I think I see the problem that Mijin is running into. Let’s keep using L and U, and say that V is the magical (and probably purely theoretical value) that the widget is worth to me, and we can clearly say that L <= V <= U.
The problem arises in saying that we’re willing to pay V, but when we see the auction go for V+1, we’re unhappy that we lost and would have been willing to pay it. Thus, we’ve just contradicted the V is the value of the item, that V is actually at least V+1 in this case and, in fact, L is equal to at least what we thought V was.
You can also work that backwards. We know we’re unwilling to pay U, but what about U-1? If we’re willing to pay U-1, then U-1 is V, if not, then U is actually at most U-1, and possibly less. As such, you’ll actually find that V=U, and if it doesn’t, then your estimate on U isn’t useful. Interestly enough, you’ll find that L is pretty much useless because anything less than V is a deal, but if you’re having trouble working down from U, it might help to work up from L instead.
This is where the store analogy is useful. I walk into a store and see a widget for sale. Obviously, if I see it for anything less than my estimate for V, great, I buy it on the spot, I’m happy because I got a deal. Similarly, if I see it for U+1, I should be happy to walk away because U is the point where I walk away; if not, my U is low and I should be willing to buy it. As such, U=V.
Or, for a more intuitive explanation, the idea of a maximum bid isn’t the point at which you feel like you’re ending up gaining value from the transaction. That idea comes from being used to shopping in stores where I see an item for $13.99 and I buy it, I may actually be willing to pay more, but that’s where it is, so I don’t have to test it. Rather, a maximum bid is the point just below which you break absolutely even simply because if you’re breaking absolutely even, why bother with the effort of the deal when you’re no better off for having done it. That’s you’re upper limit, but it’s also your personal value because, by definition, that upper limit IS your maximum.
Yes, it’s probably hard to narrow down, but I think that’s precisely because we look at the going value and add up. If it makes it easier, you might be better off just starting from your U estimate and working your way down. Are you uncomfortable spending that much for it? Yes, decrease and try again; no, bid, that’s your maximum value. The important thing to keep in mind is that you’re not necessarily paying that much, so you very well may get a deal, you just can’t depend on it.