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Topic: The peculiar auction (Read 1277 times) |
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BNC
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The peculiar auction
« on: Oct 22nd, 2003, 3:53pm » |
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Background: I got this puzzle from a non-human source. The solution I got is different than the "official" answer. First, I think they overlooked some possible exensions / complications of the problem (or, alternatively made hidden assumptions). And second, even using those assumptions, I'm getting a slightly different "result".
I would really like to hear your opinions. I will not post the complications I think exist for the time being.
You are participating in a peculiar auction. In this auction (like in normal ones), items are being auctioned, people place their bids, and the higher bidder get the item for the amount he bided. What make this auction strange are two unusual rules:
1. The second-highest bidder is also required to pay his bid, and is getting nothing in return (so, if the two higher bidders A and B offered A=$100, and B=$95, A will get the item for $100, B will get nothing for $95).
2. Every item must get at least one bid. In order to ensure that, each one of the audience members must place the initial bid on one item (one item per person, according to pre-determined order). This initial bid may, however, be as small as 1c.
It is your turn to place the initial bid on the next item. You watch in amazement as that item turns out to be .... a 1$ bill. Yes, a regular, used 1$ bill.
The question is: what should you bid to maximize your gain (or, what is the "winning" strategy).
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Sir Col
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Re: The peculiar auction
« Reply #1 on: Oct 22nd, 2003, 4:14pm » |
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I suppose you could play it safe and bid $1.
It is unlikely that anyone would bid more than $1, becasue although they are likely to win the bid, they would knowingly lose anything above the $1 they bid.
Similarly it is unlikely that anyone would bid less than $1, as they would definitely lose the bid and be paying something for nothing.
Therefore, it is likely that you will get the $1 for $1 and break even.
Bidding on anything else runs the risk of being out-bidded and paying for nothing.
This is were I think the complications arise: once you realise you're in second place, it is in your interests to win the bid. The longer the bid goes on the more you have to lose. However, you'll reason that it's worth getting something, rather than nothing, for a lot of money. The question would be, when do you stop once you've got into this fine mess?
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BNC
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Re: The peculiar auction
« Reply #2 on: Oct 22nd, 2003, 4:20pm » |
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Sir Col, the answr you gave in the 1st paragraph is the "official" answer to which I disagree...
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aero_guy
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Re: The peculiar auction
« Reply #3 on: Oct 22nd, 2003, 4:32pm » |
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My guess, BNC, is that you think the ideal answer would be 99 cents. The idea here is that no one will bother to bid $1 when they will get nothing for it.
I had figured that there would be someone willing to bid $1 just to screw you. If we consider the last part of what Sir Col reasoned we realize that any intelligent bidder realizes they run the risk of being outbid again and back and back and losing far more than the dollar. It is unlikely they would want to risk it for no gain, so you should, in fact, bid 99 cents. You can bid even lower depending on your estimation of the other bidders.
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visitor
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I'd bid 2 cents. Everyone knows now that I'm willing to bid up to $1.01, because, if that bid wins, I'll only lose 1 cent. Therefore, they know that bidding anything at all will be a losing proposition for them. They'll either bid up to $1 and lose and have to pay for nothing, or they'll have to bid over a dollar to beat me, and they'll still lose. Any bid over 2 cents is a losing proposition, and I can't imagine anyone bidding unless they hate the idea of me getting it so cheaply that they're willing to lose money on the deal to thwart my efforts. (They could bid $1.02; they'd only lose 2 cents to prevent me from making a fast .98. Of course, spite could then drive a mean and ridiculous bidding war between us).
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On second thought, it's just a little more complicated than that. If I bid 2 cents, someone could bid 4, and I'd know they're willing to go as high as 1.03, which would deter me from bidding 1.01. So I have to bid enough so that a second bid going over a dollar on my part is assured. If I bid 51 cents, anyone who bids 52 to 99 cents knows that I will lose less by rebidding than by dropping things where they stand. From a purely economic point of view they have nothing to gain. From a competitive, no-holds-barred, "let's play chicken and see who's got more guts" perspective, someone might very well bid more (once the bidding war starts, the only way to force someone out may be to bid either double his bid or a dollar more than him, but such a bid would increase my own losses). But from a 51 cent starting point, any other bidder has less to gain by entering the auction than I have to lose, so only a fool would take the chance of starting a very costly bidding war. [/hide]
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Icarus
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Re: The peculiar auction
« Reply #6 on: Oct 22nd, 2003, 7:19pm » |
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Suppose for a moment that you and someone else are both bidding on the dollar, and the other guy puts in the $1 bid in response to your bid of, say 90c. If you decide, as people seem to be arguing, that at this point you should stop bidding because you will be losing money if you bid any higher, then you still lose your 90c. But if you bid $1.05, now you only lose a nickel. I.e. if you are in second place, it makes no sense to stop bidding at $1. You lose less money if you win the bid for an amount over $1.
But... the other guy is in the same boat. He too will need to up his bid to lose less than he already has. If each of you are making small increments, and determined to keep bidding if it decreases your current debt, you will both quickly arrive at bankruptcy, having bid away your fortune to gain possession of a single dollar bill.
So... apparently, the thing to do after a betting war has begun is to increase your bid sufficiently that the other guy will loose more money by beating it than he would by sitting pat. (I am assuming that if he loses the same either way, he will bet anyway just because he likes the look of the bill.) I.e. you need to bet 1$ above his bet. This will break the bidding war. It also means you lose exactly what he bet.
Of course, the flaw is that he bet more than you did, so by bidding $1 more than his bet, you have actually increased your losses. I.e. it appears to be better for you to stand pat if outbid by someone else. You lose your bet, but betting any higher is sure to increase your losses.
Now assuming that everyone is sufficiently smart to understand all the consequences, and everyone's primary purpose is to come out with as much money as possible (no accepting a lose for other reasons), you decide "hey - I can get away with bidding 1c! Everyone knows that they would have to bid at least $1.01 to stop from entering a bidding war, so they are better off not bidding!"
So you put in your bid of 1c. Suddenly, your old logic teacher - who used to drive you crazy trying to figure out whether or not he was going to give a surprise quiz, puts in a bid of 2c. Accepting your own logic, you decline and pay in your penny, while he rakes in a whopping 98c profit!
So what's wrong with this logic?
[e]Once again my wordiness has tripped me up and let someone else in while I was working on my opus[/e]
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| « Last Edit: Oct 22nd, 2003, 7:21pm by Icarus » |
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towr
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Re: The peculiar auction
« Reply #7 on: Oct 22nd, 2003, 11:23pm » |
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A very peculiar auction indeed..
I think I'd rather pay one cent and lose it, than get into a bidding war.. and there's allways a small chance I'll still get the dollar.. (there's also a chance more than one other person will bid and I lose nothing)
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wowbagger
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Re: The peculiar auction
« Reply #8 on: Oct 23rd, 2003, 1:53am » |
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on Oct 22nd, 2003, 11:23pm, towr wrote:| I think I'd rather pay one cent and lose it, than get into a bidding war.. |
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That was my first reaction, too. And I still stick to it. After all, it's only a cent you lose and a dollar you can win, and I'm not much interested in accumulating such wealths. I don't even like the looks of your $1 bill. 
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| « Last Edit: Oct 23rd, 2003, 1:55am by wowbagger » |
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towr
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Re: The peculiar auction
« Reply #9 on: Oct 23rd, 2003, 3:47am » |
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Interestingly, on ebay people do occasional put one-dollar bills up for auction, and despite there not being a mandatory initial bid, nor a penalty for coming in second, some people still bid more than one dollar for it..
So basicly, some people are nuts. And you should keep that in mind if ever faced with the situation here..
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visitor
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Re: The peculiar auction
« Reply #10 on: Oct 23rd, 2003, 8:12am » |
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The only reason a person will bid is if he believes this will be the last bid. If I bid 1 cent, someone else can come along and bid 90 cents and feel fairly confident that this will shut me up because another bid on my part would offer me only a very small gain while exposing me to a much larger potential loss. If my first bid is 99 cents, I'm almost guaranteed to win, but I only gain a penny and I risk losing 99 cents (or more).
The higher my original bid, the more likely it is that this will be the final bid, but it also increases my risked losses and it decreases my potential reward. Is it possible to figure out a probabilistic optimum, assuming every bidder always makes the choice that is economically optimum? I know it would be rather complicated, the risk is dependent upon the risk-reward factor of your opponent, which is dependent upon what your risk-reward factor would be when you choose whether or not to rebid, which is dependent upon...
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aero_guy
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Re: The peculiar auction
« Reply #11 on: Oct 23rd, 2003, 8:21am » |
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It seems we have made the basic assumption that once two people enter in, no one else is going to bid, as they do not have any investment to protect.
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BNC
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Re: The peculiar auction
« Reply #12 on: Oct 23rd, 2003, 8:28am » |
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on Oct 23rd, 2003, 8:12am, visitor wrote:
I know it would be rather complicated, the risk is dependent upon the risk-reward factor of your opponent, which is dependent upon what your risk-reward factor would be when you choose whether or not to rebid, which is dependent upon... |
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That is basically the complication I was thinking of. To add to the confusion, I think additional factors should include your estimation of the group's inteligence, hatefulness, etc..
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Icarus
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Re: The peculiar auction
« Reply #13 on: Oct 23rd, 2003, 10:05am » |
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Aero_guy: By "other bidder", I mean whoever is currently the highest bidder other than yourself. The result is whether this is one person, or switches between many.
As for the rest: It is usual in these sorts of problems to assume that the other people are as intelligent as needed, and are strictly motivated by a simple mechanism - in this case, profit or loss. Even so, I mentioned these restrictions in my previous post.
As for the stakes being too small: Consider a similar auction, but this time your item is a hideously ugly painting. No one in their right minds would want to own it, and so its street value would normally be zero. But there is a crackpot collector who was unable to attend the auction and has offered to buy the painting from whoever owns it for a price of $1 million (he will not increase that price - so you either sell to him or throw it away). This is known to everyone. All bids at this auction are to be multiples of $10,000. Financing at the auction is such that if you win, you will be able to sell to the crackpot and use the proceeds to pay off your auction bill. Assuming full intelligence and strict monetary motivation for everyone at the auction, and that everyone will respond in the manner that produces the highest expected return, what should your strategy be?
Unlike $1, the profit from this transaction would be worth an attempt. Of course, a loss is more painful as well...
With enough conditions, this should have a deterministic rather than probable answer.
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Re: The peculiar auction
« Reply #14 on: Oct 23rd, 2003, 10:22am » |
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To knock another bidder out of the running, you don't need to bid a full dollar higher than him. A 90 cent raise, for instance, would mean that a reraise would only save himself 10 cents while any bid that barely beats yours would make it highly likely that you would rebid. So outbidding someone by 51 cents should be enough, because it would mean that a full dollar raise on his part would leave him with a smaller margin of victory than your own previous unsuccesful bid.
And, in fact, you can make do with less of a margin because a full dollar raise on his part would not actually profit him at all, even though it raises his risk.
Might I suggest an initial bid of 34 cents? Any rebid between 35 and 67 would leave the rebidder with a smaller margin of victory than your original bid, thus encouraging another bid on your part, while any bid more than 67 cents would reduce his rewards to something less than his margin of victory, and give him little reason to risk anything at all.
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Icarus
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Re: The peculiar auction
« Reply #15 on: Oct 23rd, 2003, 4:27pm » |
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Visitor, I think you and I are veiwing the motivations of the other bidders differently. From the sounds of it, you see the bidders as competing to see who comes out the farthest ahead. I don't find anything in the puzzle that suggests this. My view is that just like any other auction, the bidders are only concerned with their own outcomes and not how it relates to anyone else's. So the only question they ask in deciding is "will this mean more money for me in the end, or less?"
By the way, are you ever going to registar? For all we know you might not even be the same "Visitor" whose answers we've come to respect! 
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Re: The peculiar auction
« Reply #16 on: Oct 23rd, 2003, 7:21pm » |
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No, I don't think we're looking at it differently, but I'm thinking that you must compare your bid with his to figure out the likelihood that he's going to rebid. And if you two are using the same strategy, then, if you would rebid when he's 51 cents over you, then he will certainly rebid when he's only 49 cents behind the winning bid.
But I do think a probabilistic answer will be different than a deterministic one, because I'll bid differently if I know exactly how you will bid. In that case there is no "risk," only bidding high enough to make it unprofitable for a rebid. In that case, if I bid 49 cents, you could safely bid 99 cents, because you would gain 1 cent, and you'd know I'm not going to bid again because the only way I could make a rebid unprofitable for you is by bidding at least 1.50 (to force you to raise a full dollar or more to knock me out of the bidding). But if we only talk about probability, then your 99 cent bid would likely win, but it would only gain you a penny while there is always a slim chance that I might rebid. The odds would have to be better than 99 to one to make that bet worthwhile. I don't think any second bidder would make a first bid anywhere close to one dollar unless it was almost a sure thing, so the opening bid can be less than 50 cents (how much less, I'm not sure). And I don't think anyone would raise a bid by a full dollar because it increases the potential loss without increasing the reward (although in a deterministic solution, even one penny saved is worth it if you know for sure that this is where the auction ends).
And by the way, if you've grown to respect those answers, then it must not have been me. Actually, I will confess that I did register. (I am also Tohuvabohu, and Cho for those of you with a long memory). But I didn't like having to log in before posting. And if I stayed logged in all the time, then on those really boring days at work when I checked the board every ten minutes cuz I had nothing better to do, it would look like I was here continuously (that wouldn't look too good to my boss). Besides, once you start counting my posts, what if I end up knocking Thud off the leader board. Oh, the horror, the tragedy, the battle of meaningless posts to regain lost glory.
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Icarus
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Re: The peculiar auction
« Reply #17 on: Oct 23rd, 2003, 7:26pm » |
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Tohuvabohu AND Cho!  Alas, one of the problems of internet chatting, you can never be sure who you're talking to! I suppose your reasons are good enough, but does your boss actually check this site too, to see if you are logged in??
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I wonder: Which is larger
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