Bob: Thanks, I see what you meant - didn't realise you wanted the probabilities to remain variable. I was just pointing out that there are specific cases, with fixed choices of probabilities, where EUM leads to apparently risk averse behaviour such as maximin.
> maximin is independent of probability. I would have put it differently: maximin assumes a specific set of probabilities, namely probability of 1 for U11 and U21. This is the way minimax is used in deterministic 2 player games: assume the opponent will (with probability 1) make the move that is best for him/her. Viewed this way it's just a special case of EUM. Konrad On Thu, 21 Aug 2003, Welch, Robert wrote: > 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.
