Konrad: Yes you can find an instance where EUM picks out the maximin solution. But such a situation is unstable with respect to changes in the probabilities.
Example with two gambles neither of which dominates the other. g1 = (U11 = 5, U12 = 10, P11 = P12 = .5) has EU(g1) = 7.5 g2 = (U21 = 3, U22 = 11, P21 = P22 = .5) has EU(g2) = 7 g1 is maximin and maximizes expected utility now change the gambles so that the payoffs are the same but the probabilities are different P'11 = .6, P'12 = .4 and P'21 = .25, P'22 = .75 g'1 is still maximin as maximin is independent of probability. But EU(g'1) = 7 , EU(g'2) = 9 and g'2 maximizes expected utility. Point is, EU is continuous (in fact linear) in probability. Assume no one gamble dominates the others. Suppose that there is a configuration of the probabilities in the gambles so that EU agrees with maximin. Then I can always modify the probabilities so that EU moves away from the maximin solution. Hence, EUM cannot generally support a maximin strategy over all choices among gambles. In your chess example, I suspect that there is a dominant strategy. When there is a dominant strategy, then probability is irrelevant even in EUM. Bob. - -----Original Message----- From: Konrad Scheffler [mailto:[EMAIL PROTECTED] Sent: Thursday, August 21, 2003 3:58 PM To: Christian Borgelt; Bob Welch Cc: [EMAIL PROTECTED] Subject: Re: [UAI] Allais' paradox On Wed, 20 Aug 2003, Christian Borgelt wrote: > when I wrote the post, I thought briefly about the choice of the > utility function and whether it could be used to handle this case, > but I rejected the idea. <justification snipped> I have not really thought about how one would incorporate the variance into the utility function directly, or whether this would be a good idea. It just seemed that that would be an obvious thing to try in response to your objection. But personally, I would expect that a more likely utility in actual brains might use the probability and desirability of all (or a set of) possible outcomes, which would obviously take the variance into account. As a simple example, let's say I want to buy a car tomorrow. I don't care how much candy I can buy, if I can't buy a car I'll be devastated. A car costs $10000. I own $x. My utility function is P(x >= 10000). Or I could say my utility is a step function u(10000-x). If I own nothing, I will seek out risk; if I already own $10000 I'll be risk averse. In real life, things become more complex: I might quite like a car, but I also have to worry about whether I'll have food to eat tonight and whether I'll be able to pay the rent a month from now. But if we take all possible outcomes into account we can (in theory, at least) come up with a suitable (highly nonlinear) utility function. Bob Welch wrote: > 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. On the contrary, this is easily achieved: In the car example, if I already have $10000, all I need to do is guarantee that the worst possible outcome is nonnegative. If only one of the gambles has this property I will choose that one, no matter how small the probability of negative outcome in the other gambles. Or consider a game such as chess, where one is again trying to maximise the probability of winning (let's ignore draws for the purpose of this discussion). The above strategy is routine for a player in a good position: you don't want to risk the position being overturned, no matter how small the chance. On the other hand, in a bad position you follow the opposite strategy, seeking out risk. Konrad
