John von Neumann and Oskar Morgenstern developed a system of four axioms that they claimed any rational decision maker must follow. The major consequence of these axioms is that when faced with a decision, you should always act solely to increase your expected utility. All four axioms have been attacked at various times and from various directions; but three of them are very solid. The fourth - independence - is the most controversial.

To understand the axioms, let A, B and C be lotteries - processes that result in different outcomes, positive or negative, with a certain probability of each. For 0<p<1, the mixed lottery pA + (1-p)B implies that you have p chances of being in lottery A, and (1-p) chances of being in lottery B. Then writing A>B means that you prefer lottery A to lottery B, A<B is the reverse and A=B means that you are indifferent between the two. Then the von Neumann-Morgenstern axioms are:

• (Completeness) For every A and B either A<B, A>B or A=B.
• (Transitivity) For every A, B and C with A>B and B>C, then A>C.
• (Continuity) For every A>B>C then there exist a probability p with B=pA + (1-p)C.
• (Independence) For every A, B and C with A>B, and for every 0<t≤1, then tA + (1-t)C > tB + (1-t)C.

In this post, I'll try and prove that even without the Independence axiom, you should continue to use expected utility in most situations. This requires some mild extra conditions, of course. The problem is that although these conditions are considerably weaker than Independence, they are harder to phrase. So please bear with me here.

The whole insight in this post rests on the fact that a lottery that has 99.999% chance of giving you £1 is very close to being a lottery that gives you £1 with certainty. I want to express this fact by looking at the narrowness of the probability distribution, using the standard deviation. However, this narrowness is not an intrinsic property of the distribution, but of our utility function. Even in the example above, if I decide that receiving £1 gives me a utility of one, while receiving zero gives me a utility of minus ten billion, then I no longer have a narrow distribution, but a wide one. So, unlike the traditional set-up, we have to assume a utility function as being given. Once this is chosen, this allows us to talk about the mean and standard deviation of a lottery.

Then if you define c(μ) as the lottery giving you a certain return of μ, you can use the following axiom instead of independence:

• (Standard deviation bound) For all ε>0, there exists a δ>0 such that for all μ>0, then any lottery B with mean μ and standard deviation less that μδ has B>c((1-ε)μ).

This seems complicated, but all that it says, in mathematical terms, is that if we have a probability distribution that is "narrow enough" around its mean μ, then we should value it are being very close to a certain return of μ. The narrowness is expressed in terms of its standard deviation - a lottery with zero SD is a guaranteed return of μ, and as the SD gets larger, the distribution gets wider, and the chances of getting values far away from μ increases. So risk, in other words, scales (approximately) with the SD.