This post examines an attempt by professional decision theorists to treat an example of time inconsistency, and asks why they failed to reach the solution (i.e., TDT/UDT) that this community has more or less converged upon. (Another aim is to introduce this example, which some of us may not be familiar with.) Before I begin, I should note that I don't think "people are crazy, the world is mad" (as Eliezer puts it) is a good explanation. Maybe people are crazy, but unless we can understand how and why people are crazy (or to put it more diplomatically, "make mistakes"), how can we know that we're not being crazy in the same way or making the same kind of mistakes?

The problem of the ‘‘absent-minded driver’’ was introduced by Michele Piccione and Ariel Rubinstein in their 1997 paper "On the Interpretation of Decision Problems with Imperfect Recall". But I'm going to use "The Absent-Minded Driver" by Robert J. Aumann, Sergiu Hart, and Motty Perry instead, since it's shorter and more straightforward. (Notice that the authors of this paper worked for a place called Center for the Study of Rationality, and one of them won a Nobel Prize in Economics for his work on game theory. I really don't think we want to call *these* people "crazy".)

Here's the problem description:

An absent-minded driver starts driving at START in Figure 1. At X he

can either EXIT and get to A (for a payoff of 0) or CONTINUE to Y. At Y he

can either EXIT and get to B (payoff 4), or CONTINUE to C (payoff 1). The

essential assumption is that he cannot distinguish between intersections X

and Y, and cannot remember whether he has already gone through one of

them.

At START, the problem seems very simple. If p is the probability of choosing CONTINUE at each intersection, then the expected payoff is p^{2}+4(1-p)p, which is maximized at p = 2/3. Aumann et al. call this the *planning-optimal *decision.

The puzzle, as Piccione and Rubinstein saw it, is that once you are at an intersection, you should think that you have some probability α of being at X, and 1-α of being at Y. Your payoff for choosing CONTINUE with probability p becomes α[p^{2}+4(1-p)p] + (1-α)[p+4(1-p)], which doesn't equal p^{2}+4(1-p)p unless α = 1. So, once you get to an intersection, you'd choose a p that's different from the p you thought optimal at START.

Aumann et al. reject this reasoning and instead suggest a notion of *action-optimality*, which they argue should govern decision making at the intersections. I'm going to skip explaining its definition and how it works (read section 4 of the paper if you want to find out), and go straight to listing some of its relevant properties:

- It still involves a notion of "probability of being at X".
- It's conceptually more complicated than planning-optimality.
- Mathematically, it has the same first-order necessary conditions as planning-optimality, but different sufficient conditions.
- If mixed strategies are allowed, any choice that is planning-optimal is also action-optimal.
- A choice that is action-optimal isn't necessarily planning-optimal. (In other words, there can be several action-optimal choices, only one of which is planning-optimal.)
- If we are restricted to pure strategies (i.e., p has to be either 0 or 1) then the set of action-optimal choices in this example is empty, even though there is still a planning-optimal one (namely p=1).

In problems like this one, UDT is essentially equivalent to planning-optimality. So why did the authors propose and argue for action-optimality despite its downsides (see 2, 5, and 6 above), instead of the alternative solution of simply remembering or recomputing the planning-optimal decision at each intersection and carrying it out?

Well, the authors don't say (they never bothered to argue against it), but I'm going to venture some guesses:

- That solution is too simple and obvious, and you can't publish a paper arguing for it.
- It disregards "the probability of being at X", which intuitively ought to play a role.
- The authors were trying to figure out what is rational for human beings, and that solution seems too alien for us to accept and/or put into practice.
- The authors were
*not*thinking in terms of an AI, which can modify itself to use whatever decision theory it wants to. - Aumann is known for his work in game theory. The action-optimality solution looks particularly game-theory like, and perhaps appeared more natural than it really is because of his specialized knowledge base.
- The authors were trying to solve one particular case of time inconsistency. They didn't have all known instances of time/dynamic/reflective inconsistencies/paradoxes/puzzles laid out in front of them, to be solved in one fell swoop.

Taken together, these guesses perhaps suffice to explain the behavior of these professional rationalists, without needing to hypothesize that they are "crazy". Indeed, many of us are probably still not fully convinced by UDT for one or more of the above reasons.

**EDIT:** Here's the solution to this problem in UDT1. We start by representing the scenario using a world program:

def P(i, j):

if S(i) == "EXIT":

payoff = 0

elif S(j) == "EXIT":

payoff = 4

else:

payoff = 1

(Here we assumed that mixed strategies are allowed, so S gets a random string as input. Get rid of i and j if we want to model a situation where only pure strategies are allowed.) Then S computes that payoff at the end of P, averaged over all possible i and j, is maximized by returning "EXIT" for 1/3 of its possible inputs, and does that.

You cannot assume any α, and choose p based on it, for α depends on p. You just introduced a time loop into your example.

Indeed, though it doesn't have to be a time loop, just a logical dependency. Your expected payoff is α[p^2+4(1-p)p] + (1-α)[p+4(1-p)]. Since you will make the same decision both times, the only coherent state is α=1/(p+1). Thus expected payoff is (8p-6p^2)/(p+1), whose maximum is at about p=0.53. What went wrong this time? Well, while this is what you should use to answer bets about your payoff (assuming such bets are offered independently at every intersection), it is not the quantity you should maximize: it double counts the path where you visit both X and Y, which involves two instances of the decision but pays off only once.

You make it sound as if you expect to expect a higher utility in the future than you currently expect...

There's an old story about a motorist who gets a flat tire next to a mental hospital. He goes over to change the tire, putting the lug nuts into the wheel cap... and sees that one of the patients is staring at him from behind the fence. Rattled, he steps on the wheel cap, and the lug nuts go into the storm sewer.

The motorist is staring, disheartened, at the sewer drain, when the patient speaks up from behind the fence: "Take one lug nut off each of the other wheels, and use those to keep on the spare tire until you get home."

"That's brilliant!" says the motorist. "What are you doing in a mental hospital?"

"I'm here because I'm crazy," says the patient, "not because I'm stupid."

Robert Aumann, Nobel laureate, is an Orthodox Jew. And Isaac Newton wasted a lot of his life on Christian mysticism. I don't think theists, or Nobel laureate physicists who don't accept MWI, are stupid. I think they're crazy. There's a difference.

Remind me to post at some point about how rationalists, in a certain stage in their development, must, to progress further, get over their feeling of nervousness about leaving the pack and just break with the world once and for all.

If crazy people can nevertheless reach brilliant and correct solutions, then in what sense is their craziness an explanation for the fact that they failed to reach some solution? I really don't see what Aumann's religiousness has to do with the question I asked in this post, not to mention that he's just one person who worked on this problem. (Google Scholar lists 171 citations for P&R's paper.)

To put it another way, if we add "Aumann is religious" to the list of possible explanations I gave, that seems to add very little, if any, additional explanatory value.

Because crazy smart people don't

consistentlyreach solutions. It's not surprising when they're right, but it's not surprising when they're wrong, either. There are very few people I know such that I'm surprised when they seem to get something wrong, and the key factor in that judgment is high sanity, more than high intelligence.I'm also beginning to have a very strange thought that a reddit-derived blog system with comment upvoting and karma is just a

vastly more effectiveway of researching decision-theory problems than publication in peer-reviewed journals.Chess World Champions are sometimes notoriously superstitious, you can still rely on the consistency of their

chess moves.They really ought to be, what's the rational value in putting the time and effort into chess to become a world champion at it.

I played it semi-seriously when I was young, but gave it up when in order to get to the next level I'd have to study more than play. Most of the people I know who were good at a competitive intellectual game dropped out of school to pursue it, because they couldn't handle studying at that level for both.

I find it rather difficult to believe that pursuing chess over school is the rationally optimal choice, so I wouldn't be remotely surprised to find that those who get to that level are irrational or superstitious when it comes to non-chess problems.

"one of them won a Nobel Prize in Economics for his work on game theory. I really don't think we want to call these people "crazy".)"

Aumann is a religious Orthodox Jew who has supported Bible Code research. He's brilliant and an expert, but yes, he's crazy according to local mores.

Of course, that doesn't mean we should dismiss his work. Newton spent much of his life on Christian mysticism.

No. This statement of the problem pretends to represent the computation performed by the driver at an intersection - but it really doesn't. The trouble has to do with the semantics of alpha. Alpha is

notthe actual probability that the driver is at point X; it's the driver's estimate of that probability. The driver knows ahead of time that he's going to make the same calculation again at intersection Y, using the same value of alpha, which will be wrong. Therefore, he can't pretend that theactualpayoff is alpha x (payoff if I am at X) + (1-alpha) x (payoff if I am at Y). Half the time, that payoff calculation will be wrong.Perhaps a clearer way of stating this, is that the driver, being stateless, must believe P(I am at X) to be the same at both intersections. If you allow the driver to use alpha=.7 when at X, and alpha=.3 when at Y, then you've given the driver information, and it isn't the same problem anymore. If you allow the driver to use alpha=.7 when at X, and alpha=.7 again when at Y, then the driver at X is going to make a... (read more)

For anyone whose eyes glazed over and couldn't see how this was derived:

There are 3 possible outcomes:

The probability of missing both turns for a

pisp*p(2 turns, the samepeach time), and the reward is 1. Expected utility is probability*reward, so 2*p*1. Which is just 2*porp^2The probability of making the second turn is the probability of missing the first turn and making the second. Since

pis for a binary choice, there's only one other probability,q, of m"The authors were trying to figure out what is rational for human beings, and that solution seems too alien for us to accept and/or put into practice."

I'm not sure about that. A lot of people intuitively endorse one-boxing on Newcomb, and probably a comparable fraction would endorse the 2/3 strategy for Absent-Minded Driver.

"Well, the authors don't say (they never bothered to argue against it)"

They do mention and dismiss mystical /psychic causation, the idea that in choosing what we will do we also choose for all identical minds/algorit... (read more)

I could add some groundless speculation, but my general advice would be: Ask, don't guess!

You probably won't get answers like "I wanted to publish a paper.". But I'd bet, it would be very enlightening regardless. It'd be surprising if all of them were so extremely busy that you can't approach anybody in the area. But don't settle for PhD students, go for senior professors.

It occurs to me that perhaps "professional rationalist" is a bit of an oxymoron. In today's society, a professional rationalist is basically someone who is paid to come up with publishable results in the academic fields related to rationality, such as decision theory and game theory.

I've always hated the idea of being paid for my ideas, and now I think I know why. When you're being paid for you ideas, you better have "good" ideas or you're going to starve (or at least suffer career failure). But good ideas can't be produced on a schedu... (read more)

What I'm more interested in is: doesn't the UDT/planning-optimal solution imply that injecting randomness can improve an algorithm, which is a big no-no? Because you're saying (and you're right AFAICT) that the best strategy is to randomly choose whether to continue, with a bias in favor of continuing.

Also, could someone go through the steps of how UDT generates this solution, specifically, how it brings to your attention the possibility of expressing the payoff as a function of p? (Sorry, but I'm a bit of a straggler on these decision theory posts.)

You made a mistake here, which is assuming that when you guess you are at X, you should choose CONTINUE with probability 1, and when you guess you are at Y, you should choose EXIT with probability 1. In fact you can improve your expected payoff using a mixed strategy, in which case you can always do better when you have more information.

Here's the math. Suppose when you are at an intersection, you get a clue that reads either 'X' or 'Y'. This clue is determined by a dice roll at START. With probability .49, you get 'X' at both intersections. With probability .49, you get 'Y' at both intersections. With probability .02, you get 'X' at the X intersection, and 'Y' at the Y intersection.

Now, at START, your decision consists of a pair of probabilities, where p is your probability to CONTINUE after seeing 'X', and q is your probability to CONTINUE after seeing 'Y'. Your expected payoff is:

.02 * (p*q + 4*(p*(1-q))) + .49 * (p*p + 4*(p*(1-p))) + .49 * (q*q + 4*(q*(1-q)))

which is maximized at p=0.680556, q=0.652778. And your expected payoff is 1.33389 which is > 4/3.

In

`#lesswrong`

, me, Boxo, and Hyphen-ated each wrote a simple simulation/calculation of the absent-minded driver and checked how the various strategies did. Our results:As expected, 4/9 does distinctly worse than 1/3, which is the best strategy of the ones we tested. I'm a little surprised at Piccione & Rubi... (read more)

Wei, the solution that makes immediate sense is the one proposed by Uzi Segal in Economic Letters, Vol 67, 2000 titled "Don't Fool Yourself to Believe You Won't Fool Yourself Again".

You find yourself at an intersection, and you have no idea whether it is X or Y. You believe that you are at intersection X with probability α. Denote by q the probability of continuing to go straight rather than taking a left. What lottery do you face? Well, if you are at Y, then EXITING will yield a payoff of 4, and CONTinuing will yield a payoff of 1. If you a... (read more)

I wonder what people here think about the resolution proposed by Schwarz (2014). His analysis is that the divergence from the optimal policy also goes away if one combines EDT with the halfer position a.k.a. the self-sampling assumption, which, as shown by Briggs (2010), appears to be the right anthropic view to combine with EDT, anyway.

It's kind of an old thread, but I know people browse the recently posted list and I have a good enough understanding of what exactly the decision theorists are doing wrong that I can explain it in plain English.

First of all, alpha can only consistently be one number: 1/(1+p). And once you substitute that into α[p2+4(1-p)p] + (1-α)[p+4(1-p)], you get a peculiar quantity: (2/1+p) * [p2 + 4(-1p)p]. Where does the 2/1+p come from? Well, every time you go through the first node, you add up the expected result from the first node and the second node, and you als... (read more)

[deleted] had an error

`import java.util.*;`

public class absentMindedDriver {

static Random generator = new Random(); static int trials = 100000; static double[] utilities;

static int utility(double x) { if(generator.nextDouble() < x) { //the driver guesses return 0; } if(generator.nextDouble() < x) { //the driver guesses return 4; } return 1; //the driver missed both exits }

static double averageUtility(double x) { int sum = 0; for(int i = 0; i < trials; i++) { sum += utility(x); //iteratively generates the sum to ... (read more)

If anybody wants to try simulating this problem, here's some java code:

`import java.util.*;

public class absentMindedDriver {

static Random generator = new Random(); static int trials = 100000; static double[] utilities;

static int utility(double x) { if(generator.nextDouble() < x) { //the driver guesses return 0; } if(generator.nextDouble() < x) { //the driver guesses return 4; } return 1; //the driver missed both exits }

static double averageUtility(double x) { int sum = 0; for(int i = 0; i < tria... (read more)

I think that's twice as much as it should be, since neither your decision at X, nor at Y, contribute entirely to your utility result.

Isn't the claim in 6 (that there is a planning-optimal choice, but no action-optimal choice) inconsistent with 4 (a choice that is planning optimal is also action optimal)?

From the paper:

... (read more)