Rationality Revisited: The Ellsberg Paradox

Consider the following situation:

In an urn, you have 90 balls of 3 colors: red, blue and yellow. 30 balls are known to be red. All the other balls are either blue or yellow.

There are two lotteries:

• Lottery A: A random ball is chosen. You win a prize if the ball is red.
• Lottery B: A random ball is chosen. You win a prize if the ball is blue.

Question: In which lottery would you want to participate?

• Lottery X: A random ball is chosen. You win a prize if the ball is either red or yellow.
• Lottery Y: A random ball is chosen. You win a prize if the ball is either blue or yellow.

Question: In which lottery would you want to participate?

If you're like most people you probably chose Lottery A over B and Lottery Y over X. (If you did choose something else, skip ahead to the end section)

Did you know that under the classical models of decision theory, there is no probability distribution under which such a decision would be rational?

Why is that?

Well, to begin with we state the Axiom of Independence in Von Neumann's Model of Rationality which otherwise seem to be obviously correct.

If a person prefers event L to event M, then he always prefers L along with N to M along with N, for any event N.

This seems fairly obvious. If I like Carrots over Soya Sauce, I would definitely prefer Carrots along with Transparent Soap to Soya Sauce along with Transparent Soap.

However, in this case it appears that our decisions suddenly changed our preference for event Red to the event Blue, just as we introduced Yellow into the scene. See below:

Ellsberg Paradox

Let us formalize this bit with the Axiom of Rationality

Axiom of Rationality: A rational agent aims to maximize his expected utility.

First, we are clear that the probability of getting a red ball is $\frac{1}{3}$.

Let us say that the probability of getting a blue ball is $p$ and that of getting a yellow ball is $\frac{2}{3} - p$. Furthermore, for convenience, we assert the utility of the prize to be $1$ and not getting it $0$.

We calculated the expected utilities of lotteries A and B as follows:

$u(A) = 1 \times \frac{1}{3} + 0 \times p + 0 \times (\frac{2}{3} - p) = \frac{1}{3}$ $u(B) = 0 \times \frac{1}{3} + 1 \times p + 0 \times (\frac{2}{3} - p) = p$

Thus, if a rational agent did prefer A to B, he believes that $u(A) > u(B) \\ \implies \frac{1}{3} > p$

Now, let us set aside this result and turn to lotteries X and Y.

$u(X) = 1 \times \frac{1}{3} + 0 \times p + 1 \times (\frac{2}{3} - p) = 1-p$ $u(Y) = 0 \times \frac{1}{3} + 1 \times p + 1 \times (\frac{2}{3} - p) = \frac{2}{3}$

If a rational agent did prefer Y to X, he believes that $u(Y) > u(X) \\ \implies \frac{2}{3} > 1- p \\ \implies p > \frac{1}{3}$

But we just showed that the agent believes that $\frac{1}{3} > p$. Thus, choosing A cannot be consistent with choosing Y.

But does it mean we are really irrational? Maybe something was wrong with our axioms of rationality?

What actually drives us to do these choices is actually that we prefer the kind of surity behind those lotteries. Is there a way to formalize this idea?

The answer is yes, fortunately and we boldly propose our new axiom or rationality:

Axiom of Rationality: A rational agent aims to maximize his minimum expected utility.

This, by the way, is called The Maximin Principle of Rationality.

To account for the idea of minimum expected utility, we turn to an idea called Imprecise Probability.

This is an idea to say that we do not just believe that an event occurs with a probability $p$ but we do believe that events could have a set of possible probabilities (called a credal set) spread over an interval. Just to state an example, we do believe that heads on tails on a coin are equally likely but do we really believe the same thing about, say, a thumb tack?

Let us look at our problem in this light.

Instead of the previous precise probability distribution of $( \frac{1}{3}, p, \frac{2}{3} - p )$, we are now supposed to be working with the credal probability distribution set $\left \{ p \in [0,\frac{2}{3}] : ( \frac{1}{3}, p, \frac{2}{3} - p ) \right \}$

Similarly, we are supposed to replace the expected utilities with sets of expected utilities.

$u(A) = \left \{ \frac{1}{3} \right \} \\ u(B) = \left \{ p \in [0,\frac{2}{3}] : p \right \} \\ u(X) = \left \{ p \in [0,\frac{2}{3}] : 1 - p \right \} \\ u(Y) = \left \{ \frac{2}{3} \right \}$

Now, we are ready to calculate the minimum expected utilities for the four lotteries.

$min(u(A)) = \frac{1}{3} \\ min(u(B)) = 0$

Clearly, we should choose A over B.

$min(u(X)) = \frac{1}{3} \\ min(u(Y)) = \frac{2}{3}$

And also Y over X!

So much for the idea of imprecise probability and the maximin principle, we have seen that we are now ready to answer some really interesting question. For starters, try explaining why it is rational to buy an insurance even though the expected value is negative,