When facing a problem with uncertain outcomes, of course, we do not think in probabilistic terms. At least, the vast majority of us don't. However, ideally, we could hope that, whatever our mind does, it would respect basic principles of rationality. On the other hand, at this point, given our failure in simple logic reasoning, it should be no surprise to learn that, when our abilities for uncertain reasoning were tested, it soon became clear that we failed to follow those principles.
Early tests of EUT showed clearly that we do not reason in a way that is compatible with it, if EUT is used as a description. Those results were initially called paradoxes of decision making, namely Allais' and Ellsberg's paradoxes, despite the fact there is nothing paradoxical in them. Both experiments just showed that people sometimes do not obey the principle that choice between two bets should depend only on the aspects where bets are different and not on those where they are equal. This principle is known as Cancellation Principle.
Things might not have been so serious if that were all. But the literature is filled with different examples of mistakes we make and attempts to understand what we actually do. In 1979, Kahneman and Tversky showed that if one assumes we change the probabilities we know to a different value, by using what they called weighting functions, we can still describe our reasoning, at least in the Allais' and Ellsberg's paradoxes, using EUT with those altered probabilities.
Basically, what they observed is that, when we get close to certainties, that is, when the probability of something happening gets close to zero or one, we make decisions as if there were more uncertainty than there really is. This is actually a well observed phenomenon. Per example, people usually make bets on lotteries that shows they consider their chance of winning to be much larger than it really is. For example, if there is just one chance in 50 millions that yu will win $ 10 millions, that means that, in average, you would win 0.20 per bet. But people happily pay more than that to enter such a lottery (actually, if you do think abut utilities, the problem is far more serious since the utility of each dollar decreases as you have more). What Kahneman and Tversk proposed in their Prospect Theory is that people would actually work with a modified probability value. In the lottery example, if you think you chance is actually one in 100,000, instead of one in 50,000,000, it might make sense to pay up to $ 100.00 (if you don't correct for decreasing utility, if you do, the value would be smaller, but it still can be much higher than $ 0.20). The same effect is observed at the other extreme, where there is just a very small chance that something will NOT happen.
It was later shown in newer experiments that Prospect Theory and other options that came later can not fully explain our mistakes. Birnbaum performed a series of experiments where he shows we even disobey a principle called stochastic dominance. Stochastic dominance is basically the simple fact that we should not choose alternatives that are obviously worse in at least one aspect, while equal in all other aspects. One example he tested was a choice between the bets G and G+, given by
G
90% chance to win $96.00
10% chance to win $12.00
G +
90% chance to win $96.00
5% chance to win $14.00
5% chance to win $12.00
Clearly the option G+ is better since it is exactly the same 95% of the time and, in the remaining 5%, it pays $ 2.00 more. That is what it is meant by saying that G+ stochastically dominates G. However, what Birnbaum observed consistently in a number of choices like this is that often people would pick the worse bet!
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