Tactics To Reveal Truth

from Solomon to Clarke

Then came there two women, that were harlots, unto the king, and stood before him.

And the one woman said, O my lord, I and this woman dwell in one house; and I was delivered of a child with her in the house.

And it came to pass the third day after that I was delivered, that this woman was delivered also: and we were together; there was no stranger with us in the house, save we two in the house.

And this woman’s child died in the night; because she overlaid it.

And she arose at midnight, and took my son from beside me, while thine handmaid slept, and laid it in her bosom, and laid her dead child in my bosom.

And when I rose in the morning to give my child suck, behold, it was dead: but when I had considered it in the morning, behold, it was not my son, which I did bear.

And the other woman said, Nay; but the living is my son, and the dead is thy son. And this said, No; but the dead is thy son, and the living is my son. Thus they spake before the king.

Then said the king, The one saith, This is my son that liveth, and thy son is the dead: and the other saith, Nay; but thy son is the dead, and my son is the living.

And the king said, Bring me a sword. And they brought a sword before the king.

And the king said, Divide the living child in two, and give half to the one, and half to the other.

Then spake the woman whose the living child was unto the king, for her bowels yearned upon her son, and she said, O my lord, give her the living child, and in no wise slay it. But the other said, Let it be neither mine nor thine, but divide it.

Then the king answered and said, Give her the living child, and in no wise slay it: she is the mother thereof. (1 Kings 3:16-27, King James Version)

The problem is how to get someone to reveal by his actions information he might want to keep secret. Solomon’s threat to divide the baby is a famous solution but not a very good one because it depends on the second woman not realizing what the King is up to. If she had realized it she could, presumably would, have given the same answer as the first woman, providing Solomon with no way of knowing which one the child really belonged to.

For a better solution to the same problem in a different context, consider the problem of finding out how much property is worth in order to know how much tax it should pay. If the tax collector simply asks the owner it is in his interest to lie, to understate the value in order to reduce the tax. The solution is the self-assessed property tax. The owner must state a value for the property to be taxed at. He is then obliged to sell the property at that price to any buyer willing to pay it.

A number of objections may be made to that solution, most obviously that if the property is worth more to the present owner than its market value, perhaps for sentimental reasons, he will have to either pay tax on the higher value or risk having to sell at a price at which he doesn’t want to. That raises the interesting question of why it feels to many people as though it is just to tax someone on the market value of the property, its value to other people, but not on the additional value to the present occupant.

One advantage of the self-assessed property tax is that it solves the problem of assembling a parcel of land for a large project such as a highway or shopping center and so eliminates most of the usual justification for eminent domain, the process by which a government takes land without the assent of the owner, paying him what it claims to be the fair value. With a self-assessed property tax all parcels of land already have a price set by the owner, so the owner of a parcel needed to complete the project cannot take advantage of the situation to hold out for a high price.

My favorite historical example of an approach to the problem comes from Periclean Athens. Its solution to the problem of producing public goods — paying for its navy, for instance — was that each of the richest Athenians was obliged to produce one public good every other year, with the particular one assigned to him by the relevant official.

“You have heard that we are sending a team to the Olympics this year? Congratulations; you are the sponsor.”

Or

“See that lovely trireme at the dock. Guess who is paying her expenses this year?”

There were two ways of getting out of it. One was to show that you had already been assigned a different public good for this year or had done one in the previous year. The other was to show that there was another Athenian who was richer than you were but had not been assigned a public good for this year or done one in the previous year.

How, in Athens of 400 B.C., with no accountants, no banks, no stockbrokers, no IRS, do I prove that you are richer than I am? [For the Athenian solution, click here.]

A More Important Example

Economic efficiency, discussed in an earlier post, is increased by any change that increases total value, summed over everyone, where the value of an outcome to someone is defined by how much he is willing to pay for it.¹ You can imagine asking people how much they valued an apple and giving it to whomever said he valued it most, but that would give anyone who wanted the apple an incentive to claim to want it very much. Instead you allow apples to be sold. The apple goes to whomever values it most since he is willing to outbid anyone else. The market mechanism is a way of making it in the interest of individuals to reveal the information, their value for things, needed to produce an efficient outcome.

That is the essential insight underlying proofs of the efficiency of a competitive market. It does not solve the problem for decisions that cannot be put in that framework, such as who should be elected president. Even if we were willing to let individuals bid to have their preferred candidate elected, the sum of the amounts offered would not be the sum of the values to bidders of getting their preferred candidate. How much you are willing to pay for an apple is its value to you. How much you are willing to pay for a candidate measures his value to you times the (very small) chance that your bid will make the difference between his winning and losing.

There is an ingenious approach to solving that problem, of getting every voter to truthfully reveal the value to him of his preferred outcome, a Clarke tax. It works as follows for a decision with two possible outcomes such as a presidential election in a two party system:

Each voter states his value for his preferred alternative. The values are added up and the alternative with the largest total value is chosen. You then calculate, for each voter on the winning side, whether without his vote the other side would have won. If so, you tax him the amount of his vote that was needed to prevent that. If he claimed a value of five hundred dollars and without that the other side would have won by a hundred he owes a Clarke tax of a hundred dollars.

Suppose you actually value your preferred outcome at fifty dollars. Claiming a larger value instead only changes the outcome if, without your vote, your side would have lost by more than fifty. But if so, you end up paying more than fifty for your preferred outcome, which means you are paying more to get it than it is worth to you, like buying an apple for two dollars when it is only worth one to you. The argument works in the other direction as well; claiming a value of twenty dollars instead of fifty only changes the outcome if fifty dollars would have given you your preferred outcome and twenty does not, in which case claiming fifty would have given you a fifty dollar value for a tax of at most thirty. It follows that it is in your interest to tell the truth about how much you value your preferred outcome.

It is a very clever idea but it has a lethal flaw. I will let you see if you can spot it.

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1. This is value which, as I am about to demonstrate, is observable, not utility, which is not. I discuss the connection between the two in this post which, I think, covers the most obvious objections to the idea.