A Quantum Cognition Analysis of Daniel Ellsberg's Paradox

[bananas]

Before he became famous for leaking the Pentagon Papers, Daniel Ellsberg was an economist who studied a problem which came to be named after him - the Ellsberg Paradox.

http://www.technologyreview.com/blog/arxiv/26641/

Paradoxical Decision-Making Explained By Quantum Theory

Conventional decision theory cannot explain why humans make paradoxical choices. But quantum probability theory can, say researchers

04/13/2011

Suppose you receive the following questionnaire in an email:

Imagine an urn containing 90 balls of three different colors: red balls, black balls and yellow balls. We know that the number of red balls is 30 and that the sum of the the black balls and the yellow balls is 60. Our questions are about the situation where somebody randomly takes one ball from the urn.

- The first question is about a choice between two bets: Bet I and Bet II. Bet I involves winning '10 euros when the ball is red' and 'zero euros when it is black or yellow'. Bet II involves winning '10 euros when the ball is black' and 'zero euros when it is red or yellow'. The first question is: Which of the two bets, Bet I or Bet II, would you prefer?

- The second question is again about a choice between two different bets, Bet III and Bet IV. Bet III involves winning '10 euros when the ball is red or yellow' and 'zero euros when the ball is black'. Bet IV involves winning '10 euros when the ball is black or yellow' and 'zero euros when the ball is red'. The second question is: which of the two bets, Bet III or Bet IV, would you prefer?

This are exactly the questions sent out by Diederik Aerts and pals at the Brussels Free University in Belgium. They received replies from 59 people which broke down like this: 34 respondents preferred Bets I and IV, 12 preferred Bets II and III, 7 preferred Bets II and IV and 6 preferred Bets I and III.

That most respondents preferred Bets I and IV is no surprise. It's been verified in countless experiments since the 1960s when the situation was dreamt up by Daniel Ellsberg, a Harvard economist (who more famously leaked the Pentagon Papers later that decade).

The situation is interesting because, paradoxically, a branch of science called decision theory, on which modern economics is based, predicts that humans ought to make an entirely different choice.

<snip>

http://arxiv.org/abs/1104.1459

A Quantum Cognition Analysis of the Ellsberg Paradox
Authors: Diederik Aerts, Bart D'Hooghe, Sandro Sozzo
(Submitted on 7 Apr 2011)

Abstract: The 'expected utility hypothesis' is one of the foundations of classical approaches to economics and decision theory and Savage's 'Sure-Thing Principle' is a fundamental element of it. It has been put forward that real-life situations exist, illustrated by the 'Allais' and 'Ellsberg paradoxes', in which the Sure-Thing Principle is violated, and where also the expected utility hypothesis does not hold. We have recently presented strong arguments for the presence of a double layer structure, a 'classical logical' and a 'quantum conceptual', in human thought and that the quantum conceptual mode is responsible of the above violation. We consider in this paper the Ellsberg paradox, perform an experiment with real test subjects on the situation considered by Ellsberg, and use the collected data to elaborate a model for the conceptual landscape surrounding the decision situation of the paradox. We show that it is the conceptual landscape which gives rise to a violation of the Sure-Thing Principle and leads to the paradoxical situation discovered by Ellsberg.

Some background:

From Daniel Ellsberg's bio:

http://www.ellsberg.net/bio

He earned his Ph.D. in Economics at Harvard in 1962 with his thesis, Risk, Ambiguity and Decision. His research leading up to this dissertation—in particular his work on what has become known as the “Ellsberg Paradox,” first published in an article entitled Risk, Ambiguity and the Savage Axioms—is widely considered a landmark in decision theory and behavioral economics.

Quantum probability theory:
http://en.wikipedia.org/wiki/Quantum_probability
http://plato.stanford.edu/entries/qt-quantlog/

[Poll_Blind]

1. What a great post! I love reading this kind of stuff. Thanks!

Bookmarked to finish it off later. I had no idea about that "other side" of Ellsberg. Very cool.

PB

[NYC_SKP]

2. K&R. (nt)

[napoleon_in_rags]

3. Its all about your axioms of probability.

IF you take the philosophical stance called "max entropy" (I think its called) then you assume that given two random variables and no other information, the probability of both should be set as equal, .5 for each. Its good in a lot of science, but if you assume this then all of the bets are the same.

But we live in a culture where people try to screw us, and if information is withheld its because we are going to be screwed by it. So the bets with the most information are going to win out, so people choose bet I. People choose bet IV for the same reason: We KNOW there is 2/3 probability for black and yellow, but if we choose red and yellow, we can bet our our asses that there will be only one yellow ball in the whole mix, and somehow overdraft fees and adjustable interest rates will be involved.

[girl gone mad]

4. Very cool to know.

Thanks!

[TheStrategyExpert]

5. Why do we have to pick any of the 4?

This question is flawed, as there is no way all of the people would pick one answer, they could choose to say "indifferent", that's what I might say to the first question, and there were no pressing rules that demanded to pick one or the other even though the question attempts to get you to do that. It's like a salesman asking if you want to buy my product today or tomorrow? Well you don't actually have to pick one, you can choose to say "neither", that's a choice too. For the 2nd question, i would rather take the sure thing that had some red balls rather than take a 50/50 proposition on being forced to gamble any amount, especially when there is risk built in that you might get screwed in an unfair way, that risk portion serves as a tiebreaker since the question doesn't provide a given that the game is random and fair, which if it did, it would make for a lot more interesting question to analyze.

[DireStrike]

6. "Decision theory" sounds like so much nonsense.

From the article:

"Here's why. Decision theory assumes that any individual tackling this problem would do it by assigning a fixed probability to the chance of picking a yellow or black ball and then stick with that probability as they chose their bets. This approach leads to the conclusion that if you prefer Bet I, then you must also prefer Bet III. But if you prefer Bet II, then you must also prefer Bet IV."

Why on earth would people preserve the same evaluations of unknown components of a decision across two different situations? Or a better question, why would someone determined to set down a theory of decision making fail to understand that? There is no paradox here, the assumption is just bad.