The Allais Paradox: Why Do We ‘Win Small’ but ‘Lose Big’ in Stock Market?

If you were given a choice to invest in one of the stocks from each scenario below, which stocks would you invest?

Scenario 1:

Stock A:	100% chance to earn RM1 million
Stock B:	10% chance to earn RM5 million, 89% chance to earn RM1 million, 1% chance to earn nothing

Scenario 2:

Stock C:	11% chance to earn RM1 million, 89% chance to earn nothing
Stock D:	10% chance to earn RM5 million, 90% chance to earn nothing

Results show many prefer stock A over B, and D over C. This is because A is certain, but both C and D are risky, therefore D looks better. Why is this a ‘paradox’?

If we use U(x) to represent our preferences, then stock A over B is equal to:

     U(1mil) > 0.1U(5mil) + 0.89U(1mil) + 0.01U(0)

=> 0.11U(1mil) > 0.1U(5mil) + 0.01U(0)  ——  Eq. (A)

And if stock D over C, then:

     0.11U(1mil) + 0.89U(0) < 0.1U(5mil) + 0.90U(0)

=> 0.11U(1mil) < 0.1U(5mil) + 0.01U(0)  —— Eq. (B)

Equation (A) and equation (B) is contradictory. This is what the Allais Paradox is telling us: an individual’s decision can be inconsistent with expected utility theory. People usually prefer certainty to uncertainty, but when they are approached differently, they may prefer the uncertainty that was previously rejected.

In Prospect Theory, losses and gains were assumed to be valued differently, and thus individuals make decisions based on perceived gains instead of perceived losses.

Scenario 3:

Stock E:	100% chance to earn RM1 million
Stock F:	50% chance to earn RM2 million, 50% chance to earn nothing

Scenario 4:

Stock G:	100% chance to lose RM1 million
Stock H:	50% chance to lose nothing, 50% chance to lose RM2 million

Most people choose stock E over F, and stock H over G.

This shows that individuals will try to avoid risk when there is a prospect of sure gain, but may seek for risk when one of the options is a sure loss. And that is why sometimes it is so hard for individuals to rake in more profit or cut a small loss.

If you deduct RM2 million from the gains in stocks from scenario 3, you will get stock G’s result from stock E, and stock H from stock F. In other words, the differences between the stocks in each scenario are the same. Again, the Allais Paradox is shown.

Reference:

1. Allais Paradox, https://en.wikipedia.org

2. James Chen, Prospect Theory, www.investopedia.com