| English
"Куда идет мир? Каково будущее науки? Как "объять необъятное", получая образование - высшее, среднее, начальное? Как преодолеть "пропасть двух культур" - естественнонаучной и гуманитарной? Как создать и вырастить научную школу? Какова структура нашего познания? Как управлять риском? Можно ли с единой точки зрения взглянуть на проблемы математики и экономики, физики и психологии, компьютерных наук и географии, техники и философии?"

«Evaluating gambles using dynamics» 
O. Peters and M. Gell-Mann

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, D(n)=−x(t0), so that the corresponding growth factor is r(n)=0. The absorbing state x=0 can be reached but not escaped from. Closer inspection of Menger’s argument reveals that the issue is indeed more nuanced than he thought.

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 nmax.

2. If prizes increase as the exponential of an exponential then the expected utility change is positively divergent in the limit nmax → ∞ only for ticket fees satisfying F<x+G(1). The double-limit  results in the indeterminate form “−∞+∞.” Note that the positive divergence only happens in the unrealistic limit nmax→∞, whereas the negative divergence happens at finite F. The negative divergence is physically meaningful in that it reflects the impossibility to recover from bankruptcy under multiplicative dynamics.

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 G(n) may diverge for large 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 theory27,28 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.


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 Bernoulli5 did not actually compute the expected net change in logarithmic utility.8 Perceiving this as an error, Laplace4 corrected him implicitly without mention. Later researchers used Laplace’s corrected criterion until Menger6 unwittingly re-introduced Bernoulli’s inconsistency and introduced a new error by neglecting the second diverging term, ⟨δuB⟩. 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 book2 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.


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.


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
δuB−       loss in logarithmic utility when reducing x by F
⟨δuB+⟩    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 ·


  1. E. G. D. Cohen, “Boltzmann and statistical mechanics,” in Boltzmann’s Legacy 150 Years After His Birth, Atti dei Convegni Lincei ( Accademia Nazionale dei Lincei, Rome, 1997), Vol. 131, pp. 9–23; available at Scholar
  2. J. H. Cochrane, Asset Pricing ( Princeton University Press, 2001). Google Scholar
  3. C. Huygens, De ratiociniis in ludo aleae (On reckoning at Games of Chance) ( T. Woodward, London, 1657). Google Scholar
  4. P. S. Laplace, Théorie analytique des probabilités, 2nd ed. ( Ve. Courcier, Paris, 1814). Google Scholar
  5. D. Bernoulli, Specimen Theoriae Novae de Mensura Sortis, Translation (“Exposition of a new theory on the measurement of risk” by L. Sommer (1954)), Econometrica 22, 23–36 (1738). Scholar
  6. K. Menger, “Das Unsicherheitsmoment in der Wertlehre,” J. Econ. 5, 459–485 (1934). Scholar
  7. O. Peters, “Optimal leverage from non-ergodicity,” Quant. Finance 11, 1593–1602 (2011). Scholar
  8. O. Peters, “The time resolution of the St Petersburg paradox,” Philos. Trans. R. Soc. London, Ser. A 369, 4913–4931 (2011). Scholar
  9. O. Peters and A. Adamou, “Stochastic market efficiency,” preprint arXiv:1101.4548 (2011). URL Scholar
  10. M. Gell-Mann and J. B. Hartle, “Decoherent histories quantum mechanics with one real fine-grained history,” Phys. Rev. A 85, 062120 (2012). Scholar
  11. T. Tao, Topics in Random Matrix Theory ( American Mathematical Society, 2012). Google Scholar
  12. M. Buchanan, “Gamble with time,”Nat. Phys. 9, 3 (2013). Scholar
  13. V. Yakovenko and J. Rosser, “Colloquium: Statistical mechanics of money, wealth, and income,” Rev. Mod. Phys. 81, 1703–1725(2009). Scholar
  14. H. Morowitz, Beginnings of Cellular Life ( Yale University Press, 1992). Google Scholar
  15. O. Peters and A. Adamou, “ The evolutionary advantage of cooperation,” preprint arXiv:1506.03414 (2015). URL Scholar
  16. H. Chernoff and L. E. Moses, Elementary Decision Theory ( John Wiley & Sons, 1959). Google Scholar
  17. S. Redner, “Random multiplicative processes: An elementary tutorial,” Am. J. Phys. 58, 267–273 (1990). Scholar
  18. O. Peters, Menger 1934 revisited, preprint arXiv:1110.1578 (2011). URL Scholar
  19. P. Fermat and B. Pascal, private communication (1654). Google Scholar
  20. P. R. Montmort, Essay d’analyse sur les jeux de hazard, 2nd ed. (Jacque Quillau, Paris, 1713) (Reprinted by the American Mathematical Society, 2006). Google Scholar
  21. O. Peters and W. Klein, “ Ergodicity breaking in geometric Brownian motion,” Phys. Rev. Lett. 110, 100603 (2013). Scholar
  22. G. I. Barenblatt, Scaling ( Cambridge University Press, 2003). Google Scholar
  23. I. Todhunter, A History of the Mathematical Theory of Probability ( Macmillan & Co., 1865). Google Scholar
  24. J. von Neumann and O. Morgenstern, Theory of Games and Economic Behavior ( Princeton University Press, 1944). Google Scholar
  25. P. A. Samuelson, Foundations of Economic Analysis, enlarged edition ( Harvard University Press, 1983). Google Scholar
  26. J. L. Coolidge, An Introduction to Mathematical Probability (Oxford University Press, 1925). Google Scholar
  27. J. L. Kelly, Jr., “A new interpretation of information rate,” Bell Syst. Tech. J. 35, 917–926 (1956). Scholar
  28. T. M. Cover and J. A. Thomas, Elements of Information Theory (John Wiley & Sons, 1991). Google Scholar
  29. K. Arrow, “The use of unbounded utility functions in expected-utility maximization: Response,” Q. J. Econ. 88, 136–138 (1974). Scholar