SC has preferences which make future repetitions of his gamble relevant to a present choice he faces, given that he is resolute. As we have seen in Section 6.3, REU theory can apparently account for his preferences because it involves a risk function which introduces complementarities between different risky choices the agent faces, even when these are independent in terms of probability and utility. But this feature of REU theory means that the observations of the last section lead to a very general problem for REU theory.
Apart from the special case where r = p and REU theory reduces to expected utility theory, the risk function creates complementarities between any two risky choices an agent faces in her life. This is because the risk function is applied after utilities have already been assigned. In REU theory, just as in expected utility theory, utilities are assigned to outcomes involving different kinds of goods, occurring at different times and places. Having such a single measure allows us to express the way in which the agent trades off different kinds of goods, or the way in which she evaluates gambles that involve different kinds of goods. The risk function is applied to this single utility measure. And thus it creates complementarities between any two gambles the agent faces. We all face many risky choices of different kinds in our lives, and so this feature of REU theory has far-reaching consequences.
We can illustrate this with the transport and restaurant choice scenarios introduced above. Like any risky choices, according to REU theory, these can also be described as choices over utility gambles. Suppose that the probabilities involved in these gambles are also independent for any two occasions where I face the choice. Moreover, the agent’s preferences over the outcomes are such that they need to be modelled with utilities that are independent: Whether I took the subway one day does not affect how much added utility I get from taking it the next day, and whether I went to Steady Stan’s one day does not affect how much added utility I get from going there the next day. Further suppose that the utilities in the transport and restaurant choice are the same as those in SC’s original choice.
Now assume that I am resolute in dynamic decision problems, and can be assigned the utility and risk functions we used above. In that case, it follows from what we have said that if I think of my choices of means of transport two at a time, I will come to a different conclusion from when I think of them three at a time. In the first case I will cycle both times; in the second case I will take the subway each time. Or suppose I decide to consider my lunch decision together with my decision of what means of transport to take on two consecutive days. This will make me change my choice from cycling to taking the subway, along with eating at the riskier restaurant - even though the utilities of the possible outcomes of all these decisions are independent from one another.
The fact that REU theory creates complementarities between all the risky choices an agent faces, together with the assumption that the agent is resolute thus leads to a serious problem for REU theory. It ensures that an agent can never be justified in believing that the solution to a decision problem that falls short of taking account of all the future decisions she expects to face is going to approximate the solution to the grand-world decision problem.18 For the REU maximizer, different temporal perspectives and the corresponding formulations of choice problems may result in different recommendations regarding one
18Buchak herself (pp. 228-229) seems to think that REU maximizers should package together decisions that concern
the same kind of value, and may separate decisions that concern values that are unrelated to each other into different decision problems. However, this strategy would lead to an under-specification of decision problems according to Joyce’s rule, which Buchak also seems to be committed to. This is because the risk function creates a complementarity between risky choices even if they concern different kinds of values. REU theory itself makes risky gambles involving different types of values complementary to each other.
and the same action, even if that action will influence neither the probability nor the utility of any other outcomes she may end up with in the future. More short-term perspectives will not generally agree with the most long-term perspective.
Note that expected utility theory does not have this problem. For the expected utility maximizer, when she faces a series of gambles which are independent in terms of utility and probability, how she carves up the decision problem is not going to matter. In that case, the expected utility of the compound gamble is the same as the sum of the expected utilities. When she constructs a small world decision problem in which she faces a one-off choice of whether to accept an individual gamble, this will give her the same conclusion as the one she would draw were she to consider the series of gambles as a whole.
We have seen that a resolute REU maximizer can apparently never be justified in believing that a choice based on a decision problem which considers only a subset of the gambles she will face in her life is going to cohere with the decision she would make were she to consider the grand-world decision problem. If we think that agents are required to formulate their decision problems to ensure that there is such a coherence, then it seems like she is forced to think about her grand-world decision problem every time she makes a choice. This would make REU theory an even more demanding theory to apply than expected utility theory.
What makes things worse is that, if I am a resolute REU maximizer and I want to make the decision that I would come to were I to consider the grand-world decision problem, it seems my safest bet is to behave as if I were an expected utility maximizer. We have seen above that for a risk averse REU maximizer, as the number of repetitions of a gamble goes to infinity, the average certainty equivalent of each gamble tends to its expected utility. And so a resolute agent considering a large number of repetitions of the same gamble will choose just like an expected utility maximizer. Now of course the actual grand-world problem an agent faces will be more complicated than a large number of repetitions of the same gamble. But we can speculate that the results should be similar for large numbers of different gambles.19 And so resolute REU maximizers will in fact end up behaving approximately like expected utility maximizers.
This conclusion thus undermines the central motivation for REU theory, namely that it can account for various counter-examples to expected utility theory. Assuming the agent is resolute, and considers herself as facing the grand-world decision problem, REU theory cannot in fact cast choices like those of SC as rational. When SC says that he would reject the single gamble, he apparently fails to integrate his decision concerning that individual gamble into the grand-world decision problem — or otherwise he would accept it.20 As Benartzi and Thaler (1999) point out, he was probably already playing less favourable gambles every day by holding some of his retirement savings in stocks.
In fact, all of the toy examples that are used to motivate and illustrate REU theory, and not only SC’s case, involve agents who must have left out relevant detail from the specification of their decision problems, given they are resolute. These examples usually involve one-off choices involving outcomes that
19For one, Buchak’s repetition theorem will also apply, for instance, to the compound gamble resulting from the many
choices one makes in a typical day. Another supporting consideration is that, as long as different gambles are reasonably independent, an agent’s exposure to risk throughout her life is diversified, and the variance of the overall risk ‘portfolio’ is lower than the weighted average of the individual gambles’ variances. The lifetime gamble is in that sense less risky than the constituent gambles. The risk function makes the agent sensitive to that lower variance.
20In a similar vein, Benartzi and Thaler (1995, 1999) refer to risk aversion for small stakes gambles as “myopic loss
aversion”. What they suggest explains these preferences is that when agents are shown the rates of return for an individual gamble, they are loss averse in the way Kahnemann and Tversky (1984) describe it. And agents fail to integrate the small loss into their current wealth level, which Benartzi and Thaler interpret as an instance of “narrow framing”, as in Kahnemann and Lovallo (1993).
are described merely as the immediate goods one receives as the consequence of having made that choice. For instance, in the Allais problem described in the last chapter, the agent receives a sum of money with a particular probability as a result of her choices. Buchak shows how REU theory apparently makes sense of the choices typically made in the Allais problem, given that the agent’s risk function is convex (p.71). She describes the outcomes simply as those sums of money the agent stands to gain through the gambles she faces. This is not integrated with all of the future uncertain monetary prospects the agent faces. But if the agent is a resolute REU maximizer who faces future risky choices, and his decision problem should include everything relevant, then it should be so integrated.
Parallel claims apply to the other examples used to motivate REU theory. Given that these examples are usually presented as simple small-world problems, we can no longer be sure that on a specification of the decision problem that includes everything that is relevant to a resolute agent, REU theory can make sense of them. In fact, if these problems involve small stakes gambles and we think that the agent faces many such gambles in her life, then, as we have said before, a resolute agent should behave roughly like an expected utility maximizer.
This conclusion also raises the question of whether, quite apart from its treatment of these moti- vating examples, REU theory offers a true alternative to expected utility theory. After all, given our assumptions, REU theory and expected utility theory license the same choices. Indeed, out of the two, expected utility theory is the simpler theory, making it more attractive. One might think that there is still a difference since expected utility maximizers and resolute REU maximizers behave in the way they do for different reasons. For an REU maximizer, it is only because a risky action like accepting SC’s gamble is part of a long series of risky actions that she chooses to do it. If she faced the choice in isolation, she wouldn’t. But since we generally all face more than one risky choice in our lives, this hypothetical difference is a very remote possibility.
We may then point out that the REU maximizer’s given preferences still violate expected utility theory, even if she does not act on them. This observation, however, does not help REU theory. On our preferred way of interpreting resolute choice, the agent temporarily adjusts her preferences within a dynamic choice problem only. Now note that if the agent considers herself as facing one grand-world dynamic decision problem in her life, that may in fact amount to adjusting all her preferences.
We said that the REU representation of an agent will be based on her given preferences. But it is no longer clear what the relevance of given preferences even is if they may never manifest themselves in choice. Given preferences presumably capture how the agent would choose outside of the context of a dynamic choice problem at hand. But in the case we are considering here, the one grand-world decision problem is all there is for the agent. And in that problem, there is no difference in the actual preferences of REU maximizers and expected utility maximizers. In the light of the last chapter, it is especially important to note that a resolute REU maximizer who considers the grand-world decision problem ends up adopting separable preferences after all.
Most importantly, this means that any REU representation we may still assign to the agent’s given preferences can no longer capture what REU theory set out to capture. The distinguishing feature of REU theory was supposed to be that it introduces a risk function that represents the agent’s commit- ment to treating risks in a certain way, or her venturesome or prudent character traits. A resolute REU maximizer who acts like an expected utility maximizer in the grand-world decision problem displays no such character traits or commitments in her choices, since the given preferences that may be repre- sentable as REU maximizing never manifest themselves in choice. Under those circumstances, we do not
seem licensed to interpret any risk function we may assign to the agent as representing a disposition, commitment or character trait.
These conclusion can be avoided, however, if we give up either the assumption of resolution, or the assumption that everything that is relevant to the agent’s decision should be included in her decision problem. The next section considers the implications of relaxing these assumptions. It will turn out that relaxing the assumption of resolution, at least, is not instrumentally irrational. However, relaxing these assumptions also means that an REU maximizer’s choices will be very sensitive to her choice of temporal frame and choice behaviour in dynamic decision problems. This sensitivity again stands in the way of REU theory identifying any stable choice disposition, commitment or character trait in the face of risk.