Dawes' argument, as promised.

The context is: Dawes is explaining von Neumann and Morgenstern's axioms.

Aside: I don't know how familiar you are with the VNM utility theorem, but just in case, here's a brief primer.

The VNM utility theorem presents a set of axioms, and then says that if an agent's preferences satisfy these axioms, then we can assign any outcome a number, called its utility, written as U(x); and it will then be the case that given any two alternatives X and Y, the agent will prefer X to Y if and only if E(U(X)) > E(U(Y)). (The notation E(x) is read as "the expected value of x".) That is to say, the agent's preferences can be understood as assigning utility values to outcomes, and then preferring to have more (expected) utility rather than less (that is, preferring those alternatives which are expected to result in greater utility).

In other words, if you are an agent whose preferences adhere to the VNM axioms, then maximizing your utility will always, without exception, result in satisfying your preferences. And in yet other words, if you are such an agent, then your preferences can be understood to boil down to wanting more utility; you assign various utility values to various outcomes, and your goal is to have as much utility as possible. (Of course this need not be anything like a conscious goal; the theorem only says that a VNM-satisfying agent's preferences are equivalent to, or able to be represented as, such a utility formulation, not that the agent consciously thinks of things in terms of utility.)

(Dawes presents the axioms in terms alternatives or gambles; a formulation of the axioms directly in terms of the consequences is exactly equivalent, but not quite as elegant.)

N.B.: "Alternatives" in this usage are gambles, of the form ApB: you receive outcome A with probability p, and otherwise (i.e. with probability 1–p) you receive outcome B. (For example, your choice might be between two alternatives X and Y, where in X, with p = 0.3 you get consequence A and with p = 0.7 you get consequence B, and in Y, with p = 0.4 you get consequence A and with p = 0.6 you get consequence B.) Alternatives, by the way, can also be thought of as actions; if you take action X, the probability distribution over the outcomes is so-and-so; but if you take action Y, the probability distribution over the outcomes is different.

(If all of this is old hat to you, apologies; I didn't want to assume.)

The question is: do our preferences satisfy VNM? And: should our preferences satisfy VNM?

It is commonly said (although this is in no way entailed by the theorem!) that if your preferences don't adhere to the axioms, then they are irrational. Dawes examines each axiom, with an eye toward determining whether it's mandatory for a rational agent to satisfy that axiom.

Dawes presents seven axioms (which, as I understand it, are equivalent to the set of four listed in the wikipedia article, just with a difference in emphasis), of which the fifth is Independence.

The independence axiom says that A ≥ B (i.e., A is preferred to B) if and only if ApC ≥ BpC. In other words, if you prefer receiving cake to receiving pie, you also prefer receiving (cake with probability p and death with probability 1–p) to receiving (pie with probability p and death with probability 1–p).

Dawes examines one possible justification for violating this axiom — framing effects, or pseudocertainty — and concludes that it is irrational. (Framing is the usual explanation given for why the expressed or revealed preferences of actual humans often violate the independence axiom.) Dawes then suggests another possibility:

Is such irrationality the only reason for violating the independence axiom? I believe there is another reason. Axiom 5 [Independence] implies that the decision maker cannot be affected by the skewness of the consequences, which can be conceptualized as a probability distribution over personal values. Figure 8.1 shows (Note: This is my reproduction of the figure. I've tried to make it as exact as possible.) the skewed distributions of two different alternatives. Both distributions have the same average, hence the same expected personal value, which is a criterion of choice implied by the axioms. These distributions also have the same variance.

If the distributions in Figure 8.1 were those of wealth in a society, I have a definite preference for distribution a; its positive skewness means that income can be increased from any point — an incentive for productive work. Moreover, those people lowest in the distribution are not as distant from the average as in distribution b. In contrast, in distribution b, a large number of people are already earning a maximal amount of money, and there is a "tail" of people in the negatively skewed part of this distribution who are quite distant from the average income.[5] If I have such concerns about the distribution of outcomes in society, why not of the consequences for choosing alternatives in my own life? In fact, I believe that I do. Counter to the implications of prospect theory, I do not like alternatives with large negative skews, especially when the consequences in the negatively skewed part of the distribution have negative personal value.

[5] This is Dawes' footnote; it talks about an objection to "Reaganomics" on similar grounds.

Essentially, Dawes is asking us to imagine two possible actions. Both have the same expected utility; that is, the "degree of goal satisfaction" which will result from each action, averaged appropriately across all possible outcomes of that action (weighted by probability of each outcome), is exactly equal.

But the actual probability distribution over outcomes (the form of the distrbution) is different. If you do action A, then you're quite likely to do alright, there's a reasonable chance of doing pretty well, and a small chance of doing really great. If you do action B, then you're quite likely to do pretty well, there's a reasonable chance to do ok, and a small chance of doing disastrously, ruinously badly. On average, you'll do equally well either way.

The Independence axiom dictates that we have no preference between those two actions. To prefer action A, with its attendant distribution of outcomes, to action B with its distribution, is to violate the axiom. Is this irrational? Dawes says no. I agree with him. Why shouldn't I prefer to avoid the chance of disaster and ruin? Consider what happens when the choice is repeated, over the course of a lifetime. Should I really not care whether I occasionally suffer horrible tragedy or not, as long as it all averages out?

But if it's really a preference — if I'm not totally indifferent — then I should also prefer less "risky" (i.e. less negatively skewed) distributions even when the expectation is lower than that of distributions with more risk (i.e. more negative skew) — so long as the difference in expectation is not too large, of course. And indeed we see such a preference not only expressed and revealed in actual humans, but enshrined in our society: it's called insurance. Purchasing insurance is an expression of exactly the preference to reduce the negative skew in the probability distribution over outcomes (and thus in the distributions of outcomes over your lifetime), at the cost of a lower expectation.