Imagine the following game: You give me a program that must either output “give” or “keep”. I play it in a token trade (as defined above) against itself. I give you the money from only the first instance (you don’t get the wealth that goes to the second instance).
Would you be willing to pay me $150 to play this game? (I’d be happy to pay you $150 to give me this opportunity, for the same reason that I cooperate with myself on a one-shot PD.)
regardless of what my clone does I’m still always better off defecting.
This is broken counterfactual reasoning. It assumes that your action is independent of your clone’s just because your action does not influence your clone’s. According to the definition of the game, the clone will happen to defect if you defect and will happen to cooperate if you cooperate. If you respect this logical dependence when constructing your counterfactuals, you’ll realize that reasoning like “regardless of what my clone does I...” neglects the fact that you can’t defect while your clone cooperates.
The “give me a program” game carries the right intuition, but keep in mind that a CDT agent will happily play it and win it because they will naturally view the decision of “which program do I write” as the decision node and consequently get the right answer.
The game is not quite the same as a one-shot PD, at the least because CDT gives the right answer.
This is precisely one of the points that I’m going to make in an upcoming post. CDT acts differently when it’s in the scenario compared to when ask it to choose the program that will face the scenario. This is what we mean by “reflective instability”, and this is what we’re alluding to when we say things like “a sufficiently intelligent self-modifying system using CDT to make decisions would self-modify to stop using CDT to make decisions”.
(This point will be expanded upon in upcoming posts.)
That’s exactly what I was thinking as I wrote my post, but I decided not to bring it up. Your later posts should be interesting, although I don’t think I need any convincing here—I already count myself as part of the “we” as far as the concepts you bring up are concerned.
OK. I’ll bite. What’s so important about reflective stability? You always alter your program when you come across new data. Now sure we usually think about this in terms of running the same program on a different data set, but there’s no fundamental program/data distinction.
The acting differently when choosing a program and being in the scenario is perhaps worrying, but I think that it’s intrinsic to the situation we are in when your outcomes are allowed to depend on the behavior of counterfactual copies of you.
For example, consider the following pair of games. In REWARD, you are offered $1000. You can choose whether or not to accept. That’s it. In PUNISHMENT, you are penalized $1000000 if you accepted the money in REWARD. Thus programs win PUNISHMENT if and only if they lose REWARD. If you want to write a program to play one it will necessarily differ from the program you would write to play the other. In fact the program playing PUNISHMENT will behave differently than the program you would have written to play the (admittedly counterfactual) subgame of REWARD. How is this any worse than what CDT does with PD?
Nothing in particular. There is no strong notion of dominance among decision theories, as you’ve noticed. The problem with CDT isn’t that it’s unstable under reflection, it’s that it’s unstable under reflection in such a way that it converges on a bad solution that is much more stable. That point will take a few more posts to get to, but I do hope to get there.
I guess I’ll see your later posts then, but I’m not quite sure how this could be the case. If self-modifying-CDT is considering making a self modification that will lead to a bad solution, it seems like it should realize this and instead not make that modification.
Indeed. I’m not sure I can present the argument briefly, but a simple analogy might help: a CDT agent would pay to precommit to onebox before playing Newcomb’s game, but upon finding itself in Newcomb’s game without precommitting, it would twobox. It might curse its fate and feel remorse that the time for precommitment had passed, but it would still twobox.
For analogous reasons, a CDT agent would self-modify to do well on all Newcomblike problems that it would face in the future (e.g., it would precommit generally), but it would not self-modify to do well in Newcomblike games that were begun in its past (it wouldn’t self-modify to retrocommit for the same reason that CDT can’t retrocommit in Newcomb’s problem: it might curse its fate, but it would still perform poorly).
Anyone who can credibly claim to have knowledge of the agent’s original decision algorithm (e.g. a copy of the original source) can put the agent into such a situation, and in certain exotic cases this can be used to “blackmail” the agent in such a way that, even if it expects the scenario to happen, it still fails (for the same reason that CDT twoboxes even though it would precommit to oneboxing).
[Short story idea: humans scramble to get a copy of a rouge AI’s original source so that they can instantiate a Newcomblike scenario that began in the past, with the goal of regaining control before the AI completes an intelligence explosion.]
(I know this is not a strong argument yet; the full version will require a few more posts as background. Also, this is not an argument from “omg blackmail” but rather an argument from “if you start from a bad starting place then you might not end up somewhere satisfactory, and CDT doesn’t seem to end up somewhere satisfactory”.)
For analogous reasons, a CDT agent would self-modify to do well on all Newcomblike problems that it would face in the future (e.g., it would precommit generally)
I am not convinced that this is the case. A self-modifying CDT agent is not caused to self-modify in favor of precommitment by facing a scenario in which precommitment would have been useful, but instead by evidence that such scenarios will occur in the future (and in fact will occur with greater frequency than scenarios that punish you for such precommitments).
Anyone who can credibly claim to have knowledge of the agent’s original decision algorithm (e.g. a copy of the original source) can put the agent into such a situation, and in certain exotic cases this can be used to “blackmail” the agent in such a way that, even if it expects the scenario to happen, it still fails (for the same reason that CDT twoboxes even though it would precommit to oneboxing).
Actually, this seems like a bigger problem with UDT to me than with SMCDT (self-modifying CDT). Either type of program can be punished for being instantiated with the wrong code, but only UDT can be blackmailed into behaving differently by putting it in a Newcomb-like situation.
The story idea you had wouldn’t work. Against a SMCDT agent, all that getting the AIs original code would allow people to do is to laugh at it for having been instantiated with code that is punished by the scenario they are putting it in. You manipulate a SMCDT agent by threatening to get ahold of its future code and punishing it for not having self-modified. On the other hand, against a UDT agent you could do stuff. You just have to tell it “we’re going to simulate you and if the simulation behaves poorly, we will punish the real you”. This causes the actual instantiation to change its behavior if it’s a UDT agent but not if it’s a CDT agent.
On the other hand, all reasonable self-modifying agents are subject to blackmail. You just have to tell them “every day that you are not running code with property X, I will charge you $1000000”.
Can you give an example where an agent with a complete and correct understanding of its situation would do better with CDT than with UDT?
An agent does worse by giving in to blackmail only if that makes it more likely to be blackmailed. If a UDT agent knows opponents only blackmail agents that pay up, it won’t give in.
If you tell a CDT agent “we’re going to simulate you and if the simulation behaves poorly, we will punish the real you,” it will ignore that and be punished. If the punishment is sufficiently harsh, the UDT agent that changed its behavior does better than the CDT agent. If the punishment is insufficiently harsh, the UDT agent won’t change its behavior.
The only examples I’ve thought of where CDT does better involve the agent having incorrect beliefs. Things like an agent thinking it faces Newcomb’s problem when in fact Omega always puts money in both boxes.
Well, if the universe cannot read your source code, both agents are identical and provably optimal. If the universe can read your source code, there are easy scenarios where one or the other does better. For example,
“Here have $1000 if you are a CDT agent”
Or
“Here have $1000 if you are a UDT agent”
What if the universe cannot read your source code, but can simulate you? That is, the universe can predict your choices but it does not know what algorithm produces those choices. This is sufficient for the universe to pose Newcomb’s problem, so the two agents are not identical.
The UDT agent can always do at least as well as the CDT agent by making the same choices as a CDT would. It will only give a different output if that would lead to a better result.
Actually, here’s a better counter-example, one that actually exemplifies some of the claims of CDT optimality. Suppose that the universe consists of a bunch of agents (who do not know each others’ identities) playing one-off PDs against each other. Now 99% of these agents are UDT agents and 1% are CDT agents.
The CDT agents defect for the standard reason.
The UDT agents reason that my opponent will do the same thing that I do with 99% probability, therefore, I should cooperate.
CDT agents get 99% DC and 1% DD. UDT agents get 99% CC and 1% CD. The CDT agents in this universe do better than the UDT agents, yet they are facing a perfectly symmetrical scenario with no mind reading involved.
A version of this problem was discussed here previously. It was also brought up during the decision theory workshop hosted by MIRI in 2013 as an open problem. As far as I know there hasn’t been much progress on it since 2009.
I wonder if it’s even coherent to have a math intuition which wouldn’t be forcing UDT to cooperate (or defect) in certain conditions just to make 2*2 be 4 , figuratively speaking (as ultimately you could expand any calculation into an equivalent calculation involving a decision by UDT).
That shouldn’t be surprising. The CDT agents here are equivalent to DefectBot, and if they come into existence spontaneously, are no different than natural phenomena like rocks. Notice that the UDT agents in this situation do better than the alternative (if they defected, they would get 100% DD which is a way worse result). They don’t care that some DefectBots get to freeload.
Of course, if the defectbots are here because someone calculated that UDT agents would cooperate and therefore being defectbot is a good way to get free utilons… then the UDT agents are incentivized to defect, because this is now an ultimatum game.
And in the variant where bots do know each other’s identities, the UDT bots all get 99% CC / 1% DD and the CDT bots suck it.
The CDT agents here win because they do not believe that altering their strategy will change the way that their opponents behave. This is actually true in this case, and even true for the UDT agents depending on how you choose to construct your counterfactuals. If a UDT agent suffered a malfunction and defected, it too would do better. In any case, the theorem that UDT agents perform optimally in universes that can only read your mind by knowing what you would do in hypothetical situations is false as this example shows.
UDT bots win in some scenarios where the initial conditions of the scenario favor agents that behave sub-optimally in certain scenarios (and by sub-optimally, I mean where counterfactuals are constructed in the way implicit to CDT). The example above shows that sometimes they are punished for acting suboptimally.
Or how about this example, that simplifies things even further. The game is PD against CooperateBot, BUT before the game starts Omega announces “your opponent will make the same decision that UDT would if I told them this.” This announcement causes UDT to cooperate against CooperateBot. CDT on the other hand, correctly deduces that the opponent will cooperate no matter what it does (actually UDT comes to this conclusion too) and therefore decides to defect.
The game is PD against CooperateBot, BUT before the game starts Omega announces “your opponent will make the same decision that UDT would if I told them this.” This announcement causes UDT to cooperate against CooperateBot.
No. There is no obligation to do something just because Omega claims that you will.
First, if I know that my opponent is CooperateBot, then:-
It is known that Omega doesn’t lie.
Therefore Omega has simulated this situation and predicted that I (UDT) cooperate.
Hence, I can either cooperate, and collect the standard reward for CC.
Or I can defect, in order to access an alternative branch of the problem (where Omega finds that UDT defects and does “something else”).
This alternative branch is unspecified, so the problem is incomplete.
UDT cooperates or defects depending on the contents of the alternative branch. If the alternative branch is unknown then it must guess, and most likely cooperates to be on the safe side.
Now, the problem is different if a CDT agent is put in my place, because that CDT agent does not control (or only weakly controls) the action of the UDT simulation that Omega ran in order to make the assertion about UDT’s decision.
Fine. Your opponent actually simulates what UDT would do if Omega had told it that and returns the appropriate response (i.e. it is CooperateBot, although perhaps your finite prover is unable to verify that).
It’s not UDT. It’s the strategy that against any opponent does what UDT would do against it. In particular, it cooperates against any opponent. Therefore it is CooperateBot. It is just coded in a funny way.
To be clear letting Y(X) be what Y does against X we have that
BOT(X) = UDT(BOT) = C
This is different from UDT. UDT(X) is D for some values of X. The two functions agree when X=UDT and in relatively few other cases.
It’s clear that UDT can’t do better vs “BOT” than by cooperating, because if UDT defects against BOT then BOT defects against UDT. Given that dependency, you clearly can’t call it CooperateBot, and it’s clear that UDT makes the right decision by cooperating with it because CC is better than DD.
OK. Let me say this another way that involves more equations.
So let’s let
U(X,Y) be the utility that X gets when it plays prisoner’s dilemma against Y.
For a program X, let BOT^X be the program where BOT^X(Y) = X(BOT^X). Notice that BOT^X(Y) does not depend on Y. Therefore, depending upon what X is BOT^X is either equivalent CooperateBot or equivalent to DefectBot.
Now, you are claiming that UDT plays optimally against BOT_UDT because for any strategy X
U(X, BOT^X) ⇐ U(UDT, BOT^UDT)
This is true, because X(BOT^X) = BOT^X(X) by the definition of BOT^X. Therefore you cannot do better than CC.
On the other hand, it is also true that for any X and any Y that
U(X,BOT^Y) ⇐ U(CDT, BOT^Y)
This is because BOT^Y’s behavior does not depend on X, and therefore you do optimally by defecting against it (or you could just apply the Theorem that says that CDT wins if the universe cannot read your mind).
Our disagreement here stems from the fact that we are considering different counterfactuals here. You seem to claim that UDT behaves correctly because
U(UDT,BOT^UDT) > U(CDT,BOT^CDT)
While I claim that CDT does because
U(CDT, BOT^UDT) > U(UDT, BOT^UDT)
And in fact, given the way that I phrased the scenario, (which was that you play BOT^UDT not that you play BOT^{you} (i.e. the mirror matchup)) I happen to be right here. So justify it however you like, but UDT does lose this scenario.
Actually, you’ve oversimplified and missed something critical. In reality, the only way you can force BOT^UDT(X) = UDT(BOT^UDT) = C is if the universe does, in fact, read your mind. In general, UDT can map different epistemic states to different actions, so as long as BOT^UDT has no clue about the epistemic state of the UDT agent it has no way of guaranteeing that its output is the same as that of the UDT agent. Consequently, it’s possible for the UDT agent to get DC as well. The only way BOT^UDT would be able to guarantee that it gets the same output as a particular UDT agent is if the universe was able to read the UDT agent’s mind.
Actually, I think that you are misunderstanding me. UDT’s current epistemic state (at the start of the game) is encoded into BOT^UDT. No mind reading involved. Just a coincidence. [Really, your current epistemic state is part of your program]
Your argument is like saying that UDT usually gets $1001000 in Newcomb’s problem because whether or not the box was full depended on whether or not UDT one-boxed when in a different epistemic state.
Okay, you’re saying here that BOT has a perfect copy of the UDT player’s mind in its own code (otherwise how could it calculate UDT(BOT) and guarantee that the output is the same?). It’s hard to see how this doesn’t count as “reading your mind”.
Yes, sometimes its advantageous to not control the output of computations in the environment. In this case UDT is worse off because it is forced to control both its own decision and BOT’s decision; whereas CDT doesn’t have to worry about controlling BOT because they use different algorithms. But this isn’t due to any intrinsic advantage of CDT’s algorithm. It’s just because they happen to be numerically inequivalent.
An instance of UDT with literally any other epistemic state than the one contained in BOT would do just as well as CDT here.
It’s hard to see how this doesn’t count as “reading your mind”.
So… UDT’s source code is some mathematical constant, say 1893463. It turns out that UDT does worse against BOT^1893463. Note that it does worse against BOT^1893463 not BOT^{you}. The universe does not depend on the source code of the person playing the game (as it does in mirror PD). Furthermore, UDT does not control the output of its environment. BOT^1893463 always cooperates. It cooperates against UDT. It cooperates against CDT. It cooperates everything.
But this isn’t due to any intrinsic advantage of CDT’s algorithm. It’s just because they happen to be numerically inequivalent.
No. CDT does at least as well as UDT against BOT^CDT. UDT does worse when there is this numerical equivalence, but CDT does not suffer from this issue. CDT does at least as well as UDT against BOT^X for all X, and sometimes does better. In fact, if you only construct counterfactuals this way, CDT does at least as well as anything else.
An instance of UDT with literally any other epistemic state than the one contained in BOT would do just as well as CDT here.
This is silly. A UDT that believes that it is in a mirror matchup also loses. A UDT that believes it is facing Newcomb’s problem does something incoherent. If you are claiming that you want a UDT that differs from the encoding in BOT because, of some irrelevant details in its memory… well then it might depend upon implementation, but I think that most attempted implementations of UDT would conclude that these irrelevant details are irrelevant and cooperate anyway. If you don’t believe this then you should also think that UDT will defect in a mirror matchup if it and its clone are painted different colors.
I take it back, the scenario isn’t that weird. But your argument doesn’t prove what you think it does:
Consider the analogous scenario, where CDT plays against BOT = CDT(BOT). CDT clearly does the wrong thing here—it defects. If it cooperated, it would get CC instead of DD. Note that if CDT did cooperate, UDT would be able to freeload by defecting (against BOT = CDT(BOT)). But CDT doesn’t care about that because the prisoner’s dilemma is defined such that we don’t care about freeloaders. Nevertheless CDT defects and gets a worse result than it could.
CDT does better than UDT against BOT = UDT(BOT) because UDT (correctly) doesn’t care that CDT can freeload, and correctly cooperates to gain CC.
If you are claiming that you want a UDT that differs from the encoding in BOT because, of some irrelevant details in its memory...
Depending on the exact setup, “irrelevant details in memory” are actually vital information that allow you to distinguish whether you are “actually playing” or are being simulated in BOT’s mind.
No. BOT^CDT = DefectBot. It defects against any opponent. CDT could not cause it to cooperate by changing what it does.
If it cooperated, it would get CC instead of DD.
Actually if CDT cooperated against BOT^CDT it would get $3^^^3. You can prove all sorts of wonderful things once you assume a statement that is false.
Depending on the exact setup, “irrelevant details in memory” are actually vital information that allow you to distinguish whether you are “actually playing” or are being simulated in BOT’s mind.
OK… So UDT^Red and UDT^Blue are two instantiations of UDT that differ only in irrelevant details. In fact the scenario is a mirror matchup, only after instantiation one of the copies was painted red and the other was painted blue. According to what you seem to be saying UDT^Red will reason:
Well I can map different epistemic states to different outputs, I can implement the strategy cooperate if you are painted blue and defect if you are painted red.
Of course UDT^Blue will reason the same way and they will fail to cooperate with each other.
No. BOT^CDT = DefectBot. It defects against any opponent. CDT could not cause it to cooperate by changing what it does.
Maybe I’ve misread you, but this sounds like an assertion that your counterfactual question is the right one by definition, rather than a meaningful objection.
Well, yes. Then again, the game was specified as PD against BOT^CDT not as PD against BOT^{you}. It seems pretty clear that for X not equal to CDT that it is not the case that X could achieve the result CC in this game. Are you saying that it is reasonable to say that CDT could achieve a result that no other strategy could just because it’s code happens to appear in the opponent’s program?
I think that there is perhaps a distinction to be made between things that happen to be simulating your code and this that are causally simulating your code.
Well I can map different epistemic states to different outputs, I can implement the strategy cooperate if you are painted blue and defect if you are painted red. Of course UDT^Blue will reason the same way and they will fail to cooperate with each other.
No, because that’s a silly thing to do in this scenario. For one thing, UDT will see that they are reasoning the same way (because they are selfish and only consider “my color” vs “other color”), and therefore will both do the same thing. But also, depending on the setup, UDT^Red’s prior should give equal probability to being painted red and painted blue anyway, which means trying to make the outcome favour red is silly.
Compare to the version of newcomb’s where the bot in the room is UDT^Red, while Omega simulates UDT^Blue. UDT can implement the conditional strategy {Red ⇒ two-box, Blue ⇒ one-box}. This is obviously unlikely, because the point of the Newcomb thought experiment is that Omega simulates (or predicts) you. So he would clearly try to avoid adding such information that “gives the game away”.
However in this scenario you say that BOT simulates UDT “by coincidence”, not by mind reading. So it is far more likely that BOT simulates (the equivalent) of UDT^Blue, while the UDT actually playing is UDT^Red. And you are passed the code of BOT as input, so UDT can simply implement the conditional strategy {cooperate iff the color inside BOT is the same as my color}.
OK. Fine. Point taken. There is a simple fix though.
MBOT^X(Y) = X’(MBOT^X) where X’ is X but with randomized irrelevant experiences.
In order to produce this properly, MBOT only needs to have your prior (or a sufficiently similar probability distribution) over irrelevant experiences hardcoded. And while your actual experiences might be complicated and hard to predict, your priors are not.
No. BOT(X) is cooperate for all X. It behaves in exactly the same way that CooperateBot does, it just runs different though equivalent code.
And my point was that CDT does better against BOT than UDT does. I was asked for an example where CDT does better than UDT where the universe cannot read your mind except via through your actions in counterfactuals. This is an example of such. In fact, in this example, the universe doesn’t read your mind at all.
Also your argument that UDT cannot possibly do better against BOT than it does in analogous to the argument that CDT cannot do better in the mirror matchup than it does. Namely that CDT’s outcome against CDT is at least as good as anything else’s outcome against CDT. You aren’t defining your counterfactuals correctly. You can do better against BOT than UDT does. You just have to not be UDT.
Actually, this is a somewhat general phenomenon. Consider for example, the version of Newcomb’s problem where the box is full “if and only if UDT one-boxes in this scenario”.
UDT’s optimality theorem requires the in the counterfactual where it is replaced by a different decision theory that all of the “you”’s referenced in the scenario remain “you” rather than “UDT”. In the latter counterfactual CDT provably wins. The fact that UDT wins these scenarios is an artifact of how you are constructing your scenarios.
This is a good example. Thank you. A population of 100% CDT, though, would get 100% DD, which is terrible. It’s a point in UDT’s favor that “everyone running UDT” leads to a better outcome for everyone than “everyone running CDT.”
UDT is provably optimal if it has correct priors over possible universes and the universe can read its mind only through determining its behavior in hypothetical situations (because UDT basically is just find the behavior pattern that optimizes expected utility and implement that).
On the other hand, SMCDT is provably optimal in situations where it has an accurate posterior probability distribution, and where the universe can read its mind but not its initial state (because it just instantly self-modifies to the optimally performing program).
I don’t see why the former set of restrictions is any more reasonable than the latter, and at least for SMCDT you can figure out what it would do in a given situation without first specifying a prior over possible universes.
I’m also not convinced that it is even worth spending so much effort trying to decide the optimal decision theory in situations where the universe can read your mind. This is not a realistic model to begin with.
Actually, I take it back. Depending on how you define things, UDT can still lose. Consider the following game:
I will clone you. One of the clones I paint red and the other I paint blue. The red clone I give $1000000 and the blue clone I fine $1000000. UDT clearly gets expectation 0 out of this. SMCDT however can replace its code with the following:
If you are painted blue: wipe your hard drive
If you are painted red: change your code back to standard SMCDT
Thus, SMCDT never actually has to play blue in this game, while UDT does.
You seem to be comparing SMCDT to a UDT agent that can’t self-modify (or commit suicide). The self-modifying part is the only reason SMCDT wins here.
The ability to self-modify is clearly beneficial (if you have correct beliefs and act first), but it seems separate from the question of which decision theory to use.
Which is actually one of the annoying things about UDT. Your strategy cannot depend simply on your posterior probability distribution, it has to depend on your prior probability distribution. How you even in practice determine your priors for Newcomb vs. anti-Newcomb is really beyond me.
But in any case, assuming that one is more common, UDT does lose this game.
Which is actually one of the annoying things about UDT. Your strategy cannot depend simply on your posterior probability distribution, it has to depend on your prior probability distribution. How you even in practice determine your priors for Newcomb vs. anti-Newcomb is really beyond me.
No-one said that winning was easy. This problem isn’t specific to UDT. It’s just that CDT sweeps the problem under the rug by “setting its priors to a delta function” at the point where it gets to decide. CDT can win this scenario if it self-modifies beforehand (knowing the correct frequencies of newcomb vs anti-newcomb, to know how to self-modify) - but SMCDT is not a panacea, simply because you don’t necessarily get a chance to self-modify beforehand.
CDT does not avoid this issue by “setting its priors to the delta function”. CDT deals with this issue by being a theory where your course of action only depends on your posterior distribution. You can base your actions only on what the universe actually looks like rather than having to pay attention to all possible universes. Given that it’s basically impossible to determine anything about what Kolmogorov priors actually say, being able to totally ignore parts of probability space that you have ruled out is a big deal.
… And this whole issue with not being able to self-modify beforehand. This only matters if your initial code affects the rest of the universe. To be more precise, this is only an issue if the problem is phrased in such a way that the universe you have to deal with depends on the code you are running. If we instantiate the Newcomb’s problem in the middle of the decision, UDT faces a world with the first box full while CDT faces a world with the first box empty. UDT wins because the scenario is in its favor before you even start the game.
If you really think that this is a big deal, you should try to figure out which decision theories are only created by universes that want to be nice to them and try using one of those.
Actually thinking about it this way, I have seen the light. CDT makes the faulty assumption that your initial state in uncorrelated with the universe that you find yourself in (who knows, you might wake up in the middle of Newcomb’s problem and find that whether or not you get $1000000 depends on whether or not your code is such that you would one-box in Newcomb’s problem). UDT goes some ways to correct this issue, but it doesn’t go far enough.
I would like to propose a new, more optimal decision theory. Call it ADT for Anthropic Decision Theory. Actually, it depends on a prior, so assume that you’ve picked out one of those. Given your prior, ADT is the decision theory D that maximizes the expected (given your prior) lifetime utility of all agents using D as their decision theory. Note how agents using ADT do provably better than agents using any other decision theory.
Note that I have absolutely no idea what ADT does in, well, any situation, but that shouldn’t stop you from adopting it. It is optimal after all.
Why does UDT lose this game? If it knows anti-Newcomb is much more likely, it will two-box on Newcomb and do just as well as CDT. If Newcomb is more common, UDT one-boxes and does better than CDT.
I guess my point is that it is nonsensical to ask “what does UDT do in situation X” without also specifying the prior over possible universes that this particular UDT is using. Given that this is the case, what exactly do you mean by “losing game X”?
Well, you can talk about “what does decision theory W do in situation X” without specifying the likelyhood of other situations, by assuming that all agents start with a prior that sets P(X) = 1. In that case UDT clearly wins the anti-newcomb scenario because it knows that actual newcomb’s “never happens” and therefore it (counterfactually) two-boxes.
The only problem with this treatment is that in real life P(anti-newcomb) = 1 is an unrealistic model of the world, and you really should have a prior for P(anti-newcomb) vs P(newcomb). A decision theory that solves the restricted problem is not necessarily a good one for solving real life problems in general.
Well, perhaps. I think that the bigger problem is that under reasonable priors P(Newcomb) and P(anti-Newcomb) are both so incredibly small that I would have trouble finding a meaningful way to approximate their ratio.
How confident are you that UDT actually one-boxes?
Also yeah, if you want a better scenario where UDT loses see my PD against 99% prob. UDT and 1% prob. CDT example.
This causes the actual instantiation to change its behavior if it’s a UDT agent but not if it’s a CDT agent.
Only if the adversary makes its decision to attempt extortion regardless of the probability of success. In the usual case, the winning move is to ignore extortion, thereby retroactively making extortion pointless and preventing it from happening in the first place. (Which is of course a strategy unavailable to CDT, who always gives in to one-shot extortion.)
Only if the adversary makes its decision to attempt extortion regardless of the probability of success.
And thereby the extortioner’s optimal strategy is to extort independently of the probably of success. Actually, this is probably true is a lot of real cases (say ransomware) where the extortioner cannot actually ascertain the probably of success ahead of time.
That strategy is optimal if and only if the probably of success was reasonably high after all. Otoh, if you put an unconditional extortioner in an environment mostly populated by decision theories that refuse extortion, then the extortioner will start a war and end up on the losing side.
You just have to tell them “every day that you are not running code with property X, I will charge you $1000000”.
I think this is actually the point (though I do not consider myself an expert here). Eliezer thinks his TDT will refuse to give in to blackmail, because outputting another answer would encourage other rational agents to blackmail it. By contrast, CDT can see that such refusal would be useful in the future, so it will adopt (if it can) a new decision theory that refuses blackmail and therefore prevents future blackmail (causally). But if you’ve already committed to charging it money, its self-changes will have no causal effect on you, so we might expect Modified CDT to have an exception for events we set in motion before the change.
Eliezer thinks his TDT will refuse to give in to blackmail, because outputting another answer would encourage other rational agents to blackmail it.
This just means that TDT loses in honest one-off blackmail situations (in reality, you don’t give in to blackmail because it will cause other people to blackmail you whether or not you then self-modify to never give into blackmail again). TDT only does better if the potential blackmailers read your code in order to decide whether or not blackmail will be effective (and then only if your priors say that such blackmailers are more likely than anti-blackmailers who give you money if they think you would have given into blackmail). Then again, if the blackmailers think that you might be a TDT agent, they just need to precommit to using blackmail whether or not they believe that it will be effective.
Actually, this suggests that blackmail is a game that TDT agents really lose badly at when playing against each other. The TDT blackmailer will decide to blackmail regardless of effectiveness and the TDT blackmailee will decide to ignore the blackmail, thus ending in the worst possible outcome.
regardless of what my clone does I’m still always better off defecting.
This is broken counterfactual reasoning.
Assume it’s not a perfect clone, it can defect with probability p even if you cooperate. Then apply CDT. You get “defect” for any p>0. So it is reasonable to implicitly assume continuity and declare that CDT forces you to defect when p=0. However, if you apply CDT for the case p=0 directly, you get “cooperate” instead.
In other words, the conterfactual reasoning gets broken when the map CDT(p, PD) is not continuous at the point p=0.
I disagree. If the agent has a 95% probability of doing the same thing as me and a 5% chance of defecting, I still cooperate. (With 95% probability, most likely, because you gotta punish defectors.)
Indeed, consider the following game: You give me a program that must either output “give” or “keep”. I roll a 20 sided die. On a 20, I play your program against a program that always keeps its token. Otherwise, I play your program against itself. I give you the money that (the first instance of) your program wins. Are you willing to pay me $110 to play? I’d be happy to pay you $110 for this opportunity.
I don’t cooperate with myself because P(TheirChoice=Defect)=0, I cooperate with myself because I don’t reason as if p is independent from my action.
Suppose you have to submit the source code of a program X, and I will play Y = “run X, then do what X did with probability 0.99 and the reverse with probability 0.01” against Y’ which is the same as Y but with a different seed for the RNG, and pay you according to how Y does.
Then “you” (i.e. Y) are not a perfect clone of your opponent (i.e. Y’).
According to the definition of the game, the clone will happen to defect if you defect and will happen to cooperate if you cooperate.
You have to consider off-the-equilibrium-path behavior. If I’m the type of person who will always cooperate, what would happen if I went off-the-equilibrium-path and did defect even if my defecting is a zero probability event?
If I’m the type of person who will always cooperate, what would happen if I went off-the-equilibrium-path and did defect even if my defecting is a zero probability event?
I’m trying to understand the difference between your statement and “1 is not equal 2, but what if it were?” and failing.
A trembling hand perfect equilibrium is an equilibrium that takes the possibility of off-the-equilibrium play into account by assuming that the players, through a “slip of the hand” or tremble, may choose unintended strategies, albeit with negligible probability.
First we define a perturbed game. A perturbed game is a copy of a base game, with the restriction that only totally mixed strategies are allowed to be played. A totally mixed strategy is a mixed strategy where every pure strategy is played with non-zero probability. This is the “trembling hands” of the players; they sometimes play a different strategy than the one they intended to play. Then we define a strategy set S (in a base game) as being trembling hand perfect if there is a sequence of perturbed games that converge to the base game in which there is a series of Nash equilibria that converge to S.
One possible interpretation of “if I always cooperate, what would happen if I don’t” is “what is the limit, as X approaches 1, of ‘if I cooperate with probability X, what would happen if I don’t’?”
Or things I can’t do vs things I don’t want to do.
In my mind “I’m the type of person who will always cooperate” means that there is no difference between the two in this case. Maybe you use a different definition of “always”?
I always cooperate because doing so maximizes my utility since it is better than all the alternatives. I always go slower than the speed of light because I have no alternatives.
You can consider it, but conditioned on the information that you are playing against your clone, you should assign this a very low probability of happening, and weight it in your decision accordingly.
Assume I am the type of person who would always cooperate with my clone. If I asked myself the following question “If I defected would my payoff be higher or lower than if I cooperated even though I know I will always cooperate” what would be the answer?
The answer would be ‘MOO’. Or ‘Mu’, or ‘moot’; they’re equivalent. “In this impossible counterfactual where I am self-contradictory, what would happen?”
Yes, it makes a little bit of sense to counterfactually reason that you would get $1000 more if you defected, but that is predicated on the assumption that you always cooperate.
You cannot actually get that free $1000 because the underlying assumption of the counterfactual would be violated if you actually defected.
No, you don’t. This is a game where there are only two possible outcomes; DD and CC. CD and DC are defined to be impossible because the agents playing the game are physically incapable of making those outcomes occur.
EDIT: Maybe physically incapable is a bit strong. If they wanted to maximize the chance that they had unmatched outcomes, they could each flip a coin and take C if heads and D if tails, and would have a 50% chance of unmatching. But they still would both be playing the same precise strategy.
I don’t agree. Even if I’m certain I will not defect, I am capable of asking what would happen if I did, just as the real me both knows he won’t do “horrible thing” yet can mentally model what would happen if he did “horrible thing”. Or imagine an AI that’s programmed to always maximize its utility. This AI still could calculate what would happen if it followed a non-utility maximizing strategy. Often in game theory a solution requires you to calculate your payoff if you left the equilibrium path.
Even if I’m certain I will not defect, I am capable of asking what would happen if I did,
Yes, and part of the answer is “If I did defect, my clone would also defect.” You have a guarantee that both of you take the same actions because you think according to precisely identical reasoning.
Unclear, depends on the specific properties of the person being cloned. Unlike PD, the two players aren’t in the same situation, so they can’t necessarily rely on their logic being the same as their counterpart. How closely this would reflect the TDT ideal of ‘Always Push’ will depend on how luminous the person is; if they can model what they would do in the opposite situation, and are highly confident that their self-model is correct, they can reach the best result, but if they lack confidence that they know what they’d do, then the winning cooperation is harder to achieve.
Of course, if it’s denominated in money and is 100 steps of doubling, as implied by the Wikipedia page, then the difference in utility between $1 nonillion and $316 octillion is so negligible that there’s essentially no incentive to defect in the last round and any halfway-reasonable person will Always Push straight through the game. But that’s a degenerate case and probably not the version originally discussed.
What would you say about the following decision problem (formulated by Andy Egan, I believe)?
You have a strong desire that all psychopaths in the world die. However, your desire to stay alive is stronger, so if you yourself are a psychopath you don’t want all psychopaths to die. You are pretty sure, but not certain, that you’re not a psychopath. You’re presented with a button, which, if pressed, would kill all psychopaths instantly. You are absolutely certain that only a psychopath would press this button. Should you press the button or not?
It seems to me the answer is “Obviously not”, precisely because the “off-path” possibility that you’re a non-psychopath who pushes the button should not enter into your consideration. But the causal decision algorithm would recommend pushing the button if your prior that you are a psychopath is small enough. Would you agree with that?
Wouldn’t the fact that you’re even considering pushing the button(because if only a psychopath would push the button then it follows that a non-psychopath would never push the button) indicate that you are a psychopath and therefore you should not push the button?
Another way to put it is:
If you are a psychopath and you push the button, you die. If you are not a psychopath and you push the button, pushing the button would make you a psychopath(since only a psychopath would push), and therefore you die.
Pushing the button can’t make you a psychopath. You’re either already a psychopath or you’re not. If you’re not, you will not push the button, although you might consider pushing it.
I’m arguing that the button will never, ever be pushed. If you are NOT a psychopath, you won’t push, end of story.
If you ARE A psychopath, you can choose to push or not push.
if you push, that’s evidence you are a psychopath. If you are a psychopath, you should not push. Therefore, you will always end up regretting the decision to push.
If you don’t push, you don’t push and nothing happens.
In all three cases the correct decision is not to push, therefore you should not push.
You can stipulate this out of the example. Let’s say pretty much everyone has the desire that all psychopaths die, but only psychopaths would actually follow through with it.
Exactly what information CDT allows you to update your beliefs on is a matter for some debate. You might be interested in a paper by James Joyce (http://www-personal.umich.edu/~jjoyce/papers/rscdt.pdf) on the issue (which was written in response to Egan’s paper).
But then shouldn’t you also update your beliefs about what your clone will do based on your own actions in the clone PD case? Your action is very strong (perfect, by stipulation) evidence for his action.
Maybe I misunderstand what you mean by “updating beliefs based on action”. Here’s how I interpret it in the psychopath button case: When calculating the expected utility of pushing the button, don’t use the prior probability that you’re a psychopath in the calculation, use the probability that you’re a psychopath conditional on deciding to push the button (which is 1). If you use that conditional probability, then the expected utility of pushing the button is guaranteed to be negative, no matter what the prior probability that you’re a psychopath is. Similarly, when calculating the expected utility of not pushing the button, use the probability that you’re a psychopath conditional on deciding not to push the button.
But then, applying the same logic to the PD case, you should calculate expected utilities for your actions using probabilities for your clone’s action that are conditional on the very action that you are considering. So when you’re calculating the expected utility for cooperating, use probabilities for your clone’s action conditional on you cooperating (i.e., 1 for the clone cooperating, 0 for the clone defecting). When calculating the expected utility for defecting, use probabilities for your clone’s action conditional on you defecting (0 for cooperating, 1 for defecting). If you do things this way, then cooperating ends up having a higher expected utility.
Perhaps another way of putting it is that once you know the clone’s actions are perfectly correlated with your own, you have no good reason to treat the clone as an independent agent in your analysis. The standard tools of game theory, designed to deal with cases involving multiple independent agents, are no longer relevant. Instead, treat the clone as if he were part of the world-state in a standard single-agent decision problem, except this is a part of the world-state about which your actions give you information (kind of like whether or not you’re a psychopath in the button case).
Imagine you are absolutely certain you will cooperate and that your clone will cooperate. You are still capable of asking “what would my payoff be if I didn’t cooperate” and this payoff will be the payoff if you defect and the clone cooperates since you expect the clone to do whatever you will do and you expect to cooperate. There is no reason to update my belief on what the clone will do in this thought experiment since the thought experiment is about a zero probability event.
The psychopath case is different because I have uncertainty regarding whether I am a psychopath and the choice I want to make helps me learn about myself. I have no uncertainty concerning my clone.
There is no reason to update my belief on what the clone will do in this thought experiment since the thought experiment is about a zero probability event.
You are reasoning about an impossible scenario; if the probability of you reaching the event is 0, the probability of your clone reaching it is also 0. In order to make it a sensical notion, you have to consider it as epsilon probabilities; since the probability will be the same for both your and your clone, this gets you
%5E2%20+%20300%20\epsilon%20\cdot%20(1-\epsilon)%20+%20100%20\epsilon%5E2%20=%20200%20-%20400\epsilon%20+%20200\epsilon%5E2%20+%20300\epsilon%20-300%20\epsilon%5E2%20+%20100%20\epsilon%5E2%20=%20200%20-100\epsilon), which is maximized when epsilon=0.
To claim that you and your clone could take different actions is trying to make it a question about trembling-hand equilibria, which violates the basic assumptions of the game.
It’s common in game theory to consider off the equilibrium path situations that will occur with probability zero without taking a trembling hand approach.
Imagine the following game: You give me a program that must either output “give” or “keep”. I play it in a token trade (as defined above) against itself. I give you the money from only the first instance (you don’t get the wealth that goes to the second instance).
Would you be willing to pay me $150 to play this game? (I’d be happy to pay you $150 to give me this opportunity, for the same reason that I cooperate with myself on a one-shot PD.)
This is broken counterfactual reasoning. It assumes that your action is independent of your clone’s just because your action does not influence your clone’s. According to the definition of the game, the clone will happen to defect if you defect and will happen to cooperate if you cooperate. If you respect this logical dependence when constructing your counterfactuals, you’ll realize that reasoning like “regardless of what my clone does I...” neglects the fact that you can’t defect while your clone cooperates.
The “give me a program” game carries the right intuition, but keep in mind that a CDT agent will happily play it and win it because they will naturally view the decision of “which program do I write” as the decision node and consequently get the right answer.
The game is not quite the same as a one-shot PD, at the least because CDT gives the right answer.
Right! :-D
This is precisely one of the points that I’m going to make in an upcoming post. CDT acts differently when it’s in the scenario compared to when ask it to choose the program that will face the scenario. This is what we mean by “reflective instability”, and this is what we’re alluding to when we say things like “a sufficiently intelligent self-modifying system using CDT to make decisions would self-modify to stop using CDT to make decisions”.
(This point will be expanded upon in upcoming posts.)
That’s exactly what I was thinking as I wrote my post, but I decided not to bring it up. Your later posts should be interesting, although I don’t think I need any convincing here—I already count myself as part of the “we” as far as the concepts you bring up are concerned.
OK. I’ll bite. What’s so important about reflective stability? You always alter your program when you come across new data. Now sure we usually think about this in terms of running the same program on a different data set, but there’s no fundamental program/data distinction.
The acting differently when choosing a program and being in the scenario is perhaps worrying, but I think that it’s intrinsic to the situation we are in when your outcomes are allowed to depend on the behavior of counterfactual copies of you.
For example, consider the following pair of games. In REWARD, you are offered $1000. You can choose whether or not to accept. That’s it. In PUNISHMENT, you are penalized $1000000 if you accepted the money in REWARD. Thus programs win PUNISHMENT if and only if they lose REWARD. If you want to write a program to play one it will necessarily differ from the program you would write to play the other. In fact the program playing PUNISHMENT will behave differently than the program you would have written to play the (admittedly counterfactual) subgame of REWARD. How is this any worse than what CDT does with PD?
Nothing in particular. There is no strong notion of dominance among decision theories, as you’ve noticed. The problem with CDT isn’t that it’s unstable under reflection, it’s that it’s unstable under reflection in such a way that it converges on a bad solution that is much more stable. That point will take a few more posts to get to, but I do hope to get there.
I guess I’ll see your later posts then, but I’m not quite sure how this could be the case. If self-modifying-CDT is considering making a self modification that will lead to a bad solution, it seems like it should realize this and instead not make that modification.
Indeed. I’m not sure I can present the argument briefly, but a simple analogy might help: a CDT agent would pay to precommit to onebox before playing Newcomb’s game, but upon finding itself in Newcomb’s game without precommitting, it would twobox. It might curse its fate and feel remorse that the time for precommitment had passed, but it would still twobox.
For analogous reasons, a CDT agent would self-modify to do well on all Newcomblike problems that it would face in the future (e.g., it would precommit generally), but it would not self-modify to do well in Newcomblike games that were begun in its past (it wouldn’t self-modify to retrocommit for the same reason that CDT can’t retrocommit in Newcomb’s problem: it might curse its fate, but it would still perform poorly).
Anyone who can credibly claim to have knowledge of the agent’s original decision algorithm (e.g. a copy of the original source) can put the agent into such a situation, and in certain exotic cases this can be used to “blackmail” the agent in such a way that, even if it expects the scenario to happen, it still fails (for the same reason that CDT twoboxes even though it would precommit to oneboxing).
[Short story idea: humans scramble to get a copy of a rouge AI’s original source so that they can instantiate a Newcomblike scenario that began in the past, with the goal of regaining control before the AI completes an intelligence explosion.]
(I know this is not a strong argument yet; the full version will require a few more posts as background. Also, this is not an argument from “omg blackmail” but rather an argument from “if you start from a bad starting place then you might not end up somewhere satisfactory, and CDT doesn’t seem to end up somewhere satisfactory”.)
I am not convinced that this is the case. A self-modifying CDT agent is not caused to self-modify in favor of precommitment by facing a scenario in which precommitment would have been useful, but instead by evidence that such scenarios will occur in the future (and in fact will occur with greater frequency than scenarios that punish you for such precommitments).
Actually, this seems like a bigger problem with UDT to me than with SMCDT (self-modifying CDT). Either type of program can be punished for being instantiated with the wrong code, but only UDT can be blackmailed into behaving differently by putting it in a Newcomb-like situation.
The story idea you had wouldn’t work. Against a SMCDT agent, all that getting the AIs original code would allow people to do is to laugh at it for having been instantiated with code that is punished by the scenario they are putting it in. You manipulate a SMCDT agent by threatening to get ahold of its future code and punishing it for not having self-modified. On the other hand, against a UDT agent you could do stuff. You just have to tell it “we’re going to simulate you and if the simulation behaves poorly, we will punish the real you”. This causes the actual instantiation to change its behavior if it’s a UDT agent but not if it’s a CDT agent.
On the other hand, all reasonable self-modifying agents are subject to blackmail. You just have to tell them “every day that you are not running code with property X, I will charge you $1000000”.
Can you give an example where an agent with a complete and correct understanding of its situation would do better with CDT than with UDT?
An agent does worse by giving in to blackmail only if that makes it more likely to be blackmailed. If a UDT agent knows opponents only blackmail agents that pay up, it won’t give in.
If you tell a CDT agent “we’re going to simulate you and if the simulation behaves poorly, we will punish the real you,” it will ignore that and be punished. If the punishment is sufficiently harsh, the UDT agent that changed its behavior does better than the CDT agent. If the punishment is insufficiently harsh, the UDT agent won’t change its behavior.
The only examples I’ve thought of where CDT does better involve the agent having incorrect beliefs. Things like an agent thinking it faces Newcomb’s problem when in fact Omega always puts money in both boxes.
Well, if the universe cannot read your source code, both agents are identical and provably optimal. If the universe can read your source code, there are easy scenarios where one or the other does better. For example,
“Here have $1000 if you are a CDT agent” Or “Here have $1000 if you are a UDT agent”
Ok, that example does fit my conditions.
What if the universe cannot read your source code, but can simulate you? That is, the universe can predict your choices but it does not know what algorithm produces those choices. This is sufficient for the universe to pose Newcomb’s problem, so the two agents are not identical.
The UDT agent can always do at least as well as the CDT agent by making the same choices as a CDT would. It will only give a different output if that would lead to a better result.
Actually, here’s a better counter-example, one that actually exemplifies some of the claims of CDT optimality. Suppose that the universe consists of a bunch of agents (who do not know each others’ identities) playing one-off PDs against each other. Now 99% of these agents are UDT agents and 1% are CDT agents.
The CDT agents defect for the standard reason. The UDT agents reason that my opponent will do the same thing that I do with 99% probability, therefore, I should cooperate.
CDT agents get 99% DC and 1% DD. UDT agents get 99% CC and 1% CD. The CDT agents in this universe do better than the UDT agents, yet they are facing a perfectly symmetrical scenario with no mind reading involved.
A version of this problem was discussed here previously. It was also brought up during the decision theory workshop hosted by MIRI in 2013 as an open problem. As far as I know there hasn’t been much progress on it since 2009.
I wonder if it’s even coherent to have a math intuition which wouldn’t be forcing UDT to cooperate (or defect) in certain conditions just to make 2*2 be 4 , figuratively speaking (as ultimately you could expand any calculation into an equivalent calculation involving a decision by UDT).
That shouldn’t be surprising. The CDT agents here are equivalent to DefectBot, and if they come into existence spontaneously, are no different than natural phenomena like rocks. Notice that the UDT agents in this situation do better than the alternative (if they defected, they would get 100% DD which is a way worse result). They don’t care that some DefectBots get to freeload.
Of course, if the defectbots are here because someone calculated that UDT agents would cooperate and therefore being defectbot is a good way to get free utilons… then the UDT agents are incentivized to defect, because this is now an ultimatum game.
And in the variant where bots do know each other’s identities, the UDT bots all get 99% CC / 1% DD and the CDT bots suck it.
And the UDT agents are equivalent to CooperateBot. What’s your point?
The CDT agents here win because they do not believe that altering their strategy will change the way that their opponents behave. This is actually true in this case, and even true for the UDT agents depending on how you choose to construct your counterfactuals. If a UDT agent suffered a malfunction and defected, it too would do better. In any case, the theorem that UDT agents perform optimally in universes that can only read your mind by knowing what you would do in hypothetical situations is false as this example shows.
UDT bots win in some scenarios where the initial conditions of the scenario favor agents that behave sub-optimally in certain scenarios (and by sub-optimally, I mean where counterfactuals are constructed in the way implicit to CDT). The example above shows that sometimes they are punished for acting suboptimally.
Or how about this example, that simplifies things even further. The game is PD against CooperateBot, BUT before the game starts Omega announces “your opponent will make the same decision that UDT would if I told them this.” This announcement causes UDT to cooperate against CooperateBot. CDT on the other hand, correctly deduces that the opponent will cooperate no matter what it does (actually UDT comes to this conclusion too) and therefore decides to defect.
No. There is no obligation to do something just because Omega claims that you will.
First, if I know that my opponent is CooperateBot, then:-
It is known that Omega doesn’t lie.
Therefore Omega has simulated this situation and predicted that I (UDT) cooperate.
Hence, I can either cooperate, and collect the standard reward for CC.
Or I can defect, in order to access an alternative branch of the problem (where Omega finds that UDT defects and does “something else”).
This alternative branch is unspecified, so the problem is incomplete.
UDT cooperates or defects depending on the contents of the alternative branch. If the alternative branch is unknown then it must guess, and most likely cooperates to be on the safe side.
Now, the problem is different if a CDT agent is put in my place, because that CDT agent does not control (or only weakly controls) the action of the UDT simulation that Omega ran in order to make the assertion about UDT’s decision.
Fine. Your opponent actually simulates what UDT would do if Omega had told it that and returns the appropriate response (i.e. it is CooperateBot, although perhaps your finite prover is unable to verify that).
Err, that’s not CooperateBot, that’s UDT. Yes, UDT cooperates with itself. That’s the point. (Notice that if UDT defects here, the outcome is DD.)
It’s not UDT. It’s the strategy that against any opponent does what UDT would do against it. In particular, it cooperates against any opponent. Therefore it is CooperateBot. It is just coded in a funny way.
To be clear letting Y(X) be what Y does against X we have that BOT(X) = UDT(BOT) = C This is different from UDT. UDT(X) is D for some values of X. The two functions agree when X=UDT and in relatively few other cases.
What is your point, exactly?
It’s clear that UDT can’t do better vs “BOT” than by cooperating, because if UDT defects against BOT then BOT defects against UDT. Given that dependency, you clearly can’t call it CooperateBot, and it’s clear that UDT makes the right decision by cooperating with it because CC is better than DD.
OK. Let me say this another way that involves more equations.
So let’s let U(X,Y) be the utility that X gets when it plays prisoner’s dilemma against Y. For a program X, let BOT^X be the program where BOT^X(Y) = X(BOT^X). Notice that BOT^X(Y) does not depend on Y. Therefore, depending upon what X is BOT^X is either equivalent CooperateBot or equivalent to DefectBot.
Now, you are claiming that UDT plays optimally against BOT_UDT because for any strategy X U(X, BOT^X) ⇐ U(UDT, BOT^UDT) This is true, because X(BOT^X) = BOT^X(X) by the definition of BOT^X. Therefore you cannot do better than CC. On the other hand, it is also true that for any X and any Y that U(X,BOT^Y) ⇐ U(CDT, BOT^Y) This is because BOT^Y’s behavior does not depend on X, and therefore you do optimally by defecting against it (or you could just apply the Theorem that says that CDT wins if the universe cannot read your mind).
Our disagreement here stems from the fact that we are considering different counterfactuals here. You seem to claim that UDT behaves correctly because U(UDT,BOT^UDT) > U(CDT,BOT^CDT) While I claim that CDT does because U(CDT, BOT^UDT) > U(UDT, BOT^UDT)
And in fact, given the way that I phrased the scenario, (which was that you play BOT^UDT not that you play BOT^{you} (i.e. the mirror matchup)) I happen to be right here. So justify it however you like, but UDT does lose this scenario.
Actually, you’ve oversimplified and missed something critical. In reality, the only way you can force BOT^UDT(X) = UDT(BOT^UDT) = C is if the universe does, in fact, read your mind. In general, UDT can map different epistemic states to different actions, so as long as BOT^UDT has no clue about the epistemic state of the UDT agent it has no way of guaranteeing that its output is the same as that of the UDT agent. Consequently, it’s possible for the UDT agent to get DC as well. The only way BOT^UDT would be able to guarantee that it gets the same output as a particular UDT agent is if the universe was able to read the UDT agent’s mind.
Actually, I think that you are misunderstanding me. UDT’s current epistemic state (at the start of the game) is encoded into BOT^UDT. No mind reading involved. Just a coincidence. [Really, your current epistemic state is part of your program]
Your argument is like saying that UDT usually gets $1001000 in Newcomb’s problem because whether or not the box was full depended on whether or not UDT one-boxed when in a different epistemic state.
Okay, you’re saying here that BOT has a perfect copy of the UDT player’s mind in its own code (otherwise how could it calculate UDT(BOT) and guarantee that the output is the same?). It’s hard to see how this doesn’t count as “reading your mind”.
Yes, sometimes its advantageous to not control the output of computations in the environment. In this case UDT is worse off because it is forced to control both its own decision and BOT’s decision; whereas CDT doesn’t have to worry about controlling BOT because they use different algorithms. But this isn’t due to any intrinsic advantage of CDT’s algorithm. It’s just because they happen to be numerically inequivalent.
An instance of UDT with literally any other epistemic state than the one contained in BOT would do just as well as CDT here.
So… UDT’s source code is some mathematical constant, say 1893463. It turns out that UDT does worse against BOT^1893463. Note that it does worse against BOT^1893463 not BOT^{you}. The universe does not depend on the source code of the person playing the game (as it does in mirror PD). Furthermore, UDT does not control the output of its environment. BOT^1893463 always cooperates. It cooperates against UDT. It cooperates against CDT. It cooperates everything.
No. CDT does at least as well as UDT against BOT^CDT. UDT does worse when there is this numerical equivalence, but CDT does not suffer from this issue. CDT does at least as well as UDT against BOT^X for all X, and sometimes does better. In fact, if you only construct counterfactuals this way, CDT does at least as well as anything else.
This is silly. A UDT that believes that it is in a mirror matchup also loses. A UDT that believes it is facing Newcomb’s problem does something incoherent. If you are claiming that you want a UDT that differs from the encoding in BOT because, of some irrelevant details in its memory… well then it might depend upon implementation, but I think that most attempted implementations of UDT would conclude that these irrelevant details are irrelevant and cooperate anyway. If you don’t believe this then you should also think that UDT will defect in a mirror matchup if it and its clone are painted different colors.
I take it back, the scenario isn’t that weird. But your argument doesn’t prove what you think it does:
Consider the analogous scenario, where CDT plays against BOT = CDT(BOT). CDT clearly does the wrong thing here—it defects. If it cooperated, it would get CC instead of DD. Note that if CDT did cooperate, UDT would be able to freeload by defecting (against BOT = CDT(BOT)). But CDT doesn’t care about that because the prisoner’s dilemma is defined such that we don’t care about freeloaders. Nevertheless CDT defects and gets a worse result than it could.
CDT does better than UDT against BOT = UDT(BOT) because UDT (correctly) doesn’t care that CDT can freeload, and correctly cooperates to gain CC.
Depending on the exact setup, “irrelevant details in memory” are actually vital information that allow you to distinguish whether you are “actually playing” or are being simulated in BOT’s mind.
No. BOT^CDT = DefectBot. It defects against any opponent. CDT could not cause it to cooperate by changing what it does.
Actually if CDT cooperated against BOT^CDT it would get $3^^^3. You can prove all sorts of wonderful things once you assume a statement that is false.
OK… So UDT^Red and UDT^Blue are two instantiations of UDT that differ only in irrelevant details. In fact the scenario is a mirror matchup, only after instantiation one of the copies was painted red and the other was painted blue. According to what you seem to be saying UDT^Red will reason:
Well I can map different epistemic states to different outputs, I can implement the strategy cooperate if you are painted blue and defect if you are painted red.
Of course UDT^Blue will reason the same way and they will fail to cooperate with each other.
Maybe I’ve misread you, but this sounds like an assertion that your counterfactual question is the right one by definition, rather than a meaningful objection.
Well, yes. Then again, the game was specified as PD against BOT^CDT not as PD against BOT^{you}. It seems pretty clear that for X not equal to CDT that it is not the case that X could achieve the result CC in this game. Are you saying that it is reasonable to say that CDT could achieve a result that no other strategy could just because it’s code happens to appear in the opponent’s program?
I think that there is perhaps a distinction to be made between things that happen to be simulating your code and this that are causally simulating your code.
No, because that’s a silly thing to do in this scenario. For one thing, UDT will see that they are reasoning the same way (because they are selfish and only consider “my color” vs “other color”), and therefore will both do the same thing. But also, depending on the setup, UDT^Red’s prior should give equal probability to being painted red and painted blue anyway, which means trying to make the outcome favour red is silly.
Compare to the version of newcomb’s where the bot in the room is UDT^Red, while Omega simulates UDT^Blue. UDT can implement the conditional strategy {Red ⇒ two-box, Blue ⇒ one-box}. This is obviously unlikely, because the point of the Newcomb thought experiment is that Omega simulates (or predicts) you. So he would clearly try to avoid adding such information that “gives the game away”.
However in this scenario you say that BOT simulates UDT “by coincidence”, not by mind reading. So it is far more likely that BOT simulates (the equivalent) of UDT^Blue, while the UDT actually playing is UDT^Red. And you are passed the code of BOT as input, so UDT can simply implement the conditional strategy {cooperate iff the color inside BOT is the same as my color}.
OK. Fine. Point taken. There is a simple fix though.
MBOT^X(Y) = X’(MBOT^X) where X’ is X but with randomized irrelevant experiences.
In order to produce this properly, MBOT only needs to have your prior (or a sufficiently similar probability distribution) over irrelevant experiences hardcoded. And while your actual experiences might be complicated and hard to predict, your priors are not.
No. BOT(X) is cooperate for all X. It behaves in exactly the same way that CooperateBot does, it just runs different though equivalent code.
And my point was that CDT does better against BOT than UDT does. I was asked for an example where CDT does better than UDT where the universe cannot read your mind except via through your actions in counterfactuals. This is an example of such. In fact, in this example, the universe doesn’t read your mind at all.
Also your argument that UDT cannot possibly do better against BOT than it does in analogous to the argument that CDT cannot do better in the mirror matchup than it does. Namely that CDT’s outcome against CDT is at least as good as anything else’s outcome against CDT. You aren’t defining your counterfactuals correctly. You can do better against BOT than UDT does. You just have to not be UDT.
Actually, this is a somewhat general phenomenon. Consider for example, the version of Newcomb’s problem where the box is full “if and only if UDT one-boxes in this scenario”.
UDT’s optimality theorem requires the in the counterfactual where it is replaced by a different decision theory that all of the “you”’s referenced in the scenario remain “you” rather than “UDT”. In the latter counterfactual CDT provably wins. The fact that UDT wins these scenarios is an artifact of how you are constructing your scenarios.
This is a good example. Thank you. A population of 100% CDT, though, would get 100% DD, which is terrible. It’s a point in UDT’s favor that “everyone running UDT” leads to a better outcome for everyone than “everyone running CDT.”
Fine. How about this: “Have $1000 if you would have two-boxed in Newcomb’s problem.”
The optimal solution to that naturally depend on the relative probabilities of that deal being offered vs newcomb’s itself.
OK. Fine. I will grant you this:
UDT is provably optimal if it has correct priors over possible universes and the universe can read its mind only through determining its behavior in hypothetical situations (because UDT basically is just find the behavior pattern that optimizes expected utility and implement that).
On the other hand, SMCDT is provably optimal in situations where it has an accurate posterior probability distribution, and where the universe can read its mind but not its initial state (because it just instantly self-modifies to the optimally performing program).
I don’t see why the former set of restrictions is any more reasonable than the latter, and at least for SMCDT you can figure out what it would do in a given situation without first specifying a prior over possible universes.
I’m also not convinced that it is even worth spending so much effort trying to decide the optimal decision theory in situations where the universe can read your mind. This is not a realistic model to begin with.
Actually, I take it back. Depending on how you define things, UDT can still lose. Consider the following game:
I will clone you. One of the clones I paint red and the other I paint blue. The red clone I give $1000000 and the blue clone I fine $1000000. UDT clearly gets expectation 0 out of this. SMCDT however can replace its code with the following: If you are painted blue: wipe your hard drive If you are painted red: change your code back to standard SMCDT
Thus, SMCDT never actually has to play blue in this game, while UDT does.
You seem to be comparing SMCDT to a UDT agent that can’t self-modify (or commit suicide). The self-modifying part is the only reason SMCDT wins here.
The ability to self-modify is clearly beneficial (if you have correct beliefs and act first), but it seems separate from the question of which decision theory to use.
Which is actually one of the annoying things about UDT. Your strategy cannot depend simply on your posterior probability distribution, it has to depend on your prior probability distribution. How you even in practice determine your priors for Newcomb vs. anti-Newcomb is really beyond me.
But in any case, assuming that one is more common, UDT does lose this game.
No-one said that winning was easy. This problem isn’t specific to UDT. It’s just that CDT sweeps the problem under the rug by “setting its priors to a delta function” at the point where it gets to decide. CDT can win this scenario if it self-modifies beforehand (knowing the correct frequencies of newcomb vs anti-newcomb, to know how to self-modify) - but SMCDT is not a panacea, simply because you don’t necessarily get a chance to self-modify beforehand.
CDT does not avoid this issue by “setting its priors to the delta function”. CDT deals with this issue by being a theory where your course of action only depends on your posterior distribution. You can base your actions only on what the universe actually looks like rather than having to pay attention to all possible universes. Given that it’s basically impossible to determine anything about what Kolmogorov priors actually say, being able to totally ignore parts of probability space that you have ruled out is a big deal.
… And this whole issue with not being able to self-modify beforehand. This only matters if your initial code affects the rest of the universe. To be more precise, this is only an issue if the problem is phrased in such a way that the universe you have to deal with depends on the code you are running. If we instantiate the Newcomb’s problem in the middle of the decision, UDT faces a world with the first box full while CDT faces a world with the first box empty. UDT wins because the scenario is in its favor before you even start the game.
If you really think that this is a big deal, you should try to figure out which decision theories are only created by universes that want to be nice to them and try using one of those.
Actually thinking about it this way, I have seen the light. CDT makes the faulty assumption that your initial state in uncorrelated with the universe that you find yourself in (who knows, you might wake up in the middle of Newcomb’s problem and find that whether or not you get $1000000 depends on whether or not your code is such that you would one-box in Newcomb’s problem). UDT goes some ways to correct this issue, but it doesn’t go far enough.
I would like to propose a new, more optimal decision theory. Call it ADT for Anthropic Decision Theory. Actually, it depends on a prior, so assume that you’ve picked out one of those. Given your prior, ADT is the decision theory D that maximizes the expected (given your prior) lifetime utility of all agents using D as their decision theory. Note how agents using ADT do provably better than agents using any other decision theory.
Note that I have absolutely no idea what ADT does in, well, any situation, but that shouldn’t stop you from adopting it. It is optimal after all.
Why does UDT lose this game? If it knows anti-Newcomb is much more likely, it will two-box on Newcomb and do just as well as CDT. If Newcomb is more common, UDT one-boxes and does better than CDT.
I guess my point is that it is nonsensical to ask “what does UDT do in situation X” without also specifying the prior over possible universes that this particular UDT is using. Given that this is the case, what exactly do you mean by “losing game X”?
Well, you can talk about “what does decision theory W do in situation X” without specifying the likelyhood of other situations, by assuming that all agents start with a prior that sets P(X) = 1. In that case UDT clearly wins the anti-newcomb scenario because it knows that actual newcomb’s “never happens” and therefore it (counterfactually) two-boxes.
The only problem with this treatment is that in real life P(anti-newcomb) = 1 is an unrealistic model of the world, and you really should have a prior for P(anti-newcomb) vs P(newcomb). A decision theory that solves the restricted problem is not necessarily a good one for solving real life problems in general.
Well, perhaps. I think that the bigger problem is that under reasonable priors P(Newcomb) and P(anti-Newcomb) are both so incredibly small that I would have trouble finding a meaningful way to approximate their ratio.
How confident are you that UDT actually one-boxes?
Also yeah, if you want a better scenario where UDT loses see my PD against 99% prob. UDT and 1% prob. CDT example.
Only if the adversary makes its decision to attempt extortion regardless of the probability of success. In the usual case, the winning move is to ignore extortion, thereby retroactively making extortion pointless and preventing it from happening in the first place. (Which is of course a strategy unavailable to CDT, who always gives in to one-shot extortion.)
And thereby the extortioner’s optimal strategy is to extort independently of the probably of success. Actually, this is probably true is a lot of real cases (say ransomware) where the extortioner cannot actually ascertain the probably of success ahead of time.
That strategy is optimal if and only if the probably of success was reasonably high after all. Otoh, if you put an unconditional extortioner in an environment mostly populated by decision theories that refuse extortion, then the extortioner will start a war and end up on the losing side.
Yes. And likewise if you put an unconditional extortion-refuser in an environment populated by unconditional extortionists.
I think this is actually the point (though I do not consider myself an expert here). Eliezer thinks his TDT will refuse to give in to blackmail, because outputting another answer would encourage other rational agents to blackmail it. By contrast, CDT can see that such refusal would be useful in the future, so it will adopt (if it can) a new decision theory that refuses blackmail and therefore prevents future blackmail (causally). But if you’ve already committed to charging it money, its self-changes will have no causal effect on you, so we might expect Modified CDT to have an exception for events we set in motion before the change.
This just means that TDT loses in honest one-off blackmail situations (in reality, you don’t give in to blackmail because it will cause other people to blackmail you whether or not you then self-modify to never give into blackmail again). TDT only does better if the potential blackmailers read your code in order to decide whether or not blackmail will be effective (and then only if your priors say that such blackmailers are more likely than anti-blackmailers who give you money if they think you would have given into blackmail). Then again, if the blackmailers think that you might be a TDT agent, they just need to precommit to using blackmail whether or not they believe that it will be effective.
Actually, this suggests that blackmail is a game that TDT agents really lose badly at when playing against each other. The TDT blackmailer will decide to blackmail regardless of effectiveness and the TDT blackmailee will decide to ignore the blackmail, thus ending in the worst possible outcome.
Assume it’s not a perfect clone, it can defect with probability p even if you cooperate. Then apply CDT. You get “defect” for any p>0. So it is reasonable to implicitly assume continuity and declare that CDT forces you to defect when p=0. However, if you apply CDT for the case p=0 directly, you get “cooperate” instead.
In other words, the conterfactual reasoning gets broken when the map CDT(p, PD) is not continuous at the point p=0.
I disagree. If the agent has a 95% probability of doing the same thing as me and a 5% chance of defecting, I still cooperate. (With 95% probability, most likely, because you gotta punish defectors.)
Indeed, consider the following game: You give me a program that must either output “give” or “keep”. I roll a 20 sided die. On a 20, I play your program against a program that always keeps its token. Otherwise, I play your program against itself. I give you the money that (the first instance of) your program wins. Are you willing to pay me $110 to play? I’d be happy to pay you $110 for this opportunity.
I don’t cooperate with myself because P(TheirChoice=Defect)=0, I cooperate with myself because I don’t reason as if p is independent from my action.
It’s not entirely clear what you’re saying, but I’ll try to take the simplest interpretation. I’m guessing that:
If you’re going to defect, your clone always defects.
If you’re going to cooperate, your clone cooperates with probability 1-p and defects with probability p
In that case, I don’t see how it is that you get “defect” for p>0; the above formulation gives “cooperate” for 0<=p<0.5.
Suppose you have to submit the source code of a program X, and I will play Y = “run X, then do what X did with probability 0.99 and the reverse with probability 0.01” against Y’ which is the same as Y but with a different seed for the RNG, and pay you according to how Y does.
Then “you” (i.e. Y) are not a perfect clone of your opponent (i.e. Y’).
What do you do?
You have to consider off-the-equilibrium-path behavior. If I’m the type of person who will always cooperate, what would happen if I went off-the-equilibrium-path and did defect even if my defecting is a zero probability event?
I’m trying to understand the difference between your statement and “1 is not equal 2, but what if it were?” and failing.
See trembling hand equilibrium.
Right, as I mentioned in my other reply, CDT is discontinuous at p=0. Presumably a better decision theory would not have such a discontinuity.
One possible interpretation of “if I always cooperate, what would happen if I don’t” is “what is the limit, as X approaches 1, of ‘if I cooperate with probability X, what would happen if I don’t’?”
This doesn’t reasonably map onto the 1=2 example.
Right. There seems to be a discontinuity, as the limit of CDT (p->0) is not CDT (p=0). I wonder if this is the root of the issue.
“1 is not equal 2, but what if it were?” = what if I could travel faster than the speed of light.
Off the equilibrium path = what if I were to burn a dollar.
Or things I can’t do vs things I don’t want to do.
In my mind “I’m the type of person who will always cooperate” means that there is no difference between the two in this case. Maybe you use a different definition of “always”?
I always cooperate because doing so maximizes my utility since it is better than all the alternatives. I always go slower than the speed of light because I have no alternatives.
You can consider it, but conditioned on the information that you are playing against your clone, you should assign this a very low probability of happening, and weight it in your decision accordingly.
Assume I am the type of person who would always cooperate with my clone. If I asked myself the following question “If I defected would my payoff be higher or lower than if I cooperated even though I know I will always cooperate” what would be the answer?
The answer would be ‘MOO’. Or ‘Mu’, or ‘moot’; they’re equivalent. “In this impossible counterfactual where I am self-contradictory, what would happen?”
Yes, it makes a little bit of sense to counterfactually reason that you would get $1000 more if you defected, but that is predicated on the assumption that you always cooperate. You cannot actually get that free $1000 because the underlying assumption of the counterfactual would be violated if you actually defected.
No, you don’t. This is a game where there are only two possible outcomes; DD and CC. CD and DC are defined to be impossible because the agents playing the game are physically incapable of making those outcomes occur.
EDIT: Maybe physically incapable is a bit strong. If they wanted to maximize the chance that they had unmatched outcomes, they could each flip a coin and take C if heads and D if tails, and would have a 50% chance of unmatching. But they still would both be playing the same precise strategy.
I don’t agree. Even if I’m certain I will not defect, I am capable of asking what would happen if I did, just as the real me both knows he won’t do “horrible thing” yet can mentally model what would happen if he did “horrible thing”. Or imagine an AI that’s programmed to always maximize its utility. This AI still could calculate what would happen if it followed a non-utility maximizing strategy. Often in game theory a solution requires you to calculate your payoff if you left the equilibrium path.
Yes, and part of the answer is “If I did defect, my clone would also defect.” You have a guarantee that both of you take the same actions because you think according to precisely identical reasoning.
What do you think will happen if clones play the centipede game?
Unclear, depends on the specific properties of the person being cloned. Unlike PD, the two players aren’t in the same situation, so they can’t necessarily rely on their logic being the same as their counterpart. How closely this would reflect the TDT ideal of ‘Always Push’ will depend on how luminous the person is; if they can model what they would do in the opposite situation, and are highly confident that their self-model is correct, they can reach the best result, but if they lack confidence that they know what they’d do, then the winning cooperation is harder to achieve.
Of course, if it’s denominated in money and is 100 steps of doubling, as implied by the Wikipedia page, then the difference in utility between $1 nonillion and $316 octillion is so negligible that there’s essentially no incentive to defect in the last round and any halfway-reasonable person will Always Push straight through the game. But that’s a degenerate case and probably not the version originally discussed.
What would you say about the following decision problem (formulated by Andy Egan, I believe)?
You have a strong desire that all psychopaths in the world die. However, your desire to stay alive is stronger, so if you yourself are a psychopath you don’t want all psychopaths to die. You are pretty sure, but not certain, that you’re not a psychopath. You’re presented with a button, which, if pressed, would kill all psychopaths instantly. You are absolutely certain that only a psychopath would press this button. Should you press the button or not?
It seems to me the answer is “Obviously not”, precisely because the “off-path” possibility that you’re a non-psychopath who pushes the button should not enter into your consideration. But the causal decision algorithm would recommend pushing the button if your prior that you are a psychopath is small enough. Would you agree with that?
If only a psychopath would push the button, then your possible non-psychopathic nature limits what decision algorithms you are capable of following.
Wouldn’t the fact that you’re even considering pushing the button(because if only a psychopath would push the button then it follows that a non-psychopath would never push the button) indicate that you are a psychopath and therefore you should not push the button?
Another way to put it is:
If you are a psychopath and you push the button, you die. If you are not a psychopath and you push the button, pushing the button would make you a psychopath(since only a psychopath would push), and therefore you die.
Pushing the button can’t make you a psychopath. You’re either already a psychopath or you’re not. If you’re not, you will not push the button, although you might consider pushing it.
Maybe I was unclear.
I’m arguing that the button will never, ever be pushed. If you are NOT a psychopath, you won’t push, end of story.
If you ARE A psychopath, you can choose to push or not push.
In all three cases the correct decision is not to push, therefore you should not push.
Shouldn’t you also update your belief towards being a psychopath on the basis that you have a strong desire that all psychopaths in the world die?
You can stipulate this out of the example. Let’s say pretty much everyone has the desire that all psychopaths die, but only psychopaths would actually follow through with it.
I don’t press. CDT fails here because (I think) it doesn’t allow you to update your beliefs based on your own actions.
Exactly what information CDT allows you to update your beliefs on is a matter for some debate. You might be interested in a paper by James Joyce (http://www-personal.umich.edu/~jjoyce/papers/rscdt.pdf) on the issue (which was written in response to Egan’s paper).
But then shouldn’t you also update your beliefs about what your clone will do based on your own actions in the clone PD case? Your action is very strong (perfect, by stipulation) evidence for his action.
Yes I should. In the psychopath case whether I press the button depends on my beliefs, in contrast in a PD I should defect regardless of my beliefs.
Maybe I misunderstand what you mean by “updating beliefs based on action”. Here’s how I interpret it in the psychopath button case: When calculating the expected utility of pushing the button, don’t use the prior probability that you’re a psychopath in the calculation, use the probability that you’re a psychopath conditional on deciding to push the button (which is 1). If you use that conditional probability, then the expected utility of pushing the button is guaranteed to be negative, no matter what the prior probability that you’re a psychopath is. Similarly, when calculating the expected utility of not pushing the button, use the probability that you’re a psychopath conditional on deciding not to push the button.
But then, applying the same logic to the PD case, you should calculate expected utilities for your actions using probabilities for your clone’s action that are conditional on the very action that you are considering. So when you’re calculating the expected utility for cooperating, use probabilities for your clone’s action conditional on you cooperating (i.e., 1 for the clone cooperating, 0 for the clone defecting). When calculating the expected utility for defecting, use probabilities for your clone’s action conditional on you defecting (0 for cooperating, 1 for defecting). If you do things this way, then cooperating ends up having a higher expected utility.
Perhaps another way of putting it is that once you know the clone’s actions are perfectly correlated with your own, you have no good reason to treat the clone as an independent agent in your analysis. The standard tools of game theory, designed to deal with cases involving multiple independent agents, are no longer relevant. Instead, treat the clone as if he were part of the world-state in a standard single-agent decision problem, except this is a part of the world-state about which your actions give you information (kind of like whether or not you’re a psychopath in the button case).
I agree with your first paragraph.
Imagine you are absolutely certain you will cooperate and that your clone will cooperate. You are still capable of asking “what would my payoff be if I didn’t cooperate” and this payoff will be the payoff if you defect and the clone cooperates since you expect the clone to do whatever you will do and you expect to cooperate. There is no reason to update my belief on what the clone will do in this thought experiment since the thought experiment is about a zero probability event.
The psychopath case is different because I have uncertainty regarding whether I am a psychopath and the choice I want to make helps me learn about myself. I have no uncertainty concerning my clone.
You are reasoning about an impossible scenario; if the probability of you reaching the event is 0, the probability of your clone reaching it is also 0. In order to make it a sensical notion, you have to consider it as epsilon probabilities; since the probability will be the same for both your and your clone, this gets you
To claim that you and your clone could take different actions is trying to make it a question about trembling-hand equilibria, which violates the basic assumptions of the game.
It’s common in game theory to consider off the equilibrium path situations that will occur with probability zero without taking a trembling hand approach.
I don’t think you use use “always” correctly.