Lost memories and useless coins: revisiting the absentminded driver
Received: 1 November 2013 / Accepted: 13 February 2015 © Springer Science+Business Media Dordrecht 2015
Abstract The puzzle of the absentminded driver combines an unstable decision problem with a version of the Sleeping Beauty problem. Its analysis depends on the choice between “halfing” and “thirding” as well as that between “evidential” and “causal” decision theory. I show that all four combinations lead to interestingly different solutions, and draw some general lessons about the formulation of causal decision theory, the interpretation of mixed strategies and the connection between rational credence and objective chance.
Keywords Causal decision theory · Evidential decision theory · Diachronic rationality · Mixed strategies · Deliberation · Sleeping Beauty 1 Introduction
Sometimes it is controversial what rationality demands in a given situation. Oneboxers and two-boxers disagree on the choice to make in Newcomb’s problem, halfers and thirders disagree on the beliefs to have in the Sleeping Beauty problem. Such disagreements often trace back to different general perspectives on rationality. At the heart of Newcomb’s problem lies the divide between causal and evidential decision theory. At the heart of the Sleeping Beauty problem arguably lies a tension between evidentialism and conservatism in epistemology. In this paper, I want to look at a case that raises both of these issues, as well as several others. The case was introduced in
Piccione and Rubinstein (1997), and goes as follows.
W. Schwarz (B)
School of Philosophy, Psychology and Language Sciences,
The University of Edinburgh, 3 Charles Street, Edinburgh EH8 9AD, UK e-mail: firstname.lastname@example.org 123
An absentminded driver has to take the second exit off the highway in order to get home. If she turns off at the first exit, she reaches a desolate area and has to spend the night in her car. If she continues at both exits, she has to stay at a motel at the end of the highway. Due to her absentmindedness, she cannot tell upon arriving at an exit whether it is the first or the second (unless, of course, she knows that she turns off at the first).
Our main question is what the driver ought to do. The answer varies between evidential and causal decision theory, and even between different formulations of the latter. In addition, what the driver ought to do depends on what she ought to believe, and this, too, turns out to be controversial: we will find essentially the same two options as in the Sleeping Beauty problem. We will also see that if the driver makes her choice by tossing a coin, then her degree of belief in the two possible outcomes (heads and tails) does not always match what she knows to be the objective chance. Consequently, several widespread ideas about the role of chance in game theory and decision theory threaten to break down.
The point of this paper is not to take sides in the debate between causal and evidential decision theory or between halfing and thirding. In fact, I will argue that all four combinations give defensible answers to the puzzle if one keeps in mind the general perspective that motivates these combinations. 2 Absentmindedness and two types of expected utility
Before we begin, I should make some clarifications about the driver’s predicament.
The driver suffers from an unusual kind of absentmindedness. Her problem is not that she is likely to pass an exit without noticing it. On the contrary, she is certain to make a deliberate, rational choice at every exit she reaches. Her problem is that if she decides to stay on the highway at the first exit, then the monotony of the traffic will make her forget the whole event before she reaches the second exit, so that she arrives at that exit in the very same state of mind in which she arrived at the first exit. The two exits may look different, but the differences don’t help the driver to figure out which is which. For some reason there are no signposts, and the driver can’t leave marks to counteract the memory loss brought on by the traffic. For example, she can’t tie a knot in her handkerchief after continuing at the first exit and thus use the handkerchief to find out where she is. Throughout her journey, the driver is aware of all these facts.
Subject to the constraints of the scenario, we assume that the driver is ideally rational, and knows that she is ideally rational. We model her beliefs by a probability measure P over possible states of affairs so that the probability assigned to a state represents the degree to which she believes that the state obtains. Similarly, the degree to which she desires the different states to obtain is represented by a utility function
V . The driver would mostly like to get home, but also has a slight preference for staying at the motel over spending the night in her car. For concreteness, let’s say that
V (Car) = 0, V (Home) = 4 and V (Motel) = 1. We assume that due to the memory loss caused by the highway, her beliefs and desires are in all relevant respects the same whenever she gets to an exit. 123
It follows that if the driver’s beliefs and desires determine a particular choice as uniquely rational, then that is what she is going to do at every exit. It may therefore be reasonable for the driver to assume that whatever she does at the present exit is also what she does at the other exit (if reached). She might then reason as follows. “I can either leave the highway here or continue. If I leave, I must be at the first exit, for I know that I make the same choice at every exit, and I couldn’t be at the second exit after leaving at the first. So if I leave, I’ll end up spending the night in the car. Alternatively, if I continue on the highway, then it is clear that I’ll continue at both exits, so I’ll spend the night in the motel. That is the slightly better outcome, so