The typical case, in which there may be a practical connection between weight and probable error, may be illustrated by the two cases following of balls drawn from an urn. In each case we require the probability of drawing a white ball ; in the first case we know that the urn contains black and white in equal proportions; in the second case the proportion of each colour is unknown, and each ball is as likely to be black as white. It is evident that in either case the probability of drawing a white ball is 0.5, but that the weight of the argument in favour of this conclusion is greater in the first case.

## Friday, January 23, 2009

### Keynes and Ellsberg's Paradox

Ellsberg's Paradox is a famous conundrum in decision theory. At the Wikipedia website, they noted a reference to Keynes in the Ellsberg entry. As I have seen just about every idea attributed to Keynes, I figured this was a typical overstatement. But then I check the entry, and lo and behold, I think Keynes articulates the Ellsberg paradox pretty well. From Keynes Treatise on Probability:

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## 7 comments:

been reading an interesting article http://www.nytimes.com/2009/01/11/magazine/11Genome-t.html?pagewanted=8

and i find this lil snippet: "the probability of a single event is meaningless". would you agree?

The law of large numbers implies that probabilities are more valuable in rank ordering groups than individuals, but even though it explains groups better than individuals, predictively, you apply the information the same. That is, say you are asserting men are taller than women. As a general statement applied to the mean for women and men, it is absolutely true. As applied to individual women and men, it is true with a probability, say 70%. If you are a tall woman who takes exception to such statements, you should not feel constrained by the averages.

Thus, my average propensity, I would look at like an unconditional bayesian prior. The more I know about my individual situation, the less relevant the bayesian prior. But this does not diminish the power of the unconditional probability in other application.

Thus, I think Pinker is very misleading, because he merely notes that if you have a lot of extra information, unconditional information is not so important. But on average, such information is all over the map, and outsiders do not know the specifics. So, yeah, if you asked me if a man "Joe" was taller than a woman "Sue", I would say yes, and probably be right, even though many women are taller than many men.

unconditional priors are above my pay grade. i merely understand the basic theorem. but i think pinker is right from the individual's perspective. you think more from a high frequency trader point of view.

take HIV transmission. they say the risk of unprotected with a person of unknown status is 1/10,000. would you take the chance? knowing that you can be the "lucky winner"? and 9,999 cases after that you'll be fine? sure you are "probably"/70% safe, but still.

Pinker is likely wrong because the incremental information needs to be understood before it can be used as knowledge. We have seen in many cognitive studies where more information leads to greater error.

Dear Eric,

Keynes made fundamental contributions to decision theory 100 years ago in his 1907 and 1908 Fellowship dissertations that made up the first four parts of the A Treatise on Probability,published in 1921(TP).Keynes presented the first completely developed lower-upper interval approach to probability in chapters 15 and 17 of the TP.He had to master George Boole's original system as applied to algebra and probability in The Laws of Thought in 1854, in order to present a modified version of Boole's approach in the TP.He applied his interval-inequalities approach in chapters 20 and 22of the TP.Keynes developed the first decision weight approach in history in chapter 26 with his conventional coefficient of risk and weight,c.He also incorporated the first explicit index to measure the weight of the evidence,w,into this coefficient.A very similar index,to deal with ambiguity,was developed by Elleberg,who measured it with his rho variable.

Dear Eric:

A probabilist could explain Keynes' observation by claiming that the preference for the transparent urn simply implies that the decision maker believes that white balls are less common in the opaque urn than black balls. Ellsberg showed that this can’t be the whole reason. He did this by setting up two sets of paired gambles and showed that people always prefer the less ambiguous outcome, even though doing so implies they must have inconsistent estimates of how many balls of each kind are in the opaque urn.

Suppose balls can be red, black or yellow. Let the respective probabilities be denoted R, B, Y. A well-mixed urn has 30 red balls and 60 other balls.

We don’t know how many are black or how many are yellow. Consider the following pair of gambles:

Gamble A

Get $100 if you randomly draw red

Gamble B

Get $100 if you randomly draw black

Most everyone prefers gamble A, which obviously implies R > B. Now consider the gambles

Gamble C

Get $100 if red or yellow

Gamble D

Get $100 if black or yellow

Now the same people who prefered gamble A now prefer gamble D, which implies R + Y < B + Y, which of course implies R < B. It's the same pair of urns for both sets of gambles, so either people are believing manifestly contradictory things about the urns, or they're not making decisions according to expected utility.

Dear Scott,

You have overlooked what Subjective Expected Utility (SEU)assumes about knowledge.SEU assumes that the decision maker has a complete information set.SEU assumes that there is no uncertainty,vagueness, or ambiguity,or conflicting data .Keynes expresses this by the condition that w=1.Ellsberg expresses this by the condition that rho = 1.SEU Nis thus a special case of the general theories of decision making of Keynes and Ellsberg.The second special assumption made by SEU is that the decision maker has linear (proportional) probability preferences.Keynes,but not Ellsberg,showed that ,in general ,decision makers have non linear probability preferences.These preferences can be sub or super proportional.This part of Keynes's work has been rediscovered by Tversky and Kahneman(Prospect and Cumulative prospect theory) and by Gilboa and Schmeidler(convex and concave capacities or non additive decision weights).Neither Tversky or Kahneman or Gilboa and Schmeidler have yet realized this.

Yes.Most decision makers implicitly reject SEU theory because of its very special assumptions.Most decision makers DO NOT use SEU except in those types of decisions where they have linear probability preferences and a complete knowledge set or sample space so that they know that there is only one possible probability distribution to select.

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