The Gift of the Magi

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Not only does O Henry’s The Gift of the Magi tell us about the spirit of Christmas – its handy for economics students as well.

One of the most poignant of Christmas stories is O. Henry’s The Gift of the Magi. An impecunious New York couple, Jim and Della, have little but their love to warm their Christmas. Della has stunning locks of brown hair, and Jim’s prize possession is the gold watch he inherited from his grandfather. Della sells her hair to buy a chain for Jim’s watch. But, unbeknown to her, Jim has sold his watch to buy combs for Della’s hair. Their generous impulses lead to disaster.

Della and Jim were doubly unlucky, since when O. Henry wrote, John von Neumann and Oskar Morgenstern had not yet invented game theory. If Della or Jim had attended Princeton or MIT more recently, they would have had a much better understanding of the dilemma they faced. The Gift of the Magi exemplifies a class of non-co-operative games in which individually rational actions fail to bring about necessary co-ordination. The success of Jim’s strategy requires that Della retain her glorious hair. The success of Della’s strategy requires that Jim retain his treasured watch. But since Jim and Della are independent agents, neither can guarantee these outcomes.

Modern Americans such as Adrian Fastow and Bernie Ebbers would not fall into such a trap since it is the altruism of Jim and Della that creates the difficulty. But – and this is where the mathematics is really useful – altruism is not the root of the problem. Co-ordination problems can arise even if individuals have entirely selfish motivations.

Take the decision as to whether to drive on the left or right of the road. It does not matter what the answer is, so long as everyone reaches the same answer. But without some explicit mechanism of co-ordination, there is no reason to think they will. If Della and Jim had tried to use mobile phones to keep in touch, they might not have succeed ed: fragmented development in the US has not established a common network standard. Couples in London, Paris, Sydney or Tokyo find it easier thanks to the imposition of the GSM standard.

Regulatory solutions are often the best way to bring about co-ordination. The law tells us which side of the road to drive on, and few of us find this intrusion on our freedom burdensome. The imposed solution is stable.

But no regulation could have solved the dilemma of Della and Jim. They kept their plans from each other, hoping the element of surprise would enhance their joy. Yet communication would not necessarily have helped. If they had told each other what they planned, the difficulty of finding a mutually consistent outcome would simply have been transferred from their actions to their conversation. Jim would have insisted on making his sacrifice for Della, and Della would have done the same for Jim. It is not obvious that the outcome would have been different, or better.

In the world of non-co-operative game theory, each individual pursues the best available outcome, expecting everyone else to behave in the same way. This is the Nash equilibrium, as revealed in the Oscar-winning film of A Beautiful Mind. The difficulty revealed in The Gift of the Magi is that there may be several Nash eq uilibria. Because no one can be sure which is the more likely outcome, none of them may actually be realised.

Yet O. Henry knew much more about life than the game theorists. The economists who have analysed The Gift of the Magi assume the result was a tragedy. But it was not. The sacrifice each made for the other deepened the love, which was more important than even the few material possessions that Della and Jim enjoyed.

“The Magi,” wrote O. Henry, “were wise men. Being wise, their gifts were no doubt wise ones, possibly bearing the privilege of exchange and duplication.” The modern magi would have been expert in non-co-operative game theory utility and devotees of rational expectations, maximisers of shareholder value. But Jim, Della and O. Henry came from a different world. As the story ends: ‘In a last word to the wise of these days let it be said that of all who give gifts, these two [Della and Jim] were the wisest. They are the magi’

The thrust of O. Henry’s story is that the emotions and processes that bring material outcomes about may be more important than the outcomes themselves. That is a lesson we relearn each Christmas. Perhaps we should make a new year resolution to remember it for the rest of the year.

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