Home > The Economics Profession > Why expected utility theory is an ex-parrot (wonkish)

Why expected utility theory is an ex-parrot (wonkish)

from Lars Syll

Although the expected utility theory is obviously both theoretically and descriptively inadequate, colleagues and microeconomics textbook writers all over the world gladly continue to use it, as though its deficiencies were unknown or unheard of.

That cannot be the right attitude when facing scientific anomalies. When models are plainly wrong, you’d better replace them! As Matthew Rabin and Richard Thaler have it in Risk Aversion:

ex-Parrot

It is time for economists to recognize that expected utility is an ex-hypothesis, so that we can concentrate our energies on the important task of developing better descriptive models of choice under uncertainty.      

If a friend of yours offered you a gamble on the toss of a coin where you could lose €100 or win €200, would you accept it? Probably not. But if you were offered to make one hundred such bets, you would probably be willing to accept it, since most of us see that the aggregated gamble of one hundred 50–50 lose €100/gain €200 bets has an expected return of €5000 (and making our probabilistic calculations we find out that there is only a 0.04% risk of losing any money).

Unfortunately – at least if you want to adhere to the standard neoclassical expected utility maximization theory – you are then considered irrational! A neoclassical utility maximizer that rejects the single gamble should also reject the aggregate offer.

In his modern classic Risk Aversion and Expected-Utility Theory: A Calibration Theorem Matthew Rabin  writes:

Using expected-utility theory, economists model risk aversion as arising solely because the utility function over wealth is concave. This diminishing-marginal-utility-of-wealth theory of risk aversion is psychologically intuitive, and surely helps explain some of our aversion to large-scale risk: We dislike vast uncertainty in lifetime wealth because a dollar that helps us avoid poverty is more valuable than a dollar that helps us become very rich.

Yet this theory also implies that people are approximately risk neutral when stakes are small. Arrow (1971, p. 100) shows that an expected-utility maximizer with a differentiable utility function will always want to take a sufficiently small stake in any positive-expected-value bet. That is, expected-utility maximizers are (almost everywhere) arbitrarily close to risk neutral when stakes are arbitrarily small. While most economists understand this formal limit result, fewer appreciate that the approximate risk-neutrality prediction holds not just for negligible stakes, but for quite sizable and economically important stakes. Economists often invoke expected-utility theory to explain substantial (observed or posited) risk aversion over stakes where the theory actually predicts virtual risk neutrality.While not broadly appreciated, the inability of expected-utility theory to provide a plausible account of risk aversion over modest stakes has become oral tradition among some subsets of researchers, and has been illustrated in writing in a variety of different contexts using standard utility functions.

In this paper, I reinforce this previous research by presenting a theorem which calibrates a relationship between risk attitudes over small and large stakes. The theorem shows that, within the expected-utility model, anything but virtual risk neutrality over modest stakes implies manifestly unrealistic risk aversion over large stakes. The theorem is entirely ‘‘non-parametric’’, assuming nothing about the utility function except concavity. In the next section I illustrate implications of the theorem with examples of the form ‘‘If an expected-utility maximizer always turns down modest-stakes gamble X, she will always turn down large-stakes gamble Y.’’ Suppose that, from any initial wealth level, a person turns down gambles where she loses $100 or gains $110, each with 50% probability. Then she will turn down 50-50 bets of losing $1,000 or gaining any sum of money. A person who would always turn down 50-50 lose $1,000/gain $1,050 bets would always turn down 50-50 bets of losing $20,000 or gaining any sum. These are implausible degrees of risk aversion. The theorem not only yields implications if we know somebody will turn down a bet for all initial wealth levels. Suppose we knew a risk-averse person turns down 50-50 lose $100/gain $105 bets for any lifetime wealth level less than $350,000, but knew nothing about the degree of her risk aversion for wealth levels above $350,000. Then we know that from an initial wealth level of $340,000 the person will turn down a 50-50 bet of losing $4,000 and gaining $635,670.

The intuition for such examples, and for the theorem itself, is that within the expected-utility framework turning down a modest-stakes gamble means that the marginal utility of money must diminish very quickly for small changes in wealth. For instance, if you reject a 50-50 lose $10/gain $11 gamble because of diminishing marginal utility, it must be that you value the 11th dollar above your current wealth by at most 10/11 as much as you valued the 10th-to-last-dollar of your current wealth.

Iterating this observation, if you have the same aversion to the lose $10/gain $11 bet if you were $21 wealthier, you value the 32nd dollar above your current wealth by at most 10/11 x 10/11 ~ 5/6 as much as your 10th-to-last dollar. You will value your 220th dollar by at most 3/20 as much as your last dollar, and your 880 dollar by at most 1/2000 of your last dollar. This is an absurd rate for the value of money to deteriorate — and the theorem shows the rate of deterioration implied by expected-utility theory is actually quicker than this. Indeed, the theorem is really just an algebraic articulation of how implausible it is that the consumption value of a dollar changes significantly as a function of whether your lifetime wealth is $10, $100, or even $1,000 higher or lower. From such observations we should conclude that aversion to modest-stakes risk has nothing to do with the diminishing marginal utility of wealth.

Expected-utility theory seems to be a useful and adequate model of risk aversion for many purposes, and it is especially attractive in lieu of an equally tractable alternative model. ‘‘Extremelyconcave expected utility’’ may even be useful as a parsimonious tool for modeling aversion to modest-scale risk. But this and previous papers make clear that expected-utility theory is manifestly not close to the right explanation of risk attitudes over modest stakes. Moreover, when the specific structure of expected-utility theory is used to analyze situations involving modest stakes — such as in research that assumes that large-stake and modest-stake risk attitudes derive from the same utility-for-wealth function — it can be very misleading. In the concluding section, I discuss a few examples of such research where the expected-utility hypothesis is detrimentally maintained, and speculate very briefly on what set of ingredients may be needed to provide a better account of risk attitudes. In the next section, I discuss the theorem and illustrate its implications …

Expected-utility theory makes wrong predictions about the relationship between risk aversion over modest stakes and risk aversion over large stakes. Hence, when measuring risk attitudes maintaining the expected-utility hypothesis, differences in estimates of risk attitudes may come from differences in the scale of risk comprising data sets, rather than from differences in risk attitudes of the people being studied. Data sets dominated by modest-risk investment opportunities are likely to yield much higher estimates of risk aversion than data sets dominated by larger-scale investment opportunities. So not only are standard measures of risk aversion somewhat hard to interpret given that people are not expected-utility maximizers, but even attempts to compare risk attitudes so as to compare across groups will be misleading unless economists pay due attention to the theory’s calibrational problems …

Indeed, what is empirically the most firmly established feature of risk preferences, loss aversion, is a departure from expected-utility theory that provides a direct explanation for modest-scale risk aversion. Loss aversion says that people are significantly more averse to losses relative to the status quo than they are attracted by gains, and more generally that people’s utilities are determined by changes in wealth rather than absolute levels. Preferences incorporating loss aversion can reconcile significant small-scale risk aversion with reasonable degrees of large-scale risk aversion … Variants of this or other models of risk attitudes can provide useful alternatives to expected-utility theory that can reconcile plausible risk attitudes over large stakes with non-trivial risk aversion over modest stakes.

In a similar vein, Daniel Kahneman writes in his wonderful Thinking, Fast and Slow, that expected utility theory is seriously flawed since it doesn’t take into consideration the basic fact that people’s choices are influenced by changes in their wealth. Where standard microeconomic theory assumes that preferences are stable over time, Kahneman and other behavioural economists have forcefully again and again shown that preferences aren’t fixed, but vary with different reference points. How can a theory that doesn’t allow for people having different reference points from which they consider their options have an almost axiomatic status within economic theory?

The mystery is how a conception of the utility of outcomes that is vulnerable to such obvious counterexamples survived for so long. I can explain it only by a weakness of the scholarly mind … I call it theory-induced blindness: once you have accepted a theory and used it as a tool in your thinking it is extraordinarily difficult to notice its flaws … You give the theory the benefit of the doubt, trusting the community of experts who have accepted it … But they did not pursue the idea to the point of saying, “This theory is seriously wrong because it ignores the fact that utility depends on the history of one’s wealth, not only present wealth.”

On a more economic-theoretical level, information theory – and especially the so called the Kelly theorem – also highlights the problems concerning the neoclassical theory of expected utility. Suppose I want to play a game. Let’s say we are tossing a coin. If heads comes up, I win a dollar, and if tails comes up, I lose a dollar. Suppose further that I believe I know that the coin is asymmetrical and that the probability of getting heads (p) is greater than 50% – say 60% (0.6) – while the bookmaker assumes that the coin is totally symmetric. How much of my bankroll (T), should I optimally invest in this game?

A strict neoclassical utility-maximizing economist would suggest that my goal should be to maximize the expected value of my bankroll (wealth), and according to this view, I ought to bet my entire bankroll.

Does that sound rational? Most people would answer no to that question. The risk of losing is so high, that I already after few games played – the expected time until my first loss arises is 1/(1-p), which in this case is equal to 2.5 – with a high likelihood would be losing and thereby become bankrupt. The expected-value maximizing economist does not seem to have a particularly attractive approach.

So what’s the alternative? One possibility is to apply the so-called Kelly-strategy – after the American physicist and information theorist John L. Kelly, who in the article A New Interpretation of Information Rate (1956) suggested this criterion for how to optimize the size of the bet – under which the optimum is to invest a specific fraction (x) of wealth (T) in each game. How do we arrive at this fraction?

When I win, I have (1 + x) times more than before, and when I lose (1 – x) times less. After n rounds, when I have won v times and lost n – v times, my new bankroll (W) is

The bankroll increases multiplicatively – “compound interest” – and the long-term average growth rate for my wealth can then be easily calculated by taking the logarithms of (1), which gives

(2) log (W/ T) = v log (1 + x) + (n – v) log (1 – x).

If we divide both sides by n we get

(3) [log (W / T)] / n = [v log (1 + x) + (n – v) log (1 – x)] / n

The left hand side now represents the average growth rate (g) in each game. On the right hand side the ratio v/n is equal to the percentage of bets that I won, and when n is large, this fraction will be close to p. Similarly, (n – v)/n is close to (1 – p). When the number of bets is large, the average growth rate is

(4) g = p log (1 + x) + (1 – p) log (1 – x).

Now we can easily determine the value of x that maximizes g:

(5) d [p log (1 + x) + (1 – p) log (1 – x)]/d x = p/(1 + x) – (1 – p)/(1 – x) => p/(1 + x) – (1 – p)/(1 – x) = 0 =>

(6) x = p – (1 – p)

Since p is the probability that I will win, and (1 – p) is the probability that I will lose, the Kelly strategy says that to optimize the growth rate of your bankroll (wealth) you should invest a fraction of the bankroll equal to the difference of the likelihood that you will win or lose. In our example, this means that I have in each game to bet the fraction of x = 0.6 – (1 – 0.6) ≈ 0.2 – that is, 20% of my bankroll. The optimal average growth rate becomes

(7) 0.6 log (1.2) + 0.4 log (0.8) ≈ 0.02.

If I bet 20% of my wealth in tossing the coin, I will after 10 games on average to be times more than when I started (≈ 1.22 times more).

This game strategy will give us an outcome in the long run that is better than if we use a strategy building on the neoclassical economic theory of choice under uncertainty (risk) – expected value maximization. If we bet all our wealth in each game we will most likely lose our fortune, but because with low probability we will have a very large fortune, the expected value is still high. For a real-life player – for whom there is very little to benefit from this type of ensemble-average – it is more relevant to look at time-average of what he may be expected to win (in our game the averages are the same only if we assume that the player has a logarithmic utility function). What good does it do me if my tossing the coin maximizes an expected value when I might have gone bankrupt after four games played? If I try to maximize the expected value, the probability of bankruptcy soon gets close to one. Better then to invest 20% of my wealth in each game and maximize my long-term average wealth growth!

When applied to the neoclassical theory of expected utility, one thinks in terms of “parallel universe” and asks what is the expected return of an investment, calculated as an average over the “parallel universe”? In our coin toss example, it is as if one supposes that various “I” are tossing a coin and that the loss of many of them will be offset by the huge profits one of these “I” does. But this ensemble-average does not work for an individual, for whom a time-average better reflects the experience made in the “non-parallel universe” in which we live.

The Kelly strategy gives a more realistic answer, where one thinks in terms of the only universe we actually live in, and ask what is the expected return of an investment, calculated as an average over time.

Since we cannot go back in time – entropy and the “arrow of time ” make this impossible – and the bankruptcy option is always at hand (extreme events and “black swans” are always possible) we have nothing to gain from thinking in terms of ensembles .

Actual events follow a fixed pattern of time, where events are often linked in a multiplicative process (as e. g. investment returns with “compound interest”) which is basically non-ergodic.

Instead of arbitrarily assuming that people have a certain type of utility function – as in the neoclassical theory – the Kelly criterion shows that we can obtain a less arbitrary and more accurate picture of real people’s decisions and actions by basically assuming that time is irreversible. When the bankroll is gone, it’s gone. The fact that in a parallel universe it could conceivably have been refilled, are of little comfort to those who live in the one and only possible world that we call the real world.

Our coin toss example can be applied to more traditional economic issues. If we think of an investor, we can basically describe his situation in terms of our coin toss. What fraction (x) of his assets (T) should an investor – who is about to make a large number of repeated investments – bet on his feeling that he can better evaluate an investment (p = 0.6) than the market (p = 0.5)? The greater the x, the greater is the leverage. But also – the greater is the risk. Since p is the probability that his investment valuation is correct and (1 – p) is the probability that the market’s valuation is correct, it means the Kelly strategy says he optimizes the rate of growth on his investments by investing a fraction of his assets that is equal to the difference in the probability that he will “win” or “lose”. In our example this means that he at each investment opportunity is to invest the fraction of x = 0.6 – (1 – 0.6), i.e. about 20% of his assets. The optimal average growth rate of investment is then about 11% (0.6 log (1.2) + 0.4 log (0.8)).

Kelly’s criterion shows that because we cannot go back in time, we should not take excessive risks. High leverage increases the risk of bankruptcy. This should also be a warning for the financial world, where the constant quest for greater and greater leverage – and risks – creates extensive and recurrent systemic crises. A more appropriate level of risk-taking is a necessary ingredient in a policy to come to curb excessive risk taking.

The works of people like Rabin, Thaler, Kelly, and Kahneman, shows that expected utility theory is in deed an “ex-hypthesis.” Or as Monty Python has it:

This parrot is no more! He has ceased to be! ‘E’s expired and gone to meet ‘is maker! ‘E’s a stiff! Bereft of life, ‘e rests in peace! If you hadn’t nailed ‘im to the perch ‘e’d be pushing up the daisies! ‘Is metabolic processes are now ‘istory! ‘E’s off the twig! ‘E’s kicked the bucket, ‘e’s shuffled off ‘is mortal coil, run down the curtain and joined the bleedin’ choir invisible!! THIS IS AN EX-PARROT!!

  1. paul davidson
    July 18, 2013 at 5:06 pm

    why all this diddling around with bets where the probability of the outcomes are KNOW with certainty?

    Time is a device that prevents everything from happening at once! All economic decisions mean that the outcome of a decision occurs later in time than the point of time when the decision is made. [When later in time can be anything from a few seconds to many years].

    In an uncertain economic world — the ergodic axiom is overthrown and therefore there is no way of knowing the probability of any payout for a decision made today. In other words, probabilities can not be known about the future payout of any economic decision.

    that is why people enter into binding legal monetary contracts whenever they make economic decisions. If they are legally binding then if, or when , the other party defaults on its contractual obligation, the first party to the contract can go to the government to try to get legal enforcement of the other party’s legal obligation– and if that is not possible, the government will help the offended party to obtain whatever possible monetary payments from the defaulting party to compensate for potential damages.

    For the most part, people enter contracts because they TRUST the other party to the contract — but also know that the government will help them secure payment for whatever possible damages are experienced.

    What is so difficult to understand???

    • July 18, 2013 at 6:11 pm

      Without the ergodic axiom, there is no calculable objective probability, because the past does not provide data to calculate future probabilities. But there is still subjective probability formed from a priori reasoning which may form the basis for decision making.

      Objective probability relates to “risk”, whereas subjective probability relates to “uncertainty”. Neoclassical economics deals only with “risk”, whereas Knight, Schumpeter, Keyens and others recognised “uncertainty”, where probabilities have to be formed subjectively.

      Given subjective probabilities however formed, they have to be translated for the decision maker to “utility” for all combinations of choices and probabilities. This step is still needed in scientific decision making under uncertainty.

      Rejection of the ergodic axiom does not imply that there is no need for scientific decision making.

      • July 20, 2013 at 10:33 am

        if i understand it, this seems to be a good phrasing of the different perspectives between risk and uncertainty.

        but, my view (or hypothesis or theory) is that once phrased mathematically, the formalisms of these two approaches are equivalent, so one cannot really know who is right and whether one is dealing with objective risk or subjective uncertainty. ( except that it seems objectively uncertain which of these conceptualizations is correct—is there a deterministic master plan, or a stochastic system that creates its own distribution along. the way)

        i see this as analogous to ornstein and weiss (BAMS 1991) showing one cannot identify a stochastic or deterministic process, or the difference between marxists and capitalists, one arguing profit is neccesary and good, while others arguing its exploitation and bad.

        however, maybe there is a fundamental assymtry (a la baryon assymetry in cosmology) , and it may turn out that actually north is south, east is west, up is down, and left is right (or wrong).

  2. Egmont Kakarot-Handtke
    July 18, 2013 at 9:22 pm

    I would like to draw your attention to Jonathan Barzilai’s contribution in real-world economics review #63: Inapplicable operations on ordinal, cardinal, and expected utility. It is wonkish squared. For more see http://scientificmetrics.com/publications.html.

  3. July 19, 2013 at 11:36 am

    July 19, 2013
    I sympathise with Paul Davidson’s frustration. Simply put, Risk and Uncertainty are different, but related concepts. If all possible outcomes of a set of actions [including decisions], are known, then the calculus of probability can be used with benefit. When all possible outcomes are not known, then the calculus of probability cannot apply directly,if at all. This describes a state of uncertainty. Or as Ludwig Von Mises put it, possibly with a little help from his brother, mathematician Richard, who gave the subject much thought: “Case probability is not class probability”. The interested reader might also consult Keynes’s TREATISE on PROBABILITY. [1931]. M. Davidson’s point about monetary and forward contracts [including warranties & the various kinds of actuarially designed insurance] is quite simple [not simplistic]. Such contracts are a “Gordion Knot” solution, a kind of learning through doing, to cut through the problem of instinctive mistrust in the face of uncertainty. A kind of practical “Guide to the perplexed”. The societies & cultures [There are still lots of them] which cannot develop such instruments, usually morph into “failed nation states. This includes the many failed & bloody “experiments” with “socialist redemption”. [Paul Joseph Goebbel’s revealing phrase]. One of the reasons “planned socialist economies” have been so popular with tyrants of all stripes, down through the ages? The tyrant can decree all the outcomes & ration both the resources to achieve them, and whatever they produce. Does this sound familiar all you haters of “market fundamentalism” ? And afficionados of Marx’s “commodity fetishism” rant ? Norman L. Roth
    Please GOOGLE” [1] Norman Roth, Technos [2] Telos & Technos, Roth [3] Norman Roth, Origins of Markets.
    Yes folks, authentic free markets arise out of “spontaneous ordering”. They cannot be “designed” or decreed or “planned” by an elite of omniscient planners. That’s why frustrated control freaks hate them so much.

  4. Robert Locke
    July 19, 2013 at 11:49 am

    Norman, every “spontaneous ordering” has situational roots. That is true of the growth of free market economics, which, not accidentally, grew up in the British Empire, when London was the great Emporium. Just as Darwin’s” discoveries depended on his ability to roam rather freely within this British Emporium to do his “investigations”, so did the development of your kind of economics stem from British imperialism.

  5. Robert Locke
    July 19, 2013 at 12:27 pm

    And, to continue, Norman, in 1945 the triumph of Neo-Classical economics internationally had more to do with the unique historical situation that the US found itself in, than with free markets. In that year the Americans easily took over the mantel of the Brits because there was nothing to oppose them: Europe was in shambles, Russia ruined, China poverty stricken and entering a civil war, Latin America, undeveloped, Africa, etc. The US imposed imperially our economics everywhere. Stop talking about “spontaneous ordering,” the Americans set up the world and ran it — until those outside orbit recovered and took advantage of the American set up to enter the global economy in a big way. One might say that the two greatest economic events of the past 50 years were the recovery of Japan from total defeat, to become a mighty manufacturer, and the economic emergence of China — neither of which had much to do with the “spontaneous ordering” of the markets. Both places prospered enormously through protectionism.

    • paul davidson
      July 19, 2013 at 2:59 pm

      What about th Marshall Plan? It was the basis for the economic recovery and growth of a war devastated Europe and Japan

      As I point out in my book, The Marshall Plan was really an offspring in principle of the Keynes Plan that Keynes presented at Bretton Woods

      • A.J. Sutter
        July 22, 2013 at 1:37 pm

        Japan did receive aid from the US, but it wasn’t part of the Marshall Plan. The recovery of Japan had a lot to do with US Cold War containment policy. Part of that bargain included Japan’s getting preferential access to US markets. See generally Aaron Forsberg’s “America and the Japanese Miracle: The Cold War Context of Japan’s Postwar Economic Revival, 1950-1960” (U North Carolina Press 2000).

  6. Norman L. Roth
    August 29, 2013 at 10:32 pm

    August 29, 2013.
    Kudos to Paul Davidson for his reminder to M. Locke [and confreres] concerning their curious ‘amnesia’ about the Marshall Plan; Described by Winston Churchill as: “The most unsordid act in history”.
    On June 4th 1947 Secretary of State George C. Marshall announced that the United States was willing to provide economic assistance to ALL the war-torn nations of Europe. The Soviet reaction was a stony silence. The Soviets, however agreed to another meeting set for June 27, 1947.Molotov [a.k.a Skryabin] then demanded complete control and freedom of action over the dispersal of funds, especially to Germany. Including a demand that America inform the “Union of Soviet SOCIALIST Republics” of the exact amounts America would disperse to each recipient nation. On July 2, 1947, another date from the 1940s that “will live in infamy”, Molotov stormed out of the meeting. Stalin/Molotov then ordered their “allies” on the continent to reject all Marshall/Truman Plan assistance.
    Knee-jerk anti-Americanism and “no-nothing” posturing may still be a good starter in certain ‘constituencies’. But it doesn’t do much for the credibility, ‘eh fellas ? But didn’t Messrs. Samuelson and Solow etc. have similar problems ? In 1989. Not to mention Hugo Chavez in 2011. ;eh M. Stiglitz ?

    Norman L. Roth, Toronto, Canada, Please GOOGLE: [1] Norman Roth, Technos [2] Norman Roth, Origins of Markets [3] Telos & Technos, Roth [4] Norman Roth, Economics of Technology

    • Robert Locke
      August 30, 2013 at 7:24 am

      No amnesia about the Marshall Plan, just “knowledge” that you could tap into Norman, if you weren’t such a poor historian, about the limited effect that the Marshall Plan had on European Recovery. Get the history straight for once and stop propagandizing historians; they’ll get you every time, if you are willing to listen, because they really know what happened. Historians live by the code of objectivity, tell what really happened and in context. You have a real problem with doing that, which is why I am always skeptical about your posts — too ideological. Learn from your enemies.

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