The time interpretation of expected utility theory
TThe time interpretation of expected utility theory
O. Peters , , A. Adamou ∗ London Mathematical Laboratory, 8 Margravine Gardens, W6 8RH London, UK Santa Fe Institute, 1399 Hyde Park Road, Santa Fe, NM 87501, USA (Dated: January 12, 2018)Decision theory is the model of individual human behavior employed by neoclassical economics.Built on this model of individual behavior are models of aggregate behavior that feed into models ofmacroeconomics and inform economic policy. Neoclassical economics has been fiercely criticized forfailing to make meaningful predictions of individual and aggregate behavior, and as a consequencehas been accused of misguiding economic policy. We identify as the Achilles heel of the formalismits least constrained component, namely the concept of utility. This concept was introduced asan additional degree of freedom in the 18th century when it was noticed that previous models ofdecision-making failed in many realistic situations. At the time, only pre-18th century mathematicswas available, and a fundamental solution of the problems was impossible. We re-visit the basicproblem and resolve it using modern techniques, developed in the late 19th and throughout the20th century. From this perspective utility functions do not appear as (irrational) psychological re-weightings of monetary amounts but as non-linear transformations that define ergodic observableson non-ergodic growth processes. As a consequence we are able to interpret different utility functionsas encoding different non-ergodic dynamics and remove the element of human irrationality from theexplanation of basic economic behavior. Special cases were treated in [1]. Here we develop thetheory for general utility functions.
PACS numbers: 02.50.Ey,05.10.Gg,05.20.Gg,05.40.Jc
The first three sections are concerned with putting thiswork in context, and a brief summary of relevant aspectsexpected utility theory. The novel technical part startsin Section IV.
I. POSITIONING
The present document is concerned with decision the-ory, part of the foundation of formal economics. It istherefore worth our while to spell out where in this vastcontext we feel our contribution is located. It addressesthe most formal part of economics, something that isoften called neoclassical economics. Broadly speakingthis is the part of economics that builds simple quantita-tive models of economic processes, analyzes these modelsmathematically, and interprets their behavior by givingreal-world meaning to model variables.This approach to thinking about economic issues be-came particularly dominant in the second half of the 20thcentury. Soon after the rise of its popularity it began tobe fiercely criticized. We take these criticisms very seri-ously and interpret them as an indication that somethingis fundamentally wrong in the way we conceptualize eco-nomic problems in the neoclassical approach.It is certainly true that some of the predictions of neo-classical economics clash with observations. Paradoxes,that is, apparent internal inconsistencies stubbornly re-main in the field (examples are the St. Petersburg para-dox or the Equity Premium Puzzle). This situation may ∗ Electronic address: [email protected], [email protected] elicit different responses, for example1. we can think of it as a normal part of science inprogress. Of course there are unresolved problems– finding their solutions is the job of the economicresearcher.2. we may conclude that the pen-and-paper approachusing models simple enough for analytical solutionmakes the representation of people too simplistic.Instead of analyzing such models, it has been ar-gued, we should turn to numerical work and buildin-silico worlds of agents with more complex, morerealistic behaviour.3. we may reject the entire scientific approach,whether analytic or numerical. Proponents of thisposition argue that economic questions are funda-mentally moral, not scientific, and that a scientificapproach is bound to miss the most important as-pects of the problem.We consider all three responses valid but not mutuallyexclusive. Every discipline has open problems, and itwould be foolish to dismiss an approach only because ithas not resolved every problem it encountered. Turningto computer simulations is part of every scientific disci-pline – when simple models fail and more complex mod-els are not analytically tractable, of course we should usecomputers. Nor can we dismiss the argument that build-ing a good society entails more than building an economi-cally wealthy society, and that mathematical models onlyelucidate the consequences of a set of axioms but cannotprove the validity, let alone the moral validity, of the ax-ioms themselves. a r X i v : . [ q -f i n . E C ] J a n The treatment we present here is most informativewith respect to perspective 1. We agree with the neoclas-sical approach in the following sense: we believe that sim-ple mathematical models can yield meaningful insights.We ask precisely how the failures of neoclassical eco-nomics may be interpreted as a flaw in the formalism thatcan be corrected. Such a flaw indeed exists, buried deepin the foundations of formal economics: often expectationvalues are taken where time averages would be appropri-ate. In this sense, formal economics has missed perhapsthe most important property of decisions: they are madein time and affect the future. They are not made in thecontext of co-existing possibilities across which resourcesmay be shared. We find reflections of this missing el-ement, for instance in the criticism of “short termism”that is often levelled against neo-classical economics. In-deed, an approach that disregards time in this preciseway will result in a formalism that is overly focused onthe short term. For example, such a formalism will notprovide an understanding of the fundamental benefits ofcooperation [2].We are led by this analysis to a correction of the for-malism capable of resolving a number of very persistentproblems. The work in the present paper is part of im-plementing the correction. It also helps clarify the rela-tionship between existing work in neo-classical economicsand our own work. Overall, we propose to re-visit andre-develop the entire formalism from a more nuanced ba-sis that gives the concept of time the central importanceit must have if the formalism is to be of use to humansand collections of humans whose existence is inescapablysubject to time.
II. EPISTEMOLOGY
We begin with some remarks on rationality. Economicsis the only science that frequently states that it assumesrational behavior. The comparison with physics is illu-minating.1 a A strong though rarely articulated assumptionin physics is that observed behavior can be ex-plained, in the sense that it follows rules, laws,or tendencies that – once identified – enableus to predict and comprehend the behavior ofa given system. This assumption is a funda-mental belief. It is assumed that the world,or rather very little isolated bits of the world,can be understood. Without this assumptionit would not be sensible to try to understandthe behavior of physical systems.b When we say that we assume rational behav-ior in economics, as we do, we mean nothingelse. We assume that observed behavior canbe explained, in the sense that it follows rules,laws, or tendencies that – once identified – en-able us to predict and comprehend human be- havior.2 a In physics we proceed by specifying a model ofthe observed behavior, that is, a mathematicalanalog, our guess of the rules governing thephysical system. For instance, we might saythat electrons are point particles with mass9 . × − kg that repel one another with1 /r Coulomb force.b Similarly, in economics we proceed by speci-fying our model of human behavior. For in-stance, we might say that humans choose theaction that maximizes the expectation valueof their monetary wealth.3 a We now confront our model with observations.No observation will be exactly as predicted bythe model. No two electrons will be observedto repel each other with 1 /r Coulomb force.There are too many other electrons around,and protons and gravity and countless per-turbations. Nonetheless, the model is usefulbecause it makes more or less sensible pre-dictions of large groups of electrons. The be-havior of a billion billion billion electrons overhere and a billion billion billion electrons overthere may be well described as the behaviorof many electrons repelling each other with1 /r Coulomb force. But we may find a realmwhere the electrons behave irrationally. Forinstance, a lump of 9 . × − kg of mattershould be able to absorb any amount of en-ergy. But as it turns out, electrons bound toa nucleus only accept certain fixed amounts ofenergy. This presents a dilemma to the physi-cist. He now has a choice between i) declar-ing electrons as behaving irrationally, i.e. giv-ing up the search for an explanation, and ii)declaring his model as deficient in the regimeof interest and search for another model. Of-ten a pretty good mathematical description ofthe irrational behavior is easily found but isperceived as a mathematical trick, just a de-scription with no inherent meaning [18]. Someyears or centuries later an intuition evolves ina new context, and the previously purely for-mal model (the mathematical trick) now ap-pears as a natural part of a bigger picture.b Similarly, observations of human behavior willnot be exactly as predicted. There are toomany idiosynchratic and circumstantial fac-tors involved. No single person will be ob-served to maximize the expectation value ofhis wealth consistently. Nonetheless, an over-all tendency may be predictable – a major-ity of people may prefer a 50/50 chance ofreceiving $2 or losing $1 over no change intheir wealth. But we may find a realm wherepeople behave consistently irrationally. Per-haps few people will prefer a 50/50 chanceof winning $20,000 or losing $10,000 over nochange in their wealth. Again, the scientisthas a choice between i) giving up the funda-mental belief that made him a scientist in thefirst place and declaring humans to be irra-tional, and ii) declaring his model deficient inthe new regime and look for a better model.In the example we mentioned, a new modelwas quickly found in the early 18th century.While human behavior is not well described asmaximizing the expectation value of wealth, itis quite well described as maximizing the ex-pectation value of changes in the logarithm ofwealth. Where the logarithm comes from isunclear – the psychological label “risk aver-sion” is attached to it but that’s just a label.In essence, this is a mathematical trick thatseems to work well, just as Planck’s trick ofquantizing energy worked well. Following thedevelopment of quantum mechanics, Planck’strick doesn’t seem so strange any more. Anintuition has arisen around it. The story ofthis paper is the story of the equivalent de-velopment in decision theory. Following theformulation of the concept of ergodicity, thelogarithm – the mathematical trick that saveddecision theory – does not seem so strangeany more. We identify the use of the loga-rithm as a different model of rationality: it isrational to maximize average wealth growthover time under the null model of multiplica-tive growth; it is not rational to maximize themathematical expectation of wealth. The twomodels give similar predictions for small mon-etary amounts, but entirely different predic-tions when the amounts involved approach thescale of total disposable wealth. Maximizingthe rate of change of the logarithm of wealthnow appears as a natural part of a bigger pic-ture.Firstly, we make the methodological choice to assumethat human behavior can be understood in principle.This is sometimes called the rationality hypothesis.Secondly, we postulate a specific form of rationality,that is, we state an axiom. Our axiom is that humansmake decisions in a manner that would optimise the time-average growth rate of wealth, were those decisions tobe repeated indefinitely. In our treatment, decisions arechoices between different stochastic processes, not choicesbetween different random variables as is usually the casein decision theory.Just as the description of electrons as charged pointmasses is not a good description in all contexts, our treat-ment is not a good description of economic decisions inall contexts. For example, we expect our axiom to be apoor representation of reality if relevant time scales are short. Of course “short” is a relative term that dependson the stochastic process. Time scales are short if a typ-ical trajectory is dominated by noise. In this regime theunderlying tendencies of an individual’s decisions haveno time to emerge, and are subsumed by randomness.Having pointed out the descriptive limits of our trea-ment, we add that our theory is not normative either.We simply point out the logical and mathematical con-nections between our treatment and classsical decisiontheory. III. EXPECTED UTILITY THEORY
Expected utility theory is the bedrock of neoclassicaleconomics. It provides the discipline’s answer to the fun-damental decision problem of how to choose between dif-ferent sets of uncertain outcomes. The generality of theframework is all-encompassing. Everything in the pastis certain, whereas everything in the future comes witha degree of uncertainty. Any decision is about choosingone uncertain future over alternative uncertain futures,wherefore expected utility theory is behind the answer ofneoclassical economics to any problem involving humandecision making.To keep the discussion manageable, we restrict it tofinancial decisions, i.e. we will not consider the utility ofan apple or of a poem but only utility differences betweendifferent dollar amounts. We restrict ourselves to situa-tions where any non-financial attendant circumstances ofthe decision can be disregarded. In other words we workwith a form of homo economicus.For a decision maker facing a choice between differentcourses of action, the workflow of expected utility theoryis as follows1. Imagine everything that could happen under thedifferent actions:
Associate with any action
A, B, C... a set of possiblefuture events Ω A , Ω B , Ω C ...
2. Estimate how likely the different consequences ofeach action are and how they would affect yourwealth:
For set Ω A , associate a probability p ( ω A ) and achange in wealth ∆ w ω A with each elementary event ω A ∈ Ω A , and similarly for all other sets.
3. Specify how much these outcomes would affect yourhappiness:
Define a utility function, u ( w ) , that only dependson wealth and describes the decision maker’s riskpreferences.
4. Aggregate the possible changes of happiness for anygiven event:
Compute the expected changes in utility asso-ciated with each available action, (cid:104) ∆ u A (cid:105) = (cid:80) Ω A p ( ω A ) u ( w + ∆ w ( ω A )) − u ( w ) , and similarlyfor actions B, C ...
5. Pick the action that makes you happiest:
The option with the highest expected utility changeis the decision maker’s best choice.
Each step of this process has been criticized, but weassume that all steps are possible. This does not re-flect a personal opinion that they are unproblematic inreality but is a methodological choice. By overlookingsome undeniable but possibly solvable difficulties we areable to inspect and question aspects at a deeper level ofthe formalism. Thus we assume that all possible futureevents, associated probabilities and changes in wealth areknown, that a suitable utility function is available, andthat the mathematical expectation of utility changes isthe mathematical object whose ordering reflects prefer-ences among actions. For simplicity we also make thecommon assumption that the time between taking an ac-tion and experiencing the corresponding change in wealthis independent of the action taken.Having accepted the basic premises of expected utilitytheory we acknowledge a remaining criticism. Expectedutility theory may not be useful in practice. Of courseusefulness can only be assessed if we know what we wantto achieve. One aim of decision theory may be to gen-uinely help real people make decisions. On this score ex-pected utility theory is limited. It is designed to ensureconsistency in an individual’s choices, but judged againstcriteria other than the risk preferences of the individualthe theory may produce consistently bad choices. Forexample, decision theory is not designed to find the deci-sions that lead to the fastest growth in wealth; the deci-sions it recommends are those that maximize the math-ematical expectation of a model of the decision maker’shappiness. For a gambling addict, for instance, these de-cisions may lead to bankruptcy. Expected utility theorywill recognize the individual as addicted to gambling, andconclude that he will be happiest behaving recklessly. Itis a laissez-faire approach to decision theory. Such anapproach is not illegitimate, however its limitations mustbe borne in mind. For instance, when designing policyit is no use to recognize that a financial institution thattakes larger risks than are good for systemic stability ishappiest when doing so. For any given decision makerit requires a utility function that can only be estimatedby querying the decision maker, possibly about simplerchoices that we believe he can assess more easily. Pref-erences of the decision maker are thus an input to theformalism. The output of the formalism is also a prefer-ence, namely the action that makes the decision makerthe happiest. In other words, the output is of the sametype as the input, which makes the framework circular.It may help the decision maker by telling him which ac-tion is most consistent with other actions he has takenor knows he would take in other situations.We will interpret the basic findings of expected utilitytheory in a different light. We will remove the circularity,for better or worse, and using our model of rational be-havior show that rationality according to our axioms un-der a reasonable model of wealth dynamics is equivalent to expected utility theory with commonly used utilityfunctions. Some researchers consider this an irrelevantcontribution because in that case we might just continueusing expected utility theory. We disagree and considerour contribution an important step forward because itmotivates new questions and provides answers that arenot circular.The range of questions we can answer in this way issurprising to us. Examples are: how does an investorchoose the leverage of an investment [3]? How can weresolve the St. Petersburg paradox [4]? How can weresolve the equity premium puzzle [5]? Why do peoplechoose to cooperate [2]? Why do insurance contractsexist [6]? How can we make sense of the recent changes inobserved economic inequality [7]? Do economic systemschange from one phase to another under different taxregimes?We have variously referred to our approach as “dynam-ical” or “time-based” or as recognizing disequilibrium ornon-ergodicity. The best term to refer to our perspectivemay be “ergodicity economics” – in every problem wehave treated we have asked whether the expectation val-ues of key variables were meaningful, in particular howthey were related to time averages.
IV. TECHNICAL
We repeat our two axioms.1. Human behavior can be understood. It followsa rationale and is in that sense rational.2. We explore the following model of this ratio-nale. Humans make decisions so that the growthrate of their wealth would be maximized over timewere those decisions repeated indefinitely.We suppose that an individual’s wealth evolves overtime according to a stochastic process. This is a de-parture from classical decision theory, where wealth issupposed to be described by a random variable withoutdynamic. To turn a gamble into a stochastic process andenable the techniques we have developed, a dynamic mustbe assumed, that is, a mode of repetition of the gamble,see [1].The individual is required to choose one from a set ofalternative stochastic processes, say x ( t ) and x ∗ ( t ). Wesuppose that this is done by considering how the decisionmaker would fare in their long-time limits.At each decision time, t , our individual acts to max-imise subsequent changes in his wealth by selecting x ( t )so that if he waits long enough his wealth will be greaterunder the chosen process than under the alternative pro-cess with certainty. Mathematically speaking, there ex-ists a sufficiently large t such that the probability of thechosen x ( t ) being greater than x ∗ ( t ) is arbitrarily closeto one, ∀ ε, x ∗ ( t ) ∃ ∆ t s.t. P (∆ x > ∆ x ∗ ) > − (cid:15), (1)where 0 < ε < x ≡ x ( t + ∆ t ) − x ( t ) , (2)with ∆ x ∗ similarly defined.The criterion is necessarily probabilistic since thequantities ∆ x and ∆ x ∗ are random variables and it mightbe possible for the latter to exceed the former for any fi-nite ∆ t . Only in the limit ∆ t → ∞ does the randomnessvanish from the system.Conceptually this criterion is tantamount to maximis-ing lim ∆ t →∞ { ∆ x } or, equivalently, lim ∆ t →∞ { ∆ x/ ∆ t } .However, neither limit is guaranteed to exist. For exam-ple, consider a choice between two geometric Brownianmotions, dx = x ( µdt + σdW ) , (3) dx ∗ = x ∗ ( µ ∗ dt + σ ∗ dW ) , (4)with µ > σ / µ ∗ > σ ∗ /
2. The quantities ∆ x/ ∆ t and ∆ x ∗ / ∆ t both diverge in the limit ∆ t → ∞ and acriterion requiring the larger to be selected fails to yielda decision.To overcome this problem we introduce a montonicallyincreasing function of wealth, which we call suggestively u ( x ). We define:∆ u ≡ u ( x ( t + ∆ t )) − u ( x ( t )); (5)∆ u ∗ ≡ u ( x ∗ ( t + ∆ t )) − u ( x ∗ ( t )) . (6)The monotonicity of u ( x ) means that the events ∆ x > ∆ x ∗ and ∆ u > ∆ u ∗ are the same. Taking ∆ t > u/ ∆ t > ∆ u ∗ / ∆ t , whencethe decision criterion in (Eq. 1) becomes ∀ ε, x ∗ ( t ) ∃ ∆ t s.t. P (cid:18) ∆ u ∆ t > ∆ u ∗ ∆ t (cid:19) > − (cid:15). (7)Our decision criterion has been recast such that it fo-cuses on the rate of change r ≡ ∆ u ∆ t , (8)As before, it is conceptually similar to maximising¯ r ≡ lim ∆ t →∞ (cid:26) ∆ u ∆ t (cid:27) = lim ∆ t →∞ { r } . (9)If x ( t ) satisfies certain conditions, to be discussed be-low, then the function u ( x ) can be chosen such that thislimit exists. We shall see that ¯ r is then the time-averagegrowth rate mentioned in Section II. For the moment weleave our criterion in the probabilistic form of (Eq. 7)but to continue the discussion we assume that the limit(Eq. 9) exists. Everything is now set up to make the link to expectedutility theory. Perhaps (Eq. 9) is the same as the rate ofchange of the expectation value of ∆ u (cid:104) ∆ u (cid:105) ∆ t = (cid:104) r (cid:105) . (10)We could then make the identification of u ( x ) being theutility function, noting that our criterion is equivalent tomaximizing the rate of change in expected utility. Wenote ∆ u and hence r are random variables but (cid:104) r (cid:105) is not.Taking the long-time limit is one way of removing ran-domness from the problem, and taking the expectationvalue is another. The expectation value is simply anotherlimit: it’s an average over N realizations of the randomnumber ∆ u , in the limit N → ∞ . The effect of removingrandomness is that the process x ( t ) is collapsed into thescalar ∆ u , and consistent transitive decisions are possibleby ranking the relevant scalars. In general, maximising (cid:104) r (cid:105) does not yield the same decisions as the criterion es-poused in (Eq. 7). This is only the case for a particularfunction u ( x ) whose shape depends on the process x ( t ).Our aim is to find these pairs of processes and functions.When using such u ( x ) as the utility function, expectedutility theory will be consistent with optimisation overtime. It is then possible to interpret observed behav-ior that is found to be consistent with expected utilitytheory using the utility function u ( x ) in purely dynami-cal terms: such behavior will lead to the fastest possiblewealth growth over time.We ask what sort of dynamic u must follow so that ¯ r = (cid:104) r (cid:105) or, put another way, so that r is an ergodic observable,in the sense that its time and ensemble averages are thesame [8, p. 32].We start by expressing the change in utility, ∆ u , as asum over M equal time intervals,∆ u ≡ u ( t + ∆ t ) − u ( t ) (11)= M (cid:88) m =1 [ u ( t + mδt ) − u ( t + ( m − δt )] (12)= M (cid:88) m =1 δu m ( t ) , (13)where δt ≡ ∆ t/M and δu m ( t ) ≡ u ( t + mδt ) − u ( t +( m − δt ). From (Eq. 9) we have¯ r = lim ∆ t →∞ (cid:40) t M (cid:88) m =1 δu m (cid:41) (14)= lim M →∞ (cid:40) M M (cid:88) m =1 δu m δt (cid:41) , (15)keeping δt fixed. From (Eq. 10) we obtain (cid:104) r (cid:105) = lim N →∞ (cid:40) N N (cid:88) n =1 ∆ u n ∆ t (cid:41) (16)where each ∆ u n is drawn independently from the distri-bution of ∆ u .We now compare the two expressions (Eq. 15) and(Eq. 16). Clearly the value of ¯ r in (Eq. 15) cannot dependon the way in which the diverging time period is parti-tioned, so the length of interval δt must be arbitrary andcan be set to the value of ∆ t in (Eq. 16), for consistencywe then call δu m ( t ) = ∆ u m ( t ). Expressions (Eq. 15) and(Eq. 16) are equivalent if the successive additive incre-ments, ∆ u m ( t ), are distributed identically to the ∆ u n in (Eq. 16), which requires only that they are stationaryand independent.Thus we have a condition on u ( t ) which suffices tomake ¯ r = (cid:104) r (cid:105) , namely that it be a stochastic processwhose additive increments are stationary and indepen-dent. This means that u ( t ) is, in general, a L´evy process.Without loss of realism we shall restrict our attention toprocesses with continuous paths. According to a theo-rem stated in [9, p. 2] and proved in [10, Chapter 12] thismeans that u ( t ) must be a Brownian motion with drift, du = a u dt + b u dW, (17)where dW is the infinitesimal increment of the Wienerprocess.By arguing backwards we can address concerns regard-ing the existence of ¯ r . If u follows the dynamics specifiedby (Eq. 17), then it is straightforward to show that thelimit ¯ r always exists and takes the value a . Consequentlythe decision criterion (Eq. 7) is equivalent to the optimi-sation of ¯ r , the time-average growth rate. The process x ( t ) may be chosen such that (Eq. 17) does not applyfor any choice of u ( x ). In this case we cannot interpretexpected utility theory dynamically, and such processesare likely to be pathological.This gives our central result:For expected utility theory to be equivalent to op-timisation over time, utility must follow an additivestochastic process with stationary increments which,in our framework, we shall take to be a Brownianmotion with drift.This is a fascinating general connection. If the phys-ical reason why we observe non-linear utility functionsis the non-linear effect of fluctuations over time, then agiven utility function encodes a corresponding stochasticwealth process. Provided that a utility function u ( x ) isinvertible, i.e. provided that its inverse, x ( u ), exists, asimple application of Itˆo calculus to (Eq. 17) yields di-rectly the SDE obeyed by the wealth, x . Thus everyinvertible utility function encodes a unique dynamic inwealth which arises from a Brownian motion in utility.This is explored further below. V. DYNAMIC FROM A UTILITY FUNCTION
We now illustrate the relationship between utility func-tions and wealth dynamics. For the reasons discussedabove we assume that utility follows a Brownian motionwith drift.If u ( x ) can be inverted to x ( u ) = u − ( u ), and x ( u )is twice differentiable, then it is possible to find thedynamic that corresponds to the utility function u ( x ).Equation (17) is an Itˆo process. Itˆo’s lemma tells us that dx will be another Itˆo process, and Itˆo’s formula specifieshow to find dx in terms of the relevant partial derivatives dx = (cid:18) ∂x∂t + a u ∂x∂u + 12 b u ∂ x∂u (cid:19)(cid:124) (cid:123)(cid:122) (cid:125) a x ( x ) dt + b u ∂x∂u (cid:124) (cid:123)(cid:122) (cid:125) b x ( x ) dW (18)We have thus shown that Theorem 1.
For any invertible utility function u ( x ) aclass of corresponding wealth processes dx can be obtainedsuch that the (linear) rate of change in the expectationvalue of net changes in utility is the time-average growthrate of wealth. As a consequence, optimizing expected changes in suchutility functions is equivalent to optimizing the time-average growth, in the sense of Section IV, under thecorresponding wealth process.The origin of optimizing expected utility can be un-derstood as follows: in the 18th century, the distinctionbetween ergodic and non-ergodic processes was unknown,and all stochastic processes were treated by computingexpectation values. Since the expectation value of thewealth process is an irrelevant mathematical object toan individual whose wealth is modelled by a non-ergodicprocess the available methods failed. The formalism wassaved by introducing a non-linear mapping of wealth,namely the utility function. The (failed) expectationvalue criterion was interpreted as theoretically optimal,and the non-linear utility functions were interpreted asa psychologically motivated pattern of human behavior.Conceptually, this is wrong.Optimization of time-average growth recognizes thenon-ergodicity of the situation and computes the appro-priate object from the outset – a procedure whose build-ing blocks were developed beginning in the late 19th cen-tury. It does not assume anything about human psychol-ogy and indeed predicts that the same behavior will beobserved in any growth-optimizing entities that need notbe human.
A. Examples
Equation (18), creates pairs of utility functions u ( x )and dynamics dx . In discrete time, two such pairs wereinvestigated in [1], namely cases 1 . and 2 . below.
1. Linear utility
The trivial linear utility function corresponds to addi-tive wealth dynamics (Brownian motion), u ( x ) = x ↔ dx = a u dt + b u dW. (19)
2. Logarithmic utility
Introduced by Bernoulli in 1738 [11], the logarithmicutility function is in wide use and corresponds to multi-plicative wealth dynamics (geometric Brownian motion), u ( x ) = ln( x ) ↔ dx = x (cid:20)(cid:18) a u + 12 b u (cid:19) dt + b u dW (cid:21) . (20)In practice the most useful case will be multiplicativewealth dynamics. But to demonstrate the generality ofthe procedure, we carry it out for a different special casethat is historically important.
3. Square-root (Cramer) utility
The first utility function ever to be suggested was thesquare-root function u ( x ) = x / , by Cramer in a 1728letter to Daniel Bernoulli, partially reproduced in [11].This function is invertible, namely x ( u ) = u , so that(Eq. 18) applies. We note that the square root, in a spe-cific sense, sits between the linear function and the log-arithm: lim x →∞ x / x = 0 and lim x →∞ ln( x ) x / = 0. Sincelinear utility produces additive dynamics and logarith-mic utility produces multiplicative dynamics, we expectsquare-root utility to produce something in between orsome mix. Substituting for x ( u ) in (Eq. 18) and carryingout the differentiations we find u ( x ) = x / ↔ dx = (cid:16) a u x / + b u (cid:17) dt + 2 b u x / dW. (21)The drift term contains a multiplicative element (bywhich we mean an element with x -dependence) and anadditive element. We see that the square-root utilityfunction that lies between the logarithm and the linearfunction indeed represents a dynamic that is partly ad-ditive and partly multiplicative.(Eq. 21) is reminiscent of the Cox-Ingersoll-Ross model[12] in financial mathematics, especially if a u <
0. Simi-lar dynamics, i.e. with a noise amplitude that is propor-tional to √ x , are also studied in the context of absorbing-state phase transitions in statistical physics [13, 14].That a 300-year-old letter is related to recent work instatistical mechanics is not surprising: the problems thatmotivated the development of decision theory, and indeedof probability theory itself are far-from equilibrium pro-cesses. Methods to study such processes were only de-veloped in the 20th century and constitute much of thework currently carried out in statistical mechanics. VI. UTILITY FUNCTION FROM A DYNAMIC
We now ask under what circumstances the procedurein (Eq. 18) can be inverted. When can a utility func-tion be found for a given dynamic? In other words, whatconditions does the dynamic dx have to satisfy so thatoptimization over time can be represented by optimiza-tion of expected net changes in utility u ( x )?We ask whether a given dynamic can be mapped intoa utility whose increments are described by Brownianmotion, (Eq. 17).The dynamic is an arbitrary Itˆo process dx = a x ( x ) dt + b x ( x ) dW, (22)where a x ( x ) and b x ( x ) are arbitrary functions of x . Forthis dynamic to translate into a Brownian motion for theutility, u ( x ) must satisfy the equivalent of (Eq. 18) withthe special requirement that the coefficients a u and b u in(Eq. 17) be constants, namely du = (cid:18) a x ( x ) ∂u∂x + 12 b x ( x ) ∂ u∂x (cid:19)(cid:124) (cid:123)(cid:122) (cid:125) a u dt + b x ( x ) ∂u∂x (cid:124) (cid:123)(cid:122) (cid:125) b u dW. (23)Explicitly, we arrive at two equations for the coefficients a u = a x ( x ) u (cid:48) + 12 b x ( x ) u (cid:48)(cid:48) (24)and b u = b x ( x ) u (cid:48) . (25)Differentiating (Eq. 25), it follows that u (cid:48)(cid:48) ( x ) = − b u b (cid:48) x ( x ) b x ( x ) . (26)Substituting in (Eq. 24) for u (cid:48) and u (cid:48)(cid:48) and solving for a x ( x ) we find the drift term as a function of the noiseterm, a x ( x ) = a u b u b x ( x ) + 12 b x ( x ) b (cid:48) x ( x ) . (27)In other words, knowledge of only the dynamic is suffi-cient to determine whether a corresponding utility func-tion exists. We do not need to construct the utility func-tion explicitly to know whether a pair of drift term andnoise term is consistent or not.Having determined for some dynamic that a consistentutility function exists, we can construct it by substitutingfor b x ( x ) in (Eq. 24). This yields a differential equationfor u a u = a x ( x ) u (cid:48) + b u u (cid:48) u (cid:48)(cid:48) (28)or 0 = − a u u (cid:48) + a x ( x ) u (cid:48) + b u u (cid:48)(cid:48) . (29)Overall, then the triplet noise term, drift term, utilityfunction is interdependent. Given a noise term we canfind consistent drift terms, and given a drift term wefind a consistency condition (differential equation) for theutility function. A. Example
Given a dynamic, it is possible to check whether thisdynamic can be mapped into a utility function, and theutility function itself can be found. We consider the fol-lowing example dx = (cid:18) a u b u e − x − e − x (cid:19) dt + e − x dW. (30)We note that a x ( x ) = a u b u e − x − e − x and b x ( x ) = e − x .Equation (27) imposes conditions on the drift term a x ( x )in terms of the noise term b x ( x ). Substituting in (Eq. 27)reveals that the consistency condition is satisfied by thedynamic in (Eq. 30).A typical trajectory of (Eq. 30) is shown in Fig. 1. C a s h U t ili t y FIG. 1: Typical trajectories of the wealth trajectory x ( t ) de-scribed by (Eq. 30), with parameter values a u = 1 / b u = 1, and the corresponding Brownian motion u ( t ). Notethat the fluctuations in x ( t ) become smaller for larger wealth. Because (Eq. 30) is internally consistent, it is possi-ble to derive the corresponding utility function. Equa-tion (25) is a first-order ordinary differential equation for u ( x ) u (cid:48) ( x ) = b u b x ( x ) , (31)which can be integrated to u ( x ) = (cid:90) x d ˜ x b u b x (˜ x ) + C, (32) with C an arbitrary constant of integration. This con-stant corresponds to the fact that only changes in utilityare meaningful, as was pointed out by von Neumann andMorgenstern [15] – this robust feature is visible whetherone thinks in dynamic terms and time averages or interms of consistent measure-theoretic concepts and ex-pectation values.Substituting for b x ( x ) from (Eq. 30), (Eq. 31) becomes u (cid:48) ( x ) = b u e x , (33)which is easily integrated to u ( x ) = b u e x + C, (34)plotted in Fig. 2. This expoential utility function ismonotonic and therefore invertible, which is reflectedin the fact that the consistency condition is satisfied.The utility function is convex. From the perspectiveof expected-utility theory an individual behaving opti-mally according to this function would be labelled “risk-seeking.” The dynamical perspective corresponds to aqualitatively different interpretation: Under the dynamic(Eq. 30) the “risk-seeking” individual behaves optimally,in the sense that his wealth will grow faster than thatof a risk-averse individual. The dynamic (Eq. 30) hasthe feature that fluctuations in wealth become smaller aswealth grows. High wealth is therefore sticky – an indi-vidual will quickly fluctuate out of low wealth and intohigher wealth. It will then tend to stay there. VII. WEALTH DISTRIBUTION FROM ADYNAMIC
The dynamical interpretation of expected utility the-ory makes it particularly simple to compute wealth dis-tributions. A utility function u ( x ) implies a dynamic x ( t ), and that dynamic generates a wealth distribution P x ( x, t ). We know that u ( t ) follows a simple Brownianmotion, wherefore we know that u ( t ) is normally dis-tributed according to P u ( u, t ) = N (cid:0) a u t, b u t (cid:1) . (35)Since we know P u ( u, t ), the distribution of x is easilyderived. The wealth distribution in a large population,is P x ( x, t ) = P u ( u ( x ) , t ) dudx . (36) A. Example of a wealth distribution
The utility function (Eq. 34) corresponds to the exam-ple dynamic (Eq. 30). The wealth distribution at anytime t can be read off (Eq. 36) P x ( x, t ) = 1 (cid:112) πb u t exp (cid:32) − ( b u e x + C − a u t ) (cid:112) b u t (cid:33) b u e x , (37)which is shown in Fig. 3. The distribution is sensiblegiven what we know about the dynamic – since fluctu-ations diminish with increasing wealth many individualswill be found at high wealth (all those that have fluctu-ated away from low wealth), with a heavy tail towardslower wealth. $0 $1 $2 $3 $4 $5Cash0 U t ili t y FIG. 2: Utility function of (Eq. 34), with b u = 1 and C = 0.Optimizing the expected change in this utility function alsooptimizes time-average growth under the corresponding dy-namic (Eq. 30). An unusual utility function – like the convexfunction shown here – reflects unusual dynamics, see text. $0.0 $0.5 $1.0 $1.5 $2.0 $2.5 $3.0 $3.5 $4.0Cash0.0000.0050.0100.0150.0200.0250.0300.035 P r o b a b ili t y d e n s i t y FIG. 3: Probability density function of wealth, also known asthe wealth distribution, (Eq. 37). This distribution is gener-ated by the wealth dynamic (Eq. 30). The time is fixed to t = 5, and we use a u = 1 / b u = 1, and C = 0. VIII. UNBOUNDEDNESS OF u ( x ) The scheme outlined in Section VI is informative forthe debate regarding the boundedness of utility func- tions. A well-established but false belief in the economicsliterature, due to Karl Menger [16, 17], is that permissi-ble utility functions must be bounded. We have arguedpreviously that boundedness is an unnecessary restric-tion, and that Menger’s arguments are not valid [1, 3].Section VI implies that the interpretation of expectedutility theory we offer here formally requires unbound-edness of utility functions. Bounded functions are notinvertible, and Menger’s incorrect result therefore con-tributed to obscuring the simple natural arguments wepresent here.Of course whether u ( x ) is bounded or not is practicallyirrelevant because x will always be finite. However, fora clean mathematical formalism an unbounded u ( x ) ishighly desirable.The problem is easily demonstrated by considering thecase of zero noise. Since u ( x ) always follows a Brownianmotion in our treatment, in the zero-noise case it follows du = a u dt, (38)meaning linear growth in time. For u to be bounded,time itself would have to be bounded. Another way tosee the problem is inverting u ( x ) to find x ( u ). If werequire simultaneously linear growth of u ( t ) in time, andboundedness from above, lim x →∞ u ( x ) = U b , then x ( t )has to diverge in the finite time it takes for u ( t ) to reach U b , namely in T b = U b a u (assuming for simplicity u ( t =0) = 0).Such features – an end of time or a finite-time singu-larity of wealth – are inconvenient to carry around ina formalism. Since they have no physical meaning, forsimplicity a model without them should be chosen, i.e. unbounded utility functions will be much better. We re-peat that Menger’s arguments against unbounded utilityfunctions are invalid and we need not worry about them. IX. DISCUSSION
Expected utility theory is an 18th-century patch, ap-plied to a flawed conceptual framework established inthe 17th century that made blatantly wrong predictionsof human behavior. Because the mathematics of ran-domness was in its infancy in the 18th century, theconceptual problems were overlooked, and utility the-ory set economics off in the wrong direction. Withoutany of the arbitrariness inherent in utility functions itis nowadays possible to give a physical meaning to thenon-linear mappings people seem to apply to monetaryamounts. These apparent mappings simply encode thenon-linearity of wealth dynamics.0 [1] O. Peters and M. Gell-Mann, Chaos , 23103 (2016),URL http://dx.doi.org/10.1063/1.4940236 .[2] O. Peters and A. Adamou, arXiv:1506.03414 (2015),URL http://arxiv.org/abs/1506.03414 .[3] O. Peters, Quant. Fin. , 1593 (2011), URL http://dx.doi.org/10.1080/14697688.2010.513338 .[4] O. Peters, Phil. Trans. R. Soc. A , 4913 (2011), URL http://dx.doi.org/10.1098/rsta.2011.0065 .[5] O. Peters and A. Adamou, arXiv preprintarXiv:1101.4548 (2011), URL http://arXiv.org/abs/1101.4548 .[6] O. Peters and A. Adamou, arXiv:1507.04655 (2015),URL http://arxiv.org/abs/1507.04655 .[7] Y. Berman, O. Peters, and A. Adamou (2017), URL https://ssrn.com/abstract=2794830 .[8] P. E. Kloeden and E. Platen,Numerical solution of stochastic differential equations,vol. 23 (Springer Science & Business Media, 1992).[9] J. M. Harrison, Brownian motion of performance and control(Cambridge University Press, 2013).[10] L. Breiman, Probability (Addison-Wesley PublishingCompany, 1968). [11] D. Bernoulli, Econometrica , 23 (1738), URL .[12] J. C. Cox, J. E. Ingersoll, and S. A. Ross, Econometrica , 363 (1985), ISSN 00129682, 14680262, URL .[13] J. Marro and R. Dickman,Nonequilibrium Phase Transitions in Lattice Models.(Cambridge University Press, 1999).[14] H. Hinrichsen, Adv. Phys. , 815 (2000).[15] J. von Neumann and O. Morgenstern,Theory of games and economic behavior (PrincetonUniversity Press, 1944).[16] K. Menger, J. Econ. , 459 (1934), URL http://dx.doi.org/10.1007/BF01311578 .[17] O. Peters, http://arxiv.org/abs/1110.1578 (2011), URL http://arxiv.org/abs/1110.1578http://arxiv.org/abs/1110.1578