# 19 Ellsberg Paradox

Suppose that there is an urn with 90 balls. You are told that there are 30 blue balls in the urn and that the remaining 60 balls are either yellow or green. You are asked to compare two sets of lotteries:

Question 1: Which of the following two lotteries do you prefer?

Lottery 1: A ball is drawn from the urn and you win $1 million if the ball is blue and $0 if the ball is yellow or green.

Lottery 2: A ball is drawn from the urn and you win $1 million if the ball is yellow and $0 if the ball is blue or green.

Question 2: Which of the following two lotteries do you prefer?

Lottery 3: A ball is drawn from the urn and you win $1 million if the ball is blue or green and $0 if the ball is yellow.

Lottery 4: A ball is drawn from the urn and you win $1 million if the ball is yellow or green and $0 if the ball is blue.

You should answer the above questions before reading further.

The Ellsberg paradox asks decision makers to form preferences over two sets of lotteries. The difficulty with answering these questions is that the probabilities in the lotteries are unknown. Let \(b\) be an integer such that \(30\leq b \leq 60\) representing the number of blue balls and \(y=90-30-b\) the number of yellow balls, \(0M\) mean “0 dollars”, and \(1M\) mean “1 million dollars”. Then, the first question asks decision makers to compare the following two lotteries: \[L_1=[1M:\frac{30}{90}, 0M:\frac{b}{90}, 0M: \frac{y}{90}]\quad \mbox{vs.}\quad L_2=[0M:\frac{30}{90}, 1M:\frac{b}{90}, 0M: \frac{y}{90}].\] Many decision makers report that they strictly prefer \(L_1\) to \(L_2\) (i.e., \(L_1\mathrel{P} L_2\)). After reporting their preference between \(L_1\) and \(L_2\), decision makers are asked to compare the following two lotteries: \[L_3=[1M:\frac{30}{90}, 0M:\frac{b}{90}, 1M: \frac{y}{90}]\quad \mbox{vs.}\quad L_4=[0M:\frac{30}{90}, 1M:\frac{b}{90}, 1M: \frac{y}{90}].\]

Many decision makers report that they strictly prefer \(L_4\) to \(L_3\) (i.e., \(L_4\mathrel{P} L_3\)).

The observation of the Ellsberg paradox is the following: While there is nothing irrational about each opinion by itself, reporting both that \(L_1\mathrel{P} L_2\) *and* \(L_4\mathrel{P} L_3\) is inconsistent with expected utility theory. That is, if a decision maker ranks lotteries by their expected utility with respect to some utility function, then: \[L_1\mathrel{P} L_2\mbox{ if, and only if, }L_3\mathrel{P}L_4.\]

This means that for any rational decision maker we have the following:

\(L_1\mathrel{P} L_2\) and \(L_3\mathrel{P} L_4\) is consistent with expected utility theory.

\(L_1\mathrel{P} L_2\) and \(L_4\mathrel{P} L_3\) is not consistent with expected utility theory.

\(L_2\mathrel{P} L_1\) and \(L_3\mathrel{P} L_4\) is not consistent with expected utility theory.

\(L_2\mathrel{P} L_1\) and \(L_4\mathrel{P} L_3\) is consistent with expected utility theory.

## 19.1 The Ellsberg Preferences are Inconsistent with Expect Utility Theory

**Lemma 19.1 **If \(L_1, L_2, L_3\), and \(L_4\) are defined as in the Ellsberg paradox, then \(L_1\mathrel{P} L_2\) and \(L_4\mathrel{P} L_3\) is inconsistent with expect utility theory.

*Proof*. To see why \(L_1\mathrel{P} L_2\) and \(L_3\mathrel{P}L_4\) is inconsistent with expected utility theory, we will show that for any utility function \(u:\{0M, 1M, 5M\}\rightarrow\mathbb{R}\), it is impossible that \[EU(L_1, u) > EU(L_2, u)\quad\mbox{ and }\quad EU(L_4, u) > EU(L_3, u).\]

Suppose that \(u:\{0M, 1M\}\rightarrow\mathbb{R}\) is a utility function and that \(EU(L_1, u) > EU(L_2, u)\) and \(EU(L_4, u) > EU(L_3, u)\). We show that this leads to a contradiction. The expected utility calculations for \(L_1\) and \(L_2\) are:

\[\begin{align*} EU(L_1, u) &= EU(\left[1M:\frac{30}{90}, 0M:\frac{b}{90}, 0M: \frac{y}{90}\right], u) \\ &= \frac{30}{90} * u(1M) + \frac{b}{90} * u(0M) + \frac{y}{90} * u(0M) \\ &= \frac{30}{90} * u(1M) + \frac{b + y}{90} * u(0M) \\ \end{align*}\]

\[\begin{align*} EU(L_2, u) &= EU(\left[0M:\frac{30}{90}, 1M:\frac{b}{90}, 0M: \frac{y}{90}\right], u) \\ &= \frac{30}{90} * u(0M) + \frac{b}{90} * u(1M) + \frac{y}{90} * u(0M) \\ &= \frac{30 + y}{90} * u(0M) + \frac{b}{90} * u(1M) \\ \end{align*}\]

Since \(EU(L_1, u) > EU(L_2, u)\), we have that: \[\frac{30}{90} * u(1M) + \frac{b + y}{90} * u(0M) > \frac{30 + y}{90} * u(0M) + \frac{b}{90} * u(1M)\]

Subtracting \(\frac{30}{90} * u(1M)\) and \(\frac{30 + y}{90} * u(0M)\) from both sides of the inequality gives the following: \[\frac{b + y - 30 - y}{90} * u(0M) > \frac{b-30}{90} * u(1M)\] Simplifying the probabilities, we have that: \[\frac{b - 30}{90} * u(0M) > \frac{b-30}{90} * u(1M)\]

Now, the expected utility calculations for \(L_3\) and \(L_4\) are:

\[\begin{align*} EU(L_3, u) &= EU(\left[1M:\frac{30}{90}, 0M:\frac{b}{90}, 1M: \frac{y}{90}\right], u) \\ &= \frac{30}{90} * u(1M) + \frac{b}{90} * u(0M) + \frac{y}{90} * u(1M) \\ &= \frac{30 + y}{90} * u(1M) + \frac{b}{90} * u(0M) \\ \end{align*}\]

\[\begin{align*} EU(L_4, u) &= EU(\left[0M:\frac{30}{90}, 1M:\frac{b}{90}, 1M: \frac{y}{90}\right], u) \\ &= \frac{30}{90} * u(0M) + \frac{b}{90} * u(1M) + \frac{y}{90} * u(1M) \\ &= \frac{30}{90} * u(0M) + \frac{b + y}{90} * u(1M) \\ \end{align*}\]

Since \(EU(L_4, u) > EU(L_3, u)\), we have that: \[\frac{30}{90} * u(0M) + \frac{b + y}{90} * u(1M) > \frac{30 + y}{90} * u(1M) + \frac{b}{90} * u(0M)\]

Subtracting \(\frac{30 + y}{90} * u(1M)\) and \(\frac{30}{90} * u(0M)\) from both sides of the inequality gives the following: \[\frac{b + y - 30 - y}{90} * u(1M) > \frac{b-30}{90} * u(0M)\] Simplifying the probabilities, we have that: \[\frac{b - 30}{90} * u(1M) > \frac{b-30}{90} * u(0M)\]

But, this is impossible since we cannot have that:

- \(\frac{b-30}{90} * u(0M) > \frac{b - 30 }{90} * u(1M)\), and
- \(\frac{b - 30}{90} * u(1M) > \frac{b-30}{90} * u(0M).\)

## 19.2 The Ellsberg Preferences are Inconsistent with the Independence Axiom

**Lemma 19.2 **Suppose that \(L_1, L_2, L_3\), and \(L_4\) are defined as in the Ellsberg paradox, and that the decision maker satisfies the Compound Lottery axiom and that \((P, I)\) is a rational preference (Definition 8.3) on the set of lotteries. Then, \(L_1\mathrel{P} L_2\) and \(L_4\mathrel{P} L_3\) violates the Independence Axiom.

*Proof*. Suppose that \(L_1, L_2, L_3\), and \(L_4\) are defined as in the Ellsberg paradox, the decision maker satisfies the Compound Lottery axiom, \((P, I)\) is a rational preference (Definition 8.3) on the set of lotteries, and that the decision maker has the Ellsberg preferences:

- \(\left[1M:\frac{30}{90}, 0M:\frac{b}{90}, 0M: \frac{y}{90}\right]\mathrel{P}\left[0M:\frac{30}{90}, 1M:\frac{b}{90}, 0M: \frac{y}{90}\right]\), and
- \(\left[1M:\frac{30}{90}, 0M:\frac{b}{90}, 1M: \frac{y}{90}\right]\mathrel{P}\left[0M:\frac{30}{90}, 1M:\frac{b}{90}, 1M: \frac{y}{90}\right]\).

Assume that the decision maker satisfies the Independence Axiom. We will show that this leads to a contradiction.

We first note the following two consequences of the Compound Lottery axiom:

Since \(s(\left[\left [1M:\frac{30}{30+b}, 0M:\frac{b}{30+b}\right]:\frac{30+b}{90}, \left[0M:1\right]: \frac{y}{90}\right]) = \left[1M:\frac{30}{90}, 0M:\frac{b}{90}, 0M: \frac{y}{90}\right]\), by the Compound Lottery axiom, \[\left[\left [1M:\frac{30}{30+b}, 0M:\frac{b}{30+b}\right]:\frac{30+b}{90}, \left[0M:1\right]: \frac{y}{90}\right]\mathrel{I}\left[1M:\frac{30}{90}, 0M:\frac{b}{90}, 0M: \frac{y}{90}\right].\]

Since \(s(\left[\left [0M:\frac{30}{30+b}, 1M:\frac{b}{30+b}\right]:\frac{30+b}{90}, \left[0M:1\right]: \frac{y}{90}\right]) = \left[0M:\frac{30}{90}, 1M:\frac{b}{90}, 0M: \frac{y}{90}\right]\), by the Compound Lottery axiom, \[\left[0M:\frac{30}{90}, 1M:\frac{b}{90}, 0M: \frac{y}{90}\right]\mathrel{I}\left[\left [0M:\frac{30}{30+b}, 1M:\frac{b}{30+b}\right]:\frac{30+b}{90}, \left[0M:1\right]: \frac{y}{90}\right].\]

By first assumption of the Ellsberg preferences, \(\left[1M:\frac{30}{90}, 0M:\frac{b}{90}, 0M: \frac{y}{90}\right]\mathrel{P}\left[0M:\frac{30}{90}, 1M:\frac{b}{90}, 0M: \frac{y}{90}\right]\), and so by transitivity of strict preference and indifference, we have that \[{\small \left[\left [1M:\frac{30}{30+b}, 0M:\frac{b}{30+b}\right]:\frac{30+b}{90}, \left[0M:1\right]: \frac{y}{90}\right]\mathrel{P}\left[\left [0M:\frac{30}{30+b}, 1M:\frac{b}{30+b}\right]:\frac{30+b}{90}, \left[0M:1\right]: \frac{y}{90}\right]}.\]

By the Independence Axiom, where \(L\) is \(\left[1M:\frac{30}{30+b}, 0M:\frac{b}{30+b}\right]\), \(L'\) is \(\left [0M:\frac{30}{30+b}, 1M:\frac{b}{30+b}\right],\) \(L''\) is \(\left[0M:1\right]\), and \(r\) is \(\frac{30+b}{90}\), this implies that \[\left[1M:\frac{30}{30+b}, 0M:\frac{b}{30+b}\right]\mathrel{P}\left [0M:\frac{30}{30+b}, 1M:\frac{b}{30+b}\right].\]

Applying the Independence Axiom a second time where \(L\), \(L'\), and \(r\) are as above, and \(L''\) is \([1M:1]\), we have that \[{\small \left[\left [1M:\frac{30}{30+b}, 0M:\frac{b}{30+b}\right]:\frac{30+b}{90}, \left[1M:1\right]: \frac{y}{90}\right]\mathrel{P}\left[\left [0M:\frac{30}{30+b}, 1M:\frac{b}{30+b}\right]:\frac{30+b}{90}, \left[1M:1\right]: \frac{y}{90}\right]}.\]

To see the contradiction, note the following two consequences of the Compound Lottery axiom:

Since \(s(\left[\left [1M:\frac{30}{30+b}, 0M:\frac{b}{30+b}\right]:\frac{30+b}{90}, \left[1M:1\right]: \frac{y}{90}\right]) = \left[1M:\frac{30}{90}, 0M:\frac{b}{90}, 1M: \frac{y}{90}\right]\), by the Compound Lottery axiom, \[\left[1M:\frac{30}{90}, 0M:\frac{b}{90}, 1M: \frac{y}{90}\right]\mathrel{I}\left[\left [1M:\frac{30}{30+b}, 0M:\frac{b}{30+b}\right]:\frac{30+b}{90}, \left[1M:1\right]: \frac{y}{90}\right].\]

Since \(s(\left[\left [0M:\frac{30}{30+b}, 1M:\frac{b}{30+b}\right]:\frac{30+b}{90}, \left[1M:1\right]: \frac{y}{90}\right]) = \left[0M:\frac{30}{90}, 1M:\frac{b}{90}, 1M: \frac{y}{90}\right]\), by the Compound Lottery axiom, \[\left[\left [0M:\frac{30}{30+b}, 1M:\frac{b}{30+b}\right]:\frac{30+b}{90}, \left[1M:1\right]: \frac{y}{90}\right]\mathrel{I}\left[0M:\frac{30}{90}, 1M:\frac{b}{90}, 1M: \frac{y}{90}\right].\]

Since \(\left[\left [1M:\frac{30}{30+b}, 0M:\frac{b}{30+b}\right]:\frac{30+b}{90}, \left[1M:1\right]: \frac{y}{90}\right]\mathrel{P}\left[\left [0M:\frac{30}{30+b}, 1M:\frac{b}{30+b}\right]:\frac{30+b}{90}, \left[1M:1\right]: \frac{y}{90}\right],\) by transitivity of strict preference and indifference, we have that \[\left[1M:\frac{30}{90}, 0M:\frac{b}{90}, 1M: \frac{y}{90}\right] \mathrel{P}\left[0M:\frac{30}{90}, 1M:\frac{b}{90}, 1M: \frac{y}{90}\right].\]

But, this contradictions the second assumption about the Ellsberg preferences that \[\left[0M:\frac{30}{90}, 1M:\frac{b}{90}, 1M: \frac{y}{90}\right]\mathrel{P}\left[1M:\frac{30}{90}, 0M:\frac{b}{90}, 1M: \frac{y}{90}\right].\]