13  Allais Paradox

Suppose there is an urn with 100 balls: 89 white balls, 10 blue balls, and 1 red ball. You are asked to compare two sets of lotteries:

To make the comparison easier, here is a table summarizing the payouts of each lottery based on the color of the ball drawn. The numbers in the header represent the probabilities of selecting a ball of the given color:

0.01 (red) 0.89 (white) 0.1 (blue)
Lottery 1 \(\$1,000,000\) \(\$1,000,000\) \(\$1,000,000\)
Lottery 2 \(\$0\) \(\$1,000,000\) \(\$5,000,000\)
Lottery 3 \(\$1,000,000\) \(\$0\) \(\$1,000,000\)
Lottery 4 \(\$0\) \(\$0\) \(\$5,000,000\)
Warning

You should answer the above questions before reading further. This is known as the Allais paradox (Allais 1953).

The Allais paradox invites decision makers to express preferences over two different sets of lotteries. The first set consists of the following lotteries, where \(1M\) means “1 million dollars,” \(5M\) means “5 million dollars,” and \(0M\) means “0 dollars”:

Many decision makers report a strict preference for \(L_1\) over \(L_2\) (i.e., \(L_1 \mathrel{P} L_2\)). After expressing their preference between \(L_1\) and \(L_2\), these decision makers are then asked to compare a second set of lotteries:

Here, many decision makers report a strict preference for \(L_4\) over \(L_3\) (i.e., \(L_4 \mathrel{P} L_3\)).

The Allais paradox highlights an inconsistency: While each individual preference might seem rational on its own, expressing both \(L_1 \mathrel{P} L_2\) and \(L_4 \mathrel{P} L_3\) is inconsistent with expected utility theory. According to expected utility theory, if a decision maker ranks lotteries by their expected utility, then: \[L_1\mathrel{P} L_2\mbox{ if, and only if, }L_3\mathrel{P}L_4.\]

This implies the following for any rational decision maker:

  1. Preferring \(L_1 \mathrel{P} L_2\) and \(L_3 \mathrel{P} L_4\) is consistent with expected utility theory.
  2. Preferring \(L_1 \mathrel{P} L_2\) but not \(L_3 \mathrel{P} L_4\) is inconsistent with expected utility theory.
  3. Not preferring \(L_1 \mathrel{P} L_2\) but preferring \(L_3 \mathrel{P} L_4\) is inconsistent with expected utility theory.
  4. Not preferring \(L_1 \mathrel{P} L_2\) and not preferring \(L_3 \mathrel{P} L_4\) is consistent with expected utility theory.

The challenge is that many individuals exhibit what are known as Allais preferences, indicating they prefer \(L_1 \mathrel{P} L_2\) and simultaneously prefer \(L_4 \mathrel{P} L_3\). In the next section, we demonstrate that such preferences are not compatible with expected utility representation. Consequently, decision makers with these preferences cannot be understood as evaluating lotteries based on their expected utility. Moreover, it can be shown that these preferences lead to a violation of the Independence Axiom, a fundamental principle of expected utility theory.

13.1 The Allais Preferences are Inconsistent with Expect Utility Theory

Lemma 13.1 Suppose that \(L_1, L_2, L_3\), and \(L_4\) are defined as in the Allais paradox and that \(L_1\mathrel{P} L_2\) and \(L_4\mathrel{P} L_3\). Then, this strict preference is not expected utility representable.

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 \(u:\{0M, 1M, 5M\}\rightarrow\mathbb{R}\) 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:

\[\begin{align*} EU(L_1, u) &= EU(0.01\cdot 1M + 0.89 \cdot 1M + 0.1 \cdot 1M, u) \\ &= 0.01\times u(1M) + 0.89 \times u(1M) + 0.10 \times u(1M) \\ &= u(1M) \\ \end{align*}\]

\[\begin{align*} EU(L_2, u) &= EU(0.01\cdot 0M + 0.89 \cdot 1M + 0.1 \cdot 5M, u) \\ &= 0.01\times u(0M) + 0.89 \times u(1M) + 0.10 \times u(5M) \\ \end{align*}\]

Since \(EU(L_1, u) > EU(L_2, u)\), we have that: \[u(1M) > 0.01\times u(0M) + 0.89 \times u(1M) + 0.10 \times u(5M).\]

Subtracting \(0.89 \times u(1M)\) from both sides of the inequality gives the following: \[0.11 \times u(1M) > 0.01 \times u(0M) + 0.10 \times u(5M).\]

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

\[\begin{align*} EU(L_3, u) &= EU(0.01\cdot 1M + 0.89\cdot 0M + 0.1\cdot 1M, u) \\ &= 0.01\times u(1M) + 0.89\times u(0M) + 0.10\times u(1M) \\ &= 0.11 \times u(1M) + 0.89\times u(0M) \\ \end{align*}\]

\[\begin{align*} EU(L_4, u) &= EU(0.01\cdot 0M + 0.89\cdot 0M + 0.1\cdot 5M, u) \\ &= 0.01\times u(0M) + 0.89 \times u(0M) + 0.10 \times u(5M) \\ &= 0.90 \times u(0M) + 0.10 \times u(5M) \\ \end{align*}\]

Since \(EU(L_4, u) > EU(L_3, u)\), we have that: \[0.90 \times u(0M) + 0.10 \times u(5M) > 0.11 \times u(1M) + 0.89\times u(0M).\]

If we subtract \(0.89 \times u(0M)\) from both sides of the inequality, then we have that: \[0.01\times u(0M) + 0.10 \times u(5M) > 0.11 \times u(1M).\]

But, this is impossible since we cannot have that:

  • \(0.11\times u(1M) > 0.01 \times u(0M) + 0.10 \times u(5M)\); and
  • \(0.01 \times u(0M) + 0.10 \times u(5M) > 0.11 \times u(1M).\)

13.2 Exercises

  1. Are the Allais preferences rational? What might motivate a decision maker to hold such preferences?