The Allais paradox, is an experiment set up as follows, where you are free to chose between gambles $A$ and $B$:

(the table on wikipedia is much more readable if you prefer)

Experiment 1

Gamble 1A

\$1 million, 100% Chance Gamble 1B \$1 million, 89% Chance

\$0, 1% Chance \$5 million, 10% Chance

Experiment 2

Gamble 2A

$0, 89% Chance \$1 million, 11% Chance

Gamble 2B

\$0, 90% Chance \$5 million, 10% Chance

The paradox arises when one chooses both Gamble 1A and Gamble 2B.

This paradox is portrayed as a proof, or evidence of inconsistency in von Neumann–Morgenstern utility theory with it's status as a "rational" decision theory. In particular it is intended to show inadequacy of the independence axiom of that theory.

However, while I find that while {Gamble 1A, Gamble 2B} is an acceptable and even desirable set of choices (which would be logically inconsistent with the theory), this observation doesn't seem strong enough to show the theory is absurd -- it seems fair to argue that if one refused to accept 1% chance of gaining nothing, one would desire a 10% greater chance of winning anything (versus gaining nothing) to be the correct choice in Experiment 2. In other words, both choices {Gamble 1A, Gamble 1B} and {Gamble 2A, Gamble 2B} seem perfectly logical. In fact, if all choices were approximately the same utility, the inconsistency would disappear (which seems entirely plausible to me).

Is there another set of parameters, or another example entirely, which more convincingly shows an absurdity of the independence axiom, or are the parameters above as strong as it gets?

• What do you mean by "doesn't seem strong enough"? Inconsistent with vNM is a statement in formal logic. It is or isn't, and luckily you can check which one is the case. Sep 13 '17 at 16:07
• @denesp vNM is inconsistent as a rational decision theory if {Gable 1A,Gamble 2B} is agreed to be a set of rational decisions. This 'rationality' cannot be proved, and is an intuitive argument. Choice of axioms of any mathematical theory is ultimately arbitrary (following criteria of aesthetics and model relevance). I am asking for a maximally strong intuitive argument that would obviate the inconsisteny of vNM with our intuitive expectation to what a rational decision theory should constitute.
– Real
Sep 13 '17 at 16:45
• Perhaps there is some misunderstanding. vNM itself is not inconsistent. The observation that most people in real life choose {Gamble 1A,Gamble 2B} is inconsistent with the idea that people base their decisions on a vNM type preference. Sep 13 '17 at 16:49
• @denesp Ah yes, I were using 'inconsistency' as referring to 'inconsistency with it's premise of being a rational decision theory', not internal logical consistency. I'll make some edits to reflect that, thanks.
– Real
Sep 13 '17 at 16:52
• I don't think anyone is claiming that vNM is absurd. The Allais paradox merely shows that most people do not make decisions in a vNM framework. Sep 13 '17 at 17:29

One must first distinguish two different senses in which the Allais Paradox can be seen as a "contradiction" of vNM independence; these correspond to the two different interpretations which can be given to any model in Decision Theory. (von Neumann-Morgenstern (vNM) utility theory is one such model.) According to the descriptive interpretation, a model in Decision Theory is supposed to provide a description of how real, actual human beings actually make decisions. According to the normative interpretation, the model is supposed to tell us how an "ideal rational agent" should make decisions. To put it another way, the descriptive interpretation matters to you if you are trying to construct a predictive model of insurance markets or financial markets or some other social phenomenon where people make risky decisions; the normative interpretation matters to you if you yourself are making risky decisions, or if you are trying advise someone who is making such decisions.

The Allais Paradox certainly falsifies the descriptive interpretation of the vNM model; this simply means that, as a matter of empirical fact, most people (including, apparently, many economists) will choose 1A and 2B, which cannot be explained as maximizing the expected value of any possible vNM utility function. Whether the Allais Paradox falsifies the normative interpretation of the vNM model is a more subtle and controversial question. Basically it depends whether or not you think that {1A, 2B} is really the "correct" answer. Some people strongly feel that {1A, 2B} is correct; for them, this is evidence that vNM is not the correct normative model of decision-making under risk. Other people are initially tempted by {1A, 2B}, but once they recognize that it is inconsistent with vNM, they are willing to "bite the bullet" and reject their own intuitions in favor of the theory; their ultimate position is that the normative implications of vNM are counterintuitive, but correct. (The fact that a theory has "counterintuitive" implications does not make it wrong; modern physics is full of theories like this. Indeed, there are even mathematical theorems which are "counterintuitive", but evidently they are correct as a matter of logical necessity.)

In fact, if all choices were approximately the same utility, the inconsistency would disappear (which seems entirely plausible to me).

I don't think "approximately the same utility" is good enough ---as long as the agent has any nontrivial vNM utility function, the choice pattern {1A, 2B} is impossible. Perhaps what you mean is that the choice pattern {1A, 2B} can be reconciled with vNM theory if the agent's vNM utility function is trivial (i.e. constantly zero); in this case, the agent would be indifferent between all possible lotteries. As a purely mathematical point, this is correct. But it is not consistent with people's preferences; most people experience a strict preference for 1A over 1B, and a strict preference for 2B over 2A. So we cannot "explain away" the paradox by positing trivial utility functions.

Your post ends with the question:

Is there another set of parameters, or another example entirely, which more convincingly shows an absurdity of the independence axiom, or are the parameters above as strong as it gets?

Perhaps you will be interested in the Ellsberg Paradox. (I won't bother including a complete description here, since there is a very nice Wikipedia article about it.) The Ellsberg Paradox differs from the Allais Paradox in that the probabilities of certain events are left unspecified; the decision-maker must construct her own "subjective probabilistic beliefs" about these events. But the interesting fact is that there is no set of "subjective probabilistic beliefs" which can explain the pattern of choices most people make in the Ellsberg Paradox. So it definitely falsifies the "descriptive" validity of expected utility (EU) maximization as a model of human decision-making. More importantly, it presents a strong challenge to the "normative" validity of EU-maximization; even many economic theorists who "bite the bullet" in the Allais Paradox are willing to acknowledge that in the Ellsberg Paradox, the choice pattern which is inconsistent with EU-maximization is much more compelling than a choice pattern which is consistent with EU-maximization.

However, I should acknowledge that the Ellsberg Paradox is properly understood as a challenge to EU-maximization in a setting of "ambiguity" (where probabilities are not known a priori), whereas the Allais Paradox is already a challenge to EU-maximization even in a setting of "risk" (where all probabilities are known). It is a perfectly consistent position to accept the vNM axioms in a setting of risk (and hence, "bite the bullet" in the Allais Paradox), while rejecting EU-maximization (i.e. rejecting the Savage Theorem) in a setting of uncertainty.

Another, entirely different normative challenge to vNM independence arises in a setting of social choice. Suppose that you have an indivisible resource (e.g. a hard candy), which you must allocate to one of two claimants (e.g. two children, Alice and Bob). Suppose that, a priori, both claimaints have an equal claim to the resource (they are equally deserving, equally hungry, etc.). The resource is indivisible, so you can't "split" it between the two claimants. Question: should you allocate the resource by some random mechanism (e.g. by flipping a fair coin)? Or should you allocate it by some deterministic mechanism (e.g. give it to the older child, or the one whose name comes first in alphabetical order, or the one who is standing further to the left at the moment the decision is made)? Most people have a strong intuition that a random mechanism is more "fair" than a deterministic mechanism. So they prefer to randomize. But at the same time, they are indifferent between giving the resource to Alice and giving it to Bob (recall: Alice and Bob are equally deserving of the resource). So if A is the outcome "give it to Alice", and B is the outcome "give it to Bob", then we have a situation where the decision maker is indifferent between A and B, but strictly prefers the lottery (1/2 A, 1/2 B) to either A or B. Clearly, this normative position cannot be explained by the vNM theory. This is sometimes called the Diamond Paradox, since it was originally pointed out in a 1967 article by Peter Diamond.

• Great answer, apologies for taking long to accept. I had in mind the case of non-bounded utility, and wondered if the (normative) absurdity could be extended to bounded utility. Unbounded utility creates absurdity with risk: would you accept giving all your money for 1 in 1 trillion chance of eternal life? (also known as Pascal's mugging). This problem disappears with bounded utility, and frankly I did not find your objections convincing enough to demonstrate that (bounded) vNM utility is an entirely inadequate model, only approximately.
– Real
Feb 28 '18 at 4:28