### What to do when a transformative experience might change your attitudes to risk

A PDF of this blogpost is available here.

A transformative experience is one that changes something significant about you. In her book, Laurie Paul distinguishes two sorts of transformative experience, one epistemic and one personal. An epistemically transformative experience changes you by providing knowledge that you couldn't have acquired other than by having the experience: you eat a durian fruit and learn what it's like to do so; you become a parent and learn what it's like to be one. A personally transformative experience changes other aspects of you, such as your values: you move to another country and come to adopt more of the value system that is dominant there; you become a parent and come to value the well-being of your child over other concerns you previously held dear.

Both types of experience seem to challenge our standard approach to decision theory. Often, I must know what an experience is like before I can know how valuable it is to have it---the phenomenal character of the experience is part of what determines its value. So, if I can only know what it is like to be a parent by becoming one, I can't know in advance the utility I assign to becoming one. Yet this utility seems to be an essential ingredient I must feed in to my theory of rational choice to make the decision whether or not to do become a parent. What's more, if my values will change when I have the experience, I must decide whether to use my current values or my future values to make the choice. Yet neither seems privileged.

Elsewhere (here and here), I've argued that these problems are surmountable. To avoid the first, we simply include our uncertainty about the utilities we assign to the different outcomes in the framing of the decision problem, so that the states of the world over which the different available options are defined specify not only what the world is like, but also what utilities I assign to those states. To avoid the second, we set the global utility of a state of the world to be some aggregate of the local utilities I'll have at different times within that state of the world, and we use our global utilities in our decision-making. I used to think these solutions settled the matter: transformative experience does not pose a problem for decision theory. But then I started thinking about risk, and I've convinced myself that there is a serious problem lurking. In this blogpost, I'll try to convince you of the same.

Sapphire takes a risky step |

# The problem of choosing for changing risk attitudes

I'm going to start with a case that's a little different from the usual transformative experience cases. Usually, in those cases, it is you making the decision on your own behalf; in our example, it someone else making the decision on your behalf. At the very end of the post, I'll return to the standard question about how you should choose for yourself.

So suppose you arrive at a hospital unconscious and in urgent need of medical attention. The doctor who admits you quickly realises it's one of two ailments causing your problems, but they don't know which one: they think there's a 25% chance it's the first and a 75% chance it's the second. There are two treatments available: the first targets the first possible ailment perfectly and restores you to perfect health if you have that, as well as doing a decent but not great job if you have the second possible ailment; the second treatment targets the second ailment perfectly and restores you to perfect health if you have that, but does very poorly against the first ailment. Which treatment should the doctor choose? Let's put some numbers to this decision to make it explicit. There are two options: Treatment 1 and Treatment 2. There are two states: Ailment 1 and Ailment 2. Here are the probabilities and utilities:

$$\begin{array}{r|cc} \textit{Probabilities} & \text{Ailment 1} & \text{Ailment 2} \\ \hline P & 1/4 & 3/4 \end{array}$$

$$\begin{array}{r|cc} \textit{Utilities} & \text{Ailment 1} & \text{Ailment 2} \\ \hline \text{Treatment 1} & 10 & 7\\ \text{Treatment 2} & 2 & 10 \end{array}$$

How should the doctor choose? A natural thing to think is that, were they to know what you would choose when faced with this decision, they should just go with that. For instance, they might talk to your friends and family and discover that you're a risk-neutral person who makes all of their choices in line with the dictates of expected utility theory. That is, you evaluate each option by its expected utility---the sum of the utilities of its possible outcomes, each weighted by its probability of occurring---and pick an option among those with the highest evaluation. So, in this case, you'd pick Treatment 2, whose expected utility is $(1/4 \times 2) + (3/4 \times 10) = 8$, over Treatment 1, whose expected utility is $(1/4 \times 10) + (3/4 \times 7) = 7.75$.

On the other hand, when they talk to those close to you they might learn instead that you're rather risk-averse and make all of your choices in line with the dictates of Lara Buchak's risk-weighted expected utility theory with a specific risk function. Let me pause here briefly to introduce that theory and what it says. I won't explain it in full; I'll just say how it tells you to choose when the options available are defined over just two states of the world, as they are in the decision that faces the doctor when you arrive at the hospital.

It's pretty straightforward to see that the expected utility of an option over two states can be calculated as follows: you take the minimum utility it might obtain for you, that is, the utility it gives in the worst-case scenario; then you add to that the extra utility you would get in the best-case scenario but you weight that extra utility by the probability that you'll get it. So, for instance, the expected utility of Treatment 1 is $7 + (1/4)\times(10-7)$, since that is equal to $(1/4 \times 10) + (3/4 \times 7)$. The risk-weighted expected utility of an option is calculated in the same way except that, instead of weighting the extra utility you'll obtain in the best-case scenario by the probability you'll obtain it, you weight it by that probability after it's been transformed by your risk function, which is a formal element of Buchak's theory that represents your attitudes to risk in the same way that your probabilities represent your doxastic attitudes. Let's write $R$ for that risk function---it takes a real number between 0 and 1 and returns a real number between 0 and 1. Then the risk-weighted expected utility of Treatment 1 is $7 + R(1/4)\times(10-7)$. And the risk-weighted expected utility of Treatment 2 is $2 + R(3/4)\times(10-2)$. So suppose for instance that $R(p) = p^2$. That is, your risk function takes your probability and squares it. Then the risk-weighted expected utility of Treatment 1 is $7+R(1/4)\times (10-7) = 7.1875$. And the risk-weighted expected utility of Treatment 2 is $2+R(3/4)\times (10-2) = 6.5$. So, in this case, you'd pick Treatment 1 over Treatment 2, and the doctor should too.

So far, so simple. But let's suppose that what the doctor actually discovers when they try to find out about your attitudes to risk is that, while you're currently risk-averse---and indeed choose in line with risk-weighted expected utility theory with risk function $R(p) = p^2$---your attitudes to risk might change depending on which option is chosen and what outcome eventuates. In particular, if Treatment 2 is chosen, you'll retain your risk-averse risk function for sure; and if Treatment 1 is chosen and things go well, so that you get the excellent outcome, then again you'll retain it; but if Treatment 1 is chosen and things go poorly, then you'll shift from being risk-averse to being risk-neutral---perhaps you see that being risk-averse hasn't saved you from a bad outcome, and you'd have been better off going risk-neutral and picking Treatment 2. How then should the doctor choose on your behalf?

Let me begin by explaining why we can't deal with this problem in the way I proposed to deal with the case of personally transformative experiences above. I'll begin by saying briefly how I did propose to deal with those cases. Suppose that, instead of changing your *attitudes to risk*, Treatment 1 might change your *utilities*---what I call your local utilities; the ones you have at a particular time. Let's suppose the first table below gives the local utilities you assign before receiving treatment, as well as after receiving Treatment 2, or after receiving Treatment 1 if you have Ailment 1; the second table gives the local utilities you assign after receiving Treatment 1 if you have Ailment 2.

$$ \begin{array}{r|cc} \textit{Utilities} & \text{Ailment 1} & \text{Ailment 2} \\ \hline \text{Treatment 1} & 10 & 7\\ \text{Treatment 2} & 2 & 10 \end{array}$$

$$\begin{array}{r|cc} \textit{Utilities} & \text{Ailment 1} & \text{Ailment 2} \\ \hline \text{Treatment 1} & 8 & 1\\ \text{Treatment 2} & 4 & 4 \end{array} $$

Then I proposed that we take your global utility for a treatment at a state of the world to be some weighted average of the local utilities you'd have for that state of the world at that state were you to choose that treatment. So the global utility of Treatment 1 if you have Ailment 2 would be a weighted average of its local utility before treatment, which is 7, and its local utility after treatment, which is 1---perhaps we'd give equal weights to both selves, and settle on 4 as the global utility. So the following table would give the global utilities:

$$ \begin{array}{r|cc} \textit{Utilities} & \text{Ailment 1} & \text{Ailment 2} \\ \hline \text{Treatment 1} & 10 & 4\\ \text{Treatment 2} & 2 & 10 \end{array}$$

We'd then apply our favoured decision theory with those global utilities.

Surely we could use something similar in the present case, where the utilities don't change, but the attitudes to risk do? Surely we can aggregate your risk function at the earlier time and at the later time and use that to make your decisions? Again, maybe a weighted average would work? But the problem is that, at least if you take Treatment 1, which risk function you have at the later time depends on whether you have Ailment 1 or Ailment 2. That's not a problem when we aggregate local utilities to give global utilities, because we just need the utility of a treatment at a specific state of the world generated by the local utilities you'll have at that state of the world if you undergo that treatment. But risk functions don't figure into our evaluations of acts in that way. They specify a way that the probabilities and utilities of different outcomes should be brought together to give an overall evaluation of the option. And that means we can't aggregate risk functions held at different times within one state of the world to give the risk function we should use to evaluate an option, since an option is defined at different states of the world.

Suppose, for instance, we were to aggregate the risk function you have now and the risk function you'll have if you undergo Treatment 1 and have Ailment 1: they're both the same, so the aggregate will probably be the same. Now suppose we were to aggregate the risk function you have now and the risk function you'll have if you undergo Treatment 1 and have Ailment 2: they're different, so the aggregate is probably some third risk function. Now suppose we come to evaluate Treatment 1 using risk-weighted expected utility theory: which of these two aggregate risk functions should we use? There is no non-arbitrary way to choose.

# Choosing to limit legitimate complaints

Is this the only route we can try for a solution? Not quite, I think, but it will turn out that the other two I've considered also lead to problems. The first, which I'll introduce in this section, is based on a fascinating paper by Pietro Cibinel. I won't explain the purpose for which he introduces his account, but in that paper he proposes that, when we are choosing for others with different attitudes to risk, we should use a complaints-centred approach. That is, when we evaluate the different options available, we should ask: if we were to choose this, what complaints would the affected individuals have in the different states of the world? And we should try to choose options that minimise the legitimate complaints they'll generate.

Cibinel offers what seems to me the right account of what complaints an individual might legitimately make when an option is chosen on their behalf. He calls it the *Hybrid Competing Claims Model*. Here's how it goes: If you would have chosen the same option that the decision-maker chooses, you have no legitimate complaints regardless of how things turn out; if you would not have chosen the same option the decision-maker chooses, but the option they chose is better for you, in the state you actually inhabit, than the option you would have chosen, then you have no legitimate complaint; but if you would not have chosen the same option the decision-maker chooses and the option they choose is worse for you, in the state you actually inhabit, than the option you would have chosen, then you do have a legitimate complaint---and, what's more, we measure its strength to be how much more utility you'd have received at the state you inhabit from the option you'd have chosen than you receive from the option chosen for you.

Let's see that work out in the case of Treatment 1 and Treatment 2. Suppose you prefer Treatment 1 to Treatment 2: if the doctor chooses Treatment 1, you have no complaint, whether you have Ailment 1 or Ailment 2; if the doctor chooses Treatment 2 and you have Ailment 1, then you have a complaint, since Treatment 1 gives 10 units of utility while Treatment 2 gives only 2, and you preferred Treatment 1; what's more, your complaint has strength $10-2=8$; if the doctor chooses Treatment 2 and you have Ailment 2, then you have no complaint, since you preferred Treatment 1, but Treatment 2 in fact leaves you better off in this state.

So let's now look at the legitimate complaints that would be made by your different possible selves in the four different treatment/ailment pairs. The first table below reminds us of the risk attitudes you'll have in those four situations, while the second table records the preferences between the two treatments that those risk attitudes entail, when combined with the probabilities---25% chance of Ailment 1; 75% chance of Ailment 2.

$$ \begin{array}{r|cc} \textit{Risk} & \text{Ailment 1} & \text{Ailment 2} \\ \hline \text{Treatment 1} & R(p) = p^2 & R(p) = p\\ \text{Treatment 2} & R(p) = p^2 & R(p) = p^2 \end{array} $$

$$\begin{array}{r|cc} \textit{Preferences} & \text{Ailment 1} & \text{Ailment 2} \\ \hline \text{Treatment 1} & T_2 \prec T_1 & T_1 \prec T_2\\ \text{Treatment 2} & T_2 \prec T_1 & T_2 \prec T_1 \end{array} $$

And the following tables gives the probabilities, utilities, and strengths of complaint you'd legitimately make in each situation:

$$\begin{array}{r|cc} \textit{Utilities} & \text{Ailment 1} & \text{Ailment 2} \\ \hline \text{Treatment 1} & 10 & 7\\ \text{Treatment 2} & 2 & 10 \end{array}$$

$$ \begin{array}{r|cc} \textit{Complaints} & \text{Ailment 1} & \text{Ailment 2} \\ \hline \text{Treatment 1}& 0 & 3\\ \text{Treatment 2} & 8 & 0 \end{array}$$

Now, having given this, we need a decision rule: given these different possible complaints, which should we choose? Cibinel doesn't take a line on this, because he doesn't need to for his purposes: he just needs to assume that, if one option creates no possible complaints while another creates some, you should choose the former. But we need more than that, since that principle doesn't tell between Treatment 1 and Treatment 2. And there are many different ways you might go. I won't be able to survey them all, but I hope to say enough to convince you that none is without its problems.

First, you might say that we should pick the option whose worst possible complaint is best. In this case, that would be Treatment 1. Its worst complaint occurs in $s_2$ and has strength 3, while the worst complaint of Treatment 2 occurs in $s_1$ and has strength 8.

I see a couple of problems with that proposal, both of which stem from the fact that this proposal essentially applies the Minimax decision rule to strengths of complaint rather than utilities---that decision rule says pick the option with the best worst-case outcome. But there are two problems with Minimax make it unsuitable for the current situation.

First, it is a risk-averse decision rule. Indeed, it is plausibly the most extremely risk-averse decision rule there is, for it pays attention *only* to the worst-case scenario. Why is that inappropriate? Well, because one of the things we're trying to do is adjudicate between a risk-netural risk function, which you'll have if the doctor chooses Treatment 1 and it goes poorly, and a risk-averse risk function, which you'll have in all other situations. So using the most risk-averse decision rule seems play favourites, giving priority unduly to the more risk-averse of your possible risk functions.

The second problem is that Minimax is insensitive to the probabilities of the states. While Treatment 2 gives rise to the worst worse-case, it also gives rise to it in a less probable state of the world. Surely we should take that into account? But Minimax doesn't.

How might we take the probability of different complaints into account? The problem is that it is exactly the job of a theory of rational decision-making under uncertainty to do this---they are designed precisely to combine judgments of how good or bad an option is at different states of the world, together with judgments of how likely those states of the world are, to give an overall evaluation and from that a ranking of the options. So, if you subscribe to expected utility theory, you'll think we should minimize expected complaint strength; if you subscribe to expected utility theory with a particular risk function, you'll think we should minimize risk-weighted expected complaint strength. But which decision theory to use is precisely the point of disagreement between some of the different possible complainants in our central case. So, for instance, suppose we approach this problem from the point of view of the risk-averse risk function that you will have in three out of the four possible situations. We minimise risk-weighted expected complaint strength by maximising the risk-weighted expectation of the negative of the complaints. So the risk-weighted expected negative complaint strength for Treatment 1 is $-3 + R(1/4)\times (0-(-3)) = -2.8125$, while for Treatment 2 it is $-8+R(3/4)\times (0-(-8)) = -3.5$. So Treatment 1 is preferable. But now suppose we approach the problem from the point of view of the risk-neutral risk function you will have in one of the four possible situations. So the expected negative complaint strength for Treatment 1 is $(1/4)\times 0 + (3/4)\times (-3) = -2.25$, while for Treatment 2 it is $(1/4)\times (-8) + (3/4)\times 0 = -2$. So Treatment 2 is preferable. That is, the different selves you might be after the treatment disagree on which is to be preferred from the point of a complaints-centred approach. And so that approach doesn't help us a great deal.

# Putting risk attitudes into the utilities

Let's try a different approach. According to Buchak's approach, your attitudes to risk are a third component of your mental state that is combined with your utilities and your probabilities using her formula to give an overall evaluation for an option. Your attitudes to risk don't affect your utilities for different outcomes; instead, they tell you how to combine those utilities with your probabilities to give the evaluation. But perhaps it's more natural to say that our attitudes to risk are really components that partly determine the utilities we assign to different outcomes as a result of different options. And once we do that, we can just use expected utility theory to make our decisions, knowing that our attitudes to risk are factored in to our utilities. Indeed, it turns out that we can do that in a way that perfectly recovers the preferences given by risk-weighted expected utility theory with a particular risk function (see here and here).

Suppose $R$ is your risk function. And suppose an option $o$ gives utility $u_1$ in state $s_1$, which has probability $p_1$, and utility $u_2$ in state $s_2$, which has probability $p_2$. And suppose $u_1 < u_2$. Then$$REU(o) = u_1 + R(p_2)(u_2 - u_1)$$But then$$REU(o) = p_1\left ( \frac{1-R(p_2)}{p_1}u_1 \right ) + p_2\left ( \frac{R(p_2)}{p_2} u_2 \right )$$So if we let $\frac{1-R(p_2)}{p_1}u_1$ be the utility of this option at state $s_1$ and we let $\frac{R(p_2)}{p_2} u_2$ be the utility of this option at state $s_2$, then we can represent you as evaluating an option by its expected utility, and maximising risk-weighted expected utility with the old utilities is the same as maximising expected utility with these new ones. The attraction of this approach is that it transforms a chance of risk attitudes into a change in utilities, and that's something we know how to deal with.

But I think there's a problem. When we transform the utilities to incorporate the risk attitudes, so that maximising expected utility with respect to the transformed utilities is equivalent to maximising risk-weighted expected utility with respect to the untransformed utilities, they don't seem to incorporate them in the right way. Let's suppose we have the following decision:

$$\begin{array}{r|cc} \textit{Probabilities} & s_1 & s_2 \\ \hline P & 1/2 & 1/2 \end{array}$$

$$\begin{array}{r|cc} \textit{Utilities} & s_1 & s_2 \\ \hline o_1 & 9 & 1\\ o_2 & 4 & 5 \end{array}$$

So it's really a choice between a risky option ($o_1$) with higher expected utility and a less risky one ($o_2$) with lower expected utility. It's easy to see that, if $R(p) = p^2$, then$$REU(o_1) = 1+(1/4)\times (9-1) = 3 < 4.25 = 4 + (1/4)\times (5-4) = REU(o_2)$$even though $EU(o_1) > EU(o_2)$.

Now, you'd expect that we'd incorporate risk into the utilities in this case by simply inflating the utilities of $o_2$ on the grounds that this option is less risky, and maybe deflating the utilities of $o_1$. But that's not what happens. Here are the new utilities once we've incorporated the attitudes to risk in the way described above:

$$\begin{array}{r|cc} \text{Probabilities} & s_1 & s_2 \\ \hline P & 1/2 & 1/2 \end{array}$$

$$\begin{array}{r|cc} \text{Utilities} & s_1 & s_2 \\ \hline o_1 & 9\frac{R(1/2)}{1/2} = 4.5 & 1 \frac{1-R(1/2)}{1/2} = 1.5\\ o_2 & 4\frac{1-R(1/2)}{1/2} = 6 & 5 \frac{R(1/2)}{1/2} = 2.5 \end{array}$$

Notice that the utilities of the worst-case outcomes of the two options are inflated, while the utilities of the best-cases are deflated. These don't seem like the utilities of someone who values less risky options, and therefore values outcomes more when they are the result of a less risky outcome. And indeed they just seem extremely strange utilities to have. Suppose you pick option the less risky option, $o_2$, in line with your preferences. Then, in state $s_2$, even though you've picked the option you most wanted, and you're in the situation that gets you most of the original utility---5 units as opposed to the 4 you'd have received in state $s_1$---you'd prefer to be in state $s_1$---after all, that gives 6 units of the new utility, whereas $s_2$ gives you only 2.5.

So, in the end, I don't think this approach is going to help us either. Whatever is going on with individuals who set their preferences using probabilities, utilities, and a risk function in the way Buchak's theory requires, they are not maximizing the expectation of some alternative conception of utility that incorporates their attitudes to risk.

# Choosing for others and choosing for yourself

It's at this point that I can't see a way forward, so I'll leave it here in the hope that someone else can. But let me conclude by noting that, just as the problem arises for the doctor choosing on behalf of the patient in front of them, so it seems to arise in exactly the same way for the individual choosing on behalf of their future selves. Instead of arriving at the hospital unconscious so that the doctor must choose on your behalf, suppose you arrive fully conscious and have to choose between the two treatments based on the different outcomes the doctor describes to you and the probabilities that you assign to each based on the doctor's expertise. Then you face the same choice on behalf of your future selves in this example that the doctor faced on their behalf in the original case. And it seems like the same moves are available, and they're problematic for the same reason: transformative experiences that might lead to changes in your attitudes to risk pose a problem for decision theory.

Interesting, thanks. It strikes me that it would be a pretty odd transformative event if it did not change one's risk preferences: indeed the examples you give (e.g. becoming a parent or moving country) are famously associated with changing one's attitude to risk.

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