Christian Variance is a property reserved for probability distributions over real valued domains only. Thus it is a mixture of value and probability. Consider a gamble as a deviation from income or wealth. Under the expected utility hypothesis, an individual is risk neutral for infinitesimally small gambles. He/she is risk averse if willing to an pay insurance premium to avoid a large gamble. In other words, the graph of the utility function lies everywhere below its tangent plane; ie, its concave. Risk aversion can be measured in the degree of concavity of the utility. In your example there is a clear choice between gamble A and B that will depend on the concavity of U. Note that expected utility can result in risk loving (convex U) and sometimes risk loving / sometimes risk averting.
There are some risk averse behaviors that are independent of probability. The maxi-min strategy is one. Consider a set of possible compound gambles each with a set of possible outcomes. The maxi-min strategy looks at each gamble for the worst possible outcome and chooses the gamble which guarantees the best among the worst outcomes, irrespective of the probability. This extremely risk averse strategy is not possible with expected utility. Bob. - -----Original Message----- From: Christian Borgelt [mailto:[EMAIL PROTECTED] Sent: Wednesday, August 20, 2003 9:35 AM To: [EMAIL PROTECTED] Subject: Re: [UAI] Allais' paradox Gordon Hazen writes: > >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 > > This is simply false. For example, if payoffs are normally distributed > then expected utility depends on both the mean and the variance of the > normal distribution in question. Variance definitely can be taken into > account by the theory! Dear Gordon, if it is the case that expected utility takes the variance into account, then it should be able to handle Allais' and Ellsberg's paradox easily. Would you please tell me how it does? The problem seems to be that you talk about a different situation. Please consider the following: You have two options, A and B. If you choose option A, you get a payoff that is normally distributed with (positive) mean \mu_A and variance \sigma_A^2. If you choose option B, you get a payoff that is normally distributed with (positive) mean \mu_B and variance \sigma_B^2. Let \mu_A = c_1 *\mu_B, c_1 > 1, and \sigma_A^2 = c_2 \sigma_B^2, c_2 > 1. So option A leads to the higher expected value, but also to the higher variance in the outcome. How does expected utility prescribe to decide between the options A and B, depending on the values of c_1 and c_2? > No one claims that risk aversion is irrational - that is not what the > Allais or Ellsberg paradoxes are saying. Risk aversion consists in > preferring a payoff of x to a random payoff with mean x. Such preferences > are universally agreed to be rational and can be successfully accommodated > by expected utility theory using any concave utility function over payoffs. What are Allais' and Ellsberg's paradox saying then (in your opinion)? And again, would you please tell me how expected utility theory handles these paradoxes? For me it is perfectly rational what people do in these situations, but expected utility theory seems to prescribe something else. > >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. > > Nowhere in the axioms underlying expected utility theory is there any > assumption that a decision is to be made repeatedly. The axioms deal > explicitly with one-time decisions. It happens that the axioms imply that > the utility of a gamble g with payoffs x is numerically equal to the > expected value of the utilities u(x) one realizes from the > gamble. However, the theory does not appeal to repeated choice to reach > this conclusion. To criticize the theory on this ground is to criticize a > particular misinterpretation of the theory. This criticism does not apply > to the theory when it is properly interpreted. I did not mean that expected utility refers to multiple decisions, but only that people make a difference between a single decision instance and a set or sequence of such decision instances. It seems to me that the problem is in the expected value/expected utility, which is the only relevant magnitude only if the variance can be neglected. And one way to get rid of the variance is to consider a large number of decision instances, so that it vanishes due to the law of large numbers. How does expected utility theory get rid of it, if it only refers to single decision instances? Maybe it is now a little bit clearer what I meant. Regards, Chris
