I attended an auction where three virtually identical items were
auctioned in a row - all were signed, cancelled checks from a deceased
Hollywood starlet, all in the same condition, each with the same
pre-sale estimate from the auction house. The opening bid for each was
$800. The first went for $1200, the second for $1100, and the third
for $1400. What is the correct bidding strategy in such a situation?
Assume that the number of interested bidders is unknown - if one knew
that there were only three interested bidders, one's strategy would be
vastly different than with an unknown number of potential bidders. 
The winner of the second auction came off best, and the third one
worst.  What should he (winner #3) have done differently?  It seems
that this might be a rather complicated question for game theorists.

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Clarification of Question by pdq2-ga on 31 Mar 2005 14:12 PST

Regarding what ssmithfl said regarding observing the number of
interested bidders in the room - these days that is not practical,
because some bids come in on the phone bank in  real time (manned by
auction house personnel); other bids are placed early by absentee
bidders, and some auction houses have simultaneous real time bidding
on the internet.

The third item is clearly the riskiest in terms of price.  

If there are only 3 bidders interested then the 3rd item will sell for
$800 (unless someone else sees an opportunity to profit from buying
the item then reselling it after seeing the other items sell for
more).  So $800 is a potential price, but not a likely price.

My personal strategy would be first to determine what I'm willing to
pay for the item.  Then I'd bid up to 15% less than that in the first
item then up to 7.5% less in the second item then up to the full value
in the third item.
In this case if the value to me was $1300 those numbers would have been 
Item 1: $1100
Item 2: $1200 (This item would likely have been won somewhere between
$1100 and $1200)
Item 3: $1300

**Notice that if the value to me had been $1200 then I probably would
not have won the item even though I probably could have won it at the
$1200 price if I bid that every time.  But winning it at that price
kills any chance of winning it at a cheaper price... which you might
call your elasticity of risk.  How much are you willing to potentially
pay extra (above what you might win the item for) to avoid missing the
chance of not purchasing the item at all.

The most important thing to remember in an auction is that the item
has a value to you... and you need to know what that value is before
tossing out bids.

pdq2-ga,  that is a really interesting question, and that is Jack's
theory, not that mine is better:

Obviously, the bidders all knew that they would have a second and
third chance during the auction of the first check, and the range of
their personal limits was originally around $ 1200.  the most eager
bidder got the item at that price, while the others considered that
they still had a couple of chances and didn't go higher. With the
second item, despite there being at least three bidders still
interested, they let one get the item for $ 1100, maybe thinking that
that or $ 1200 was the limit each of the others had set for
themselves, since they had not topped the winning bid on the first
item  - but, of course, at that time they were all hoping for a second
and third chance, so that assumption was a weak one.
I think that one of them should have bid $ 1200 for the second item in
the hope that the other bidders would pass, still in the hope that
they could get the last item.  If that had been the bidder that bought
the last item for $ 1400, he may have been chagrinned if it then went
for less, but he got his item cheaper than he eventually did.  But in
fact, his bid on it was preceeded by a bid of $ 1300, so that would
not have to be the case.  That person might have bid $ 1300 to top his
on the second item, and if no one bid further, he might assume that he
could then get the third item for that price, or maybe for his $ 1200.
BUT, if the $ 1300 bidder did bid on the second item and got it, that
would suggest to all that that was a fair price, maybe exciting the
other bidders to bid that high on the last item.  We know that they
did not then bid higher, but of course none of them knew that before
hand. But it seems like this strategy would not have been wrong  -to
me :)

But as you know, professional bidders at auctions set themselves a
limit and stick to it, whereas others can become emotional and in this
situation  - in my scenario -  might conclude after seeing the first
item go for $ 1100 and the second for $ 1200 or $ 1300, considered
that the last one was worth at least that much.  They did not top the
final $ 1400 bid, but they might have if the second item had gone for
more than the first one.

That is not very conclusive.
Now let's see what game theorists say ...

I agree that the 3rd item is the riskiest of the 3, as your example
demonstrated, because it has the greatest range of potential outcomes.
Here's my line of thought on the dynamic bidding strategy, as opposed
to jack's method of setting %'s as the guideline. It's interesting to
note that we seem to agree on the final result (round 2) as confirmed
by your real life example.

An important variable to consider would be the number of bidders in
the first round, if you are in a position to observe the actual
auction as the bids are made. Those bidders are the ones most likely
to represent real "final demand" for the product, and likely are not
"arbitrageurs." They are also likely to bid in all rounds, at least up
to the price they bid in round 1. You should carefully note where they
drop out in round 1, and see how that compares to your personal
"bottom line" number.

As you are aware and jack notes, if only 3 people, including you, are
bidding at the first round, then the 3rd auction gives you the
opportunity to perhaps buy without any other bids against you, thus at
the opening bid. This could be confirmed in part in the 2nd round by
noting if only 2 people, again including you, are then actively
bidding.

The winner in the first round is likely the one participant who (1)
"must have" the item, (2) no matter what the cost. That round also
sets a "market value" for the 2 remaining rounds.

If 2 or more people (other than you) are bidding in the second round,
then at least 1 person is likely to continue to bid against you in the
final round, at least up to the 2nd round price, possibly the 1st
round price, and possibly more. This would tend to argue that you
should buy in the second round if possible, so long as the price is
less than round 1.

The first round thus establishes a value that will allow
"arbitrageurs" to enter the market, those who want to buy at a price
less than the first round in the hope of reselling at or near that
first round price. Those participants will tend to bid up the price
the most in the second round (i.e., be the "most active"), at least
approaching the first round price. They will not be as large a factor
in the third round, because of the observation in the final paragraph.

If this kind of bidding does not appear, the second round may likely
give you a good price...unless other, new, bidders appear near the end
of the round, which should give you pause to consider whether you want
to risk waiting for the 3rd round, with its new variable mentioned
below.

Unfortunately, there is one missing variable in rounds 1 and 2: the
person who "must have" the item, no matter what the cost, but who also
is willing to "roll the dice" and wait until the final round. That
person, by being coy, makes it impossible to know if the first or
second round is the "best" price or not.

Assuming then at least 3 other bidders in round 1, I think round 2
would offer the best "value" vs "risk" and I would purchase in that
round at any price less than round 1, assuming that your "motive" is
"real demand" and not "arbitage."

All of this, of course, is subject to jack's wise observation in the
preceding comment that the most important variable is within your own
control...what you are willing to pay. You should have a firm grasp of
this number in your mind before you raise the paddle in the first
place, tempered by your observations in round 1 of what others are
willing to pay. The bottom line...are you one of those bidders who
"must have it" at any price? If so, I still think round 2 is your best
price.

All this falls in the category of "shooting from the hip" - these are
observations only, not supported by any specific research results. I
leave those for the folks who do "answer" your question. Good luck!

Some people get carried away at auctions so it's hard to determine a
strategy. Junk often sells for a higher price than expected whilst
great products can often be bought competitively.

It's a lottery.

Probono, I agree that there is a big element of luck involved.

However, to call it a lottery implies that you will either win or lose
based on this luck factor.  But the wonderful thing about an auction
is that you cannot lose if you keep your head in the game.  If you
know what the item is worth to you then you won't bid higher than that
amount, so there are 2 worst case scenarios:
1) you pay exactly what the item is worth to you ... in this case, you
gain an item worth $X and lose $X which is not a loss or a gain.
2) you don't get the item at the auction ... in this case, you gain
nothing and lose nothing which again is not a loss or a gain.

So in both of the worst case scenarios, you lose nothing.

However there is 1 other scenario that can occur:
3) you pay less than the item is worth to you ... in this case you
gain an item worth $X and lose $X-$Y for a gain of $Y.

So in an auction, you stand to gain $Y and you stand to lose nothing
as long as you keep in mind how much the item is worth to you.

Glad other auction aficionados have posted. Hate to have us all shot
down by a profi. :)
An glad the ssmithfl-ga also thinks the second round is the best opportunity.
I  would like to point out at that since the third item went to $
1400, there were at least two bidders, therefore, at least four
interested parties at the time of the bidding for the first item,
maybe not bidding  - unless the auctioneer was "calling bids off the
back wall" in the last round ... But we will exclude that possibility
here, though it does happen.

The real imponderable - as I think we all recognize - is that
subjective emotions can upset any logical analysis, even a sudden new
bidder that suddenly discovers a desire to have a check signed by
"starlet" because he has seen the other two going for real money.

Which reminds me of two stories:

The one about a young couple that sent Picasso a $ 10 check with a
letter explaining that they had no money but would just love it if he
did a sketch on the back of the check when he signed and cashed it. 
The story says that he did, obviously back when cashed checks were
returned with monthly bank statement.

The other was about the politician asked to sign his name on a dollar
bill he lost on bet, and did so, then asking what the winner of the
bet was going to do with it.  He was going to give it to his child as
a souvenir.  "Oh, in that case, I should have paid the bet with a
check."

I don't disagree that the 2nd round might be the best... but just to
point it out, all I said about it is that it's less risky than the 3rd
round and that in this particular case it was the most likely one to
win with my betting strategy (as it probably would be with most
betting strategies since it was the cheapest selling round).  We must
not let 1 example determine our theory, but we can apply our theory to
1 example for analysis.

In 1 other example (the only auction I've ever taken part in), I
played in "casino night" for the seniors at my college.  We gambled
for several hours with fake money then there was an auction at the end
of the night.  I did quite well for myself, everyone started with $100
and I ended with about $450...  But just before the auction I took a
quick survey of the room to see how much money people had (and being
college seniors, everyone wanted to show off the cards they were
playing with).  I noticed that too many people had too much money for
the prizes to go at reasonable prices.

So anyways, the first item came up for bid... it was a nice IKEA chair
(probably valued at about $100 in real cash).  The bidding went up to
about $200 and I decided to join in.  I got alot of "boooo"s and "quit
raising the prices" when I bid, but I knew I was getting a good deal
based on the amount of money in the room.  I paid $250 (fake money of
course) for that chair and I still have and enjoy it today.  Not a
single item the rest of the night sold for under $1,000, and I had
major bragging rights for bidding wisely dispite everyone hating me
for "raising the prices".

Sure college seniors with fake money will bid differently than middle
aged adults in a classy auction, but it's a fun story.  And I've heard
it said many times that the first item in an auction tends to be the
most underpriced as people are still getting a feel for the crowd.

In my opinion there is only one stategy for buying at an auction.  If
you really want the item, decide beforehand what you are prepared to
pay and bid up to that amount.  This is true no matter how many times
identical items are offered.

If you are looking for a bargain, the same strategy applys.  Decide on
your price beforehand and stick to it.

If you win the auction you should be pleased.