GIVE THE OLD MAN A BREAK

Here is the simple, shameful truth:  I don't much like politics.  It is infuriating, frustrating, desperately important, and I cannot really do anything about it.  What I like is ideas, which are simple, pure, eternal, and as satisfying as a Baroque fugue.  I talk about politics because I think I should, but I would rather talk about ideas.

So today, I am going to talk about an elegant much-misunderstood idea:  the idea of a zero-sum game.  Now, as it happens, I have already done just that, nine years ago, but it was on my other blog, which most readers of this blog do not even know exists.  So, herewith, from that blog, my formal explanation of the concept of a zero-sum game, very lightly edited.  Even if it does not grab you, perhaps you will be able to appreciate its peaceful clarity:

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, using R to mean “prefers or is indifferent between,” 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.

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.

    This a very powerful axiom.  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), with the probabilities p summing to 1.

    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 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.

I want now to take some time to make sure that everyone understands just how strong these assumptions are, and also exactly how to interpret them. The first point to understand is in a way the hardest. You might think that our subject, A, decides how she feels about all of these simple and compound lotteries by carrying out expected utility calculations and then saying to herself, "Well, since this one has a greater mathematical expectation than that one, I prefer this one to that one." You might think that, because, good heavens, how else could she possibly decide which she prefers to which? But if you thought that [which of course none of you does], you would be WRONG, WRONG, WRONG! TOTALLY WRONG, WRONG, WRONG! That would be, to use correctly a phrase that these days is almost always used incorrectly, begging the question. It would be assuming what is to be proved, and thus arguing in a circle. What von Neuman actually supposes is that our subject, A, looks at the outcomes O1, O2, etc and decides how she feels about them. She ranks them in order of her preference. She then looks at the infinitude of simple lotteries and compound lotteries and decides how she feels about them as well. She merges this all in her mind into a single complete, transitive ordering of all of those outcomes and simple lotteries and compound lotteries. Then von Neuman posits that her preferences, thus arrived at, in fact obey the six Axioms. If that is so, then, von Neuman shows, her preferences can be represented AS THOUGH she were carrying out expected utility calculations in her head in accordance with the axioms.

    We are talking here about an enormously powerful set of idealizing and simplifying assumptions, as powerful in their way as the assumptions economists have to make before they can talk about continuously twice differentiable production functions [which they need in order to prove their nifty equilibrium theorems.] Let me draw on something I said earlier to show you just how powerful these Axioms are. Look at Axiom V, the transitivity axiom, and let us recall the eye doctor example. 

Suppose that the lotteries A is comparing are big Amusement Park wheels, on which are marked off different sized wedges [each defined by two radii], each one of which is associated with one of the outcomes in the set, O. It would be no problem at all to construct a whole series of wheels, each of which is such a tiny bit different from the one next to it that when A is shown the two wheels together, she looks at them and says, I am indifferent between those two lotteries." But suitably arranged, the series of wheels might very slowly, indiscernibly, alter the size of the wedges associated with two prizes or outcomes, Oi and Oj, until, if we were to show A the first and the last in the series, she would look at them and say, flatly, I prefer the one on the left to the one on the right. Whoops. No transitivity! Axiom V rules out any such state of affairs.

    Well, you can think about each one of the Axioms and see whether you can imagine a situation in which the assumption of that Axiom clearly requires something very strong and even counterintuitive. But rather than go on about that, I am going to take the next step.

    We are now ready to extend our notion of strictly opposed preference orders. Recall that we describe the preference orders of A and B over a set of outcomes, O, as "strictly opposed" when A prefers Oi to Oj or is indifferent between them if and only if B prefers Oj to Oi or is indifferent between them. We will describe the preference orders of A and B over the infinite set of lotteries, simple and compound, over the set of outcomes, O, as "strictly competitive" when A prefers Lottery L1 to Lottery L2 or is indifferent between them if and only if B prefers L2 to L1 or is indifferent between them. This means that A and B not only rank all of the outcomes in exactly opposite ways. They also rank all of the lotteries, simple or compound, over those outcomes in exactly opposite ways.

    In this very specific set of circumstances [where all six axioms apply to both A's preferences and B's preferences, and A and B have strictly competitive preferences], we can normalize the utility functions of A and B so that for any lottery, L, simple or compound, over the set of outcomes, O, the sum of the utility index assigned to L by A's utility function and the utility index assigned to L by B's utility function is a constant. This is what is meant by saying that a game played by A and B is a constant sum game.

    Rather than grind out an algebraic proof, I will offer a simple, intuitive proof that should be easy to grasp. We shall use u(L) to mean the utility that A's utility function assigns to L, and u'(L) to mean the utility that B's utility function assigns to L. Now, we are permitted arbitrarily to let A's most preferred outcome, O1, have a utility of 1, and A's least preferred outcome have a utility of 0. Since A and B have strictly opposed preferences for outcomes, B's most preferred outcome is On and his least preferred outcome is O1. We are permitted to set B's utility for On equal to 1 and for O1 equal to 0. So the utility assignments of both A and B can be portrayed as lying along a line that runs between 1 and 0.

    No matter what lottery, L, we have chosen, we know from the Axioms that it is equivalent, for A, to some lottery over just O1 and On whose probability weights are u and (1-u) for some u. Think of that as a point somewhere on the line running between 1 and 0. [Remember that for the best and worst alternatives, O1 and On, the point is an endpoint of the line.] The same thing is true for B. We are now going to prove that the point on the line representing A's utility for L and the point on the line representing B's utility for L are the same point. To prove this, we will assume the contrary and derive a contradiction with our assumption that A and B have strictly opposed preferences. So, let us choose a point representing u(L) and a different point representing u'(L), and then choose some point that lies between those two points, which we shall call S. Here is a picture of the situation. The line runs from 1 to 0 for A, and from 0 to 1 for B:

                                                                                                                     1                                                                                   0

         |---------u(L)---------------S-------------u'(L)-------------|

         0                                                                                   1

    The point S represents a lottery, Ls, with weights S for On and (1-S) for O1. Now, just from looking at the diagram, we can see the following:

    (i) A prefers L to Ls, because L puts greater weight on O1 than Ls does. [u(L) is closer to the 1 than S is].

    (ii) B prefers L to Ls, because L puts greater weight on On [his favorite] than Ls does. [u'(L) is closer to his 1 than S is.]

    But this means that A and B do not have strictly opposed preferences, since they both prefer L to Ls. And this contradicts the assumption. So no matter which lottery L we choose, there cannot be a point S between u(L) and u'(L), which means they are the same point.

    But if they are the same point, then A's utility is u and B's utility is u' = (1-u), regardless of which lottery, L, we choose. and:

    u + u' = u + (1-u) = 1 

   

    Now, B's utility function is invariant under an affine (linear) transformation. So let us introduce the following affine transformation:

    u'' = u' - 1    

    What this does is to re-label B's utility assignments so that instead of running from 1 to 0, the run from 0 to -1. This means that A's and B's utilities for any arbitrary lottery L are no longer u and (1-u).  Instead, they are now u and -u. And the sum of u and -u is zero.

    THIS, AND ONLY THIS, IS WHAT IS MEANT BY SAYING THAT A GAME PLAYED BY A AND B IS A ZERO-SUM GAME.