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.