## So you think you’re rational? I bet you’re not!

15 May, 2012 at 15:24 | Posted in Economics, Statistics & Econometrics, Theory of Science & Methodology | 4 CommentsAs yours truly has repeatedly argued over the years (e.g. here) probability, risk and uncertainty are tricky concepts when applied to reality.

In neoclassical economics *rational choice under risk* is usually equated with agents’ behaviour fulfilling the axioms of *expected utility theory*. Developed originally by **John von Neumann** and **Oskar Morgenstern** in 1947, the theory basically says that utility can be conceived of as a weigted sum of utilities in each “state of nature” and where the weights are given by probabilities.

Consider the case where you take a bet on flipping an unbiased coin and you get a utility equivalent to €100 if Heads comes up and loose the equivlent of €110 if Tails comes up. Being “rational” you do not accept the bet since its expected utility is negative [(0.5 x 100) -(0.5 x 110) = -5].

**But is this really “rational”?** You know that you get *either* €100 *or* loose €110, and certainly not -5 (on the inadequacy of basing decisions on expectation values and “ensemble averages” in bettings, see my article on the Kelly criterion here). And – most importantly – if Heads or Tails comes up in a single throw, is not a question of *risk*, but of *uncertainty*, since probability – and risk – according to the theory is only measurable when repeated an infinite (or at least a “large”) number of times.

After all, accepting the bet under uncertainty may very well be “rational”, since the setting – the probabilistic nomological machine – is not the one that expected utility theory presupposes. Getting the expected value in repeated betting is irrelevant for decisions on single bets.

Let’s take another example to really rub it in how inadequate expected utility theory is:

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.

Although the expected utility theory is obviously 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! Or as

Or as Matthew Rabin and Richard Thaler have it:

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.

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I figured something was wrong with this example, so I set the problem up in matlab. Using a utility function of log(10100+x), where x is the outcome of the lottery, the expected utility is monotonically increasing in the number of draws. ()

The reason is of course that when you increase the number of draws, you create a more stable portfolio, as the draws are independent of each other. This is first year stuff.

So expected utility theory is definitely consistent with your example above. To be honest, I would expect more from a professor than creating these kind of strawman arguments over and over again.

(if you want to check for yourself, the expected utility of 100 draws is 9.6174, and of one draw 9.2251)

Comment by pontus— 18 May, 2012 #

“Loading the dice” one can prove whatever one wants. I don’t know how it is in Cambridge, but I have the impression that my students learned that long before they attended university.

Your comment totally misses the point. A neoclassical utility maximizer that rejects the single gamble should also reject the aggregate offer – something that is, to say the least, totally counterintuitive. Instead of far-fetched evasive manouvres it would be more interesting if you tried to tackle the questions raised in the posts you comment on.

Comment by Lars P Syll— 18 May, 2012 #

Well, I certainly haven’t “loaded the dice”. It’s your dice! I haven’t rigged any game here. I’m playing by your rules! Stipulate any other, and I’ll play by them too.

Now as I said, a neoclassical utility maximizer that would have rejected the single gamble would not necessarily reject the aggregate gamble. In fact, he might as well accept it. So what on earth are the “questions raised in the posts”? I don’t get it. To me, it just seems as someone too eager to portray neoclassical economics as something stupid, something which it’s not, even if it takes straightout lying (or incompetence) to get there.

This is not a far-fetched “evasive maneuver”. It’s a clarification of what neoclassical theory is about.

So please explain to me, and your readers, in what way I’m wrong! The expected utility of the repeated bet is simply higher than the expected utility of the one-shot bet.

Comment by pontus— 19 May, 2012 #

I have a new post up today – read it!

Comment by Lars P Syll— 19 May, 2012 #