Economists Do It With Models

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An Incentives Puzzle For You…

March 17th, 2009 · 17 Comments

Today I have been working on a post regarding the auto industry bailout, and it made my head hurt so much that I figured I would give you something easier to digest in the meantime. My primary area of research is in incentives, both in the workplace and in education, and I plan to do a feature on the site regarding different incentive programs along with their intended and unintended consequences. Setting up an incentive scheme generally requires thinking a lot about the motivations and potential behavior of the person that you are trying to incent (and the other players in the game) rather than just focusing on the result that you are trying to get from him or her. To illustrate the need for this type of thinking, I present a particular type of auction. (This example was first given to me in Max Bazerman’s negotiations class at HBS.)

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?

I will post an analysis tomorrow, so you can think about this over your St. Patrick’s Day green beer. Sláinte!

Tags: Incentives

17 responses so far ↓

  • 1 LL Cool A // Mar 17, 2009 at 3:28 pm

    I suppose this entirely depends on the risk-aversion profile of the person playing this game…

  • 2 Tom // Mar 17, 2009 at 7:39 pm

    I think you need to look to gambling models, since it follows the same winner-takes-all outcome; then compensate for the intangibles you find in casino gambling (eg. herd mentality, being a showoff, etc.). Those intangibles still reward the losers.

    You might construct non-monetary intangibles to entice the potential runner-ups in your scenario (eg. giving them an “atta boy,” letting them speak at the awards banquet, etc.).

    Absent the intangibles, I might risk $4 for an all-or-nothing shot at $20.

  • 3 Dan L // Mar 18, 2009 at 9:31 pm

    I agree with LL Cool A. Isn’t the risk-aversion profile the most important part of the answer? In fact, if you take a simplified model, with a simultaneous single-bid auction among 2 people instead of a live auction, couldn’t your bid essentially be a measure of your risk-tolerance?

    If you take away the risk issue, this could be transformed into a (possibly tractable) math problem: 2 people play the simplified single-bid auction version of your game repeatedly. They play the game indefinitely, and whoever has more money wins. What strategy should you employ in order to win the game? Is there any strategy that is strong enough to not lose to any other strategy?

    Of course, my game is a gross distortion of the original question and has almost applicability to the real world. On the other hand, it holds out hope for a closed form solution.

    But going to back to real-world applicability to an incentive scheme, wouldn’t the more likely scheme involve *all* of the losers having to pay rather than just second place?

  • 4 Andrew // Mar 18, 2009 at 10:44 pm

    Its either $1, or $20.

  • 5 Adam // Mar 19, 2009 at 7:08 am

    Just a thought… If you commit yourself to winning, you are actually incented to bid much higher than $20. Once you have exceeded the $20 mark, you are still incented to outbid your opponent each turn in order to reduce your net loss.

    With that said, I wouldn’t bid.

  • 6 hcpark // Mar 19, 2009 at 7:58 am

    I agree with Adam. If you are 2nd place with a bid of $19.99, someone bids $20. Then you will bid 20.01, etc. So the real question, I think, comes down to:
    1. Will someone actually start the bid?
    2. Will the current 1st and 2nd place bidders realize what’s happening and “stop the madness” at a low enough loss number?
    My guess is that it heavily depends on the “structure” of the situation… does the auctioneer create a sense of excitement that clouds the possible endgame for participants? What is the nature of the bidder group (friends, family, econ class, complete strangers with no future interactions, etc.).

  • 7 Isaac Petit-Frere // Mar 19, 2009 at 10:44 am

    It would be stupid to bid anything no matter what the risk aversion profile of the individual is. That is assuming that each player is only allowed to bid once and that bids are placed in sequential order. The first bidder is guaranteed to lose with probability 1 if he bids less than $20. He is guaranteed to win with probability 1 if he bids $20 and up. BUT… He will not bid for more than the reward (unless he’s an idiot and decides that it the money is better when you win it, ie. Ebay, lol). So therefore he will bid $20 for $20 or not participate at all. The same goes for everyone after.

  • 8 Isaac Petit-Frere // Mar 19, 2009 at 10:51 am

    More specifically, no one after will bid anything and so the first bidder wins.

  • 9 Dan L // Mar 19, 2009 at 4:27 pm

    I agree with a lot of the above, but not all of it. If you just think of two people playing the game against each other, then it’s sort of a slow-death version of chicken. It’s essentially equivalent to a simultaneous single-bid auction (in which both players pay the lower bid), because each person decides in their own minds how high she is willing to go.

    On the other hand, this seems to be at odds with the following analysis (which is just a rephrasing of what Adam and hcpark said): every time you have a chance to bid, you have a chance to pay a penny for a chance at $20, and since sunk costs are sunk, it’s like playing the game afresh every time. My brain hurt trying to resolve this paradox, but I’m going to stick with my analysis from the first paragraph because it seems more clear to me somehow.

    In one-off games like these, the risk profile is always important, because you have no idea what the other person is going to do. For example, Adam and Isaac don’t want to bid anything. I guess that if I play against them, I will win $19.99. You’d have to be extremely risk-averse to not even bid a penny. I believe it is also foolish to let the auction get all the way up to $20. (I’m assuming that we are ignoring irrational eBay-type behavior mentioned by Isaac, or else the question definitely can’t be answered, except through experiment.)

    Again, to make it into a tractable math problem (and thus not a real world problem), we can remove the risk-tolerance factor by playing the game an infinite number of times. We then have to define what it means for there to be a best strategy. We could look for a strategy that has nonnegative expectation value against any opposing strategy, but I doubt that one exists (and could probably prove it). So the next thing to ask would be what strategies are at least reasonable. I looked for a strategy which, if employed by both players, would constitute a Nash-type equilibrium. If my math is right, I think I found a candidate, but I don’t know whether the solution is interesting. For example, I wouldn’t be surprised if there were no stable equilibria at all. The game has a “divergent” feel to it.

    Disclaimer: The dynamic may change a lot with more players. I haven’t thought about that at all.

  • 10 Adam // Mar 19, 2009 at 6:36 pm

    I think the only equiplibrium is for first bid to be $20. You remove incentive to continue bidding, but there is a net gain of zero.

    Beyond that, mathematically, I can think of three scenarios:

    Scenaio 1: Currency is continuous:
    Because, you only bid for a gain, you can set first bid at 19.99. Well second bid could realize gain with a winning bid of 19.995. If we allow for continuity in our currency, it may continue for infinity without reaching 20.00 (i.e. each bid could split the delta between the previous bid and $20).

    Scenario 2: Currency is discrete, with bid increments set to multiples of 0.01.
    Scenario 2a: First Bid is 19.99. Game over. First bidder realizes 1 penny and second bidder has no incentive to bid.
    Scenario 2b: First Bid is 19.98 or less. Second bidder stands to gain with a bid of 19.99. However, second bidder has now put first bidder at risk and created incentive to manage loss. Or, recognizing the concept of sunk cost, there is now a condition where there is always gain in putting an additional sum of money, less than the $20, at risk each additional turn.

    In reality, we know this wouldn’t happen. So someone will have to create a clever application of rationality/risk aversion.

    Great dialogue though.

    If you do not allow for continuity and bid increments must be a multiple of 0.01, then if the first bid is at 19.99, you have also reached an equilibrium. The second bid would be of no value.

  • 11 Adam // Mar 19, 2009 at 6:44 pm

    I really like hcpark’s first question. I think that is one way of building in “rationality”. If you have perfect information, will someone place the first bid?

    I think the answer is “yes” once certain conditions are met. Personally, if bid increments are set to 0.01, then I would not bid; because its not worth my time. I’d rather comment on hypothetical situations on websites. If the bid increment was $5.00, then I would bid first with an opening bid of $15.00. (I assume my opponent does not dislike me and will not up my bid out of spite 🙂

  • 12 Dan L // Mar 19, 2009 at 11:47 pm

    Sorry, I keep trying to change the question into a very different problem (without even adequately describing what that problem is), and then working on my own problem. So let me share my thoughts on the actual question asked.

    For the 2 player version of the original problem, I think that Adam is essentially right if you assume that we have 2 logical risk-averse robots playing the game. Specifically, the robots follow (at least) the following two rules:

    (1) Their bidding strategies never allow them to place bids of $20 or more, under the reasoning that they would never allow themselves to get in a situation with no possible upside. By “bidding strategy,” I mean a complete set of rules for exactly what you will bid based on what your opponent bids.

    (2) They assume that their opponents follow rule 1.

    Under these conditions, the first bidder will bid $19.99 and the auction will stop there. (Quick justification: Each player always has to deal with the possibility that the next bid could be $19.99. Bidding $19.99 is the only way to avoid that.)

    However, I want to point out that rule 1 may not be as innocuous as it looks. It actually smells a little fishy to me.

    Also note that the solution above does not exist in the continuous version of the problem. Also, I do not believe that the trivial solution (no one bids) is stable, assuming logical robots who know that their opponent is also logical (although I’m not sure that’s a meaningful statement). Semi-convincing proof: If the logical thing to do is to bid nothing, then the first bidder knows that the second bidder will bid nothing. It follows that the first bidder knows that he will not be outbid, and therefore he should bid something. Contradiction.

    As an aside, does this discussion really have anything to do with incentives research? I mean, in real life, I seriously doubt anyone behaves even remotely like logical robots. I consider myself logical, and I almost never behave like a logical robot. Do these gedanken experiments actually help to construct real incentive schemes? I would think that there’s no substitute for hard data. If the theory really does fit the data, then I guess you could use the theory to extrapolate. But the extrapolation should still be modest. For example, this weird $20 auction could be a stand-in for a workplace contest, in which the bids are labor, and he who labors most wins a prize or bonus. Economically, the problems are nearly equivalent, but I would not be surprised if they were induced significantly different behaviors.

  • 13 AlanS // Mar 20, 2009 at 10:53 am

    I believe the teaching point with this exercise is that it all depends on what the other bidders are doing. This answer cannot be provided in a vacuum, just like incentives.

    It seems to me that with a large number of bidders, the bidding would regularly exceed $20 by only two bidders. Once the bids get right near $20 it becomes less about potential profit and more about potential loss.

    For example, let’s say bidder A bids $19.98 (assuming no partial cents for ease here), and bidder B bids $19.99 for the potential profit of $.01. If bidding stopped now, bidder A would loose $19.98, so it is in his best interest to bid $20, so he would lose nothing. Then it is in B’s interest to bid $20.01, so he would only lose $.01. Now the game has switched. The one with the ability to bid the most stands to lose the least.

    I can see the comparison to incentives now. They cannot be considered in a vacuum. The profit or the loss potential of the alternative always has to be considered. If the resource is worthwhile, there is always going to be an alternative course of action with its own consequences.

  • 14 Vincent // Mar 21, 2009 at 11:28 am

    If you think about it in the context of the auto industry, If you are going broke and there is no bailout you cut your losses and shut down (don’t start bidding). If there is a possibility for a bailout then you keep digging your hole (You start the bidding). When it looks like the hole you dig will be deeper than the bailout can fill (You bid $20). You start to hope for more bailouts. (You bid over $20) because if there is no second bailout you are broke either way. It seems to me that a company does not have the same self preservation as an individual. But if the individual is broke and is allowed to start bidding than why stop? If there needs to be fewer auto companies but they are all broke which one stops and quits seeking a bailout?
    Kinda convoluted but …..

  • 15 Eric Napier // Mar 22, 2009 at 10:00 pm

    Sounds like the dollar auction, covered well in Prisoner’s Dilemma by William Poundstone. If you’re playing with rational bidders, an open of $19.99 is sensible.

  • 16 Dan L // Mar 24, 2009 at 10:12 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 details of the bidding history are completely irrelevant. (The biggest change here is the issue of tie-bids, but this is rare and may not affect the solution.)

  • 17 Dan L // Mar 24, 2009 at 10:14 am

    Please delete this comment and the one above. It was supposed to go under a different post.

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