Expected utility theory is transmogrifying truth

17 June, 2015 at 16:40 | Posted in Economics | 6 Comments

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.

Daniel Kahneman writes — in 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?

41kgYr0Fs2L._SY344_BO1,204,203,200_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 Kelly criterion — 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 criterion — 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 as much as before, and when I lose (1 – x) times as much. After n rounds, when I have won v times and lost n – v times, my new bankroll (W) is

(1) W = (1 + x)v(1 – x)n – v T

[A technical note: The bets used in these calculations are of the “quotient form” (Q), where you typically keep your bet money until the game is over, and a fortiori, in the win/lose expression it’s not included that you get back what you bet when you win. If you prefer to think of odds calculations in the “decimal form” (D), where the bet money typically is considered lost when the game starts, you have to transform the calculations according to Q = D – 1.]

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. Alternatively, we see that the Kelly criterion implies that we have to choose x so that E[log(1+x)] — which equals p log (1 + x) + (1 – p) log (1 – x) — is maximized. Plotting E[log(1+x)] as a function of x we see that the value maximizing the function is 0.2:

kelly2

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 have 1.0210 times more than when I started (≈ 1.22).

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 criterion 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 and “parallel universe.”

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 criterion 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 2 % (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 Kelly and Kahneman show that expected utility theory is indeed transmogrifying truth.

6 Comments »

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  1. 1. To suppose that utility is just money seems to me an ‘Aunt Sally’ argument. (Although I agree with the conclusion.)

    2. People generally suppose that it is sensible to attempt to maximize long-run expected utility, and that normally maximizing immediate expected utility will achieve this. As you say, it doesn’t always. But this still leaves open the possibility that maximizing long-run expected utility might be sensible. (But is it?)

    An instructive example for either would be ‘investing’ in the stock market with a view spending the gains in 2025.

  2. I don’t think that there is anything inherently wrong with taking limited models as partial explanations that are not only partial (admitting there are other significant factors) but also imperfect (admitting the assumptions are simplifications). Many economists may admit this, but then some go on to ignore it and treat the models as comprehensive causal explanation of how things actually are and operate. That is where the illegal moves enter logically. There is nothing wrong with modeling anything anyway one chooses to do so providing one obeys the rules. The problems begin with making logical jumps from a model to what it purportedly shows and supports.

    Overreach in model interpretation is especially insidious when assumptions are constructed ideologically to generate models that advance ideological conclusions. To do so knowingly is simply sophistry, and it is mendacious. This is what Paul Romer seems to be aiming at in his condemnation of “mathiness.” “If you can’t dazzle them with brilliance, baffle them with bullshit.” ― comedian W.C. Fields. Maybe he got the inspiration for it from P. T. Barnum, — or Edward Bernays.

    On the other hand, some models are simply wrong because the assumptions are not merely (over) simplifications but are just flat out wrong. For example, economists regularly confuse the existing floating rate monetary system with the previous fixed rate system based on gold convertibility at the international level. Reinhart & Rogoff did this in the now infamous study on the effect of public debt.

    Models that are not stock-flow consistent are also defective, as are models that get operations wrong.

    Even international trade expert Paul Krugman still denies endogenous money and bases his analysis on a natural rate and loanable funds. He also finally admitted that he didn’t realize the import of the distinction between a currency issuer and currency users as it applies to currency sovereigns like the US, UK, Canada, Japan, and Australia compared with the nations of the EZ that are currency users. He has also come around to realizing that “bond vigilantes” don’t control the yield curve or force the central bank to change its desired policy rate.

    But a whole raft of economists haven’t yet grasped endogenous money, the difference between the current and previous monetary system, and the difference between a sovereign currency issuer and users of a currency that they do not issue. In fact, some are even in denial of it.

  3. Reblogged this on Forwardeconomics.

  4. This is where the heterodox go completely off the rails. When studying actual human behavior, scientists discover problems with Neoclassical economics. The problem is clear: human behavior, even in the aggregate, cannot be accurately described using mechanical laws of the sort that govern plantetary orbits.
    .
    But even the great critics of the orthodox are blinded by theory. They misread the work of scientists as supporting the conclusion that mechanical laws are still just fine, we just need to modify them. It’s like noticing that orbits aren’t perfect circles.
    .
    But it’s not the difference between ellipses and circles, it’s the difference between Newton and Darwin, or mineral and animal.
    .
    The Kelly criterion is less wrong, but it is still wrong. Wrong. Wrong. Wrong.

    • No, the kelly criterion is just a way to maximize returns given uncertainty. It has been used successfully in gambling and in markets.

      You seem to be saying economics is impossible, which is quite an unscientific statement.

  5. A real problem is when they sidestep the problem of risk by assuming that agents have a time based discount for utility. First this doesn’t make sense, because why would am agent with rational expectations discount utility at different times? The know exactly how much utility they will get at each time so there is no logical reason to value utility at different times differently. If you add money to this model it totally falls apart because an agent can exchange money for future consumption at a discounted rate, and then sell less future consumption to get back the same amount of money.

    It seems to me that this is a kludge to get Kelly like behavior out of the expected utility model to act like the Kelly model. But In a Kelly model the discount rate comes from actual uncertainty, in a discounted utility model no uncertainty is necessary.

    This has the very bad side effect of making the discount rate on utility determine the interest rate in the model. So you get a completely confused financial system from the start in any model that uses discounted utility.


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