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.

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.

Of course, humans don’t generally think like that, which is why most people prefer Bets I and IV (and why modern economic theory has served us so badly in recent years).

At the heart of the Ellsberg paradox are two different kinds of uncertainties. The first is a probability: the chance of picking a red ball versus picking a non-red ball, which we are told is 1/3. The second is an ambiguity: the chance of the non-red ball being black or yellow which is entirely uncertain.

Conventional decision theory cannot easily handle both types of uncertainty. But various researchers in recent years have pointed out that quantum theory can cope with with both types and what’s more, can accurately model the patterns of answers that humans come up with.

We looked at an example a couple of years ago that showed how quantum probability theory can explain other paradoxical behaviours in humans called the conjunction and disconjunction fallacies.

Now Aerts and pals have done the same for the Ellsberg paradox by creating a model of the way humans think about this problem and framing it in terms of quantum probability theory.

In fact, these guys go further. The point out that humans can also think in a way that is consistent with decision theory and therefore that this thinking must employ classical logic. So both classical and quantum logic must both be at work at some level in human thought.

Maybe.

The big surprise is that quantum theory works at all. Just why quantum probability theory should explain the strange workings of the human mind, nobody is quite sure. Neither is it yet clear how quantum probability theory will help to mould new ideas about economics and broader human behaviour.

But that’s why there is so much excitement over this new approach and why you’re likely to hear much more about it in future.

Ref: arxiv.org/abs/1104.1459: A Quantum Cognition Analysis of the Ellsberg Paradox

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