Believe those who are seeking the truth. Doubt those who find it. Andre Gide

Sunday, March 29, 2009

King Solomon's Dilemma and Behavioral Economics

When the tale of King Solomon's dilemma was first told to me as a kid, I was (like most people, no doubt) left marvelling at Solomon's brilliant solution to a rather difficult predicament.

But then I grew up and made the unfortunate choice of pursuing a graduate degree in economics. My mind was left rotted to the point where I could no longer appreciate what most other people continued to believe was the self-evident wisdom of Solomon.

The problem with Solomon's "solution" is that it adopts what in modern parlance would be labeled a "behavioral approach." In other words, the solution relies heavily on the assumption that people are "irrational" in a particular sense. It turns out to be easy to be a wise philosopher king when one assumes that everyone else is irrational. Perhaps this is why so many aspiring philosopher kings today want to replace conventional economic theory with what they call "behavioral economics."

Let's think about this. The "mechanism" (game) designed by Solomon proposes to split the baby in two (sounds "fair" at least). One women screams out "No! Let the other have the whole baby instead." The other woman coldly agrees to the solution. The real mother is revealed in the obvious manner. What is not so obvious is why the false mother could not have anticipated this outcome; a more clever woman would have simply mimicked the behavior of the true mother. Instead, the false mother fails to make this calculation (and instead adopts a simple "behavioral" strategy; which is just a fancy label for irrational behavior).

Now, perhaps there really are "irrational" people like the false mother. But would you be willing to stake a baby's life on this assumption? Even if this mechanism worked out one time, could we reasonably expect it to work in the future (would people not learn from the outcome and tailor their strategies accordingly?). If you believe that people are fundamentally irrational in this sense, then you will make a fine behavioral economist (and a poor philosopher king).

So what is the solution to Solomon's dilemma?

One approach might be to adopt the Coase theorem, which states that if transaction costs are zero, then an arbitrary assignment of property rights will lead to the efficient solution. That is, Solomon could just have assigned the baby at random to one or the other woman. If it fell into the hands of the false mother, the true mother (who presumably values the baby more) could then purchase the baby (from the one who values it less). In other words, if there are gains to trade (as would obviously exist in this case), then these gains will be realized--if transaction costs are zero.

The problem with this approach is that transaction costs are obviously not zero (these costs could arise, for example, if the true value of the baby by both women is private information). Moreover, this "solution" violates what most people would consider to be a principle of "fairness" (why should the true mother pay for her own baby?). The Coase theorem is a fascinating theorem, but it should not be applied as a solution to the problem at hand; the theorem simply states what one could expect to happen IF transaction costs are zero. In fact, the Coase theorem should be interpreted as explaining precisely why various institutions emerge to handle the problem of resource allocation in a world where transaction costs are not zero.

One such solution was offered by Solomon. But I have already highlighted the problem with his proposed institution (or mechanism). Another possible solution was offered by William Vickery: a sealed-bid second-price auction (or a Vickery auction). Assume, as seems reasonable in this case, that only the two mothers know the true value they attach to the baby. A Vickery auction would have both mothers submitting sealed bids for the baby. The woman with the highest bid would then win the auction, but pay the second-highest bid.

This solution is clever because the amount that either woman expects to pay is independent of their actual bid. Accordingly, neither one of them have an incentive to misrepresent how much they really value the baby. If the true mother values the baby more, she will win the auction (it would not be rational for the false mother to bid more than what the baby is worth to her).

Clever indeed. But there is still a problem associated with this solution. In particular, it requires that the true mother actually pay for her baby. Leaving issues of "fairness" aside, a more relevant problem may be that this mother does not have the resources to make the requisite payment. (It is absolutely critical that the payment be forthcoming; if Solomon could not credibly commit to collecting the payment, then rational players will understand this limitation and alter their strategies accordingly).

One solution might be to let the women offer themselves as indentured servants. This sounds feasible and has the desirable property that the true mother gets her baby (she would presumably be happy to offer herself as Solomon's servant, if it means getting her baby). While this solution has its drawbacks, it seems to dominate Solomon's solution--something that risks having the baby split in two.

But is it possible to design a mechanism that "does the right thing" without any cost to the true mother? Several solutions have been proposed in the literature; but each with its own peculiar drawbacks. But I recently came across one proposed solution that seems quite clever; see Bid and Guess: A Nested Solution to King Solomon's Dilemma, by Cheng-Zhong Qin of UC Santa Barbara.

The idea as presented in Qin's paper seems a little more complicated than it needs to be (but I could be wrong). The basic idea, as I see it, is to have the women play a "participation game" just before playing a standard Vickery auction. We could set up the mechanism as follows.

First, Solomon informs the women of the Vickery auction that will be used to allocate the baby. Second, he informs each woman that the price of participating in the Vickery auction will be a half-life of servitude in some miserable occupation. The women are then asked to submit envelopes with ballots that are marked "yes" or "no" (yes, I am willing to participate; no I am not). If both women submit "yes," then the Vickery auction is played. If only one woman submits "yes," then the baby is allocated to her for free (the auction is not played). If neither woman submits "yes," then the baby is disposed of in some manner (perhaps in the King's service).

Now, put yourself in the place of first, the true mother and second, the false mother. How would you play the game? Would you say "yes" or "no?"

Theory suggests that the true mother will say "yes" to the participation game (she knows that she will get the baby if the auction is played; she will pay one half-life of servitude for participation, and the other half-life in payment for the baby). Likewise, the false mother will say "no." Why submit to a half-life of servitude when she knows that she will inevitably lose the subsequent auction? The false mother will rationally bow out of the bidding; she will choose not to participate. And the baby is allocated for free to the true mother.

Of course, this assumes that the people playing this game are "rational" in the sense that they understand the rules of the game and in the sense that they can anticipate how others are likely to play it. One of the great strengths of assuming rationality in this form is that the assumption can be applied as a general condition that prevails in any resource allocation problem. Its weakness is that people may not always possess this assumed degree of rationality.

But the alternative--the "behavioral approach"--suffers from an even greater problem. In particular, the policymaker must be aware of precisely how people are irrational in each and every given circumstance (a great loss in generality). There are an infinite number of ways in which people might be irrational; and the behavioral theorist is forced to choose among an infinite number of "behavioral rules" that he or she believes captures this irrationality in a plausible manner. The only hope that a behavioral theorist has for developing a general theory is in discovering that people are irrational in some systematic manner. But if the theorist can identify this systematic pattern of irrationality, it seems hard to know why people cannot discover it for themselves too. But then, it seems clearly in the interest of aspiring philosopher kings prefer to think of themselves as being systematically more rational than the subjects they study.


  1. "In particular, the policymaker must be aware of precisely how people are irrational in each and every given circumstance "

    in fact, isn't this the very definition of the word "Wisdom"?

  2. Gabby: I suppose so. But I prefer the wisdom of Socrates (as relayed to us by Plato): the recognition of the limitations to one's knowledge.

  3. i dunno, i seems more like a sly discussion of spiritual versus material love, than a resource allocation experiment

  4. Hapa: your contribution would have been much more useful had you bothered to offer some support of your views on this matter. An unsubstantiated opinion is not worth very much.

  5. what's to support? a mighty elder in a religious text acts on "the knowledge" that real love is selfless love and everyone applauds his wisdom. hello, parable.

    (are you mad at my comment or my blog? gabby and i had maybe a couple inches difference in utility.)

  6. that was from hapa. the authentication gizmo is hiding my identity!

  7. In King Solomon's story one women is rational and the other is irrational, so King Solomon presents an irrational solution with the foresight that at least one woman is irrational. People who steal babies are not rational. The real mother will not go along with the irrational decision but the false mother (irrational) does. In your scenario, you assume both participants are completely rational which is false.

  8. Love...priceless. For everything else, there's Mastercard.;-)

  9. Gabby said exactly what I want to say, but in real world we would very much like to avoid resorting to a wise ruler. The fact that the story of Solomon stands out shows that we do not have similar wise kings most of the time. Therefore we need wise rules, and not rulers.

    "Of course, this assumes that the people playing this game are "rational" in the sense that they understand the rules of the game and in the sense that they can anticipate how others are likely to play it. One of the great strengths of assuming rationality in this form is that the assumption can be applied as a general condition that prevails in any resource allocation problem. Its weakness is that people may not always possess this assumed degree of rationality."

    Assign them lawyers. Rationality and job creation in one go.

  10. People are systematically irrational, read the book "Predictably Irrational" by Dan Ariely. I don't agree with all his prescriptions, but the analysis is very interesting. I suppose if people knew these tricks they would be less likely to be fooled by them and he even tested that theory in a few of the examples cited, but let's put this in context for a moment -- who doesn't know drug tests use placebos in their studies? Yet it still works. That's no different. That Plato quote is interesting, but what is equally interesting is the extent to which intelligent people irrationally project their rationality on inferior minds. People who only deal with thoughtful people on a daily basis forget the average human is pretty stupid.

  11. i guess we're all a little cranky

  12. More wild-ass conjecture from me..

    .. and this probably isn't allowed because preferences are just assumed to exist, but I can imagine say everyone is endowed with preferences at birth, but the preferences are unsorted. One possible endowment is that we're indifferent to everything at birth. As more information is attained and processed, they are sorted, possibly into tentative positions. As time passes, the preferences become stable. This is not unlike a person growing up and "discovering themselves".

    So it seems to me the question of behavioural versus rational is really a question about where preferences come from and when they become stable? It's plausible that it would be hard to predict someone whose preferences are still stabilizing. It would be hard to model, but doesn't invalidate it, and this wouldn't invalidate rationality since at some point it stabilizes, conditional on information acquisition and processing. Is this right? Maybe I'm missing something.

  13. A fascinating problem !
    I may sound a little pedantic, but William Vickery was Canadian-born and his name was actually Guillaume Vicquerie, though his name is generally anglicized (sparing us the pain to speak of Vicquerie auctions, Vicquerie payoffs and so on). Truth should be restored however : imagine Italian economists come to dominate the field and we all have to speak of Giuseppe Stigelizze !

  14. One unsatisfying point about the proposed rational solution is that in order for it to deliver the "fair" outcome for sure, it assumes that the true mother knows the baby is hers. If, on the other hand, she is uncertain (few-day-old babies look alike) - and her uncertainty may increase if the false mother puts on a great show of emotion - she may chose not to participate in the auction, and the baby may end up with the false (but sufficiently convincing/hysterical) mother.

  15. funny essay David, I was imagining you as a child, hearing your alternate-universe King Solomon story about how he designed a second-price auction, asked for sealed bids and awarded to the highest bidder at the second highest price...

    definitely wouldn't have made a story to be remembered and told to children for 1000 years (or whatever) - maybe that's why those behavioral papers get published in QJE these days and reported in popular press?

    (btw do VICKREY auctions work with 2 players?) and yes, his name is actually Vickrey and he's a Victoria BC native.

  16. Let's clarify a few things.

    First, Solomon knew in advance to whom he wished to allocate the baby. The true mother. The 'revelation mechanism' is designed to reveal information where one party is being less than fully forthcoming.

    Second, in the real world, society has decided to strongly discourage markets in babies. Why? For "behavioural reasons"? Because humans are intrinsically irrational? Because actual markets would fail to recognize full intrinsic value?

    Or because markets in babies would cause a whole series of other problems: commitment to many years of nurturing, security.

    This argument should strike a chord with David because he offered a brilliant theoretical explanation as to why social welfare recipients cannot borrow against future streams of social assistance in a JPE article entitled A Theory of Inalienable Property Rights, Journal of Political Economy, April 2002, 110(2):382-393

    Now I understand the danger of assuming that people are infinitely irrational, or perhaps better said, incapable of updating their information set.

    Let's take tobacco smoking parents of children, for example. Assume the parent or parents in question are well educated so they understand the risks and dangers associated with one of the most deadly, highly-addictive recreational drugs in common use. Clearly, such a habit strongly signals a less than firm commitment to the children.

    Suppose we decide to punish these egotistical parents by confining them temporarily to a medieval stockade where they alternatively receive rotten tomatoes and Singapore-style caning as punishment.


    Clearly there is no point. As all well socialized economists know De gustibus no es disputandum. (Tastes are beyond dispute.) Well-educated tobacco-smoking parents will not change. They may simply alter their behaviour to avoid detection.

    The other alternative is to confiscate children of tobacco-smoking parents and hand them over to the state. But that could turn out horribly disruptive and expensive. There is no guarantee that a non-tobacco smoking employee of the state would make a better parent than a tobacco smoker.

    So the state offers a series of other partial solutions. High taxes, which are routinely avoided through trafficking. Regulated graphic, ugly packaging that smokers can choose to ignore. Hard to enforce restrictions on minors purchasing tobacco.

    The obstacles and difficulties should make everybody want to throw up their hands in despair and immediately seek employment as tenured profs in well-financed para-public universities.

    Ultimately through all of this, there is a discussion that occurs at the socio-political level and slowly but surely fewer and fewer people, included well educated parents consume tobacco.

    Si, se puede.

  17. here's my rendition of the king solomon "game"

    Assume the following:
    1. both mothers are risk-averse expected utility maximizers
    2. the 'false' mother has a Bernouli utility over the baby, i.e. as if it is divisible (like money), that is U(b)>U(1/2b)>0 where b is the baby. The 'true' mother instead has U(b)>0=U(1/2b).
    3. each mother has only 2 strategies: accept (A) the baby division or reject it (R)
    4. the payoffs are as follows: if (A,A) then each gets 1/2b; if one or both mother disagrees then they assume (or the king tells them) that they'll get a lottery with which they get b with prob. 1/2 and 0 otherwise.
    5. (this is a technicality to assume away indifference) - the false mother thinks the other player may play A with some positive (could be very small) probability.
    6. this is a one shot game and the mothers don't know much about the king

    now let's solve it: for the true mother the dominating strategy is to play R (as the fable suggests). the false mother (she's risk averse) prefers 1/2b for sure to a lottery with the same expected payoff, thus a (strictly if assumption 5) or weakly otherwise dominating strategy for her is to play A (as the fable suggests).

    now the King (like all rulers) reveals his lack of commitment (i.e. he does not offer the promised/assumed lottery) and instead awards the baby to the (revealed) true mother.

  18. Alex:

    I suppose this will work, although it violates the standard (possibly inappropriate) assumption of common knowledge. In particular, you assume that the women do not understand that the king lacks commitment.

    Your argument could be made consistent with full rationality by assuming that the women have a prior defined over the king's commitment power; and that their prior is that he can commit.

    Of course, the king's solution can only work once. At least, if we assume that people will update their beliefs concerning the king's apparent lack of commitment.

  19. I agree, I did say I assume a one-shot game and yes, this is the prior the women start with.

    and how do you know it didn't work only once indeed, I haven't heard that he did this many times with many women :-)

  20. Well, it has been my experience that women learn very quickly and they never forget.

    But then again, I'm no King Solomon!

  21. Mr. Alexander: Maybe I am misunderstanding, but in your "game", is it crucial that the false mother have utility for 1/2 a baby? So this only works if the King knows there is one of the two women who have such a utility?

  22. Pani - yes, correct. But I thought that was implicit in the king Solomon story - the king knew there was one woman of each type - he just needed to figure out who is who. This is called a 'screening' problem in economics.

  23. Oh I see. Thank you. I'm getting so much out of this blog thread!

  24. Alex: Mixed strategies where there is some a priori probability of the type of King who would chop the baby in half might be sufficient to support the socially desired outcome.

    It would correspond to situations where authorities fail to commit credibly to enforcement. Enforce on occasion.

  25. is it essential that the participation fee f be so great? wouldn't any f>0 work?

  26. The problem with the proposed solutions is that the two women are bidding on different things. One woman is bidding on the life of her child, the other is bidding on destroying the life of the true mother. As long as the true mother does not receive the child, the false mother gets what she wants hence being ok with killing the child. I believe the best test would be to give the child at random to either of the mothers and measuring the quality of response to the child by both. Then reassigning the child to the mother with the most attachment to the child.

  27. If I was king solomon I would simply take the baby to a back room, come back out and tell the two women that I killed the baby. The real mother would scream and cry while the false mother wouldnt care at all. When the scene would cool down I would bring the screaming true mother to the back to show her her Living baby. NO ONE WOULD LEARN FROM MY JUDGMENT because it would be between me and the true mother. Everyone else would think the baby to be truly dead. Another patient strategy would be to keep the baby until the baby favors the appearance of the True mother :-)