2. The Time Consistency Problem

Piccione and Rubinstein (Reference Piccione and Rubinstein1997) point out that there seems to be a tension between optimal planning and time consistency in the absentminded driver problem. Suppose the driver must plan what to do before he leaves the house of his friend. Since he cannot make his choice conditional on reaching the first or second exit, a strategy would simply be: continue if you see an exit, leave the highway if you see an exit, or possibly a randomization over these two possibilities. Let us first focus on the case without randomization. Then, the best strategy from an ex ante perspective would be to continue if you see an exit. Now, suppose the driver finds himself on the highway and sees an exit. Given his strategy to always continue, it seems reasonable for him to believe that he is at the first exit with probability 0.5. But then, the expected utility of leaving the highway would be 2, which is more than what he would get by continuing. Hence, the driver would be tempted to change his plan once he finds himself on the highway. In other words, the optimal plan from the ex ante perspective is not time consistent.

This tension between optimal planning and time consistency will persist if we allow for randomization. Suppose the driver plans to use the same randomization device every time he sees an exit, which induces him to continue with probability $\pi $ and to leave with probability $1 - \pi $ at both exits. Then, the ex ante expected utility from this plan would be ${\pi ^2} \cdot 1 + \pi (1 - \pi ) \cdot 4 + (1 - \pi ) \cdot 0$ , which is maximized when $\pi = {2 \over 3}$ . Suppose again that the driver finds himself on the highway and sees an exit, and let $\beta $ be the probability that he assigns to being at the first exit. Given his strategy, it seems reasonable for him to believe that the probability of being at the second exit is $\beta \cdot {2 \over 3}$ , which means that $\beta = 0.6$ . Given these beliefs, however, the expected utility of leaving the highway is higher than that of continuing, and hence the driver would again be tempted to change his plan to leaving the highway with probability 1. That is, the probabilistic ex ante optimal plan is not time consistent, either.

Existing approaches to the absent-minded problem in the literature all try to resolve the time-consistency issue, but present serious drawbacks, which we will discuss in more detail below (see section 6): either (i) act-state independence is violated; or (ii) the states are not mutually exclusive; or finally (iii) the driver is separated into two agents that choose independently. As a consequence, none of these approaches can be fully reconciled with the standard Savage (Reference Savage1972) framework. In the next section, we present an alternative proposal where a minimal reformulation of the problem allows us to avoid all of these issues, making it consistent with the Savage framework.

A common feature among the approaches in the literature is what we will call the ‘no news assumption’ (also called no news, no change assumption in Baratgin and Walliser (Reference Baratgin and Walliser2010)), i.e. that the driver does not learn any new information upon reaching an exit. We challenge this assumption in the following two sections, and use this to motivate two alternative approaches in these sections. In the first scenario we present a framework that restores the act-state independence in Savage (Reference Savage1972). The second scenario still violates act-state independence, and thus requires a revision of the Savage framework, but we argue that the reasoning supporting our solution there is independently plausible and better motivated than other alternatives.

3. Reasoning about Your Degree of Absentmindedness

In this section we offer an alternative approach that is completely in line with the Savage set-up. We do not explicitly incorporate time into our analysis, but rather allow the driver to reason about his degree of absentmindedness. This will enable us to provide a Savage-style model in which the events about which the driver is uncertain at the ex ante stage are mutually exclusive, and where the driver’s possible acts – to continue or to leave the highway at an exit – do not influence the state. Moreover, the driver is not separated into different agents.

More precisely, at the ex ante stage the driver believes that he may either be absentminded (i) at no exit, (ii) at the first exit only, (iii) at the second exit only, or (iv) at both exits. If the driver is absentminded at an exit then, as a consequence, he will continue at this exit even if he planned to leave there, and he will forget about this exit for the remainder of the journey. Moreover, the driver is perfectly aware of this before he starts the journey. The four mutually exclusive events listed above correspond to the states ${\omega _0},{\omega _1},{\omega _2}$ and ${\omega _{12}}$ , respectively. The decision problem at the ex ante stage may therefore be summarized by Table 1.

Table 1.

Reasoning about your degree of absentmindedness

To understand the consequences of the various acts at the different states, assume first that the driver chooses to continue. Then, no matter what the state is, he would always continue at every exit he reaches, either because he wants to, or because he is absentminded at that exit, which automatically makes him continue there. As a consequence, he would certainly reach the motel.

Suppose next that the driver plans to leave, and that the state is either ${\omega _0}$ or ${\omega _2}$ , that is, the driver is not absentminded at the first exit. Then he would already leave at the first exit, since he is not absentminded there, and reach the scary road.

Now assume that the driver plans to leave and the state is ${\omega _1}$ , which means that he is absentminded at the first, but not at the second, exit. Then, he would continue at the first exit since he is absentminded there, but leave at the second exit, at which he is not absentminded. As a result, he would reach his home.

Finally, suppose that the driver plans to leave and the state is ${\omega _{12}}$ , meaning that the driver is absentminded at both exits. Then, he would still continue at both exists since he is absentminded at both of these. As a consequence, he would reach the motel.

The states in our model are fundamentally different from those in Schwarz (Reference Schwarz2015). In the latter paper, the state describes the exit at which the driver currently finds himself, and, in case he is at the first exit, a description of what the driver would do at the second exit in case it is reached. In our set-up, the state merely describes the exit(s) at which he believes to be absentminded. It does not describe at which exit he is, nor what he would do at the second exit. In fact, the latter is determined by the combination of his act and his state of absentmindedness at the second exit.

Note that in our model there is no dependence between acts and states. Indeed, the plan to continue or not at an exit has no influence on the fact whether the driver is absentminded at a particular exit or not. Let $P(\omega )$ denote the prior probability that the driver assigns to state $\omega $ at the ex ante stage. Then, the ex ante expected utilities of continue and leave would be $EU(cont) = 1$ and $EU(leave) = 4P({\omega _1}) + 1P({\omega _{12}})$ , respectively. Hence, from an ex ante perspective it is optimal for the driver to leave the highway at an exit precisely when $4P({\omega _1}) + P({\omega _{12}}) 1$ , that is, when he deems it sufficiently likely that he will be absentminded at the first exit but not at the second one.

Now assume that $P({\omega _{12}}) \lt 1$ , the driver sees an exit, and reasons about whether to leave or not. Then he must conclude that he is not absentminded at both exits, and hence ${\omega _{12}}$ is no longer possible. If he uses general Bayesian updating with respect to the event $E = \{ {\omega _0},{\omega _1},{\omega _2}\} $ to revise his prior belief, then his updated belief $P( \cdot |E)$ will be given by

$$P({\omega _0}|E) = {{P({\omega _0})} \over {1 - P({\omega _{12}})}},P({\omega _1}|E) = {{P({\omega _1})} \over {1 - P({\omega _{12}})}},P({\omega _2}|E) = {{P({\omega _2})} \over {1 - P({\omega _{12}})}}.$$

Hence, it will still be optimal to leave the highway at an exit precisely when $4P({\omega _1}) + P({\omega _{12}}) 1$ , just like at the ex ante stage. In other words, there is no time inconsistency in this model.

Indeed, if the driver sees an exit, then he receives the informative message that the state ${\omega _{12}}$ , at which he would be absentminded at both exits, is no longer possible. The evidence received in this model is thereby uncentred, because it does not reveal anything about the time period. However, this feature does not play an essential role in our analysis here. The message received could, in principle, be a reason for revising his optimal plan, but in this case the driver will stick to the optimal plan he chose before receiving the message.

The deeper reason for why the driver’s preferences before and after seeing an exit remain the same lies in Savage’s sure-thing principle. This principle states that, whenever the decision maker chooses the same optimal plan, no matter whether an event E is realized or not, then the decision maker must also choose that same optimal plan without knowing whether E is realized or not. Now, take the event E to be the state ${\omega _{12}}$ . Clearly, if the driver knows that ${\omega _{12}}$ applies, then he would be indifferent between his two choices, as both would lead him to the motel. In particular, both choices are optimal for the driver if he knows that the state is ${\omega _{12}}$ . Suppose, without loss of generality, that the driver would prefer to continue if he knows that ${\omega _{12}}$ does not apply. As continuing is also optimal for the driver if he knows that ${\omega _{12}}$ applies, it follows by the sure-thing principle that the driver would still (weakly) prefer to continue before he knows whether ${\omega _{12}}$ applies or not. In other words, his optimal choice before and after seeing an exit would be the same. This is in sharp contrast with many of the existing approaches, which do lead to decisions that are time inconsistent.

Important in our model is that the driver, upon seeing an exit, learns that he cannot be absentminded at both exits, and thus that the state ${\omega _{12}}$ is no longer possible. In other words, the driver receives an informative message when seeing an exit, which means that the no news, no change assumption does not apply. The rationality criterion of no news, no change in Baratgin and Walliser (Reference Baratgin and Walliser2010) asserts that the decision maker should not change his beliefs if no relevant news comes in. This criterion is implied by the principle of strict conditionalization in Teller (Reference Teller1973), but also by temporal conditionalization in Talbott (Reference Talbott1991), which states that a belief is updated in Bayesian way if and only if a new message is received. It also follows from the strong conservation principle in Walliser and Zwirn (Reference Walliser and Zwirn2002), which stipulates that when a message is already known according to the prior probability, then the posterior probability must remain unchanged.

4. Reasoning with Centred Possibilities

The solution we discussed in the previous section relies on a reformulation of the absentminded driver problem. In the original story, the driver knows that he will forget having passed any exit, even though he is aware of being at an exit at the time he reaches it. The second solution that we now turn to explore keeps the original formulation of the problem, and addresses one of the two concerns that we identified: namely, that in the formulation of the problem, the states are not mutually exclusive.

We begin by observing that there are three possible histories, terminating at the Motel $(A)$ , at Home $(B)$ , or at the Scary road $(C)$ , and the probability of each of these histories can be expressed in terms of the probability $\pi $ that the driver continues at each time he arrives at an exit (Table 2). ^{Footnote 1}

Table 2.

The uncentred worlds model

If the utility of each history is: $U(A) = 1$ , $U(B) = 4$ , $U(C) = 0$ , then the expected utility is maximized at $\pi = {2 \over 3}$ . However, this way of representing the problem does not take into account the driver’s uncertainty about what the time period is (and therefore what exit he is approaching). But, of course, this makes a difference to what the driver should do: if he is at the first exit, then he should prefer to continue, while if he is at the second exit, then he should exit. We cannot express what exit the driver is approaching using the model in Table 2, since that does not model the different time periods at which the driver could be located. In other words, the simple model in Table 2 does not capture one important dimension of the driver’s uncertainty, that is which time period he is at.

To address this issue, we enrich the model to add the driver’s time location. This will allow us to represent the driver’s uncertainty regarding both the history and the time period at which he is located. The resulting refined model is summarized in Table 3.

Table 3.

The centred world model

As before, $\pi $ represents the probability that the driver continues when reaching an exit, which we call the driver’s ‘choice probability’, and it determines the overall probability of each possible complete history (as before, A for the history ending at the motel, B for the one ending at home, and C for the one ending up at the scary road). In addition to the choice probability, we add parameters to capture the driver’s uncertainty about the time period. Since we are interested in the driver’s decisions at two successive points, we focus on two times: $t1$ for when the driver reaches the first exit, and $t2$ for when he either reaches the second exit or is on the way to the scary road, having already taken the first exit at $t1$ . It is important to note that $t1$ and $t2$ in our model refer to time periods, not spatial locations, as the spatial location of the driver can indeed be different, at the same time point, for different histories. This gives rise to six centred possibilities, represented in Table 3. In this model, the states represent centred possibilities that are identified by a time point and the history within which they are located. So, for instance, $At2$ is the centred possibility where the driver is at $t2$ and within the history ends up at the motel, $Bt2$ is the centred possibility where the driver is at $t2$ and within the history ends up at home, and $Ct2$ is the centred possibility where the driver is at $t2$ and within the history ending at the scary road. In this model, $\alpha $ , $\beta $ and $\gamma $ stand for the probability that it is time $t1$ , given that we are in history $A,B$ or C, respectively. We can imagine that this is set on the basis of some randomizing mechanism by nature, where both of the time points within each history have equal probability. From now on, unless otherwise noted, we will assume that $\alpha = \beta = \gamma = {1 \over 2}$ . ^{Footnote 2}

When the driver approaches an exit, the only thing he learns is that he hasn’t yet reached the Scary Road; in other words, he learns that it is not the second time period in history C, as otherwise he would have already abandoned the highway and reached the Scary Road. Out of the six possible centred locations, only five are compatible with his current evidence as he approaches the exit. He could be at the first time period, in which case each of the three histories is possible, as he could ultimately reach any of the final destinations depending on his next choices. Or he may already be at the second time period, and in this case he’d be on the road to either Home or the Motel, but certainly not the Scary Road, since he would have already passed that exit. We can represent the driver’s evidence set, as he approaches an exit, as the set $E = \{ At1,At2,Bt1,Bt2,Ct1\} $ .

The model in Table 2 did not allow us to formulate questions concerning which time period the driver is located at. But relative to the model in Table 3 we can now calculate the probability that the driver is at the first time period, and so approaching the first exit:

$$Pr(t1|E) = {{Pr(t1\& E)} \over {Pr(E)}} = {{1/2} \over {{{1 - \pi } \over 2} + \pi }} = {1 \over {1 + \pi }}$$

Note that $Pr(t1|E) = 1/2$ only if $\pi = 1$ , so only if the driver is certain to continue at each exit the probability of being at the first one is $1/2$ . But we cannot fix the probability of being at the first exit without knowing the value of $\pi $ , given that the driver is absentminded.

As $\pi $ represents the choice probability, which is decided when the driver reaches an exit, its value is fixed then. ^{Footnote 3} At the time of choice, that is when he reaches an exit and his evidence is E, the driver should fix a value of $\pi = x$ that maximizes expected utility, given the payoffs associated with each history:

$$E{U_E}(\pi = x) = P{r_E}(A|\pi = x)U(A) + P{r_E}(B|\pi = x)U(B) + P{r_E}(C|\pi = x)U(C)$$

To find the value of x that is optimal, we need to know the conditional probability of each history, conditional on fixing the value of $\pi $ . We know the following:

$$Pr(At1|E) = Pr(At2|E) = {{{\pi ^2}} \over {\pi + 1}};$$

$$Pr(Bt1|E) = Pr(Bt2|E) = {{\pi - {\pi ^2}} \over {\pi + 1}};$$

$$Pr(Ct1|E) = {{1 - \pi } \over {\pi + 1}};{\rm{and}}\ Pr(Ct2|E) = 0.$$

From this, it follows that the expected utility above is maximised at $\pi = \sqrt {{7 \over 3}} - 1$ , or approximately 0.53. This value differs from most other solutions in the literature, and is also different from the value of $\pi $ according to the ex ante optimal strategy, which remains ${2 \over 3}$ also in our approach.

Before moving on, two remarks will be worth making. We could in fact recover the ${2 \over 3}$ solution in our framework in at least two ways, that however have independent drawbacks (on which more below). First, we could recover the ${2 \over 3}$ solution if, instead of general Bayesian updating, we calculated the driver’s posterior probabilities when reaching an exit by using imaging, an updating scheme that has been discussed by Lewis (Reference Lewis1976) and Cozic (Reference Cozic2011) in the context of a related problem, known as the Sleeping Beauty problem. ^{Footnote 4} In this case, upon seeing an exit, the prior probability of $Ct2$ is shifted to the “most similar” state that is still possible, which is $Ct1$ . As a result, the driver’s expected utility is maximised at $\pi = {2 \over 3}$ . In fact, this is independent of how the parameters $\alpha $ , $\beta $ and $\gamma $ are chosen. A second way in which the ${2 \over 3}$ solution could be recovered in our model is by changing the value of the parameter $\gamma $ to 1. However, a significant problem with both these approaches would be to open the door to diachronic inconsistency, as we argue in the next section.

5. Dutch Strategies

In this section we show that, within the centred worlds model of section 4, general Bayesian updating is both necessary and sufficient for being invulnerable against diachronic Dutch strategies.

5.1. Definition of Dutch Strategies

In general, a Dutch strategy is a system of bets that guarantees a sure loss from an ex-ante perspective, but that the decision maker is nevertheless willing to accept, given his conditional beliefs, upon receiving new information. See de Finetti (Reference de Finetti1937) for an early discussion of this concept. It thus points at a ‘discrepancy’ between the decision maker’s ex ante beliefs and his conditional beliefs. For traditional decision problems, Dutch strategies are typically called Dutch books, and for those scenarios it is well-known that a Dutch strategy will never be accepted precisely when the decision maker revises his beliefs by Bayesian updating (Teller Reference Teller1973; Skyrms Reference Skyrms1987).

The absentminded driver problem, as we have seen, is not a traditional decision problem. However, Dutch strategies can still be defined for this type of problem (Hitchcock Reference Hitchcock2004). For example, consider a system of two bets, where both bets concern the event of reaching a safe harbour (the motel or the driver’s home) versus reaching the scary road. The first bet, ${B_1}$ , is offered ex ante, whereas the second bet, ${B_2}$ , is offered after the driver leaves the friend’s house, every time he reaches an exit. Note that, if a safe harbour is eventually reached, then the bet ${B_2}$ will be offered twice on the highway without the driver knowing it. Indeed, if the driver would realize that the bet ${B_2}$ is offered for the second time on the highway, then he would know that he is at the second exit, contradicting the assumption that he is absentminded. As such, when deciding whether or not to accept the bet ${B_2}$ , the driver focuses only on the expected payoff from accepting this bet now, since he has no immediate control over his betting decision at the other time period. Such a system of bets can be summarized by Table 4.

Table 4.

A Dutch book for the absentminded driver problem

Here, $a,b,c$ and d are (possibly negative) payoffs. Hence, if the driver accepts the bet ${B_1}$ , then he receives a if a safe harbour is reached, and b if the scary road is reached. Similarly, if he accepts the bet ${B_2}$ , then he receives c if a safe harbour is reached, and d if he reaches the scary road. If the payoff is negative, say $ - e$ with $e \gt 0$ , then by receiving this payoff we actually mean that the driver must pay the amount e. Recall that, if a safe harbour is reached, then the bet ${B_2}$ will be offered twice, at both exits of the highway. Hence, if you accept ${B_2}$ at both occasions, the amount c will be received twice too, resulting in a total amount of $2c$ . This system of bets is called a Dutch strategy if $a + 2c \lt 0$ and $b + d \lt 0$ . In this case, the total payoff from accepting the two bets, no matter whether you reach a safe harbour or the scary road, will always be negative.

When will the driver be willing to accept the bets ${B_1}$ and ${B_2}$ ? Precisely when the expected payoff from accepting the bet is greater than zero. It is important that the bet ${B_1}$ is evaluated with respect to the ex ante beliefs, whereas the driver uses his conditional beliefs to evaluate the bet ${B_2}$ . Suppose that the driver, ex ante, assigns probabilities p and $1 - p$ to reaching a safe harbour and reaching the scary road, respectively. Then, his expected payoff from accepting the bet ${B_1}$ would be

$$EU({B_1}) = p \cdot a + (1 - p) \cdot b.$$

Assume that his conditional beliefs, upon seeing an exit on the highway, assign probabilities q and $1 - q$ to these two events. Then, the expected payoff from accepting the bet ${B_2}$ on the highway would be

$$EU({B_2}) = q \cdot c + (1 - q) \cdot d.$$

Therefore, the driver would be willing to accept the system of bets if $EU({B_1}) \gt 0$ and $EU({B_2}) \gt 0$ .

5.2. Immunity against Dutch strategies

We will now show that the driver is immune against Dutch strategies precisely when he revises his beliefs by Bayesian updating. Before doing so, we first characterize Bayesian updating in terms of the relationship between the ex ante probability p and the conditional probability q assigned to a safe harbour.

As before, let the three destinations motel, house and scary road be denoted by $A,B$ and C, respectively, and let $At1,At2,Bt1,Bt2,Ct1,Ct2$ be the centred possibilities of reaching each of these destinations, indexed by time. We have seen that seeing an exit can be identified with the event

$E = \{ At1,At2,Bt1,Bt2,Ct1\} $ , where the possibility $Ct2$ has been ruled out. If we adopt the notation from Table 3, with $\alpha = \beta = \gamma = {1 \over 2}$ , then the ex ante probability assigned to a safe harbour is

$$p = \pi $$

whereas the conditional probability assigned to a safe harbour, obtained from Bayesian updating after seeing an exit, is

$$q = {\pi \over {1 - {1 \over 2}(1 - \pi )}} = {{2p} \over {1 + p}}.$$

Therefore,

$${q \over {1 - q}} = 2 \cdot {p \over {1 - p}}.$$

This relationship characterizes Bayesian updating.

The following result states that the driver will only be immune against Dutch strategies if he revises his beliefs by Bayesian updating.

Theorem 5.1. Suppose that ex ante, the driver assigns probability p and $1 - p$ to reaching a safe harbour versus reaching the scary road, and that on the highway he assigns probabilities q and $1 - q$ to these two events. Assume that both p and q are strictly between 0 and 1. Then, there is no Dutch strategy of the form above that the driver is willing to accept, if and only if,

$${q \over {1 - q}} = 2 \cdot {p \over {1 - p}},$$

that is, if and only if the driver revises his beliefs by Bayesian updating.

The condition in this theorem is visualized in Figure 2, and the formal proof is given in the Appendix. The theorem thus asserts that, if we adopt the model with centred possibilities as proposed in section 4, then the solution we put forward in that section is the only plan that (a) is optimal for the driver once he sees an exit, and (b) makes him immune against Dutch strategies. Any other belief revision rule, including imaging, allows for the design of a Dutch strategy that the driver is willing to accept.

Figure 2.

Immunity against Dutch strategies.

6. Related Literature

7. Time Consistency Revisited

A common concern in the analysis of the absentminded driver problem is time consistency. As we recalled in sections 2 and 6, a recurring motivation for the solutions that have been discussed in the literature is to resolve the apparent inconsistency between the driver’s optimal strategy ex ante (before he starts the journey) and when he reaches an exit. This poses a problem because it is implicitly assumed that the driver does not learn anything new between these times – call this the no news assumption. He always knew that he was going to reach an exit, and he should not be surprised when he does. Therefore, the driver should not update his beliefs upon reaching an exit, and the discrepancy between his optimal strategies when considered ex ante vs. when reaching an exit (what we will call interim) cannot be explained by a change in the probabilities assigned to the relevant states of the world. Starting from the no news assumption, most of the solutions in the literature (see section 6) attempt to resolve the issue of time inconsistency by looking for a strategy that could be characterized as rational, according to some modified notion of optimality, both ex ante and interim.

Contrary to this received approach, our analysis highlights that the no news assumption is incorrect. The driver learns some new relevant information upon arriving at an exit, namely that $Ct2$ is not the case at this point. Since this is a possibility that the driver eliminates when he reaches an exit, he should update on this information. Failing to do so, moreover, would lead to diachronic inconsistency and vulnerability to a Dutch strategy. Therefore, the driver’s beliefs ex ante and upon reaching an exit differ, but are not time inconsistent. On the contrary, the difference results from applying standard Bayesian updating upon learning a new piece of evidence. Our analysis explains why the driver’s changing beliefs, and the resulting change in optimal strategies, are in fact time consistent, eliminating the need to resolve the apparent discrepancy that motivated previous solutions presented in the literature.

Still, one may pose the following objection to our analysis, based on the idea of forming a commitment. Suppose that the driver has the possibility, ex ante, of forming a commitment, that is, selecting a strategy that will be binding for him whenever he reaches an exit. Based on our analysis, the optimal strategy for the driver ex ante would be to continue with a probability of ${2 \over 3}$ . However, if he were to commit to this strategy, he would regret doing so at the moment he reaches an exit, when the optimal strategy after updating would be to continue with probability 0.53. Under these conditions, would it be rational for the driver to choose to commit to the ex ante optimal strategy? Or should he commit to the strategy he knows will be optimal for him after updating, even though it is not optimal at the time he makes the commitment?

Neither option seems rational. If the driver commits to the ex ante optimal strategy, he will come to regret it, based on his own beliefs at the time when he needs to implement the strategy. Choosing to commit to the other strategy, on the other hand, would be irrational from the ex ante perspective. Thus, if the driver has the option to commit, it seems that neither of the available strategies (the ex ante optimal one or the interim optimal one) are rational to commit to.

For a rational agent who plans to update her beliefs via conditioning, and does not expect to forget any of her experiences, forming a commitment is redundant, since her present and future (expected) optimal strategies are identical (see, for instance, Perea Reference Perea2012, Lemma 8.14.9). However, if the agent expects her future self to become irrational, then it may be rational to form a commitment now in order to prevent herself from making a bad choice in the future, as in the story of Ulysses tying himself to the mast of his ship in order to escape the sirens’ call (van Fraassen Reference van Fraassen1995). Thus, forming a commitment in the face of an expected deviation, in the future, from what is the present optimal strategy, may be rational if the agent expects to become irrational by the time of making a decision.

Interestingly, the case of absentmindedness illustrates how forming a commitment is not always either redundant or necessary to prevent anticipated irrationality. Sometimes, it is simply not possible to rationally commit. Although his future beliefs deviate predictably from his ex ante beliefs, the driver’s future self is not irrational. Upon reaching an exit, the driver arrives at the new beliefs by conditioning on newly acquired evidence, as we have seen in section 4, and committing to the ex ante optimal strategy would make him vulnerable to a Dutch Strategy, as we have shown in section 5. So, although the driver expects that he will deviate from his ex ante optimal strategy when he will make a decision in the future, he can’t justify committing to the ex ante optimal strategy on the basis of his own future irrationality.

Absentmindedness means that the future driver cannot locate himself in time. When he approaches an exit, he is uncertain whether it is the first or the second one. This happens predictably and therefore his evidence at any time that he approaches an exit is the same. But this also means that when he considers, ex ante, what his own beliefs will be later during the journey, on the highway, he is not referring to his future self at a definite point in time. Later here refers both to his future self approaching the first exit, and his future self approaching the second exit (if he gets that far). ^{Footnote 6} If the driver were expected to defer to his rational, future self, this would raise the question: to his future self at which point in time?

Ex ante, the driver can only be certain that he will hit the first exit (since that exit is reached no matter what the final destination), at which point he will update his beliefs and optimal strategy in the way that we have described. So, a natural answer to the previous question is to defer to the driver’s future self at the only time he is certain to reach an exit, that is $t1$ . Interpreted in this way, later picks out $t1$ , but it is not a stopping time (Schervish et al., Reference Schervish, Seidenfeld and Kadane2004). A (possibly random) time T is called a stopping time when, at any point in time, the agent knows whether T has already occurred or not. An example of a stopping time would be the day of your son’s first birthday (assuming that you do not forget what day it is!), or the first time a coin that you are about to toss repeatedly comes up heads, assuming that you observe and don’t forget any of the previous tosses. Since the driver does not know he has reached the first exit when he is there, ${t_1}$ is not a stopping time in our case. Moreover, learning that later, so interpreted, has arrived would give the driver information that changes his optimal strategy: if he knew that he was approaching the first exit, then it would be optimal for him to continue.

Under these conditions (that is, later not being a stopping time, and the additional information that later has arrived being relevant to the driver’s other beliefs), Schervish et al. (Reference Schervish, Seidenfeld and Kadane2004) show that the so-called Principle of Reflection does not apply. ^{Footnote 7} This principle, introduced by van Fraassen (Reference van Fraassen1984), says that for any two times ${t_1} \lt {t_2}$ , and for any event A, your credence in A at ${t_1}$ , conditional on your credence in A at ${t_2}$ being r, should also be r. That is: ${P_{{t_1}}}(A|{P_{{t_2}}}(A) = r) = r$ . Since Reflection does not apply to our case, the absent-minded driver’s ex ante beliefs do not reflect his expected later beliefs (Schervish et al., Reference Schervish, Seidenfeld and Kadane2004). Therefore, he cannot rationally commit ex ante to a strategy that is optimal later, since his own beliefs ex ante should not rationally reflect the beliefs he will have later.

Anticipating his own absentmindedness, the driver can only make rational decisions at the moment when it is required. If he is required to pick a strategy ex ante, and prevented from changing it along the way, then he should pick the ex ante optimal strategy, which differs from what he would choose on the highway. In this scenario, the driver is effectively occupying the role of a passenger, since he has no control over the direction travelled by the car during the journey. If, instead, the driver is allowed to make choices while on the highway, then he should opt for the interim optimal strategy, deviating from the choice probability that he would have picked ex ante. This is not a sign of dynamic inconsistency, since the change in beliefs is the result of rational updating, and commitment is not appropriate in this case.