4 rooms

OK I'll explain;
New series in which people who believe they have a valuable artefact
get a chance to sell it to four of the country's leading buyers.
(tonight for instance there was a Francis Bacon canvas and
a Norman Foster table)
They can meet the buyers in 4 seperate rooms in turn ,who make an offer that
is kept secret from the other buyers.
They can choose the order of the rooms to visit.
But, once an offer is rejected , there's no going back...

I think the first offer should be rejected.
but I wouldn't wait till the last offer.

Do the buyers know the order the rooms are visited?

If they don't then take an offer at random.

If they do then take offer 1 or 2 at random.

The buyers don't know the order, and they must have their offer
written down before the first bid is made.

if the buyers knew the order, the last would offer a penny.
if they did not, then a random pick is a one in four of
being the highest, and a one in four of being the lowest.
i would reject the first bid and listen to the second,
if it was lower then i know the first was not lowest and second offer
is now a one in three of being lowest.
if it was higher then i know the first is still a one in four of being lowest
i would keep going until i got an offer bigger than the previous one.
theres a one in twenty chance of finishing up with the lowest.

should be one in twenty four chance   last line
i just read that again and confused myself.

so basically just keep rejecting prices until you get an uptick and take it?

I like it - as you say, only if the prices are offered in the order 4-3-2-1 do you end up with the worst price (where 4 is best offer and 1 is worst)

looking at all 24 combos it's even better:

sequence -  selection

1234 - 2
1243 - 2
1324 - 3
1342 - 3
1423 - 4
1432 - 4
2134 - 3
2143 - 4
2314 - 3
2341 - 3
2413 - 4
2431 - 4
3124 - 2
3142 - 4
3214 - 4
3241 - 4
3412 - 4
3421 - 4
4123 - 2
4132 - 3
4213 - 3
4231 - 3
4312 - 2
4321 - 1

Total 10 * 4, 8 * 3, 5 * 2, 1 * 1

Nice - 3/4s chance of getting top price or second best, 1/4 of getting worst or second worst.

Anyone got a better strategy than MTW's?

well done longbridge ,
I can understand that better than my own ramblings,
i only needed to write last two lines.

with each offer you are given new information
and the probability changes.
a bit like the monty hall paradox on here a few years ago

Well done wallet and longbridge,
my way(not going to the last room)
would give me a 2/3 chance of top 2 offers
and a 1/3 of the bottom 2.
I shall use your strategy when I go on the show to flog my Picasso.