The Dilemma of a Postmodern King Solomon

Divinely intelligent people don’t need to study economics. King Solomon, of biblical fame, solved a timeless economic problem having no training at all. The problem is described in the famous account from 1 Kings 3:5-14, and is often told to illustrate Solomon’s great wisdom. As the story goes, two women appear before Solomon with a baby boy, both claiming to be his true mother. To resolve the dispute, Solomon calls for a sword to be brought before him and decrees that the child be cut in half with one half going to each woman. Upon hearing the brutal verdict, the true mother shrieks and insists that the child be given to the other woman rather than have him meet such a cruel fate. Solomon immediately awards the baby to the distraught woman, as only the child’s true mother would rather he be raised by another woman than have any harm inflicted upon him.

The brilliance of Solomon’s tactic is evident in its ability to identify the woman who values the child most who in this case is also the child’s true mother. The economic task of assigning an indivisible object to the party that values it most remains a necessity to this day. Examples include the assignment of equipment among employees, the granting of exclusivity over an article of intellectual property, or the allocation of a marital asset of sentimental value during divorce proceedings. In such settings, it is often the case that the involved parties know the identity of the one who values the disputed object most, but the arbiter or decision maker does not.

For this exercise, let’s consider the allocation of a family heirloom whose ownership is disputed amongst two siblings, a brother and a sister. Solomon’s tactic is useless to us since anyone familiar with the story will know how best to respond, and we’re left with the dispute still intact. It is true that Newman did successfully employ Solomon’s tactic to resolve a dispute over a bike in an episode of Seinfeld, but his success relied on the ignorance of Elaine. For this reason, we need improve upon Solomon’s tactic and devise a mechanism to allocate efficiently (read “to the party who values it most”) the heirloom even when the parties to the dispute anticipate that the mechanism will be used.

Invoking the Nobel-Prize-worthy theory of Ronald Coase, proponents of the Chicago School of economics would argue that this dilemma is no dilemma at all as we can simply grant ownership at random. The efficient allocation will then be arrived at through mutually-beneficial trade. To see this, suppose that by the flip of a coin, the heirloom is awarded to the un-deserving heir. Since the deserving heir values the heirloom more than her sibling, she can offer her brother enough to make trade worthwhile so the object ultimately ends up in her hands. In the event the coin flip favors the deserving heir, the un-deserving heir will be unable to sufficiently compensate his sister for giving up the heirloom and it will stay in the hands of his more deserving sister. Either way, the heirloom ends up in the hands of the deserving sister and our objective has been met.

The problem is, when bargaining is required to achieve the efficient allocation, both sides have an incentive to misrepresent their willingness to pay and willingness to accept respectively. Agreement may only be reached after significant delay and at a greater expense to all involved. Since it requires agreement between two sides already prone to disagreement, random assignment is really no solution at all.

A better approach would assign the heirloom without any additional interaction between the disputing parties. We might try asking each side, in private, how much the heirloom is worth to them, but how would we know if we are getting an honest answer? In an artful use of game theory, William Vickrey devised a mechanism called a second-price auction that provides the correct incentive for both parties to truthfully reveal their valuation. The second-price auction works like the familiar sealed-bid auction except that instead of paying the proposed bid, the winning bidder pays an amount equal to the next highest bid. Vickrey showed that each party, when required to participate in a second-price auction, can do no better than to bid their respective valuation regardless of what they think the other will bid. To see this, consider the bidding decision of either sibling, say the sister. Since the price she pays upon winning is independent of her bid, her bid serves as the maximum price at which she can still win the auction. At any price above her valuation, she prefers to lose the auction since her payment would exceed here valuation. At any price below her valuation she prefers to win since her valuation would exceed her payment. By bidding her valuation, she is guaranteed to always win when doing so makes her better off and always lose when winning would have made her worse off.

By having the siblings participate in a second-price auction, the party with the greatest value for the heirloom will place the highest bid and be awarded the heirloom. This is efficient except when the deserving party is constrained to bid no more than her income allows and she loses to her less-deserving brother. To bypass this problem, we need an allocation rule that does not involve any form of payment. The trick is to screen out the un-deserving party by having the disputing parties play a game designed in such a way so as to be unprofitable only for the un-deserving party. If the un-deserving party, who knows he is un-deserving, refuses to participate in such a game, then the deserving party identifies herself as such by indicating her willingness to participate. And with one side to the disagreement dropping his case, the game is never actually played, so nary a payment is ever made.

In a paper to be presented next month at the International Conference on Game Theory, professors Cheng-Zhong Qin & Chun-Lei Yang of UC Santa Barbara and Academia Sinica in Taipei respectively design such a game. Their mechanism, dubbed “bid and guess,” is a variant on the celebrated second-price auction. In this game, both parties are asked to submit sealed bids along with a guess of the other party’s bid. When the envelopes are opened by the arbiter, the rules of the game determine who gets the item, how much they pay, and the size of the entry fee paid by each participant. As in the second-price auction, the high bidder gets the item and pays an amount equal the next-highest bid. The size of the entry fee is calculated to be some fraction of the difference between the bidder’s guess and his opponents actual bid. To insure that income does not constrain the bidders, the arbiter can offer either party a loan at no interest.

Qin & Yang show that within the bid-and-guess game, both parties can do no better than to bid their respective valuations and make a guess that has them paying a positive entry fee. Since the less deserving party necessarily loses the auction while still incurring an entry fee, he finds it unprofitable to participate at all. Thus, if he knows the bid and guess mechanism will be employed by the arbiter, he will drop the dispute before it ever gets to that point. The deserving party will nonetheless agree to participate since her total payment including the entry fee can never exceed her valuation. Using this mechanism, a hypothetical Solomon can resolve all property disputes efficiently without any divine knowledge of human nature.

Now I know you’re saying, “No one will ever use anything as ridiculous as bid and guess to resolve a property dispute,” and you’re right. But that’s not the point. The point is, by understanding the incentives created by various mechanisms, you can understand the logic behind the institutional arrangements that we see in the real world. A good example of this way of thinking is in the allocation of spectrum licenses by the FCC to wireless service providers. From the standpoint of efficiency, licenses should be awarded to those companies capable of providing the best service at the lowest cost. Since these companies are likely to earn the greatest profits, they should also place the greatest value on owning a license. Thus, the problem facing the FCC is similar to that of the arbiter in the dispute over the heirloom. Initially, administrative hearings were convened to identify the companies most deserving of the licenses. When this process became too cumbersome, lotteries were instituted. Since entry was free and open to the public, these lotteries attracted many non-serious applicants giving rise the type of problems already discussed with respect to random assignment. The economically sensible solution came in 1994 with the implementation of an auction mechanism that awarded each license to the highest bidder with the added feature of permitting companies to bid on multiple licenses simultaneously. By allowing companies to bid on several licenses at once, the simultaneous ascending auction made it possible for companies to take advantage of cost savings incurred when providing cellular service to Tacoma is cheaper for someone also providing service to Seattle. These auctions have allocated licenses faster than ever before while raising millions for the federal government. And who did the FCC consult to design such auctions? Why mathematical economists of course.