To refresh your memory, here is the original problem:
I am auctioning off a normal $20 bill. (Normal auction- live, open bid, etc.) The highest bidder in the auction pays his bid and gets the $20 bill. The unique part about this auction is that the second highest bidder also pays his bid, but gets nothing. How much would you bid for this $20 bill? Are there certain circumstances that are required for you to make this bid?
Point 1: I noticed in some of your comments that you were making random assumptions to get to your conclusions- for example, assuming you can only bid once, etc. Haven’t you ever heard the line “when you assume, you make an ass out of u and me?” Just checking.
Let’s start by outlining the incentives at each step:
Say you are the current high bidder. So if the auction were to end, you would pay your bid and get $20. However, the second place bidder (who likely bid just a little bit less than you did) has to pay his bid and gets nothing. Therefore, at least for the purposes of this step, the second place bidder has an incentive to up his bid to outbid you in order to get the $20. Let’s put some numbers to this example:
Iteration 1: You bid $15, second place is $14
Iteration 2: The other guy bids $16 (technically, he could bid $15.01, since I didn’t stipulate bid increments, but let’s make things simple here), since he would rather pay $16 to get $20 than pay $14 to get nothing.
Iteration 3: You bid $17, since you would rather pay $17 to get $20 than pay $15 to get nothing.
So far, nothing seems particularly strange about the auction, since shouldn’t the price of a $20 bill get bid up to $20?
Well, let’s think what happens at each stage when the price gets to $20.
Iteration 1: You bid $20 for the $20 bill.
Iteration 2: The guy that had previously bid $19 now bids $21. Okay, that’s the part that gets weird. Why would anyone bid $21 to get $20? Well, think about it- if you are the second bidder at $19, you will pay $19 to get nothing, for a net loss of $19. On the other hand, if you bid $21 and win, you will pay $21 for the $20 bill, which will give you a net loss of $1. And aren’t small losses better than big ones?
Iteration 3: You bid $22, since a net loss of $2 is less bad than a net loss of $20.
And so on…note that once the bid goes above $20, there will only be two people left with an incentive to keep going in the auction.
So what is going on here? This seems like a financial game of chicken, which I will lovingly call “Who is going to run out of cash first?” There are three levels of reasoning to be had here, and people seem to vary in where they fall in terms of what level they get to:
Level 1: I understand my own incentives at each step (smaller losses and bigger gains are better, and I act accordingly in each iteration).
Level 2: I understand the other bidders’ incentives at each step.
Level 3: I understand that the overall game is comprised of many steps, and thinking only one step ahead is not necessarily enough.
People usually get Level 1. Levels 2 and 3 are more iffy.
Now the solution…to your potential dismay, the solution falls under the heading of “it depends on what you believe about the other bidders.”
The explanation from the traditional neoclassical economist:
One of the general assumptions that economists make is that people are profit-maximizing, rational, forward-looking individuals. IF (and I would argue that this is a big if) we believe this to be true, how should we behave in the auction? One simple option is to be the first bidder with a bid of $20. There is then not an incentive for anyone to outbid you (unless they want to punish you at their own potential expense, which is generally a situation left to the behavioral economists), so you get the $20 bill for $20 and the game is over. Effective, but not really profitable, though you would at least be sparing others cost and humilation. Is there another option that is likely to be successful here?
Looking one iteration at a time, we see that the second-place bidder (the underdog, if you will) always has an immediate incentive to outbid. However, since the problem is symmetric, if one looks at the overall game rather than a single iteration, it is clear that the outbidding is mutually destructive, since it leads to a game of chicken with no clear end. The economist might then think that there is a first-mover advantage, since why would a second bidder ever want to come into the picture and cause a bidding war? The economist’s natural inclination is then to bid the smallest amount possible and threaten to start a bidding war if anyone else comes in.
There is a problem with this strategy in that this threat is not credible. Since everyone in the economist’s world is a perfectly rational human being, including the economist, the economist should realize that the other people in the game will realize that it’s not in the economist’s best interest to enter a bidding war either, since there is no end in sight for it and the bidders get more committed at each step. (People are more likely to cut their losses if those losses are small, whereas they tend to irrationally try to avoid large losses.) It’s sort of like saying “I am going to set this $100 bill that I had in my pocket on fire if you don’t do what I say.” Who is going to believe that you are actually going to make good on that, since it is clearly not in your best interest? So if the economist is going to be too smart for a bidding war, then why not be the second bidder? Or the third? Or the fourth? You can see the problem here.
As such, if you believe everyone to be perfectly rational, you should see that they are not going to take your first-mover (or last-mover, for that matter) threat seriously, and it’s probably best to stay out of the auction entirely.
Glad we got that out of the way. As a behavioral economist, I am willing to consider that sometimes people exhibit cognitive biases, make mistakes, etc. When you take into account that the people you are competing with in the auction may not be playing “perfectly”, you could potentially have an opportunity for profit. So, some solutions from the behavioral economist:
- If you think that others don’t understand the Level 1 thought process of the game, by all means bid until the price gets to $20, since the other players aren’t going to think to bid over $20 in order to avoid a larger loss.
- If you think that others are going to take your threat of a bidding war seriously, feel free to try to be the first bidder and make the smallest bid possible. (You could also make this threat after the first bid, but it seems les likely to be successful since there is at least one person with an immediate incentive to outbid you.)
(Note that these solutions don’t have much to do with risk aversion, even though a lot of you mentioned it in your comments. The only real risk here is the risk of not knowing your opponent well enough.) So what is the overall takeaway you are supposed to get from this? Well, there are a few:
- Think not only about your own incentives, but the incentives of those around you.
- Know your opponent and his level of sophistication and rationality. Then act accordingly.
- Think of the bigger picture and not just of the individual steps. Sometimes a “greedy” strategy – one that is the best at easch single step – is not the best strategy overall.
- Think through a situation rather than just going with your initial gut reaction.
Words to live by, I promise.