Evidential Decision Theory is flawed, but its flaws are not fully understood. David Lewis (

Evidential Decision Theory is flawed, but its flaws are not fully understood. David Lewis (

We offer a new case which shows that

Both Evidential and Causal Decision theory agree you should maximize expected utility. The difference between them arises from how they calculate expected utility. The standard informal way to cash out this difference is as follows: According to

To illustrate their differences, we begin with the familiar:

The Evidential Decision Theorist tells you to one-box. One-boxing is strong evidence you’ll get $1M, whereas two-boxing is strong evidence you’ll only get $1,000.

The Causal Decision Theorist says you should take both boxes. Either the money is in the opaque box or it isn’t. It’s too late to do anything about that now. And either way, you cause a better result by taking both.

Before diagnosing whether

For simplicity, we’ll formulate _{1} over _{2}, then _{1} and _{2} are elements of distinct outcomes. We measure the desirability of an outcome with a real-valued function

To capture the difference between the two theories, we follow Gallow (

This formulation of

In Newcomb’s problem,

This divergence famously led David Lewis (

Admittedly, one can get oneself into the mood where it seems strange to consider Pr(

While there’s much more to say theoretically, we don’t think that

In N

Structurally,

Whatever the merits or demerits of

Therefore, causal decision theorists cannot charge

This does not mean that

To show that invoking Pr(

Consider:

John obviously shouldn’t pay. However,

If John doesn’t pay, then he’ll believe to degree.9 that the coin landed tails, and he’ll be tortured.

If John pays, then he knows the sequence he observes over the two rounds is (or will be) either

By paying, John is playing the ostrich. He’s merely changing the

C

Note that this is a different sort of case from others where

_{1}, the experimenters tell you they will reveal whether the money is in the opaque box unless you pay them $1. At _{2}, you’ll get to decide whether to one-box or two-box. The predictor is highly reliable both at determining whether you will pay not to know what’s in the opaque box and whether you’ll one-box or two-box at _{2}. If it was predicted that you’d ultimately take only the contents of the opaque box, then the opaque box contains $1,000,000. Otherwise, it contains nothing. What should you do?

Suppose you know at _{1} that you’ll follow _{1} and _{2}. Then you know that if at _{2}, you are certain there’s nothing in the opaque box, you’ll two-box. And you know that if you’re certain there’s a million in the opaque box, you’ll also two-box. So, given that you know the contents of the box, you’ll two-box no matter what at _{2}. However, the predictor is very reliable, so at _{1}, you think that if you decide not to pay the experimenters, it’s highly likely you’ll learn there’s nothing in the opaque box. On the other hand, if you aren’t certain what’s in the opaque box at _{2}, _{1}.

This case _{1} to stop yourself from choosing an act at a different time that you now foresee as sub-optimal. If you had your druthers at _{1}, you’d avoid paying and commit your _{2}-self to one-boxing no matter what. But you don’t have that option. Instead, it’s worth a small fee to avoid letting your later self decide differently from how you’d like.

A further virtue of this case—although inessential for the main point of news-management—is that it does not involve any strange prediction, as in N^{,}

Our reasoning that John would think himself less likely to be tortured conditional on paying the $1 than conditional on not paying the $1 is plausible, but not beyond criticism. John’s situation involves possible memory loss and attendant self-locating uncertainty, just as Adam Elga’s (

The Relevance Limiting Thesis does not merely muddy the waters; it invalidates our reasoning. Given the Relevance Limiting Thesis, John’s credence that he will avoid torture conditional on paying the $1 is no greater than his credence that he will avoid torture conditional on his not paying.

Let’s look at the details of why the Relevance Limiting Thesis invalidates our reasoning. In our case, we have a sequence of states of the world: _{0} is either _{1} and _{2} are either red lights or green lights. We will index _{1}_{2} is the world where the coin lands Heads, a red light blinks first, and a green light blinks second.

The agent has uncertainty both over which world is actual and over which center he occupies. So, we’ll write Pr(_{0} _{1} _{2} _{0} _{1} _{2} and him currently occupying center _{1} in _{1}_{2}) is his probability that he’s seeing the red light flash for the first time in the world where the coin lands tails and the light flashes red both times.

According to the Relevance Limiting Thesis, upon seeing red, the agent only rules out the _{1}_{2}-worlds. It provides him with no _{1}_{2}-world relative to an _{1}_{2}- or _{1}_{2}-world. Put differently: the agent takes seeing red

One way to make this concrete is to appeal to the most common form of the Relevance Limiting Thesis, known as Compartmentalized Conditionalization (CC).

According to CC, Pr(_{0} _{1} _{2} _{0} _{1} _{2} _{0} _{1} _{2}), where #(_{0} _{1} _{2}) is the number of times the agent has total evidence _{0} _{1} _{2}. For instance, if ‘red’ refers to the evidence the agent has when he has observed a red light, #(red,_{1}_{2}) = 2.

To see that paying is sub-optimal, we need only calculate John’s subjective probabilities for being tortured (equivalently, for the coin landing tails) conditional on paying or not paying given that he observes red.

First, consider the policy of not paying, which we abbreviate p̄. John’s subjective probability here is:

The second line follows given the Relevance Limiting Thesis in general (and from CC in particular). Note that

Furthermore, we can verify

Putting this all together, we have:

To calculate the conditional probability of torture given John pays upon seeing red, we use the same derivation to see that:

So, if John follows both the Relevance Limiting Thesis and

Our analysis of T

The Relevance Limiting Thesis matters for our initial statement of T

Michael Titelbaum’s (_{1} will be bright and the light he sees at _{2} will be dim, and if the coin lands Tails then the light he sees at _{1} will be dim and the light he sees at _{2} will be bright. Since the brightness of the light is guaranteed to vary across times, even cases in which John sees two red lights or two green lights will not contain duplicate experiences, and thus the Relevance Limiting Thesis will not apply. Whatever sort of light John sees, he can rule out worlds in which he never sees that sort of light and renormalize his credences in the worlds in which he does see that sort of light.

Most problems in the epistemology of self-locating belief are not so easily avoided. As we mentioned, the main controversies involve how self-locating evidence affects credences in uncentered propositions. And the crux of the controversies is how confirmation works between worlds that contain different numbers of agents (or different quantities of experience for some agent)._{1} or _{2}, this self-locating uncertainty is entirely pedestrian—like not being sure exactly what time it is under ordinary circumstances. All major views regarding the epistemology of self-locating belief will validate the following calculations.

To see why the technicolor trick works, we’ll assume without loss of generality that John sees a dim red light, which we abbreviate dr.

We’ll write the results of the first coin toss (which determines whether John gets tortured) as either _{1} or _{1} and the second coin toss as _{2} or _{2} and use upper and lower case letters to denote bright or dim lights, respectively. So _{1}_{2}_{1} _{2} denotes the fact that both coins landed heads, the first light was bright red, and the second light was dim green.

Suppose John will pay upon seeing a red light (dim or not). Then:

The second equality follows by the definition of conditional probability and the fact that observing a dim red light is guaranteed in each of the worlds considered.

Next we calculate:

So,

On the other hand, if John doesn’t pay, a similar calculation reveals that:

A tedious calculation shows that

Given that the cost of payment is trivial, John prefers paying to not paying if he follows

(The astute reader may notice that John has more possible options now, such as paying if the red light is dim but not when it’s bright or vice versa. We won’t go through the calculations here, but John will prefer paying upon seeing

A traditional motivation for

The causal decision theorist can of course retort that the evidentialist is looking at the wrong reference classes. Of people who walk into a room with only a thousand dollars, causalists do better. And of people who walk into a room with a million and a thousand dollars, causalists also do better. From the

In this case, though, there’s no good sense in which

In N

Consider an analogous situation: You suffer from infrequent but very painful migraine headaches. There’s a biotech company that can predict when you’ll get migraines, and it notifies you about upcoming migraines the day before they happen. But if you pay them extra, they’ll also randomly tell you that you’re going to get a migraine even when you won’t. That way the news value of being told that you’re going to get a migraine won’t be as bad. It’s obviously irrational to pay to get bad news more often in order to make each instance of bad news less bad. By paying you don’t get to have any fewer headaches, so it’s not worth anything. And indeed,

It is a common view that the correct decision theory mandates the maximization of expected utility.

Several such arguments have been attempted regarding

Arntzenius (

Wells (

A further advantage of T

The most prominent problem cases for

For a more precise characterization which differentiates causation from causal dependence, see Hedden (

Some formulations of

By ‘upstream’, we mean not downstream.

For defenses of one-boxing, see Spohn (

See Levinstein and Soares (

If the predictor runs a simulation of your decision procedure, then the simulation still would have likely output

There are actually many different versions of

Of course, one could criticize

Cohesive Decision Theory (

As Ahmed (

Note that a ratifiability requirement would plausibly alter

See Arntzenius (

This case involves the possibility of memory loss, which some consider to be a rational failing. We don’t share this view, but those who do may consider a variant of the case in which John has a twin, and both twins are sure that they will make the same choices. In this variant, the relevant issues are reproduced without the possibility of memory loss.

Soares and Fallenstein (

One could revise the procedure, replacing the single Green light flash with a sequence of a Green light, a Green light, and a Red light and replacing the single Red light flash with a sequence of a Green light, a Red light, and a Red light. It’s natural to think that seeing a Green light is evidence that the coin landed Heads and that seeing a Red light is evidence that the coin landed Tails. But since it’s certain that John will see at least one Green light and at least one Red light, according to the Relevance Limiting Thesis the flashes give him no evidence at all. This peculiar consequence is often taken as an argument against the Relevance Limiting Thesis. See also Weintraub (

In the framework of time-slice epistemology these amount to the same thing. See Hedden (

For discussions of how to update in the face of centered evidence, see Bostrom (

See Lewis (

To see why, note that this is essentially the same question as the proponent of Compartmentalized Conditionalization asks: those who see red at least once and pay are tortured just as often as those who see red at least once and don’t pay.

Ahmed and Price (

(1) The average return of being a non-payer exceeds that of being a payer.

(2) Everyone can see that (1) is true.

(3) Therefore not paying foreseeably does better than paying.

(4) Therefore

T

For recent alternatives to expected utility theory, see Buchak (

For more on this point, see Horgan (

See Ahmed and Price (

For more on this point see Ahmed (

Thanks to John Hawthorne, Vince Conitzer, and an audience at the Formal Rationality Forum at Northeastern University. Special thanks to Caspar Oesterheld who provided insightful comments and pushed us on the Relevance Limiting Thesis. Ben Levinstein’s research was partly supported by Mellon New Directions grant 1905-06835.