FEW just ended, and it was just as exciting as ever. I think it was a bit more international, a bit more interdisciplinary, and substantially larger (in terms of audience) than either of the past two years. Anyway, as seems to happen when I’m at conferences, I’ve got some more ideas to blog about. This one was something I thought of when writing my comments for Katie Steele’s paper on decision theory and the Allais paradox. To start out, here is a presentation of the Allais paradox: In the first decision, one has a 90% chance of $1,000,000 and is choosing whether the other chances should be a 1% chance of $0 and a 9% chance of $5,000,000, or a 10% chance of $1,000,000. The second decision is just the same, but with a 90% chance of $0. The principle of independence suggests that whether the 90% chance is of $1,000,000 or of $0 should make no difference when choosing between the different 10% options – yet many seemingly-rational people choose the flat $1,000,000 in the first case and the 9% chance of $5,000,00 in the second.
There are several principles of decision theory that seem fairly intuitive I have abstracted from the general notion of “independence”, and they seem to lead directly to expected utility theory:
- The value of a gamble is a real number.
- Nothing besides probabilities and utilities of outcomes is relevant for the value of a gamble.
- The value of a gamble is a weighted sum of several contributions.
- Each contribution is associated with a particular possible outcome.
- The weight of each contribution is proportional to the probability of the corresponding outcome.
- The value of each contribution is proportional to the utility of that outcome.
From these principles, it is easy to see that (wherever possible) the value of a gamble is equal to its expected utility (modulo some scalar multiple that applies equally to all gambles).
These principles seem fairly plausible, but of course there are reasons one might question each. The first principle underlies most traditional decision theory of any sort, even though there are traditional examples (the St. Petersburg game) that seem to contradict it, and one can also come up with actions where the intuitive preference relation between them can’t possible be represented by real numbers.
The second principle can be questioned in cases where one already has a package of gambles, and one wants to amortize risk. That is, if purchasing insurance is to be considered rational, it will be because we care not just about the probabilities and payoffs of the insurance gamble, but also about the fact that we get positive payoffs when something bad happens, and negative payoffs when good things happen.
The third principle is probably fairly easy to question, but I don’t know of any natural way to reject it.
The fourth principle can be rejected in one natural way to deal with risk-aversion. For instance, in addition to the “local” factors associated with each outcome, one can add a “global” factor associated with the variance or standard deviation of the payoffs of the gamble. We might need to be careful when adding this factor to make sure that we don’t violate more fundamental constraints (like the principle of dominance – that if gamble A always has a better payoff than gamble B in every state, then one should prefer A to B). In introducing such a factor, we’ll have to figure out just what extra factors might be relevant, and how to weight them, which is at least one reason why this option is much less attractive than standard expected utility theory, though it has obvious appeal for dealing with risk-aversion.
I don’t know if there’s any reason one might rationally reject the fifth principle, though presumably it will have to be relaxed somehow if one is to rationally prefer gamble A to gamble B if they are identical, save for a much higher payoff for A than B on a state of probability 0.
The most natural way to relax the sixth may be in conjunction with relaxing the second. Some other factor beyond utility of an outcome may be considered. Another way to relax it would be to make the contribution of an outcome depend not only on its utility, but also on how things could have turned out otherwise on the same gamble. In an example by Amartya Sen cited by Katie, cracking open a bottle of champagne when receiving nothing in the mail may make quite a different contribution to an overall gamble when one could have received a serious traffic summons in the mail than when one could have received a large check in the mail.
To make it more clear that this doesn’t only arise when the experience of the event is different, we can consider a version of the Allais paradox with memory erasure. That is, the payoffs are just the same, but in addition to the cash, one has one’s memory erased and replaced with the memory of making a gamble with a sure outcome of whatever it is one received. Thus, there is nothing worse about the $0 when one could have had a guaranteed $1,000,000 than about the $0 when one only had a chance of making money. Since we seem to make the same decisions anyway, it seems that a counterfactual factor of what could have happened otherwise (rather than an emotional factor) must be factored into the value of the gamble.
These methods of relaxing the fourth and sixth principle seem to do different violations to the notion of independence (the sixth maintaining a kind of locality, and the fourth adding a global factor), but it turns out that the procedures in each can model exactly the same decisions as the other. I think the only way to decide between them will be by finding replacements for these principles and seeing which has more natural restrictions.
I hadn’t thought much about violations of independence before reading Katie’s paper, but I think they might be quite plausible. However, it’s interesting to see how some very strong version of principles like independence lead directly to expected utility, in a way that avoids standard representation theorems and the laws of large numbers.