Saturday, April 4, 2009

More math

Last Sunday I wrote a short piece on three concepts from mathematics that are frequently misunderstood, but have a great effect on our understanding of current events and what people are saying and writing about them. For the record, those are: so-called exponential growth, curve fitting, and nonlinearity.

It's occurred to me that I left one important math misunderstanding out, so I'll swing back and put in here. That concept is one from game theory, the idea of zero-sum vs. positive-sum vs. negative-sum games. These are words that are thrown around a lot, and, as invariably occurs when amateurs pick up the terms of a specialized field, they are abused a lot as well.

In short, the nature of the sum (zero, positive, negative) comes from the inherent nature of the game being played, and refers to the total resources of the players. (It's important to stress that the word "game" here has a specialized meaning, not restricted to fun activities, but to any human endeavor that has an uncertain outcome.)

Casino games are negative-sum games, at least for the players, because they never return as much to the players as the players put in. Two important points here: 1) Obviously, if one counts the casino as a player in the game, the money stays the same - it must be a zero-sum game - but it is customary to exclude the rule-setter as a player; and 2) The game is the whole system between player and casino, so it is not a refutation to point out that some players do walk away ahead.

Private poker games are, generally, zero-sum games. Each of five people walk in with $100, and, at the end of the evening, $500 walks out the door; we just don't know ahead of time in whose hands it will be. It is not impossible, but fairly unlikely, that everyone will leave with exactly what they came in with. That's why we don't see poker as a productive activity, even though in the usual case one or two people do walk away richer.

Now let's take the poker game, but add the idea that some wealthy person has decided to subsidize the action, say by throwing an extra dollar into each pot. If the players engage in 100 hands during the evening, the total they will walk away with will be $600. We have just defined a positive-sum game, one in which, theoretically, every player could come away better off (even if this doesn't ever actually happen).

Here's the thing. While everyone might come out better off, it's more likely that one person will walk away with $600, the other four all losing $100. Telling the latter four that it's a positive-sum game doesn't really do anything for them at all, they still lost all their money. In fact, they may even resent this game more, because there was an extra $100 up for grabs and they got none of it.

I'll give just one example of the way this is used in political and economic discussion, but there are others. Very often, someone will support free trade by saying, usually pretty smugly in an attempt to forestall further discussion, that free trade is a positive-sum game. The clear inference we're to draw is that everyone will be better off. But that's not what "positive-sum" means at all, it only means there will be more wealth in the system than there otherwise would have been - that's all.

It's the old question of growing the pie vs. splitting the pie, which is more of a political issue than an economic one. Free trade, for all of its pie-growing capabilities, is not "good" if it brings unacceptable distributional effects. Whichever you come down on that issue, it is clearly more complicated than citing "positive-sum" and sitting back, convinced the discussion is over. We need to be smarter than that.

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