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An Incentive Puzzle Solution (Or Solutions)…

March 23rd, 2009 · 11 Comments

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.)
  • etc…

(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.

Tags: Incentives

11 responses so far ↓

  • 1 allan // Mar 23, 2009 at 6:14 am

    A comment and 2 questions:

    I’ve heard some interesting discussion (too lazy to pull up sources) mapping the commitment game to counter-insurgency situations like Iraq and Gaza. Both sides view sunk costs as actual costs in reputation, so even though the net expenditures (in life, devastation, international reputation) might be greater than the value of victory, both sides have the incentive to keep bidding once they start playing. Note that in Iraq, the solution seems to have been–as it often is in game theoretical situations–change the payoffs of various players with actual side payments.

    Q1) What does the literature say about behavioral responses to the Commitment Game? I’ve only seen anecdotal evidence that says that people will bid lots, that some of the very high bidding is ego driven, and that people can sometimes be taught not to bid. Can you summarize any interesting experiments?

    Q2) I’ve been poking around a bit in all-pay auctions [researching crowdsourcing] and am curious if there are any interesting behavioral distinctions between a both-pay auction and an all-pay auction.

  • 2 Michael Lumley // Mar 23, 2009 at 8:13 am

    Here are my thoughts:

    Your game also depends on how many players there are (2 vs. 500), and how much money is at stake. (From my perspective, a game for $20 billion looks significantly different from a game for $20.)

    Have you given any thought to the possibility of outside agreements between the players (i.e. I’ll give you $5 to stop bidding)?

    My more complete thoughts are below…

  • 3 Dan L // Mar 23, 2009 at 12:40 pm

    Actually, there does seem to be a perfect strategy when playing with a group of rational robots, but it requires you to step outside the bounds of the game a bit. The first bidder can bid one penny and make a promise to outbid any other bid that is $20 or less. He can make this promise credible by giving $20 to mafia-bot to hold in escrow, and mafia-bot will break your legs if you violate your promise. You pay mafia-bot a small fee for his troubles. Your opponents don’t even have to be especially rational for this strategy to work. This can also be slightly amended to a near-perfect strategy for non-first bidders, assuming none of the earlier bidders have already employed a similar strategy.

    Also, I guess this is semantics, but I disagree that a person’s assessment of the behaviors of others has nothing to do with risk. Since the behaviors of others is unknown, it has a LOT to do with risk. It’s even more uncertain than dealing with a known probability distribution, which is a classic example of risk.

    The problem is still interesting for the case of logical robots who cannot communicate outside of the game. The $19.99 bid seems to be the best solution so far, but it still has a slightly odd assumption built into it (as I wrote earlier), so I remain slightly skeptical.

    A $20 bid is NEVER rational, because it has no possible upside and an improbable but nonzero possible downside.

  • 4 Dan L // Mar 24, 2009 at 10:13 am

    Edit to previous post: In the first paragraph, I forgot to mention the obvious—that mafia-bot is with you at the auction, verifying your story, and that mafia-bot is respected by the community as a robot of his word. (Or less colorfully, you could bring to the auction some sort of legally binding contract that has the same effect as mafia-bot.)

    In the last paragraph I meant to say that a $20 *opening* first bid is never rational.

    New content:

    By the way, while I find this puzzle very interesting (it keeps popping into my head whenever I am idle), your original post was a total tease. I was expecting more of a real solution. Of course your bidding strategy depends heavily on whether the other bidders are morons or not. But the interesting question remains: What should you do if your opponents are all reasonably intelligent, rational people? This question has not been adequately answered. The $19.99 solution is not really a true solution to the problem.

    And since you belittled my earlier comments, I’ll just say in my defense that when you are faced with a difficult problem, it makes sense to first solve an easier version of the same problem. Sometimes the solution to the easier problem sheds light on the harder problem; sometimes it doesn’t. For your problem, a reasonable simplification of the problem is to first consider the 2-player version. Another reasonable alteration of the problem is to consider the continuous version of the problem, since penny increments are very small compared to $20. Another alteration is to consider a simultaneous single-bid auction. This might seem like a gross corruption of the original problem, but one way of thinking about the problem is that the only number that matters is the highest number that you are willing to bid, since the actual bidding history is completely irrelevant. (The biggest change here is the issue of tie-bids, but this is rare and may not affect the solution.)

  • 5 Tom // Mar 24, 2009 at 12:41 pm

    i’m quite facinated with the potential out comes of actually applying this test to a group of people and i plan to speak with my econ professor to see if this will be possible. should he allow it i will post the results should anyone be interested.

    i agree with you, Dan, regarding most of what you said. the one thing i disagree with however is your position on an opening bid of $20. from where i’m standing an opening bid of 20 would be the only way to get the 20 without having to start a bidding war and invariably starting losses. if you are the first bidder and bid 20, then, and this is assuming your playing with rational and self interested people, no one else will bid. this leaves you exactly where you were at the begining and doesn’t invite others to join because they reap no benefit, only loss. if you disagree however i would be interested in hearing your own rational.

  • 6 econgirl // Mar 25, 2009 at 1:08 am

    Since you seem curious, I watched this auction with a group of HBS doctoral students and also with a group of MBA students. The doctoral students got over $20, but not by that much. The MBA students got up to a few hundred dollars. I should make it clear that Max donates the profits to charity. 🙂

  • 7 Tom // Mar 25, 2009 at 6:48 am

    oh. in that circumstance, while i find the MBA students actions commedable, i also think it represents an additional variable and therefor a different situation then the origional.

    the doctoral students seem to represent that intended scenario. i’m somewhat suprised that a group as knowlegable as that wold have gone over. then again, as dan said, your never playing with a group of rational robots.

  • 8 econboy // Apr 1, 2009 at 11:00 am

    Actually bidding could have gone higher than $20 without ever incurring a loss.

    Lets say for example that you bid $20, and I wanted to bid higher to win the auction. I could bet $20 today and $1 a year from now. The value of the $1 a year from now is greater than $0 today. Thus I could win the bet by bidding over $20 and still not lose money if the $20 is invested at a greater than .5% interest rate. Of course you could then bet $20 today and $1 a year from now and another $1 two years from now thereby outbidding me. The point is that it is possible for the bidding to go on inifitely without a the high bidder ever feeling that they could lose money on the deal. The bidders stop bidding on the actual $20 bill and start bidding on what return they can get for the cash.

  • 9 suman // Apr 2, 2009 at 5:08 am

    1. In winner-take-all system second highest bidder and others gets nothing. It depends on the system.
    2. All bidders are homogeneous ( in terms of cost ) the ultimate solution in the repeated bid is $20 (if psychological value for each bidder is same to $ 20) .
    3. Regarding the solution to this problem, I think Game Theory explains it best. Example: Like, Duopoly market
    4. If all bidders should bear different cost, the solution is the lowest difference between $20 bill and individuals cost.

    If there is no cost, all other things are same, the solution reaches to $ 20.

  • 10 Model Economist // Apr 2, 2009 at 6:23 am

    It’s completely irrational that the bidding would go over $40, not only would it be counterproductive in the fact that at that point you have a 100% chance of losing money (which would be any number past the $20 itself) you have also passed the point of losses that double your possible initial profit (at best after $40 of bidding you lose $20). From a purely mathemical standpoint knowing that with X number of bidders you have a 1/X chance of winning the bid, it would be in a person’s best interest to walk out without even placing the first bid knowing that someone would eventually bid at minimum $19.99.

  • 11 Ben // Apr 10, 2009 at 6:34 pm

    “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”

    just a quibble. that was my first thought when I first encountered this problem in grad school. but I was quickly informed this is wrong.

    it can’t be rational to stay out entirely, because that’s not a Nash equilibrium. (think about it, if everyone else is staying out, then it is optimal for me to enter)

    my classmate Michael Grubb (now at MIT so you may have run into him), worked out the equilibrium. even assuming rational actors, it is still optimal to bid with probability p, where p depends on how many players, etc.

    it is an interesting problem. i use it in class. Another classmate of mine, Ron Siegel, analyzed the general class of this game (all-pay auctions) for his job market paper and wound up at Northwestern.

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