Marriage Markets
Babylon, Baghdad, Blacksburg
Once a year in every village all the maidens as they came to marriageable age were collected and brought together into one place, with a crowd of men standing round. Then a crier would display and offer them for sale one by one, first the fairest of all; and then when she had fetched a great price he put up for sale the next comeliest, selling all the maidens as lawful wives. Rich men of Assyria who desired to marry would outbid each other for the fairest; the commonalty, who desired to marry and cared nothing for beauty could take the ill-favoured damsels and money therewith; for when the crier had sold all the comeliest, he would put up her that was least beautiful, or crippled, and offer her to whosoever would take her to wife for the least sum, till she fell to him who promised to accept least; the money came from the sale of the comely damsels, and so they paid the dowry of the ill-favoured and the cripples. But a man might not give his daughter in marriage to whomsoever he would, nor might he that brought the girl take her away without giving security that he would indeed make her his wife. And if the two could not agree, it was a law that the money be returned. Men might also come from other villages to buy if they so desired. (Herodotus 1.196 on the Babylonian marriage market
People talk a lot about inequality of income but there are other forms of inequality. In a society where most women get married and whom they marry largely determines the rest of their lives, inequality in the characteristics that men value in a wife, most obviously physical attractiveness, may be more important than inequality of wealth. An attractive woman has her choice of husbands, an ugly woman may be unable to get an offer from even one. In a society with bride price and dowry, the parents of an attractive woman can collect a sizable bride price while other parents, if they want to marry off their daughter, may have to provide a sizable dowry.
Herodotus describes an economist’s solution to this particular form of inequality. The attractive woman sells for a high price but the price is paid not to her in the form of a particularly desirable husband or to her parents as bride price but to the pool of cash that will be used to buy husbands for the less attractive brides. Wealth redistribution on the marriage market.
There are two problems. One is that there is nothing to guarantee that the market will clear, that the total collected for the more desirable brides will be the amount needed to buy husbands for the less desirable. It might be less, in which case some remain unmarried. It might be more.
The other problem is more complicated. Pairing up couples is not merely a question of who gets the better partners, who the worse, not solely what economists describe in other contexts as a question of distribution. It is also a matter of who fits with whom. I once estimated that the woman to whom I am still married, more than forty years after I met her, was about a one in a hundred thousand catch — for me. I did not have to bid her away from competing suitors by promising to obey her lightest whim or even to wash all the dishes, because when I met her there were no competing suitors — I was the first man she ever dated. For some odd reason not every man, hearing a woman give a clear and correct explanation of a point in calculus, falls in love on the spot.
Matching up husband and wife is not merely a problem of distribution. It is also a problem of allocation, getting the pairing right, forming couples each member of which has the characteristics the other values. The Babylonian marriage market does half the job, since how much each man is willing to bid for a particular bride depends in part on how well she matches what he wants. But there is no bidding from the other side, nothing to make it easier for a woman to end up with the husband she prefers, although the requirement that the transaction be cancelled “if the two could not agree” sets a limit to how much she disprefers him. The auction makes sense only in a world where every woman has about the same preferences in potential marriage partners.
Contrast that to the way we do it. Marriage is by mutual assent of the partners. Each marriage includes an implicit price in the form of its implicit terms, how much each is expected to do, how much control each will have over joint decisions such as where to live or how many children to have and how to bring them up. If I value the woman I am courting more than my rivals I will be willing to offer her a higher price for her hand. If she values me more than she values them, she will be willing to accept a lower price from me than from them. Seen through the simplifying lens of economics it is a market transaction; like other market transactions, it tends to allocate in a way that maximizes the summed value to the parties. But like other market transactions and unlike the Babylonian auction, it does nothing to eliminate the inequality that comes from the fact that some people are more desirable marriage partners than others.
The market model of marriage has, however, at least one serious problem, illustrated in the next story.
200 Dinars
A certain vizier was going about the country taking petitions, giving alms, dealing with what must be dealt with. A poor woman came to him with a petition for alms. He gave her an order for two hundred dinars, which she took to his paymaster. The paymaster, astonished that his master would have ordered so large a sum, took the writing to the Vizier, asking it if was in truth his. The vizier answered that it was. He had intended to write two hundred dirhem, but since it was the will of Allah that he wrote dinar for dirham, gold for silver, the money should be paid out as it was written.
Some days later, he received a petition from a poor man, saying that since the Vizier had given his wife two hundred dinars she now considered herself too rich to be married to a poor man like him and was threatening to force him to divorce her. He asked that the vizier would appoint someone in authority to keep her from doing so.
The vizier considered the matter briefly and then wrote out an order to pay the man two hundred dinar. (Based on an account in The Table Talk of a Mesopotamian Judge, a translation by D. S. Margoulieth of a book written by al-Muhassin ibn Ali al-Tanukhi in the tenth century.)
Under Islamic law a husband could divorce his wife but a wife could not divorce her husband. The wife in the story, having received a windfall due to the Vizier’s carelessness in giving her gold instead of silver — a dinar was worth about ten times as much as a dirhem — wants to get rid of her husband, still poor. She cannot divorce him. She can however make being married to her sufficiently unpleasant to force him to divorce her. For the Vizier, enormously more wealthy and powerful, a matching donation to the husband is an easier solution than trying to force the wife to maintain her side of the implicit contract of her marriage.
The story, written and set in the Islamic world of a thousand years ago, reveals a problem with the market model of our system for allocating marriage partners. A man can try to persuade a woman to marry him by offering her especially favorable terms, promising to wash all the dishes and live near her friends and family instead of his, and similarly for the same situation with partners reversed. But once they are married many of the terms of the contract cannot be enforced even if the authorities want to enforce them, because performance cannot be observed from outside the marriage. No wife, so far as I know, has ever gotten more favorable terms in a divorce settlement by showing that her husband was less deferential to her preferences than he had promised. No husband, so far as I know, has ever gotten more favorable terms by arguing to the court that his wife deliberately cooked, or made love, badly. There is still the possibility of enforcing the terms by the threat to leave the marriage, at least in a society with easy divorce. But that threat becomes less believable as the marriage partners acquire joint assets, most obviously children, best enjoyed jointly.
[For more examples of literary works with economic insight see Embedded Economics]
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As I’m sure you know, Prof. Friedman, this issue inspired a problem: given N eligible heterosexual bachelors and N eligible heterosexual spinsters, each with a preference list of the N potential spouses (spice?), (a) is there a pairing stable against divorce, that is, if A is paired to B but prefers C, then C prefers their spouse D to A (my first unironic use of the singular “their”!), (b) if there is, can it be computed in finite time? in polynomial time? and (c) if so, how—by what algorithm?
And the answers are (a) yes, (b) yes and yes, and (c) by the HIgh School Dance Algorithm, as follows.
For the first dance, each swain asks his favorite maiden. If asked by exactly one swain, a maiden chooses and dances with him. If asked by more than one, she chooses her favorite among them, dances with him, and rejects the rest. All the unchosen swains and maidens sit out the dance.
For every subsequent dance, each swain who danced the previous dance returns to his partner and asks her for this dance too; each swain who didn’t now approaches his favorite maiden who has not explicitly rejected him for some earlier dance and asks her for this dance. If asked by exactly one swain, a maiden chooses and dances with him. If asked by more than one, she chooses her favorite among them, dances with him, and rejects the rest. Her favorite need not be her partner on the previous dance. All the unchosen swains and maidens sit out the dance.
Iterate until everyone is dancing.
It is a simple exercise for the interested student to prove that this algorithm always terminates, does so in polynomial time, and produces a final pairing that is stable against divorce.
"I once estimated that the woman to whom I am still married, more than forty years after I met her, was about a one in a hundred thousand catch — for me."
Amen.