Envy Solves The Allais Paradox

Maurice Allais won the Nobel prize for stuff that is never really read anymore, but his curious Allais paradox has endured because it's both simple and baffling.

It arises when comparing participants' choices in two different experiments, each of which consists of a choice between two gambles, A and B. The payoffs for each gamble in each experiment are as follows:

Experiment 1				Experiment 2
Gamble 1A		Gamble 1B		Gamble 2A		Gamble 2B
Winnings	Chance	Winnings	Chance	Winnings	Chance	Winnings	Chance
$1 million	100%	$1 million	89%	Nothing	89%	Nothing	90%
		Nothing	1%	$1 million	11%
		$5 million	10%			$5 million	10%

Several studies involving hypothetical and small monetary payoffs, and recently involving health outcomes, have supported the assertion that when presented with a choice between 1A and 1B, most people would choose 1A. Likewise, when presented with a choice between 2A and 2B, most people would choose 2B. Personally, I would make those choices too, and haven't met anyone who wouldn't.

This is inconsistent with expected utility theory. According to expected utility theory, the person should choose either (1A and 2A) or (1B and 2B).

The problem comes from basic utility theory, because if you plug in a utility function like

U(x)=-exp(-aW) or W^(1-a)/(1-a)

Where W is your wealth and 'a' is your relative risk aversion coefficient, it never works; that is, they can't generate the preferences for 1A and 2B. The Wikipedia page on this outlines the simple proof, which is irrefutable. Something is wrong, and there have been several solutions, all ad hoc. For example, Kahneman and Tversky's Prospect theory allows one to weight outcomes arbitrarily, and so can accomodate this, but the weightings that solve one paradox imply nonsensical other outcomes, such as the simultaneous desire to prefer gambling and insurance, which was the initial motivation for prospect theory (see here).

Allais highlighted that the problem was probably with the independence axiom of von Neumann-Morgenstern utility functions, which basically is the one that implies we don't care about our peers, just our own wealth. For fun, I tried applying a relative utility function to the gambles in the Allais paradox (back to the envy meme). The key is that decision makers imagine the gamble in a world where their neighbor is presented with the same option, so you have to imagine them having been given this absurdly generous gamble as well, and contemplating the relative squalor or richness in the various states. Basically, applying U(x/y), where y is your neighbor's wealth (who is offered the same gamble), as opposed to U(x), where your utility is independent of your neighbor.

If you assume you and your neighbor start with $100k in wealth, then at a certain level of risk aversion (>3.02), with exponential utility, your preferences are for 1A and 2B! This applies to either averaging the two states, or a 'maximin' approach that maximizes the minimum utility among all the states. At really low risk aversion, you prefer the higher expected value choice 'B' in both gambles, and with sufficiently large risk aversion you prefer choice 'A' in both gambles.

It's not really important what the precise numbers are, but it's useful to understand this paradox has a solution within relative utility that is not as ad hoc as prospect theory. Exponential utility is very common because the expected utility for normally distributed wealth has a closed form solution, and this is very convenient, an innocuous simplification for exposition in many applications. However, as it implies 'constant absolute risk aversion', where everyone treats $1 of wealth variance the same, this is not desirable, because then the rich and poor would allocate the same dollar amount to risky assets, which we don't find realistic. Thus, most researchers like to use 'constant relative risk aversion utility functions' like x^(1-a)/(1-a), and there the trick does not work--same paradox as before.

So, it's not as if relative utility neatly provides an insanely robust solution to a prominent game-theoretic paradox, but it does provide a reasonable solution. The exponential utility function is pretty common, and I suspect this result holds across many other specifications. Further, it highlights Maurice Allais's initial intuition was correct, that the wrong assumption was that utility is independent of what others have.

You can download my Excel sheet with the example worked out here.