The argument sounds plausible. If the logarithm specifies the value attached to money, like another currency, then there is no intuitive reason why it should be qualitatively different from a linear function. But the logarithm encoding multiplicative dynamics provides us with additional intuition: multiplicative dynamics imply an absorbing boundary. Unlike under additive dynamics it is impossible to recover from bankruptcy, and this is a qualitative difference. In the coin-toss example in Fig. 2 bankruptcy cannot occur, but in the general gamble under multiplicative dynamics, bankruptcy is possible if at least one possible outcome, * n^{∗}*, say, leads to the loss of one’s entire wealth,

*, so that the corresponding growth factor is*

**D(n**^{∗})=−x(t_{0})

**r(n**^{∗})=**0**. The absorbing state

*can be reached but not escaped from. Closer inspection of Menger’s argument reveals that the issue is indeed more nuanced than he thought.*

**x=0**We separate out the first term, for the smallest payout, and write the expected utility change as

This form motivates the following evaluation of the three steps in Menger’s argument:

1. Apart from turning exponential wealth changes into linear utility changes, logarithmic utility also imposes a no-bankruptcy condition. Bankruptcy becomes possible at * F=x+G(1)*. Reflecting this, the limit is negatively divergent for any

*.*

**n**_{max}2. If prizes increase as the exponential of an exponential then the expected utility change is positively divergent in the limit * n_{max} → ∞* only for ticket fees satisfying

**F****<**

*. The double-limit results in the indeterminate form “*

**x+G(1)****−∞+∞.**” Note that the positive divergence only happens in the unrealistic limit

*, whereas the negative divergence happens at finite*

**n**_{max}→∞*. The negative divergence is physically meaningful in that it reflects the impossibility to recover from bankruptcy under multiplicative dynamics.*

**F**3. In such games, logarithmic utility does not predict that the player will want to pay any finite ticket fee. Instead, it predicts that the player will not pay more than * x+G(1)*, irrespective of how

**may diverge for large**

*G(n)**n*. This is qualitatively different from behavior predicted by Huygens’ criterion (linear utility), where under diverging expected prizes no ticket fee exists that the player would not be willing to pay. Logarithmic utility, carefully interpreted, resolves the class of problems for which Menger thought it would fail.

Despite a persisting intuitive discomfort, renowned economists accepted Menger’s conclusions and considered them an important milestone in the development of utility theory. Menger implicitly ruled out the all-important logarithmic function that connects utility theory to information theory^{27,28} and provides the most natural connection to the ergodicity argument we have presented. Menger also ruled out the linear function that corresponds to Huygens’ Criterion, which utility theory was supposed to generalize.

Requiring boundedness for utility functions is methodologically inapt. It is often stated that a diverging expected utility is “impossible” [Ref. 16, p. 106], or that it “seems natural” to require all expected utilities to be finite [Ref. 29, p. 28–29]. Presumably, these statements reflect the intuitive notion that no real thing can be infinitely useful. To implement this notion in the formalism of decision theory, it was decided to make utility functions bounded. A far more natural way to implement the same notion would be to recognize that money amounts (and quantities of anything physical, anything money could represent) are themselves bounded, and that this makes any usefulness one may assign to them finite, even if utility functions are unbounded. There is no need to place bounds on *u(x)* if x itself is bounded.

**V. SUMMARY AND CONCLUSION**

Our method starts by recognizing the inevitable non-ergodicity of stochastic growth processes, e.g., noisy multiplicative growth. The specific stochastic process implies a set of meaningful observables with ergodic properties, e.g., the exponential growth rate. These observables make use of a mapping that in the tradition of economics is viewed as a psychological utility function, e.g., the logarithm.

The dynamic approach to the gamble problem makes sense of risk aversion as optimal behavior for a given dynamic and level of wealth, implying a different concept of rationality. Maximizing expectation values of observables that do not have the ergodic property of Section I cannot be considered rational for an individual. Instead, it is more useful to consider rational the optimization of time-average performance, or of expectation values of appropriate ergodic observables. We note that where optimization is used in science, the deep insight is finding the right object to optimize (e.g., the action in Hamiltonian mechanics, or the entropy in the microcanonical ensemble). The same is true in the present case—deep insight is gained by finding the right object to optimize—we suggest time-average growth. Laplace’s Criterion interpreted as an ergodic growth rate under multiplicative dynamics avoids the fundamental circularity of the behavioral interpretation. In the latter, preferences, i.e., choices an individual would make, have to be encoded in a utility function, the utility function is passed through the formalism, and the output is the same as the input: the choices an individual would make.

We have repeated here that Bernoulli^{5} did not actually compute the expected net change in logarithmic utility.^{8} Perceiving this as an error, Laplace^{4} corrected him implicitly without mention. Later researchers used Laplace’s corrected criterion until Menger^{6} unwittingly re-introduced Bernoulli’s inconsistency and introduced a new error by neglecting the second diverging term, ⟨**δu _{B}−**⟩. Throughout the twentieth century, Menger’s incorrect conclusions were accepted by prominent economists although they noticed, and struggled with, detrimental consequences of the (undetected) error for the developing formalism.

We have presented Menger’s argument against unbounded utility functions as it is commonly stated nowadays. This argument is neither formally correct (it ignores the negative divergence of the logarithm) nor compatible with physical intuition (it ignores the absorbing boundary). Laplace’s Criterion—contrary to common belief—elegantly resolves Menger-type games.

Logarithmic utility must not be banned formally because it is mathematically equivalent to the modern method of defining an ergodic observable for multiplicative dynamics. This point of view provides a firm basis on which to erect a scientific formalism. The concepts we have presented resolve the fundamental problem of decision theory, therefore game theory, and asset pricing. Cochrane’s book^{2} is important in this context as it sets out clearly that all of asset pricing can be derived from the “basic pricing equation”—precisely the combination of a utility function and expectation values we have critiqued here. Cochrane further argues that the methods used in asset pricing summarize much of macroeconomics. The problems listed there as those of greatest importance to the discipline at the moment can be addressed using the modern dynamic perspective.

In presenting our results, we have made a judgement call between clarity and generality. We have chosen the most general problem of decision theory, but have treated it specifically for discrete time and wealth changes. Gambles that are continuous in time and wealth changes can be treated along similar lines,^{7} as can the specific St Petersburg problem.^{8} We have contrasted purely additive dynamics with purely multiplicative dynamics. A generalization beyond purely additive or multiplicative dynamics is possible, just as it is possible to define utility functions other than the linear or logarithmic function. This will be the subject of a future publication. The arguments we have outlined are not restricted to monetary wealth but apply to anything that is well modeled by a stochastic growth process. Applications to ecology and biology seem natural.

**ACKNOWLEDGMENTS**

We thank K. Arrow for discussions that started at the workshop “Combining Information Theory and Game Theory” in 2012 at the Santa Fe Institute, and for numerous helpful comments during the preparation of the manuscript. O.P. would like to thank A. Adamou for discussions and a careful reading of the manuscript and D. E. Smith for helpful comments.

**NOMENCLATURE
**

D net payout from a gamble

F fee for a ticket in a lottery

G prize in a lottery

n integer specifying an outcome

N number of parallel realizations of a gamble

n* integer specifying an outcome that leads to bankruptcy

nτ outcome that occurs in round τ

nmax number of possible outcomes

ν index specifying one parallel realization of a gamble

p p(n) is the probability of outcome n

r growth factor. If outcome n occurs,

t time

T number of sequential rounds of a gamble

t0 time before the first round of the gamble

τ index specifying one sequential round of a gamble

δt duration of one round of a gamble

u utility function

δu change in utility in one round of a gamble

uB Bernoullis logarithmic utility function

δu

_{B}− loss in logarithmic utility when reducing x by F

⟨δu

_{B}+⟩ expectation value of gains in logarithmic utility at zero ticket price

uC Cramers square-root utility function

x wealth

xν wealth in realization ν

x⎯⎯T finite-time average wealth

δX change in wealth in one round of a gamble.

⟨⋅⟩ expectation value of ·

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