Posts on: Decision Theory
Assume that for any proposition A there is a proposition \(\Box A\) saying that A ought to be the
case. One can imagine an agent – call him Frederic – whose only basic
desire is that whatever ought to be the case is the case. As a result,
he desires any proposition A in proportion to his belief that it ought
to be the case:
\[\begin{equation*}
(1)\qquad V(A) = Cr(\Box A).
\end{equation*}
\]
Let w be a maximally specific proposition. Such a "world" settles all
descriptive and all normative matters. In particular, w entails either
\(\Box w\) or \(\neg \Box w\). Suppose w entails \(\Box w\). Does Frederick desire to live in
such a world? Yes. On the assumption that w is actual, the entire world
is as it ought to be. That's what Frederick wants. So he desires w.
Standard decision theory studies one-shot decisions, where an agent faces a single choice. Real decision problems, one might think, are more complex. To find the way out of a maze, or to win a game of chess, the agent needs to make a series of choices, each dependent on the others. Dynamic decision theory (aka sequential decision theory) studies such problems.
There are two ways to model a dynamic decision problem. On one approach, the agent realizes some utility at each stage of the problem. Think of the chess example. A chess player may get a large amount of utility at the point when she wins the game, but she plausibly also prefers some plays to others, even if they both lead to victory. Perhaps she enjoys a novel situation in move 23, or having surprised her opponent in move 38. We can model this by assuming that the agent receives some utility for each stage of the game. The total utility of a play is the sum of the utilities of its stages.
Decision theory textbooks often distinguish between decision-making under risk and decision-making under uncertainty or ignorance. The former is supposed to arise in situations where the agent can assign probabilities to the relevant states, the second in situations where they can't.
I've always found this puzzling. Why would a decision maker be unable to assign probabilities (even vague or indeterminate ones) to the states? I don't think there are any such situations.
I haven't looked at the history of this distinction, but I suspect it comes from von Neumann, who (I suspect) had no concept of subjective probability. If the only relevant probabilities are objective, then of course it may happen that an agent can't make their choice depend on the probability of the states because these probabilities may not be known.
Let's return to my recent explorations into the formal structure of reasons. One important approach that I haven't talked about yet is that of Dietrich and List, described in Dietrich and List (2013a), Dietrich and List (2013b), and Dietrich and List (2016).
Gallow (2023) spells out an interventionist theory of counterfactuals that promises to preserve two apparently incompatible intuitions.
Suppose the laws of nature are deterministic. What would have happened if you had chosen some act that you didn't actually choose? The two apparently incompatible intuitions are:
(A1) Had you chosen differently, no law of nature would have been violated.
(A2) Had you chosen differently, the initial conditions of the universe would not have been changed.
Rejecting one of these intuitions is widely thought to spell trouble for Causal Decision Theory. Gallow argues that they can both be respected. I'll explain how. Then I'll explain why I'm not convinced.
When I recently taught Newcomb's Problem in an undergraduate class, opinions were – of course – divided. Some students were one-boxers, some were two-boxers. But few of the one-boxers were EDTers. I hadn't realised this in earlier years. Many of them agreed, after some back and forth, that their reasoning also supports one-boxing in a variation of Newcomb's Problem in which both boxes are transparent. In this version of the case, EDT says that you should two-box.
The argument students gave in support of one-boxing is that committing to one-boxing would make it likely that a million dollars is put into the opaque box.
This line of thought is most convincing if we assume that you know in advance that you will face Newcomb's Problem, before the prediction is made. It is uncontroversial that if you can commit yourself to one-boxing at this point, then you should do it.
By "committing", I mean what Arntzenius, Elga, and Hawthorne (2004) call "binding". By committing yourself to one-box, you would effectively turn your future self into an automaton that is sure to one-box. Your future self would no longer make a decision, based on their information and goals at the time. They would simply execute your plan.
In a comment on an old blog post, a person called "D" brought up a nice puzzle for Causal Decision Theory. Here's (my version of) the scenario.
You have just taken over as one of three pilots on a spaceship that is on its way to Betelgeuse. The spaceship's flight operations are largely automatised. The only input needed from the pilots is the destination. At present, the only available destinations are Betelgeuse and a service station on a nearby moon. (Other destinations could not be safely reached with current fuel levels.)
Why should you maximize expected utility? A well-known answer – discussed, for example, in McClennen (1990), Cubitt (1996), and Gustafsson (2022) – goes as follows.
There are many alleged counterexamples to expected utility theory: Allais's Paradox, Ellsberg's Paradox, Sen (1993)'s polite agent who prefers the second-largest slice of cake, Machina (1989)'s mother who prefers fairness when giving a treat to her children, and so on. In all these cases, the preferences of seemingly reasonable people appear not to rank the options by their expected utility.
Those who make these claims generally assume that utility is a function of material goods. In Allais's Paradox, for example, the possible "outcomes" (of which utility is a function) are assumed to be amounts of money. As has often been pointed out, the apparent violations of expected utility theory all go away if the outcomes are individuated more finely – if, for example, we distinguish between an outcome of getting $1000 as the result of a risky gamble and an outcome of getting a sure $1000. See, for example, Weirich (1986), or Dreier (1996).
When something is good, or desirable, or a reason, then this is usually because it has some good (desirable, etc.) features. The thing may also have bad features, but if the thing is good then the good features outweigh the bad features. How does this weighing work? I'd like to say that the total goodness of a thing is always the sum of the goodness of its features. This "additive" view seems to be unpopular in both ethics and economics. I'll try to defend it.
I first need to state the view more precisely.
To begin, I assume that there are ultimate bearers of value. If we're talking about personal desire, this means that there are some things an agent desires "intrinsically" or "non-derivatively". Being free from pain might be a common example. If you desire to be free from pain then this is typically not because you really desire something else, and you think being free from pain is either a means to the other thing or evidence for the other thing. You simply desire being free from pain, and that's the end of the story.
In Jordan Howard Sobel's papers on decision theory, he generally defines the (causal) expected utility of an act in terms of a special conditional that he calls "causal" or "practical". Concretely, he suggests that
\[
(1)\quad EU(A) = \sum_{w} Cr(A\; \Box\!\!\to w)V(w),
\]
where 'A □→ B' is the special conditional that is true iff either (i) B is the case and would remain the case if A were the case, or (ii) B is not the case but would be the case as a causal consequence of A if A were the case (see e.g. Sobel (1986), pp.152f., or Sobel (1989), pp.175f.).
Good
(1967) famously "proved" that the expected utility of an informed
decision is always at least as great as the expected utility of an
uninformed decision. The conclusion is clearly false. Let's have a look
at the proof and its presuppositions.
Suppose you can either perform one of the acts
A1…An now, or learn the answer to some question E
and afterwards perform one of A1…An. Good argues
that the second option is always at least as good as the first. The
supposed proof goes as follows.
Here is a case where a plan maximises expected utility, you are sure that you are going to follow the plan, and yet the plan tells you to do things that don't maximise expected utility.
Middle Knowledge. In front of you are two doors. If you go through the left door, you come into a room with a single transparent box containing $7. If you go through the right door, you come into a room with two opaque boxes, one black, one white. Your first choice is which door to take. Then you have to choose exactly one box from the room in which you find yourself. A psychologist has figured out which box you would take if found yourself in the room with the two boxes. She has put $10 into the box she thinks you would take, and $0 into the other.
Two recent papers – Oesterheld and Conitzer (2021) and Gallow (2021) – suggest that CDT gives problematic recommendations in certain sequential decision situations.
In my recent posts on decision theory, I've assumed that friends of CDT should accept a "ratifiability" condition according to which an act is rationally permissible only if it maximises expected utility conditional on being chosen.
Sometimes no act meets this condition. In that case, I've assumed that one should be undecided. More specifically, I've assumed that one should be in an "stable" state of indecision in which no (pure) option is preferable to one's present state of indecision. Unfortunately, there are decision problems in which no act is ratifiable and no state of indecision is stable. I'm not sure what to say about such cases. And I wonder if whatever we should say about them also motivates relaxing the ratifiability condition for certain cases in which there are ratifiable options.
The eighth and final chapter of Evidence, Decision and Causality asks whether the actions over which we deliberate should be evidentially independent of the past. It also presents a final alleged counterexample to CDT.
A few quick comments on the first topic.
It is often assumed that there can be evidential connections between what acts we will choose and what happened in the past. In Newcomb's Problem, for example, you can be confident that the predictor foresaw that you'd one-box if you one-box, and that she foresaw that you'd two-box if you two-box. Some philosophers, however, have suggested that deliberating agents should regard their acts as evidentially independent of the past. If they are right then even EDT recommends two-boxing in Newcomb's Problem.
One might intuit that any rationally choosable plan should be rationally implementable. In the previous post, I discussed a scenario in which some forms of CDT violate that principle. In this post, I have some more thoughts on how this can happen. I also consider some nearby principles and look at the conditions under which they might hold.
Throughout this post I'll assume that we are dealing with ideally rational agents with stable basic desires. We're interested in the attitudes such agents should take towards their options in simple, finite sequential choice situations where no relevant information about the world arrives in between the choice points.
Pages 201–211 and 226–233 of Evidence, Decision and Causality present two great puzzles showing that CDT appears to invalidate some attractive principles of dynamic rationality.
First, some context. The simplest argument for two-boxing in Newcomb's Problem is that doing so is guaranteed to get you $1000 more than what one-boxing would get you. The general principle behind this argument might be expressed as follows:
Could-Have-Done-Better (CDB): You should not choose an act if you know that it would make you worse off than some identifiable alternative.
Why should you take both boxes in Newcomb's Problem? The simplest argument is that you are then guaranteed to get $1000 more than what you would get if you took one box. A more subtle argument is that there is information – about the content of the opaque box – of which you know that if you had that information, then you would rationally prefer to take both boxes. Let's have a closer look at this second argument, and at what Arif says about it in chapter 7 of Evidence, Decision, and Causality.
The argument is sometimes presented in terms of an imaginary friend. Imagine you have a friend who has inspected the content of the opaque box. No matter what the friend sees in the box, she would advise you to two-box. You should do what your better-informed friend advises you to do. In the original Newcomb scenario, you don't have such a friend. But you don't need one, for you already know what she would say.
Chapter 7 of Evidence, Decision and Causality looks at arguments for one-boxing or two-boxing in Newcomb's Problem. It's a long and rich chapter. I'll take it in two or three chunks. In this post, I will look at the main argument for one-boxing – the only argument Arif discusses at any length.
The argument is that one-boxing has a foreseeably better return than two-boxing. If you one-box, you can expect to get $1,000,000. If you two-box, you can expect $1000. In repeated iterations of Newcomb's Problem, most one-boxers end up rich and most two-boxers (comparatively) poor.
Chapter 6 of Evidence, Decision and Causality presents another alleged counterexample to CDT, involving a bet on the measurement of entangled particles.
The setup is Bohm's version of the Einstein, Podolsky, Rosen experiment, as described in Mermin (1981) (see esp. pp.407f.).
We have prepared a "source" S that, when activated, emits two entangled spin 1/2 particles, travelling towards causally isolated detectors A and B. The detectors contain Stern-Gerlach magnets whose orientation is controlled by a switch with three settings (1, 2, 3). When the switches on the two detectors are on the same setting, the magnets have the same orientation. Detector A flashes 'y' if the measured spin is along the magnetic field and 'n' otherwise. Detector B uses the opposite convention, flashing 'n' if the measured spin is along the magnetic field.
Chapter 5 of Evidence, Decision and Causality presents a powerful challenge to CDT (drawing on Ahmed (2013) and Ahmed (2014)).
Imagine you have strong evidence that a certain deterministic system S is the true system of laws in our universe. You also believe that questions about what you should do are meaningful even in a deterministic world. Now consider the following two decision problems.
Chapter 4 of Evidence, Decision and Causality considers whether there are any "realistic" Newcomb Problems – and in particular, whether there are any such cases in which EDT gives obviously wrong advice.
Arif goes through some putative examples and rejects most of them. The only realistic Newcomb Problems he admits are versions of the Prisoners' Dilemma (as suggested in Lewis (1979)). Here EDT recommends cooperation while CDT recommends defection. Neither answer is obviously wrong.
Chapter 3 of Evidence, Decision and Causality is called "Causalist objections to CDT". It addresses arguments suggesting that while there is an important connection between causation and rational choice, that connection is not adequately spelled out by CDT.
Arif discusses two such arguments. One is due to Hugh Mellor, who rejects the very idea of evaluating choices by the lights of the agent's beliefs and desires. I'll skip over this part because I agree with Arif's response.
The other argument is more important, because it touches on an easily overlooked connection between rational choice and rational credence.
Consider the "Psycho Button" case from Egan (2007).
How odd. I'm in the office. I'm not terribly exhausted. I have some time to read and think and write. Where do I start?
Here's a book that I've long wanted to read carefully, but never got around to: Arif Ahmed's Evidence, Decision and Causality (Ahmed (2014)). I'll work my way through it, and post my reactions. This first post covers the preface, the introduction, and the first two chapters.
The book is an extended defence of Evidential Decision Theory. When I read a text with whose conclusion I disagree, I often find that the discussion already starts off on the wrong foot, with dubious presuppositions about the topic and how to approach it. Not so here. I'm largely on board with how Arif frames the disagreement between Evidential Decision Theory (EDT) and Causal Decision Theory (CDT). I like his broader philosophical outlook – his positivism, his distrust of metaphysics, his conviction that decision-makers should see themselves as part of the natural world. It should to be interesting to see where we'll end up disagreeing.
1. Suppose you have strong evidence that L are the true laws of nature, where L is
a system of deterministic laws. You also have strong evidence that the universe
started in the exact microstate P. Your have a choice of either affirming or
denying the conjunction of L and P. You want to speak truly. What should you do?
Intuitively, you should affirm. But what would happen if you denied?
Since L is deterministic, L & P either logically entails that you affirm, or it
logically entails that you don't affirm. Let's consider both possibilities.
My "Objects
of Choice" paper has now appeared in Mind.
The paper asks how we should understand an agent's decision-theoretic options.
That is, what are the things whose expected utility we are supposed to maximize?
I think the question is a lot harder than often assumed. For example, I argue
that it won't do to say that the options are certain "willings" or "intentions",
as some authors have suggested.
Suppose we want our decision theory to not impose strong constraints on
people's ultimate desires. You may value personal wealth, or you may value being
benevolent and wise. You may value being practically rational: you may value
maximizing expected utility. Or you may value not maximizing expected
utility.
This last possibility causes trouble.
If not maximizing expected utility is your only basic desire, and you
have perfect and certain information about your desires, then arguably (although
the argument isn't trivial) every choice in every decision situation you can
face has equal expected utility; so you are bound to maximize expected utility
no matter what. Your desire can't be fulfilled.
I'm generally happy with Causal Decision Theory. I think two-boxing is
clearly the right answer in Newcomb's problem, and I'm not impressed by any of
the alleged counterexamples to Causal Decision Theory that have been put
forward. But there's one thing I worry about. It is what exactly the theory
should say: how it should be spelled out.
Suppose you face a choice between two acts A and B. Loosely speaking, to
evaluate these options, we need to check whether the A-worlds are on average
better than the B-worlds, where the "average" is weighted by your credence on
the subjunctive supposition that you make the relevant choice. Even more
loosely, we want to know how good the world would be if you were to choose A,
and how good it would be if you were to choose B. So we need to know what else
would be the case if you were to choose, say, A.
A common assumption in economics is that utilities are reducible to choice
dispositions. The story goes something like this. Suppose we know what an agent
would choose if she were asked to pick one from a range of goods. If the agent
is disposed to choose X, and Y was an available alternative, we say that the
agent prefers X over Y. One can show that if the agent's choice
dispositions satisfy certain formal constraints, then they are "representable"
by a utility function in the sense that whenever the agent prefers X over Y,
the function assigns greater value to X than to Y. This utility function is
assumed to be the agent's true utility function, telling us how much the agent
values the relevant goods.
I recently refereed Eliezer Yudkowsky and Nate Soares's "Functional Decision Theory" for a philosophy journal. My recommendation was to accept resubmission with major revisions, but since the article had already undergone a previous round of revisions and still had serious problems, the editors (understandably) decided to reject it. I normally don't publish my referee reports, but this time I'll make an exception because the authors are well-known figures from outside academia, and I want to explain why their account has a hard time gaining traction in academic philosophy. I also want to explain why I think their account is wrong, which is a separate point.
I've been reading about objective consequentialism lately. It's
interesting how pervasive and natural the use of counterfactuals is in
this context: what an agent ought to do, people say, is whichever
available act would lead to the best outcome (if it were
chosen). Nobody thinks that an agent ought to choose whichever act
will lead to the best outcome (if it is chosen). The
reason is clear: the indicative conditional is information-relative,
but the 'ought' of objective consequentialism is not supposed to be
information-relative. (That's the point of objective
consequentialism.) The 'ought' of objective consequentialism is
supposed to take into account all facts, known and unknown. But while
it makes perfect sense to ask what would happen under condition
C given the totality of facts @, even if @ does not imply C, it
arguably makes no sense to ask what will happen under condition
C given @, if @ does not imply C.
Why maximize expected utility? One supporting consideration that is occasionally mentioned (although rarely spelled out or properly discussed) is that maximizing expected utility tends to produce desirable results in the long run. More specifically, the claim is something like this:
(*) If you always maximize expected utility, then over time you're likely to maximize actual utility.
Since "utility" is (by definition) something you'd rather have more of than less, (*) does look like a decent consideration in favour of maximizing expected utility. But is (*) true?
There's something odd about how people usually discuss iterated
prisoner dilemmas (and other such games).
Let's say you and I each have two options: "cooperate" and
"defect". If we both cooperate, we get $10 each; if we both defect, we
get $5 each; if only one of us cooperates, the cooperator gets $0 and
the defector $15.
This game might be called a monetary prisoner dilemma, because
it has the structure of a prisoner dilemma if utility is measured by
monetary payoff. But that's not how utility is usually
understood.
Suppose you prefer $105 today to $100 tomorrow. You also prefer $105 in 11 days to $100 in 10 days. During the next 10
days, your basic preferences don't change, so that at the end of that
period (on day 10), you still prefer $105 now (on day 10) to $100 the
next day. Your future self then disagrees with your earlier self about
whether it's better to get $105 on day 10 or $100 on day 11.
In economics jargon, your preferences are called time
inconsistent. Time inconsistency is supposed to be a failure of
ideal rationality.
The decision-theoretic concept of preference is linked to the concepts
of subjective probability and utility by the expected utility
principle:
(EUP) A rational agent prefers X to Y iff the expected
utility of X exceeds the expected utility of Y.
Economists usually take preference to be the more basic concept and
interpret the EUP as an implicit definition of the agent's utilities
(and sometimes also her probabilities).
In his "Dicing
with Death" (2014), Arif Ahmed presents the following scenario as
a counterexample to causal decision theory (CDT):
You are thinking about going to Aleppo or staying in
Damascus. Death has predicted where you will be and is waiting for
you there. For a small fee, you can delegate your choice to a coin
toss the outcome of which Death can't predict.
Tossing the coin promises to reduce the chance of death from about
1 to 1/2. Nonetheless, CDT seems to suggest that you shouldn't toss
the coin. To illustrate, suppose you are currently completely
undecided and thus give equal credence to Death being in Aleppo and to
Death being in Damascus. Then you're 50 percent confident that if you
were to stay in Damascus, you would survive; similarly for
going to Aleppo. You're also 50 percent confident that you would
survive if you were to toss the coin, but in that case you'd have
to pay the small fee. So it's not worth paying.
In decision theory, the available options are often glossed informally
as the acts the agent can perform, or the propositions she can make
true. But this yields implausible results in cases where an agent has
doubts about what she can do.
For example, assume Bob suspects that the button in front of him
functions as a light switch, as in fact it does. Then Bob can turn
on the light by pressing the button. But if he is not certain that
the button is a light switch, decision theory should consider the
consequences of pressing the button if it has some other function. So
turning on the light by pressing the button should not count as
an option.
In my recent post on Interventionist Decision Theory, I suggested that causal interventionists
should follow Stern and move from a Jeffrey-type definition
of expected utility to a Savage-Lewis-Skyrms type definition. In that case, I also suggested that they could avoid various problems arising from the concept of an intervention by construing the agent's
actions as ordinary events. In conversation,
Reuben Stern convinced me that things are not so easy.
Causal models are a useful tool for reasoning about causal
relations. Meek
and Glymour 1994 suggested that they also provide new resources to
formulate causal decision theory. The suggestion has been endorsed by
Pearl
2009, Hitchcock
2016, and others. I will discuss three problems with this proposal
and suggest that fixing them leads us back to more or less the
decision theory of Lewis
1981 and Skyrms
1982.
Decision theory comes in many flavours. One of the most important
but least discussed divisions concerns the individuation of
outcomes. There are basically two camps. One side -- dominant in
economics, psychology, and social science -- holds that in a
well-defined decision problem, the outcomes are exhausted by a
restricted list of features: in the most extreme version, by the
amount of money the agent receives as the result of the relevant
choice. In less extreme versions, we may also consider the agent's
social status or her overall well-being. But we are not allowed to
consider non-local features of an outcome such as the act that brought
it about, the state under which it was chosen, or the alternative acts
available at the time. This doctrine doesn't have a name. Let's call
it localism (or utility localism).
Decision theory says that faced with a number of options, one
should choose an option that maximizes expected utility. It does not
say that before making one's choice, one should calculate and compare
the expected utility of each option. In fact, if calculations are
costly, decision theory seems to say that one should never calculate
expected utilities.
Informally, the argument goes as follows. Suppose an agent faces a
choice between a number of straight options (going left, going
right, taking an umbrella, etc.), as well as the option of calculating
the expected utility of all straight options and then executing
whichever straight option was found to have greatest expected
utility. Now this option (whichever it is) could also be taken
directly. And if calculations are costly, taking the option directly
has greater expected utility than taking it as a result of the
calculation.
I'm currently teaching a course on decision theory. Today we discussed
chapter 2 of Jim Joyce's Foundations of Causal Decision Theory,
which is excellent. But there's one part I don't really get.
Joyce mentions that Savage identifies acts with functions from states
to outcomes, and that Jeffrey once suggested representing such
functions as conjunctions of material conditionals: for example, if an
act maps S1 to O1 and S2 to O2, the corresponding proposition would be
(S1 → O1) & (S2 → O2). According to Joyce, this
conception of acts "cannot be correct" (p.62). That's the part I don't
really get.
Decision theoretic representation theorems show that one can read
off an agent's probability and utility functions from their
preferences, provided the latter satisfy certain minimal rationality
constraints. More substantive rationality constraints should therefore
translate into further constraints on preference. What do these
constraints look like?
Here are a few steps towards an answer for one particular
constraint: a simple form of the Principal Principle. The Principle
states that if cr is a rational credence function and ch=p is the
hypothesis that p is the chance function, then for any E in the domain
of p,
In The Logic of Decision, Richard Jeffrey pointed out that
the desirability (or "news value") of a proposition can be usefully
understood as a weighted average of the desirability of different ways
in which the proposition can be true, weighted by their respective
probability. That is, if A and B are incompatible propositions,
then
(1) Des(AvB) = Des(A)P(A/AvB) + Des(B)P(B/AvB).
So desirabilities are affected by probabilities. If you prefer A
over B and just found out that conditional on their disjunction, A is
more likely then B, then the desirability of the disjunction goes
up. That seems right.
Let's say that an act A is subjectively better than an
alternative B if A is better in light of the agent's information; A is
objectively better if it is better in light of all the
facts. The distinction is easiest to grasp in a consequentialist
setting. Here an act is objectively better if it brings about more
good -- if it saves more lives, for example. A morally conscientious
agent may not know which of her options would bring about more
good. Her subjective ranking of the options might therefore go by the
expectation of the good: by the probability-weighted average of the
good each act might bring about.
When we face a decision and work out what we should do, we gain information about what we will do. Taking into account this information can in turn affect what we should do. Here's an example.
(I) In front of you are two opaque boxes, one black one white. You can
open one of them and keep whatever is inside. Yesterday, a perfect (or
almost perfect) predictor tried to predict what you would choose. If
she predicted that you'd take the black box, she put a million dollars
in the white box and two dollars in the black box. If she predicted
that you'd take the white box, she put a thousand dollars in the black
box and one dollar in the white box. Which box do you open?
Let's say that at the beginning of your deliberation, you are
completely undecided, giving 50 percent credence to the hypothesis
that you'll end up opening the black box. Standard formulations of
causal decision theory then say that opening the white box has greater
expected payoff: since there's a 50 percent probability that it
contains a million, the expected payoff is 500000.50, which is a lot
more than what you could possibly find in the black box. However, choosing to open the white box would
provide you with highly relevant information: it would reveal
that the predictor has (almost certainly) put only one dollar in the white box and a thousand in the black box. As
a rational decision-maker you should take that information into
account. Many putative "counterexamples" to causal decision
theory, such as those
in Richter
1985 and Egan
2007, are based on this observation.
Lewis, in "Causal Decision Theory" (1981, p.308):
Suppose we have a partition of propositions that distinguish worlds
where the agent acts differently ... Further, he can act at
will so as to make any one of these propositions hold, but he cannot
act at will so as to make any proposition hold that implies but is
not implied by (is properly included in) a proposition in the
partition. ... Then this is the partition of the agent's
alternative options.
That can't be right. Assume I "can act at will so as to make hold"
the proposition P that I raise my hand. Let Q be an arbitrary fact
over which I have no control, say, that Julius Caesar crossed the
Rubicon. Then I can also act at will so as to make P & Q true. (By
raising my hand, I make it true, by not raising it I make it false.)
So, by Lewis's definition, P is not an option, since I can act at will
so as to make a more specific proposition P & Q true (a
proposition that implies but is not implied by P). By the same
reasoning, all my options must entail Q. So they don't form a
partition: they don't cover regions of logical space where Q is
false.
In a large election, an individual vote is almost certain to make
no difference to the outcome. Given that voting is inconvenient and time-consuming,
this raises the question whether rational citizens should bother to
vote.
It obviously depends on the citizen's values. For a completely
selfish person, the answer may well be 'no'. Different election
outcomes typically don't matter too much for an ordinary citizen's
selfish interests; and a miniscule chance of a medium-sized gain does
not offset the cost in time and inconvenience.
Suppose you have a choice between two options, say, raising your
arm and lowering your arm. To evaluate these options, we should
compare their outcomes: what would happen if you raise your arm, what
if you don't? But we don't want to be misled by merely evidential
correlations. Your raising your arm might be evidence that your twin
raised their arm in a similar decision problem yesterday, but since
you have no causal control over other people's past actions, we should
hold them fixed when evaluating your options. Similarly, your choice
might be evidentially relevant to hypotheses about the laws of nature,
but you have no causal control over the laws, so we should hold them
fixed. But now we have a problem. The class of facts outside your
causal control is not closed under logical consequence. On the
contrary, if the laws are deterministic then facts about the distant
past together with the laws logically settle what you will do. We
can't hold fixed both the past and the laws and vary your choice.
In
"Gandalf's
solution to the Newcomb problem" (2013), Ralph Wedgwood
proposes a new form of decision theory, Benchmark Theory, that
is supposed to combine the good parts of Causal and Evidential
Decision Theory.
Like many formulations of Causal Decision Theory, Benchmark Theory
(BT) assumes a privileged partition of
states that are outside the agent's causal control. Like
Evidential Decision Theory, BT only considers the probability of these
states conditional on a given act A. However, what is weighted by the
conditional probabilities P(S_i/A) is not the absolute utility of S_i
& A, but the comparative utility of S_i & A, which is
determined by the difference between the absolute utility U(S_i &
A) and the average absolute utility U(S_i & A') for all options
A'. (This average is the benchmark B(S_i).) So the degree of
choiceworthiness of an act A is given by
There's an exciting new theory in cognitive science. The theory began
as an account of message-passing in the visual cortex, but it quickly
expanded into a unified explanation of perception, action, attention,
learning, homeostasis, and the very possibility of life. In its most
general and ambitious form, the theory was mainly developed by Karl
Friston -- see
e.g. Friston
2006, Friston
and Stephan 2007,
Friston
2009,
Friston
2010,
or the
Wikipedia page on the free-energy principle.
Luc Bovens and Wlodek Rabinowicz (2010
and 2011)
present the following puzzle:
Three people are each given a hat to put on in the
dark. The hats' colours, either black or white, has been decided by
three independent tosses of a fair coin. Then the light goes on and
everyone can see the hats of the two others, but not their own. All of
this is common knowledge in the group.
Let's call the three players X, Y and Z. There are eight possible
distributions of hat colours, each with probability 1/8:
It is well-known that humans don't conform to the model of rational
choice theory, as standardly conceived in economics. For example, the
minimal price at which people are willing to sell a good is often much
higher than the maximal price at which they would previously have been
willing to buy it. According to rational choice theory, the two prices
should coincide, since the outcome of selling the good is the same as
that of not buying it in the first place. What we philosophers call
'decision theory' (the kind of theory you find in Jeffrey's Logic
of Decision or Joyce's Foundations of Causal Decision
Theory) makes no such prediction. It does not assume that the
value of an act in a given state of the world is a simple function of
the agent's wealth after carrying out the act. Among other things, the
value of an act can depend on historical aspects of the relevant
state. A state in which you are giving up a good is not at all
the same as a state in which you aren't buying it in the first place,
and decision theory does not tell you that you must assign equal
value to the two results.
To what extent are the beliefs and desires of rational agents
determined by their actual and counterfactual choices? More precisely,
suppose we are given a preference order that obtains between a
possible act A and a possible act B iff the relevant agent is disposed
to choose A over B. Say that a pair (C,V) of a credence function C and
a utility (desirability) function V fits the preference order
iff, whenever A is preferred over B, then A has higher expected
utility than B by the lights of (C,V). Now, to what extent does a
rational preference order constrain fitting credence-utility
pairs?
Imagine you're a hedonist who doesn't care about other people, nor
about your past or your distant future. All you care about is how much
money you can spend today. Fortunately, you're on a pension that pays
either $100 or $1000 every day, plus an optional bonus. How much you
get is determined as follows. Every morning, a psychologist shows up
to study your brain. Then he puts two boxes in front of you, one
opaque, the other transparent. You can choose to take either both boxes or
only the opaque one. The transparent box contains a $10 bill. The
opaque box contains nothing if the psychologist has predicted that you
will take both boxes; if he has predicted that you will take one box,
it contains $100. The psychologist's predictions are about 99%
accurate. The content of your boxes is your bonus payment. In addition, you get your
ordinary payment, which is either $100 or $1000 depending on how many
boxes you took the previous day: if you took both, you now get $1000,
otherwise $100. The ordinary payment is given to you before the psychologist
studies your brain, so by the time you choose between the two boxes, you already
know how much you received. What do you do?
Professor Procrastinate has to make an important phone call. The
call is long overdue because Procrastinate has been playing Farmville
all week. The problem is that Procrastinate values current pleasure
higher than future pleasure. So when he applies his decision theory,
he finds that it is better to play some more Farmville now and make
the phone call later instead of making the call now: it doesn't matter much
whether the call is delayed by a few more hours, and this way the
immediate future will be much more pleasant.
Here is an attempt at an argument against formulating causal decision theory in
terms of counterfactuals (loosely following up on the discussion in the previous
post). The point seems rather obvious, so it is probably old. Does anyone know?
Suppose you would like to go for a walk, but only if it's not
raining. Unfortunately, it is raining heavily, so you have
almost decided to stay inside. Then you remember Gibbard and
Harper's paper "Counterfactuals and two kinds of expected
utility".
One of the novelties in Richard Jeffrey's "Logic of Decision"
(1965) was to unify the space over which probabilities and values are
defined: both probability and desirability are distributed over the
space of possible worlds, of ways things might be. By contrast, in
earlier theories like that of Savage, probabilities were defined over
states (or events) and utilities over
consequences, which were taken to be distinct kinds of
things. Technically, this difference between Savage and Jeffrey isn't
terribly important as long as anything an agent may care about can be
found in the set of 'consequences'. However, the distinction and the
labeling in Savage's treatment carries a danger to overlook the
complexity of human values. This has, I believe, led to a number of
serious mistakes.
A time traveler offers you a game. You can toss a fair coin. If it lands heads, you win $2; if it lands tails, you lose $1. The time traveler informs you that all fair coins tossed today will land tails. (He knows, because he's seen all the results before traveling back in time.) Do you play?
Suppose you decide to toss. Trusting the time traveler, you can then be confident that you will lose $1. You would not have lost anything if you hadn't tossed, so the alternative option would have been better. It seems that you've made the wrong decision.
Hey there. I've been a bit busy moving house, sitting in the garden, watching the falling leaves, etc. I've also thought some more about the absentminded driver. Here's something odd: on a certain interpretation of this case, we get a an unstable decision problem that remains interestingly unstable even when mixing (randomization) is allowed.
Some background. A decision problem is unstable if a decision to do one thing
inevitably makes another thing preferable. In a classic example, Death, who is very good at predicting people's whereabouts, has predicted where you will be tomorrow and awaits you there. Should you stay where you are (in Damascus) or flee to Aleppo?
A curious aspect of the Sleeping Beauty debate is the role of Dutch Books. At first sight, it looks as if Dutch Book considerations support thirding (see e.g. Hitchcock 2004). However, as Halpern 2006 shows, Beauty can also be Dutch Booked if she is a thirder. Some have argued that these arguments might fail because in Sleeping Beauty type cases, credences and betting odds can come apart (see e.g. Bradley and Leitgeb 2006). I disagree. Instead, I will argue that her vulnerability to Dutch Books doesn't show that Beauty is irrational -- at least not if she is a halfer.
I finally found the decision theory puzzle that I posted recently in a series of papers by Reed Richter from the mid 1980s. I'm not convinced by Richter's treatment though, and I'm still somewhat puzzled.
Here is Richter's version:
Button: You and another person, X, are put in separate rooms where each of you faces a button. If you both push the button within the next 10 minutes, you will (both) receive 10 Euros. If neither of you pushes the button, you (both) lose 1000 Euros. If one of you pushes and the other one doesn't, you (both) get 100 Euros.
What would you do? Most people, I guess, would push the button. After all, if you don't push it, there is a high risk of losing 1000 Euros. For how how can you be certain that X won't do the same? On the other hand, if you push the button, the worst possible outcome is a gain of 10 Euros.
This is probably old, so pointers to the literature are welcome. Consider this game between Column and Row:
| | C1 | C2 |
|---|
| R1 | 0,0 | 2,2 |
|---|
| R2 | 2,2 | 1,1 |
|---|
What should Column and Row do if they know that they are equally rational and can't communicate with one another? The game doesn't have a Nash equilibrium has no unique Nash equilibrium, nor is there a dominant strategy (Thanks Marc!), so perhaps there is no determinate answer.
Searching. Mary is in the park, looking for Fred. She recognizes Fred's friend Ted some distance away on the left. Knowing that Fred is often in the park with Ted, she turns that way.
Waiting. Alexandre is waiting for Veronique in a cafe. He's been waiting for several hours now, and is doubtful that Veronique will ever show up. Nevertheless, he thinks it is worth waiting some more.
Mary and Alexandre are acting rationally here, even though Mary does not know that Fred is to the left, and Alexandre does not know that Veronique will ever show up. Even if it turns out that both were wrong, I wouldn't blame them for their decisions.
Suppose you and I both face a choice between several different
options. Say, we both have to pick a ball out of a bag of 100
balls. We win a prize if we make the same choice. But we have no means
to communicate. Moreover, our only relevant interest is to win the prize, otherwise we are completely indifferent about the options.
If one of the options is somehow salient, say one ball is red and all the others white, most people will choose that one. And
wisely so, as many people following this strategy win the prize,
whereas hardly anyone picking a white ball does. However, is this a
rational decision among perfectly rational agents who know of
each other's rationality and preferences? (I also assume that the
agents know that they make exactly the same judgements about
salience.)
There's something odd about Albert's reasoning:
If that stranger's predictions are true, he probably is a time
traveler. I want him to be a
time traveler. Therefore I should try to make his predictions
come true.
The problem is that by trying to make the predictions come true,
Albert decreases the evidential support their truth lends to the
claim that the stranger is a time traveler.