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From: Hal Finney <hal.domain.name.hidden>

Date: Wed, 17 Jul 2002 18:49:04 -0700

Wei wrote:

*> Here's a simplified thought experiment that illustrates the issue. Two
*

*> copies of the subject S, A and B, are asked to choose option 1 or option
*

*> 2. If A chooses 1, S wins a TV (TV), otherise S wins a worse TV (TV2). If
*

*> B chooses 1, S wins a stereo, otherwise S wins TV. S prefers TV to TV2 to
*

*> stereo, but would rather have a TV and a stereo than two TVs. The copies
*

*> have to choose without knowing whether they are A or B.
*

OK, I understand now that the utilities below are the utilities for A

and B when S gets the various items. So U(TV) is the utility for A for

S to get a TV, which is the same as the utility for B since they are

identical copies.

*> According to my incorrect analysis, SSA would imply that you choose option
*

*> 2, because that gives you .5*U(TV2) + .5*U(TV) > .5*U(TV) + .5*U(stereo)
*

*> since U(TV2) > U(stereo). I argued that you should consider yourself A and
*

*> B simultaneously so you could rationally choose option 2, because
*

*> U({TV,stereo}) > U({TV2, TV}).
*

Yes, that makes sense.

*> However taking both SSA and game theory
*

*> into account implies that option 2 is rational. Furthermore, my earlier
*

*> suggestion leads to unintuitive results in general, when the two players
*

*> do not share the same utility function.
*

I know you meant to write that game theory implies that option 2 is

irrational.

*> The game theoretic analysis goes like this. There are two possible
*

*> outcomes with pure strategies (I'll ignore mixed strategies for now).
*

*> Either A and B both choose 1, or they both choose 2. The first one is a
*

*> Nash equilibrium, the second may or may not be. To understand what this
*

*> means, suppose you are one of the players in this game (either A or B but
*

*> you don't know which) and you expect the other player to choose option 1.
*

*> Then your expected utility if you choose option 1 is .5*U({TV,stereo}) +
*

*> .5*U({TV,stereo}). If you choose option 2, the expected utility is
*

*> .5*U({TV2,stereo}) + .5*({TV,TV}) which is strictly less. So you have no
*

*> reason not to choose option 1 if you expect the other player to choose
*

*> option 1. Whether or not the second possible outcome is also a Nash
*

*> equilibrium depends on whether U({TV2,TV}) > .5*U({TV2,stereo}) +
*

*> .5*({TV,TV}). But even if it is, the players can just coordinate ahead of
*

*> time (or implicitly) to choose option 1 and obtain the better equilibrium.
*

If option 2 is also a Nash equilibrium, that is better than option 1,

right? This is why option 2 was preferred under the first analysis.

However I see that under this reasoning there are utility assignments

which make option 1 be a Nash equilibrium while option 2 is not, hence

option 1 would be preferred in those cases, despite the earlier reasoning

which would choose option 2.

I have a problem with this application of game theory to a situation where

A and B both know that they are going to choose the same thing, which I

believe is the case here. Let me make this more specific by assuming that

A and B (and S) are deterministic computational systems. Their needs

for randomness are met by an internal pseudo random number generator.

When S is duplicated to form A and B, the PRNG state is duplicated as

well, so that A and B are running exactly the same deterministic program.

This is the situation which most sharply appeals to the intuition that A

and B should be thought of as "the same person". They are two instances

of the same deterministic calculation, with exactly the same steps being

executed for both.

Under these circumstances, I don't see how considerations of Nash

equilibria can arise. These require implicitly assuming that the other

side may choose a different value than yourself. But with the setup I

give, it is physically impossible for that to happen. The other player

has no more freedom to behave differently than does an image in a mirror.

Likewise with the amnesiac prisoner's dilemma, if the amnesia is provided

in the manner I have described, so that both parties are running exactly

the same program and both know that they are doing so, it seems perfectly

reasonable to choose to cooperate. There are actually fewer degrees

of freedom than the game matrix implies; only two possible outcomes,

rather than four. And the best of the two possible outcomes is when

both parties cooperate rather than defect.

This approach suggests a question with regard to the causal interpretation

of Newcomb's paradox. First, as something of a digression, suppose

it turns out that the experimenter's eerie accuracy in the Newcomb

setup is because he has a time machine. After the subject's decision

is made to choose one or two boxes, the experimenter goes back in time

and fills the boxes appropriately. In this case, it seems to me that

the causalist may decide that taking one box is the preferred outcome,

because his choice does *cause* the filling of the boxes. The effect

takes place earlier in time, but given that there is a time machine in

the picture, we have to accept reversed causality. OTOH the causalist may

reject this reasoning, arguing along his usual lines that the boxes have

already been filled, and taking two has to give him more than taking one.

I don't know which conclusion he would choose.

But more relevantly, suppose that the experimenter's secret is as

follows. Let the subject be a deterministic computational system as in

the APD and other examples above. What the experimenter does is to run

the computation forward until it makes a choice. Then he rewinds the

computation to the state it was in at the beginning, and fills the boxes.

Now he runs the computation forward again, where it will make the same

decision (being deterministic) and so the "prediction" is always correct.

Of course, this description is not much different from standard variants

where the experimenter is an alien with a perfect grasp of human

psychology, or God, able to predict with perfection what people will do.

But by making it concrete in terms of deterministic computations, it

allows for a different view from the causal perspective.

Specifically, when the subject is asked to make his decision, he knows

that he will be put into this state twice; once when the experimenter

was running him to find out what he'd do, and again when the actual

choice was made. From the point of view of shared minds, the subject

must view himself as being in a superposition of these two states.

The point is that in one of those two states, his decision does in

fact have a causal effect on the outcome. It is the direct effect of

his decision that lets the experimenter fill the boxes. So from his

subjective perspective, where he doesn't know if this is the first or

second run, he can at least figure that there is a 50% chance that his

decision has a causal effect on the outcome. It seems to me that this

might be enough to justify choosing one box even from a causal analysis.

Hal Finney

Received on Wed Jul 17 2002 - 18:49:35 PDT

Date: Wed, 17 Jul 2002 18:49:04 -0700

Wei wrote:

OK, I understand now that the utilities below are the utilities for A

and B when S gets the various items. So U(TV) is the utility for A for

S to get a TV, which is the same as the utility for B since they are

identical copies.

Yes, that makes sense.

I know you meant to write that game theory implies that option 2 is

irrational.

If option 2 is also a Nash equilibrium, that is better than option 1,

right? This is why option 2 was preferred under the first analysis.

However I see that under this reasoning there are utility assignments

which make option 1 be a Nash equilibrium while option 2 is not, hence

option 1 would be preferred in those cases, despite the earlier reasoning

which would choose option 2.

I have a problem with this application of game theory to a situation where

A and B both know that they are going to choose the same thing, which I

believe is the case here. Let me make this more specific by assuming that

A and B (and S) are deterministic computational systems. Their needs

for randomness are met by an internal pseudo random number generator.

When S is duplicated to form A and B, the PRNG state is duplicated as

well, so that A and B are running exactly the same deterministic program.

This is the situation which most sharply appeals to the intuition that A

and B should be thought of as "the same person". They are two instances

of the same deterministic calculation, with exactly the same steps being

executed for both.

Under these circumstances, I don't see how considerations of Nash

equilibria can arise. These require implicitly assuming that the other

side may choose a different value than yourself. But with the setup I

give, it is physically impossible for that to happen. The other player

has no more freedom to behave differently than does an image in a mirror.

Likewise with the amnesiac prisoner's dilemma, if the amnesia is provided

in the manner I have described, so that both parties are running exactly

the same program and both know that they are doing so, it seems perfectly

reasonable to choose to cooperate. There are actually fewer degrees

of freedom than the game matrix implies; only two possible outcomes,

rather than four. And the best of the two possible outcomes is when

both parties cooperate rather than defect.

This approach suggests a question with regard to the causal interpretation

of Newcomb's paradox. First, as something of a digression, suppose

it turns out that the experimenter's eerie accuracy in the Newcomb

setup is because he has a time machine. After the subject's decision

is made to choose one or two boxes, the experimenter goes back in time

and fills the boxes appropriately. In this case, it seems to me that

the causalist may decide that taking one box is the preferred outcome,

because his choice does *cause* the filling of the boxes. The effect

takes place earlier in time, but given that there is a time machine in

the picture, we have to accept reversed causality. OTOH the causalist may

reject this reasoning, arguing along his usual lines that the boxes have

already been filled, and taking two has to give him more than taking one.

I don't know which conclusion he would choose.

But more relevantly, suppose that the experimenter's secret is as

follows. Let the subject be a deterministic computational system as in

the APD and other examples above. What the experimenter does is to run

the computation forward until it makes a choice. Then he rewinds the

computation to the state it was in at the beginning, and fills the boxes.

Now he runs the computation forward again, where it will make the same

decision (being deterministic) and so the "prediction" is always correct.

Of course, this description is not much different from standard variants

where the experimenter is an alien with a perfect grasp of human

psychology, or God, able to predict with perfection what people will do.

But by making it concrete in terms of deterministic computations, it

allows for a different view from the causal perspective.

Specifically, when the subject is asked to make his decision, he knows

that he will be put into this state twice; once when the experimenter

was running him to find out what he'd do, and again when the actual

choice was made. From the point of view of shared minds, the subject

must view himself as being in a superposition of these two states.

The point is that in one of those two states, his decision does in

fact have a causal effect on the outcome. It is the direct effect of

his decision that lets the experimenter fill the boxes. So from his

subjective perspective, where he doesn't know if this is the first or

second run, he can at least figure that there is a 50% chance that his

decision has a causal effect on the outcome. It seems to me that this

might be enough to justify choosing one box even from a causal analysis.

Hal Finney

Received on Wed Jul 17 2002 - 18:49:35 PDT

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