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The dollar auction is a famous game that shows how seemingly logical choices can end up with very illogical results. Wikipedia says the best strategy is not to play, which is probably true in most instances. But what if all the players are perfect logicians?


Rules:

  • The game is played by a group of perfect logicians all of which have, say, exactly $31.41 in their pocket. All the rules are common knowledge, including how much money they all have.

  • At anytime, any player may bid any amount up to $31.41 as long as it is strictly larger than the current largest bid.

  • Once the game ends (with either someone bidding \$31.41 or everyone declining to play further), the highest bidder pays their bid and recieves \$1 in return. The second highest bidder pays their bid as well and receives nothing.

  • Players never do anything to lower their expected winnings, but players would prefer winning over not winning even if the dollar amount is the same (for example, winning with a bid of \$1 is better than not playing at all)

  • If multiple players bid at the same time, then the bid is given to a random bidder. (Edit: For example, if Alice, Bob and Carol all bid immediately at the start 10, 20, and 30 cents respectively, and Bob is randomly chosen, then the state of the game is Bob has a bid of 20 cents and Alice and Carol have no bid whatsoever)

  • They are not allowed to collude or even communicate beyond placing their bids.

What happens in the dollar auction described above?

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  • 1
    $\begingroup$ And these perfect logicians are all down on their luck and need every penny, right? $\endgroup$ – Lily Potter Apr 14 '18 at 4:03
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    $\begingroup$ Do the rules need to specify that agreeing to split the winnings is forbidden? $\endgroup$ – Zomulgustar Apr 14 '18 at 11:53
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What happens:

Everyone immediately tries to bid 41 cents. No one bids after that.

Why:

Like many of these game theory puzzles, it helps to analyze things from the end.

Clearly, the auction ends once the highest bid is \$31.41. I claim the auction will end whenever the highest bid reaches \$30.41.

Suppose the Tyler has the highest bid of 30.41. The only person who would try to outbid would be the second highest bidder, and the highest they could rationally outbid with is 30.40. But then the Tyler would be wise to bid 31.41. This guarantees that Tyler wins the auction, and doing so costed nothing; they increased their bid by a dollar, and won a dollar they wouldn’t have won. This is ignoring their previous bid of 30.41, which is a sunk cost. Therefore, no one will want to outbid, as their bid win them nothing and lost them money.

This same logic implies that the auction will end at 29.41. If anyone tries to outbid, then the response will be to 30.41, which ends the auction by the previous paragraph. Continuing this logic all the way down to the bottom, the auction ends if the current highest bid is 0.41. Therefore, bidding 41 cents is a good idea, as it ends the auction and gives the bidder a profit of 59 cents.

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  • $\begingroup$ What do you exactly mean by "current bid"? Is it the bid of the highest bidder, or of the second highest bidder? $\endgroup$ – Reinier Apr 14 '18 at 17:38
  • $\begingroup$ @Reinier Your comment has made me realize my argument has some flaws, I will have to fix it later though... $\endgroup$ – Mike Earnest Apr 14 '18 at 17:48
  • $\begingroup$ Nice -- that's that's the answer I got as well! $\endgroup$ – Tyler Seacrest Apr 14 '18 at 19:39
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    $\begingroup$ @athin Because they would be spending 1.01 for the chance to win 1.00. This is irrational. $\endgroup$ – Mike Earnest Apr 15 '18 at 1:57
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    $\begingroup$ @deepthought The phrase "The only person who would try to outbid would be the second highest bidder" was really only there to help make the situation seem less complicated by focusing your attention on one bidder. In the 0.41 situation, multiple people could outbid you, but the same logic applies. No matter who outbids, you will jump to 1.41 and win. $\endgroup$ – Mike Earnest Oct 4 '18 at 19:59
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I believe

As soon as the game starts, all the logicians will attempt to bid $1. One of them will be randomly chosen as the selected bid, and then nobody else shall join the game, and the game will end.

Why:

I am assuming the logicians has this as their priority list:
1) Try to maximize their profits (and as such, avoid negative profits)
2) Prefer winning over losing

This means that, for everyone, winning a \$1 bid is better than not bidding at all, even if your profit is 0.

However, nobody will start a bid at > 1, as this will put their profits from 0 to negative, violating rule 1. On the other hand, bidding any amount X < 1 will end up as a loss of X for you, because it is guaranteed that other logicians will bid $1 (to fulfill their own rule 2) and outbid you.

Additionally,

"If multiple players bid at the same time, then the bid is given to a random bidder."
Considering this rule, there is no risk to trying to bid $1 as soon as the game starts. If you're lucky you will manage to get the bid, if you're not then there will be no consequences to you either.

As such,

I believe that, assuming all of the logicians are perfectly logical, they will all bid $1 right at the start, one of them will be randomly chosen as the selected bid, and the game will end there with said character 'winning'. Nobody receives any loss or profit.

In other words, this game is simply an over-complicated test of luck to see who among the logicians will be chosen :P

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    $\begingroup$ I tried to clarify the “at the same time” bit. Does the current wording make it clear? $\endgroup$ – Tyler Seacrest Apr 14 '18 at 5:28
  • $\begingroup$ My bad, just saw the reply now. That clears it up, yeah! Edited my answer to fit the new description. $\endgroup$ – votbear Apr 14 '18 at 12:49
  • $\begingroup$ "Players never do anything to lower their expected winnings" doesn't that mean that they will start all bidding 1 cent. In any case, they know they won't lose because the bidding is assigned randomly. $\endgroup$ – Mhmd Apr 14 '18 at 18:12
  • $\begingroup$ Suppose I make a bid of 99 cents, then announce that if anyone outbids me, my next bid will be \$31.41. Even if I'm bluffing, bidding \$1 in response at best leaves you with a profit of zero. And if I'm not bluffing, then you'll lose at least \$1. Now what do you do? $\endgroup$ – Acccumulation Apr 14 '18 at 18:14
  • $\begingroup$ @Acccumulation - no communication :-) $\endgroup$ – deep thought Oct 4 '18 at 20:11
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they will all bid one cent

and then

knowing that any over-bid will lead to a bidding war where they will lose more than they can hope to gain they will all decline.

because

with two people bidding the best strategy is always to outbid your opponent by less than a dollar, so being first, even with a bid of only one cent is sufficient to scare further bidders off.

if instead the greater of simultaneous bids was used instead of a random any one

the first bit would be $1

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