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Ayale-com.419 net.math utzoo!decvax!yale-com!wagar Wed Apr 21 16:41:55 1982 Resolving Newcombe's Paradox

Given a choice between B or A+B, some people choose B, and some choose both. Everyone already knows they are right, and they really are, but only because they are answering different questions.

The crux of the problem is a failure to define terms. The term 'maximize' is just thrown out, with the expectation that people will define it to suit themselves. Having done this, they cannot understand how anyone could choose differently.

Consider the following equivalent paradox:

The probability of rain is 30% today. Should you a) carry an umbrella b) not carry an umbrella Maximize your gain.

Some people will choose a, some b. Neither is 'right', because, for some people, 'maximize' means "don't get wet", while for others it means "don't carry an umbrella if you don't have to". In Newcombe's paradox, the choice the listener implicitly makes is between breaking the law of cause and effect and breaking the laws of probability. Neither of these is actually a law, of course. Both, however, seem to be, because in 'real' situations they always seem to hold. But since one law must get broken to answer the question, people decide which one they can't bear to break, and then go ahead and ignore the other, allowing Newcombe to laugh in everyone's face.

As soon as the definition of 'maximize' is provided in the problem, the right choice is obvious, and the paradox vanishes. Suppose one makes no assumptions about these so-called 'natural laws'. Then maximize has no meaning. One can choose B, or one can choose both, and one receives some quantity of money as a result. With either choice, several things can happen, and the amount of money you have at the end of the day could be greater with either. You have absolutely no basis for deciding which of the possibilities will happen, and you can't refer to 'average gains' because average has no meaning, either.

Suppose, however, it is agreed that maximize is to be interpreted under the assumption: "the course of a set of events sets a pattern for the course of future events." Then everyone chooses box B. Suppose it instead means, "assume that only related events can have an effect on each other, i.e. given 'independent' acts a and b, whether b is performed separately, or with a, its outcome will be the same." Then everyone chooses both boxes.

The moral is, "Don't take the obvious for granted in a controversy." Every real controversy I've ever seen can be resolved simply by insuring that both parties agree on the definitions of the terms they are using. Otherwise, they aren't even arguing about the same thing (controversies based on differing reasoning schemes are also possible, but I've never seen a serious one.) There is one crucial 'exception' that clouds matters, and that is that people enter into disagreements with different definitions for the purpose of life. This is where net.theology comes in, and while such discussion can affect people's ideas on this matter, any definition for the purpose of life is 'right', because it is true by definition. A quiz to emphasize this:

The purpose of life is: a) life, liberty, and the pursuit of happiness, in appropriate ratios. b) maximize probability of survival of the individual and the species, in appropriate ratios. c) to build an ordered state based on Communism or some other utopian philosophy.

Now, knowing all this, should you take box B or both? A logical approach is the following. Bring yourself to accept the fact that these two laws of nature will be revoked during the choosing period. Consider whatever happens then to be magical, and hope the the 'law of nature' will be restored when the choosing is over so that you can spend your money. The only basis for choice then is considering how people have fared with each choice, assuming both laws are restored after choosing. One notices that A+B people have about $2,000, and B people have about $999,000. The choice is now obvious.

-Steve Wagar

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