Hi all! It seems to me that the problem with expected utility as a theory of rational decision making is that it does not properly take into account the variances of the outcomes for the different options, especially if the decision is to be made in only few instances, maybe even only once. I have not followed the discussion really closely, but it seems to me that the only reference to this is the notion of "risk aversion" that turned up a few times. My personal problem is that I cannot see risk aversion as irrational, even if this means choosing the option with the lower expected value.
In other words, expected utility theory works, as its name says, with the expected utility of an option. However, the expected utility is the only relevant value only if I am offered to make the decision several, (or actually quite a lot of) times. Then the law of large numbers says that the average utility of the outcomes in these different instances converges in probability to the expected utility. This basically means that I can neglect the variance, as it goes to zero with a large number of instances. But if I can make the decision only few times, or if I can make it even only once, the variance cannot be neglected and information about this variance becomes relevant. In my opinion, the paradoxes only show that human beings take this information about the variances into account - and I see nothing irrational in this. It is just that expected utility theory neglects this information and thus is not sufficient as a proper model of rational decision making (descriptive as well as normative). To make my point clear, I would like to quote (or actually translate from the German/Swiss original, as I do not own an English translation) from Max Frisch's novel "Homo Faber". The setting: Elisabeth Piper, daughter of Hanna Piper, was bitten by a snake and has been admitted to a hospital. The narrator of the story is Walter Faber, an engineer with a completely rationalistic world view. "Did you know", I ask, "that the rate of mortality from snake bites is only three to ten percent?" I was surprised. Hanna does not think much of statistics, that I discovered soon. [...] "You and your statistics!" she says. "If I had a hundred daughters, each of them bitten by a viper, then yes! Then I would loose only three to ten daughters. Surprisingly few! You are absolutely right." Her laughter with this. "I have only a single child!" she says. The point is: A single decision instance is different from a (large) set or sequence of instances of the same type. The frequentist (or empirical) interpretation of probability explicitely acknowledges this, for instance, by refraining from saying that there is a certain probability that a particular smoker gets lung cancer - it only says that he/she belongs to a "risk group", in which a certain (higher) percentage of people gets lung cancer. By this I am not saying that the subjective (or personalistic) interpretation of probability is wrong or inappropriate. I just would like to point out a problem connected with it, which should be handled with care. It appears to me that it could be checked whether my view is correct by considering the following settings: a) One can choose in a single instance between the two lotteries of Allais' paradox. b) One can do this 5 to 10 times, each decision independent of any other. c) One can do it 100 or 1000 times, again each decision independent of any other. I assume that with a higher number of times one can choose between the lotteries, it will turn out that in a certain percentage of the cases people will go for the sure payoff and in the rest for the higher expected value (which, however, is connected to a higher risk), thus achieving a certain expected value to variance (or risk) relation. And this is what I believe people are actually optimizing - and I think that they are acting completely rationally if they do so, simply because the law of large number does not really apply to most decision situations of individual real life. Best regards, Chris
