working with multiple buyers, and that finding the right buyer (that is, the one who will pay you $1,000 more than some other buyer) requires time and effort. When considering whether to accept an offer on the table or hold out for a better offer, the seller implicitly or explicitly considers the transaction costs (search, delay, processing, risk, privacy, etc.).
This implies non-zero arbitrage, multiple offers, and market inefficiency.
Forgive me for inferring too much from your statement if I am, but this is exactly the kind of situation in which the zero arbitrage principle doesn't apply. if I make a bunch of offers to buy a car for less than it's worth, then all I need to do is find one person who is selling that car and not shopping around. There's no reason to expect any market forces to govern that transaction at all.
There is no reason to believe that selling your car for whatever the first person you find offers is likely to give you a fair price.