As I relayed in my last post, in "Theory of Games and Economic Behavior", von Neumann and Morgenstern defended the validity of using utility to predict economic outcomes despite the fact that cross-individual utility cannot apparently be measured or apprehended directly:

"The [ . . . ] difficulties of the notion of utility, and particularly of the attempts to describe it as a number, are well known [ . . . ] It is sometimes claimed in economic literature that discussions of the notions of utility and preference are altogether unnecessary, since these are purely verbal definitions with no empirically observable consequences, i.e., entirely tautological. It does not seem to us that these notions are qualitatively inferior to certain well established and indispensable notions in physics, like force, mass, charge, etc. That is, while they are in their immediate form merely definitions, they become subject to empirical control through the theories which are built upon them" [von Neumann and Morgenstern, 8-9]

"[W]e wish to describe the fundamental concept of individual preferences by the use of a rather far-reaching notion of utility. Many economists will feel that we are assuming far too much [ by purporting to treat utility synthetically ], and that our standpoint is a retrogression from the more cautious modern technique of 'indifference curves' [ . . . ]" [von Neumann and Morgenstern, 16]

And they construct a theoretical framework for doing so:

"[A] numerical utility is dependent upon the possibility of comparing differences in utilities. This may seem - and indeed is - a more far-reaching assumption than that of a mere ability to state [subjective] preferences. [ . . . ]

Let us for the moment accept the picture of an individual whose system of preferences is all-embracing and complete, i.e. who, for any two objects or rather any two imagined events, possesses a clear intuition of preference.

More precisely we expect him, for any two alternative events which are put before him as possibilities, to be able to tell which of the two he prefers.

It is a very natural extension of this picture to permit such an individual to compare not only events, but even combinations of events with stated probabilities.

By a combination of two events we mean this: Let the two events be denoted by B and C [which are] 50%-50%. Then the "combination" is the prospect of seeing B occur with a probability of 50% and (if B does not occur) C with the (remaining) probability of 50%. [ . . . ]

It is clear that if [the individual] prefers A to B and also to C, then he will prefer [A] to the above combination as well; similarly, if he prefers [both] B [and] C to A, then he will prefer the combination [ 50% B, 50% C ] [ to A ], too. But if he should prefer A to, say, B, but at the same time C to A, then any assertion about his preference of A against the combination contains fundamentally new information. Specifically: if he now prefers A to the 50-50 combination of B and C, this provides a plausible base for the numerical estimate that his preference of A over B is in excess of his preference of C over A." [von Neumann and Morgenstern, 18] [emphases mine]

However, there is a confusion here.

Von Neumann and Morgenstern are trying to anchor a theory of utility - a quantity that had been most canonically dealt with in the analysis of public markets, which have a legible, monetary component:

"[Our] standpoint [ . . . ] will be mainly opportunistic. We wish to concentrate on one problem - which is not that of the measurement of utilities and of preferences [ . . . ] We shall therefore assume that the aim of all participants in the economic system, consumers as well as entrepreneurs, is money, or equivalently a single monetary commodity. This is supposed to be unrestrictedly divisible and substitutable, freely transferable and identical, even in the quantitative sense, with whatever 'satisfaction' or 'utility' is desired by each participant." [von Neumann and Morgenstern, 8]

But they are also trying to do something radically novel, and anchor a basic theory of how subjective [in the sense of being relativistic, not in the sense of being conscious] preferences - which are generally private to each individual - drive decisions:

"Historically, utility was first conceived as quantitatively measurable, i.e. as a number. [ . . . ] [E]very measurement - or rather every claim of measurability - must ultimately be based on some immediate sensation, which possibly cannot and certainly need not be analyzed any further. In the case of utility the immediate sensation of preference - of one object or aggregate of objects as against another - provides this basis. [ . . . ] All this is strongly reminiscent of the conditions existant at the beginning of the theory of heat: that too was based on the intuitively clear concept of one body feeling warmer than another, yet there was no immediate way to express significantly by how much, or how many times, or in what sense." [von Neumann and Morgenstern, 16]

Like I wrote in my last post, the concept of expected utility as an assignation of value to forecasted states bounded by their probability originated with Daniel Bernoulli to solve the St. Petersburg paradox.

Suppose, wrote Nicolaus, you are offered a lottery. The terms of the lottery are as follows:

You pay down initial stakes $N.

Then a fair coin is flipped, until it comes up H.

However many times «num_runs» the coin comes up H, you are awarded .

So if the coin comes up T on the 1st throw, you're awarded $$ 2^{0}$ , or $$ 1$ . If it comes up T on the 2nd throw, you get $$ 2^{1}$ , or $$ 2$ . If it doesn't come up T until the 4th throw, you're awarded $$ 2^{3}$ , or $$ 8$ .

Nicolaus's question was: what $N initial stakes should you be willing to offer in this lottery?

If you just multiply out the expected values here according to Jakob's formula, to try and get your expected value, you get

$$ (p_{1} = 1 / 2) * (a_{1} = 1)$ $[= $ (1 / 2)]$
$+ $ (p_{2} = 1 / 4) * (a_{2} = 2)$ $[= $ (1 / 2)]$
$+ $ (p_{3} = 1 / 8) * (a_{3} = 4)$ $[= $ (1 / 2)]$
$+ $ (p_{4} = 1 / 16) * (a_{4} = 8)$ $[= $ (1 / 2)]$

. . . and so on. As you get closer and closer to evaluating "all" the possibilities, your largest payout grows just as quickly as [the probability of the event you're currently computing], shrinks - that is, your payout goes by a factor of 2 every time, just as the probability of your event goes by a factor of 1/2 every time. So you'll continue doing the sum forever, 1/2 + 1/2 + 1/2, and never converge on an expected value.

It makes sense that «utility», in the wake of Daniel Bernoulli's coinage, became an object or atom of the theory of markets. The motivating concern for this likelihood-bounded assignation of value in the first place, was a thought-exploit conjecture relying specifically on an offer of unbounded monetary value. The hypothetical bookie offers us a potentially uncapped dollar value; as part of our thought-experiment-local suspension of disbelief, we leave out of our minds the inflationary effects that sufficiently large dollar values would have on the economy, and such things, and instead condition on the possibility of being offered unbounded units of market exchange.

Treating «expected received-units-of-market-exchange» [Bernoulli's «expected utility»] as necessarily shrinking faster than the underlying probabilities of [the events to which they are assigned] [i.e., bounded] is a perfectly reasonable solution to this paradox, and a reasonable bedrock, going forward, for the analysis of public markets.

But much proceeding analysis - including, importantly, von Neumann and Morgenstern's - has failed to distinguish between Bernoulli's «utility [bounded expected received-units-of-market-exchange]», and what von Neumann and Morgenstern were pointing at when they compared the subjective experience of preference to the subjective experience of heat.

If we can evaluate the terminal expected «valence» [or quantity-of-preference] of a foreseen event privately, in our own mind, without needing to take on faith any claimed number-of-publically-legible-units-of-market-exchange at which we would value the whole thing, then this «expected valence» doesn't necessarily need to be bounded.

Imagine a bookie offering you 128 units of valence - that is, value-in-the-world according to your subjective personal preferences - for an event about to occur with 1/256 probability. How many units of valence should you be willing to pay up front to play this game?

As von Neumann/Morgenstern mused, this question doesn't compute whichever set of disbeliefs you try to suspend:

"[The immediate sensation of preference] permits us only to say when for one person one utility is greater than another. It is not in itself a basis for numerical comparison of utilities for one person nor of any comparison between different persons [of course the authors, on the basis of assuming within-individual preferences over lotteries-of-events, go on to synthesize a numerical basis for the comparison of 'utilities' for one person - but not, in general, between different persons]." [von Neumann and Morgenstern, 16]

How can I give the bookie 1, or 2, units of value-to-me? How can the bookie reliably promise to deliver me 128 units of what I value, unless they can prove to me that they understand what I value in detail? Contrast with my ability to simply show the bookie that I have a $1 bill, or 2 $1 bills, or 1K satoshis or 2K satoshis in my Bitcoin wallet - and the bookie's ability to do the same. The «valence» version of the St. Petersburg paradox is over before we begin it, because without market valuation to anchor us in a common context, the bookie and I have no shared terms in our common economic language. So «valence» need not be bounded.

Another thing follows of what von Neumann and Morgenstern analyzed, not having been «utility». While it may necessarily be true that each complete and coherent set of preferences over simple lotteries, implies a unique set of preferences over complex lotteries, and actions, it is not the case that each complete and coherent private set of preferences over simple lotteries, implies a unique set of shareable market valuations over complex lotteries or actions. Thus, my «valence» might be VNM-coherent just from my private preferences conforming to VNM's axioms - but my «utility» is only VNM-coherent if the preferences I can legitimately express as market decisions so conform.