Ninth Installment

At long last, we are ready to state the six assumptions about someone's preferences, or Axioms, as von Neuman and Morgenstern call them, the positing of which is sufficient to allow us to deduce that the person's preferences over a set of outcomes can be represented by a Cardinal Utility Function. There is a very great deal of hairy detail that I am going to skip over, for two reasons. The first is that I want there to be someone still reading this when I get done. The second is that it is just too much trouble to try to get all this symbolism onto my blog. You can find the detail in Luce and Raiffa. O.K., here we go.

    Assume there is a set of n outcomes, or prizes, O = (O1, O2, ...., On)

AXIOM I: The individual has a weak preference ordering over O, with O1 the most preferred and On the least preferred, and this ordering is complete and transitive. Thus, for any Oi and Oj, either Oi R Oj or Oj R Oi. Also, If Oi R Oj, and Oj R Ok, then Oi R Ok. [I told you we would use that stuff at the beginning.]

AXIOM II: [A biggie] The individual is indifferent between any Compound Lottery and the Simple Lottery over O derived from the Compound Lottery by the ordinary mathematical process of reducing a compound lottery to a simple lottery [as I did for the example].

    This a very powerful axiom, and we have already met something like it in our discussion. In effect, it says that the individual has neither a taste for nor an aversion to any distribution of risk. The point is that the Compound Lotteries may exhibit a very broad spread of risk, whereas the Simple Lottery derived from them by the reduction process may have a very narrow spread of risk. Or vice versa. The individual doesn't care about that.

AXIOM III: For any prize or outcome Oi, there is some Lottery over just the most and least preferred outcomes such that the individual is indifferent between that Lottery and the outcome Oi. A Lottery over just the most and least preferred outcomes is a Lottery that assigns some probability p to the most preferred outcome, O1, and a probability (1-p) to the least preferred outcome, On, and zero probability to all the other outcomes. Think of this as a needle on a scale marked 0 to 1. You show the person the outcome Oi, and then you slide the needle back and forth between the 1, which is labeled O1 and the 0 [zero] which is labeled On. Somewhere between those two extremes, this Axiom says, there is a balancing point of probabilities that the person considers exactly as good as the certainty of Oi. Call that point Ui. It is the point that assigns a probability of Ui to O1 and a probability of (1 - Ui) to On.

    We are now going to give a name to the Lottery we are discussing, namely the Lottery [UiO1, (1- Ui)On]. We are going to call it Õi . Thus, according to this Axiom and our symbolism, the player A is indifferent between Oi and Õi.

    If you have good mathematical intuition and are following this closely, it may occur to you that this number between 1 and 0, Ui, is going to turn out to be the Utility Index assigned to Oi in A's cardinal utility function. You would be right.

    This Axiom is essentially a continuity axiom, and it is very, very powerful. It implies a number of important things. First, it implies that A does NOT have a lexicographic preference order. All of the outcomes are, in A's eyes, commensurable with one another, in the sense that for each of them, A is indifferent between it and some mix or other of the most and the least preferred outcomes. It also implies that we can, so far as A's preferences are concerned, reduce any Lottery, however complex, to some Simple Lottery over just O1 and On. The Axiom guarantees that there is such a Lottery. Notice also that this Axiom implies that A is capable of making infinitely fine discriminations of preference between Lotteries. In short, this is one of those idealizing or simplifying assumptions [like continuous production functions] that economists make so that they can use fancy math.

AXIOM IV. In any lottery, Õ can be substituted for Oi. Remember, Axiom III says that A is indifferent between Õi and Oi. This axiom says that when you substitute Õi for Oi in a lottery, A is indifferent between the old lottery and the new one. In effect, this says that the surrounding or context in which you carry out the substitution makes no difference to A. For example, the first lottery might assign a probability of .4 to the outcome Oi, while the new lottery assigns the same probability, .4, to Õi. [If you are starting to get lost, remember that Õi is the lottery over just O1 and On, such that A is indifferent between that lottery and the pure outcome Oi.]

AXIOM V. Preference and Indifference among lottery tickets are transitive relations. So if A prefers Lottery 1 to Lottery 2, and Lottery 2 to Lottery 3, then A will prefer Lottery 1 to Lottery 3. Also, if A is indifferent between Lottery 1 and Lottery 2, and is indifferent between Lottery 2 and Lottery 3, then A will be indifferent between Lottery 1 and Lottery 3. This is a much stronger Axiom than it looks, as we shall see presently.

    If you put Axioms I through V together, they imply something very powerful, namely that for any Lottery, L, there is a lottery over just O1 and On, such that A is indifferent between L and that lottery over O1 and On. We need to go through the proof of this in order to prepare for the wrap up last axiom.

    Let L be the lottery (p₁O1, p₂O2, ...., p_nOn).

    Now, for each Oi in L, substitute Õi. Axioms III and IV say this can be done.

    So, using our previous notation, where xIy means A is indifferent between x and y,

    (p₁O1, ..., p_nOn) I (p₁Õ1, ..., p_nÕn) so, expanding the right hand side,

    (p₁O1, ..., p_nOn) I (p₁[U1O1, (1-U1)On]), ...., (p_n[UnOn, (1-Un)On) or, multiplying

    (p₁O1, ..., p_nOn) I ([p₁U1 + p₂U2 + ... + p_nUn]O1, [p₁{1-U1} + .... + p_n{1-Un}On]) or

    (p₁O1, ..., p_nOn) I ([p₁U1 + p₂U2 + ... + pnUn]O1, [p₁{1-U1} + ... + p_n{1-Un}]On)

    if we let p = p₁U1 + p₂U2 + ... p_nUn then we have:

    (p₁O1, ..., p_nOn) I (pO1, (1-p)On) In other words, the lottery, L, with which we started is indifferent to a lottery just over the best and worst outcomes, O1 and On.

AXIOM VI The last axiom says that if p and p' are two probabilities, i.e., two real numbers between 1 and 0, then: (pO1, [1-p]On) R (p'O1, [1-p']On) if and only if p ≥ p'

    This Axiom says that the individual [A in our little story] prefers [or is indifferent between] one lottery over the best and the worst alternatives to another lottery over those same two alternatives if and only if the probability assigned to O1 in the first lottery is equal to or greater than the probability assigned to O1 in the second lottery.

    Now, let us draw a deep breath, step out of the weeds, and remember what we have just done. First, we started with a finite set of outcomes, O = (O1, O2, ...., On). Then we defined a simple lottery over the set O as a probability distribution over the set O. Then we defined a compound lottery as a lottery whose prizes include tickets in simple lotteries. At this point, we introduced five AXIOMS or assumptions about the preferences that our sample individual A has over the set of outcomes and simple and compound lotteries of those outcomes. These are not deductions. They are assumptions. Then we showed that these five Axioms, taken together, imply a very powerful conclusion. Finally, we introduced a sixth Axiom or assumption about A's preferences.

    That is where we are now. von Neuman now takes the last step, and shows that if someone's preferences obey all six Axioms, then that person's preferences can be represented by a cardinal utility function over those outcomes that is invariant up to an affine (linear) transformation. I am not going to go through the proof, which consists mostly of substituting and multiplying through and gathering terms and all that good stuff. Suffice it to say that when von Neuman gets all done, he has shown that one way of assigning utility indices to the outcomes in O in conformity with the six Axioms is to assign to each outcome Oi the number Ui [as defined above]. This is then "the utility to A of Oi." Remember that this is just one way of assigning A's utility indices to the outcomes in the set O. Any affine transformation of those assignments will serve just as well.

    All of this has to be true about A's preferences in order for us to say that A's preferences can be represented by a cardinal utility function.