Math Problem -- Optimal Auction Bid

Suppose there is an auction for an obscure license plate than you know that only you and your arch competitor Mr. X would be interested in. You would be indifferent to paying $1,000 for it. You don't know how much value Mr. X would put on it, but you feel his bid could be anywhere from $0 to $2,000, distributed uniformly.

How much should you big to maximize your expected happiness, measured as the difference between $1,000 and what you bid, if you win.

Free beer to the first answer and explanation I deem correct.

If you ignore ties and get zero happiness for losing, then I get 500.

$1333

If this is on eBay get a sniping program and let it get you the plate.

Max (x/2000)*(1000-x) (Pr winning x Happiness factor)
Same as maximize x*(1000-x)
Same as minimize x*(x-1000)
Same as minimize x^2-1000*x
dy/dx = 2x-1000
This is zero when x = 500

I should have said that it is blind auction, meaning everyone submits a secret bid and the item goes to the highest bid.

$500 is the correct answer. Since I asked for an answer and solution, Charlie gets the beer. Please add it to your running total.

What about the other bidder who is presumed to know of your interest and to be vindictive... how much more should be added to counter his added incentive to deprive you of your win rather than to just merely get the plate?

I would bid the maximum I feel comfortable paying. In this case, $1000, right? I mean, isn't that how auctions work? I'm not gonna bid more than it's worth to me.

Quote: TigerWu

I would bid the maximum I feel comfortable paying. In this case, $1000, right? I mean, isn't that how auctions work? I'm not gonna bid more than it's worth to me.

If it's worth $1,000 to you, then getting it for $1,000 has no value, as it's an even trade. You'd basically be trading $1,000 for $1,000. You're the same off whether you bought it for $1,000 or didn't buy it at all.

Wouldn't the truly optimal bid be $501, given that you know the mathematical optimal bid is $500 and you'd want to outdo anyone else that's mathematically inclined?

Quote: Romes

Wouldn't the truly optimal bid be $501, given that you know the mathematical optimal bid is $500 and you'd want to outdo anyone else that's mathematically inclined?

In that case, the super-duper optimal bid would be $502, given that the truly optimal bid is $501 since the mathematical optimal bid is $500.

Or maybe it's $503, given the super-duper optimal bid would be $502, given that the truly optimal bid is $501 since the mathematical optimal bid is $500.....

Or.....

But who's gonna think THAT far into it?? =D

If RS was there my bid would be $502, and I'd definitely smash his $501 bid.

Quote: RS

If it's worth $1,000 to you, then getting it for $1,000 has no value, as it's an even trade. You'd basically be trading $1,000 for $1,000. You're the same off whether you bought it for $1,000 or didn't buy it at all.

The question isn't about value, though.... it's about maximizing happiness.

If I really want that license plate, and getting it is going to make me happy, then I am going to bid the maximum amount I can afford, which in this case is $1000.

However, for the purposes of the poll, I voted that I didn't understand the question, because judging by everyone else's answers, it seems that I don't.

Quote: Romes

Wouldn't the truly optimal bid be $501, given that you know the mathematical optimal bid is $500 and you'd want to outdo anyone else that's mathematically inclined?

That was my answer as well.

With the rival's potential bid being uniformly distributed, each $1 bid increases your probability of winning equally. The way I am reading the problem, it seems like $500 maximizes happiness, but $501 would give you nearly as much potential happiness with the greatest probability of outbidding your rival.

Quote: gamerfreak

That was my answer as well.

With the rival's potential bid being uniformly distributed, each $1 bid increases your probability of winning equally. The way I am reading the problem, it seems like $500 maximizes happiness, but $501 would give you nearly as much potential happiness with the greatest probability of outbidding your rival.

The OP says he could bid from $0 to $2,000, distributed uniformly. I don't know exactly what that means, but I take it to mean he will bid a random amount between $0 and $2,000.

Quote: Wizard

$500 is the correct answer. Since I asked for an answer and solution, Charlie gets the beer. Please add it to your running total.

Why do I resort to magic?!?

Quote: RS

The OP says he could bid from $0 to $2,000, distributed uniformly. I don't know exactly what that means, but I take it to mean he will bid a random amount between $0 and $2,000.

Yea, I took it as an unknown amount, with the probability being equal for every amount between $0 and $2000. So it would follow that each dollar amount you bid would increase your chances of winning the auction equally. My math skills need work, though.....

great explanation charlie.

500 gives you a probable happiness unit of 125.

bidding 501 gives you a probable happiness unit of 124.9995, as does 499. so 500 is the optimal answer.

PS wizard, your poll answers are always hilarious.

Quote: StrangeMage

...500 gives you a probable happiness unit of 125.

bidding 501 gives you a probable happiness unit of 124.9995...

I believe the question is flawed given it asked us to put ourselves in a "real life" situation essentially but it doesn't take in to account real life strategies/logic. While I love and praise math, I think this is one of the times math would lead you astray. That extra .0005 is not worth it when you'll win X more amount of the time by simply bidding $501. Yes, there will come a line in which it DOES become not worth it to up your bet, but the factor here not calculable is your opponent... whom if he loved the piece as equally as you, then he'd more than likely try to do the same logic we're doing to figure out the correct bid amount. Thus, if we respect our opponent (another factor not measurable in current context) then we'd respect that he would "probably" come to a similar conclusion, thus the extra $1 could come in handy... and again before someone says "then he'll bid an extra dollar, you bid two..." so on and so forth I'll re-iterate that there is a point of diminishing returns when compared to value lost to % win. The calculation now seems to only produce a happiness value, but it's based on bidding against a computer, not a human.

$501 for life.

Quote: Romes

thus the extra $1 could come in handy... and again before someone says "then he'll bid an extra dollar, you bid two..."

This was another vague part of the problem, auctions can have different formats/parameters.

As it's written, I assumed the outcome to be like a silent type auction where the interested parties submit their best offer, and the highest is accepted.

Quote: gamerfreak

This was another vague part of the problem, auctions can have different formats/parameters.

As it's written, I assumed the outcome to be like a silent type auction where the interested parties submit their best offer, and the highest is accepted.

Yeah, as a silent auction I'd more than expect to see if someone valued this at $1k that someone would be willing to bid up to like $700-$800 for it. Silent ones like this are a lot like telling 2 people buying a home they're in "competition." More than not you'll find a party that goes with their max offer if they really like the home.

Quote: Wizard

his bid could be anywhere from $0 to $2,000, distributed uniformly.

i think this implies there's no logic to your competitor's bidding. he is equally as likely to bid $1 dollar as he is to bid $1999

i wonder why wizard picked a license plate to be the item of auction.

Quote: TigerWu

If I really want that license plate, and getting it is going to make me happy, then I am going to bid the maximum amount I can afford, which in this case is $1000.

$1,000 wasn't what you could afford, it is your indifference point to buying it. If you saw it in a store for that price you would truly be indifferent to buying it, because it is worth exactly $1,000 to you. So by bidding $1,000 you gain no happiness regardless of what the other guy bid.

The only way to increase your happiness is to bid less than $1,000 and hope the other guy bids less than you. The less you bid, the more happiness you stand to gain if you win, but you run a greater risk of being outbid.

Quote: StrangeMage

i think this implies there's no logic to your competitor's bidding. he is equally as likely to bid $1 dollar as he is to bid $1999

i wonder why wizard picked a license plate to be the item of auction.

Because, with obscure license plates, it can be very difficult to estimate where sniper bids will come in. Besides, the math is easier with a uniform distribution of your competitors bid.

Quote: Wizard

Because, with obscure license plates, it can be very difficult to estimate where sniper bids will come in. Besides, the math is easier with a uniform distribution of your competitors bid.

why not make the auction item a strategy chart for m21? :P

i suppose if you wanted to make it more complicated, you could make mr. X's bid a normal distribution with a mean of 1000 and a standard deviation of 200 or something

Mathematically it depends upon

- whether the amount of happiness you can otherwise purchase with an amount of money is linear, i.e. $1,000 purchases twice as much happiness as $500. It is convenient to assume this relationship is linear, but usually it is not true.

- whether there is a non-trivial probability/prospect to make a future purchase of an essentially identical item at a potentially lower cost (given that your immediate competitor for the rather esoteric item will have already gotten his)
- In this case, an immediate purchase of the item essentially forecloses your opportunity in the future to buy it for less -i.e., forecloses a future opportunity to more optimally maximize happiness. So, there is an imputed "future opportunity cost."

Quote: Romes

Silent ones like this are a lot like telling 2 people buying a home they're in "competition." More than not you'll find a party that goes with their max offer if they really like the home.

When I was house shopping, another interested party and their real estate agent showed up at a place we were touring. I don't think that was an accident.

Quote: gamerfreak

When I was house shopping, another interested party and their real estate agent showed up at a place we were touring. I don't think that was an accident.

I don't either. It's such a joke. If/when I sell my house, all I'd have to do is have my friend call in an offer (can be a complete crap offer) but then I legally have to disclose to the other buyers they're in competition... Jackpot.

Quote: Romes

I don't either. It's such a joke. If/when I sell my house, all I'd have to do is have my friend call in an offer (can be a complete crap offer) but then I legally have to disclose to the other buyers they're in competition... Jackpot.

I have family members who are owners of a real estate company and I understand the rules fairly well. You are not accurately describing the modern real estate business.

If you make an offer on a house, and there is a competing offer, the real estate agents will disclose to you that there is another offer, and whether it is higher, the same or lower than your offer, and whether there are substantial differences in terms - contingent or non-contingent offer, etc. You cannot get jacked around the way you describe just because there is a lower offer.

It requires a deposit to make an offer, which discourages frivolous offers.

The way real estate agents make money is not by colluding to jack up the offers by some small percentage - it is by rapid turnover, by getting the houses sold quickly and with a minimum of problems so they can move onto representing the next house. People pick real estate agents by word of mouth - and if their clients are not happy/delighted with how an agent represented them, their income will suffer. So, they are incentivized to treat their clients fairly.

Quote: gordonm888

I have family members who are owners of a real estate company and I understand the rules fairly well. You are not accurately describing the modern real estate business.

If you make an offer on a house, and there is a competing offer, the real estate agents will disclose to you that there is another offer, and whether it is higher, the same or lower than your offer, and whether there are substantial differences in terms - contingent or non-contingent offer, etc. You cannot get jacked around the way you describe just because there is a lower offer.

Yeah no, they do not have to and do not give you details about the other persons offer. Hell, if I made an offer to someone it would be in the offer contract that they can not disclose my details and the intricacies of my offer. To further that, when I bought my house 9 months ago, I had this happen and they would not tell me anything other than there was another offer. Perhaps this varies by state, but where I am, they do not tell you anything other than "You're in competition, would you like to modify your offer?" Sellers dream.

And to boot, I'm pretty sure it's illegal to disclose my offers information (especially if it's written in the contract or agreed upon prior) because I went through some back and forth with a couple about their property. In the end we settled on a decent number that I was very happy with. They verbally accepted my offer, and both agents heard them do so. The next day they had another showing, which after accepting they can not continue to show the house. They did anyways and TOLD them my offer and told them to beat it and they could have it. I checked with numerous local experts and it was definitely something I could have sued them for (on multiple fronts), but alas I was just pissed and moved on to the next (and ended up with twice the house at the same price in a better neighborhood so perhaps that helped me get over it too). Oh, and I found out the other offer they hijacked me with fell through. Mmmmmmm karma.

Quote: gordonm888

It requires a deposit to make an offer, which discourages frivolous offers.

Kind of false. You do have to sign a check to the reality company for 1% of your down payment / closing costs, but that check is COMPLETELY WORTHLESS unless your offer is accepted. Thus my buddy could offer X and put down a $10k check on the table and it wouldn't matter because no one would ever touch that so long as I didn't accept his offer. So while I said he could write a crap offer, that doesn't even have to be the case. He could write a good offer, and so long as I don't accept it, then it doesn't matter. Even if you live in a state where you tell them "lower/higher/details/etc" he could write a great offer you could tell them all the great details.

Quote: gordonm888

The way real estate agents make money is not by colluding to jack up the offers by some small percentage - it is by rapid turnover, by getting the houses sold quickly and with a minimum of problems so they can move onto representing the next house. People pick real estate agents by word of mouth - and if their clients are not happy/delighted with how an agent represented them, their income will suffer. So, they are incentivized to treat their clients fairly.

Oh I agree with this entirely, that still doesn't stop the "competition" situation in my state.

Wouldn't the correct answer be $500 minus the buyer's auction fees

Where is the negative happiness due to losing the item after being outbid factored in? In addition, isn't the pain of being outbid by a small amount more severe than being outbid by a large amount?

Quote: Romes

Yeah no, they do not have to and do not give you details about the other persons offer. Hell, if I made an offer to someone it would be in the offer contract that they can not disclose my details and the intricacies of my offer. To further that, when I bought my house 9 months ago, I had this happen and they would not tell me anything other than there was another offer. Perhaps this varies by state, but where I am, they do not tell you anything other than "You're in competition, would you like to modify your offer?" Sellers dream.

Wow, this is a "hot button issue with you, I guess. And no wonder, the specific situations you describe are outrageous -even criminal. I would have at least filed a complaint with the State real estate board.

Just a month ago my father-in-law made a full price offer on a house, contingent upon his selling his own house. His agent subsequently communicated back that there was another full price offer on that same house and that the other offer was not contingent on anything. My father-in-law opted not to raise his offer nor remove the contingency -he began searching for another house. he found one and I am helping him move in in a couple of days. I never said that they communicated the exact amount of the other person's offer (unless it is a full price offer - which is a special category of offer.)

I imagine that there are indeed state-to-state differences and that Anyone's Mileage May Vary depending upon the people you are dealing with. Some people are crooks. Sorry you've had such bad experiences.