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The Ellsberg Paradox

January 13th, 2013 by dimpledbrain

The Dilemma:

Suppose you have 2 urns in front of you, Urn A and Urn B . Both urns contain red marbles and black marbles. Urn A has 50 of each colour, making it to a total of 100 marbles, but Urn B’s mixture is unknown. What we know however is that it also contains 100 marbles.

If I offer you a bet, say RM100 per correct  guess, which colour will you bet on:

(a) A red marble from Urn A or a black marble from Urn A

(b) A red marble from Urn B or a black marble from Urn B

Let’s recap. 2 urns with 100 marbles each. Urn A has 50 of each colour but Urn B’s distribution is unknown to you.

Write down the answers to (a) and (b) above.

How about the following?

(c) A red marble from Urn A or a red marble from Urn B

(d) A black marble from Urn A or a black marble from Urn B

Most people prefer to bet on Urn A for both (c) and (d) and here lies the paradox.

If I choose a red draw from Urn A, I must rightly or wrongly believe that Urn B has more black marbles. It then follows that I would (rationally) choose a black draw for (d).

If Urn A has 50 marbles of each colour and I choose to draw Red from it, I must believe that Urn B has more black marbles in it. Note that we don’t know the mixture in Urn B but we can infer from our decision on Urn A. The opposite is also true for Black.

But most people will choose to draw a black marble from Urn A again for (d) because of the certainty of information. It has 50 marbles of each colour and I assume that because I know its distribution, I am safe.

In his book Uncertainty, Jonathan Fields explains that we play it safe because we fear being criticised by others. We toe the line for fear of being ridiculed by others, to avoid social judgement. We believe there is safety in numbers. Herd thinking. Which is not wrong by itself before you cry blue murder. It’s always ok to be a living zombie, do as what you have been conditioned to do.

I thought that that’s a very interesting piece of knowledge so I proceeded to ask the author himself in one of the book discussions led by him – if there’s anything that we can learn from children on this then? I don’t remember getting a satisfactory answer from him, but hey, I’m sure that some books will be published on this, sooner or later- how to be childlike but at the same time avoid being childish.

On the other end of the spectrum, some people become total contrarians. If you choose A, I’ll choose B. That is also wrong. See, a little knowledge is always dangerous. If we follow this line of thinking, sometimes we’ll be right, sometimes we’ll be wrong. But I wonder if we will ever realize that we have been constantly fooled by randomness?

The immortal writer, Khalil Gibran, explains the situation beautifully:

“Your soul is oftentimes a battlefield, upon which your reason and your judgment wage war against your passion and your appetite.

Your reason and your passion are the rudder and the sails of your seafaring soul.

If either your sails or your rudder be broken, you can but toss and drift, or else be held at a standstill in mid-seas.

For reason, ruling alone, is a force confining; and passion, unattended, is a flame that burns to its own destruction.

Therefore let your soul exalt your reason to the height of passion, that it may sing;

And let it direct your passion with reason, that your passion may live through its own daily resurrection, and like the phoenix rise above its own ashes.”

The moral of the story:

(1) Do not be paralyzed by social judgement and so take a little more risks in life. They say happiness is about wanting what you want, not having what you want.

(2) Do not be too extreme on either spectrum, but apply moderation in everything, including moderation itself.

 

…like the phoenix rise above its own ashes…

Beautiful.

 

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