Author's Note: Please let me know in the comments exactly what important background material I have missed, and *exactly* what I have misunderstood, and please try not to mind that everything here is written in the academic voice.

## Abstract: Timeless Decision Theory often seems like the correct way to handle many game-theoretical dilemmas, but has not quite been satisfactorily formalized and still handles certain problems the wrong way. We present an intuition that helps us extend Causal Decision Theory towards Timeless Decision Theory while adding rigor, and then formalize this intuition. Along the way, we describe how this intuition can guide both us and programmed agents in various Newcomblike games.

## Introduction

One day, a Time Lord called Omega drops out of the sky, walks up to me on the street, and places two boxes in front of me. One of these is opaque, the other is transparent and contains $1000. He tells me I can take either the opaque box alone, or both boxes, but that if and only if he predicted using his Time Lord Science I would take just the opaque box, it contains $1,000,000. He then flies away back to the his home-world of Gallifrey. I know that whatever prediction he made was/will be correct, because after all he is a Time Lord.

The established, gold-standard algorithm of Causal Decision Theory fails to win the maximum available sum of money on this problem, just as it fails on a symmetrical one-shot Prisoner's Dilemma. In fact, as human beings, we can say that CDT fails miserably, because while a programmed agent goes "inside the game" and proceeds to earn a good deal less money than it could, we human observers are sitting outside, carefully drawing outcome tables that politely inform us of just how much money our programmed agents are leaving on the table. While purely philosophical controversies abound in the literature about the original Newcomb's Problem, it is generally obvious from our outcome tables in the Prisoners' Dilemma that "purely rational" CDT agents would very definitely benefit by cooperating, and that actual human beings asked to play the game calculate outcomes as if forming coalitions rather than as if maximizing personal utility -- thus cooperating and winning. Even in the philosophical debates, it is generally agreed that one-boxers in Newcomb's Problem are, in fact, obtaining more money.

While some have attempted to define rationality as the outputs of specific decision algorithms, we hold with the school of thought that rationality means minimizing regret: a rational agent should select its decision algorithms in order to win as much as it will know it could have won ex-post-facto. Failing perfection, this optimum should be approximated as closely as possible.

Yudkowsky's Timeless Decision Theory approaches this problem by noting that many so-called decisions are actually outcomes from concurrent or separated instantiations of a single algorithm, that Timeless Decision Theory itself is exactly such an algorithm, and that many decisions (that actually are decisions in the sense that the algorithm deciding them is a utility-maximizing decision-theory) are acausally, timelessly connected. Agents running TDT will decide not as if they are determining one mere assignment to one mere variable in a causal graph but as if they're determining the output of the computation they implement, and thus of every logical node in the entire graph derived from their computation. However, it still has some kinks to work out:

Yudkowsky (2010) shows TDT succeeding in the original Newcomb’s problem. Unfortunately, deciding exactly when and where to put the logical nodes, and what conditional probabilities to place on them, is not yet an algorithmic process.

How would TDT look if instantiated in a more mature application? Given a very large and complex network, TDT would modify it in the following way: It would investigate each node, noting the ones that were results of instantiated calculations. Then it would collect these nodes into groups where every node in a group was the result of the same calculation. (Recall that we don’t know what the result is, just that it comes from the same calculation.) For each of these groups, TDT would then add a logical node representing the result of the abstract calculation, and connect it as a parent to each node in the group. Priors over possible states of the logical nodes would have to come from some other reasoning process, presumably the one that produces causal networks in the first place. Critically, one of these logical nodes would be the result of TDT’s own decision process in this situation. TDT would denote that as the decision node and use the resulting network to calculate the best action by equation 1.1.

The bolding is added by the present authors, as it highlights the issue we intend to address here. Terms like "timeless" and "acausal" have probably caused more confusion around Timeless Decision Theory than any other aspect of what is actually an understandable and reasonable algorithm. I will begin by presenting a clearer human-level intuition behind the correct behavior in Newcomb's Problem and the Prisoner's Dilemma, and will then proceed to formalize that intuition in Coq and apply it to sketch a more rigorously algorithmic Timeless Decision Theory. The formalization of this new intuition avoids problems of infinite self-reference or infinite recursion in reasoning about the algorithms determining decisions of oneself or others.

## Timeless decisions are actually entangled with each-other

The kind of apparent retrocausality present in Newcomb's Problem makes no intuitive sense whatsoever. Not only our intuitions but all our knowledge of science tell us that (absent the dubious phenomenon of closed timelike curves) causal influences always and only flow from the past to the future, never the other way around. Nonetheless, in the case of Newcomb-like problems, it has been seriously argued that:

the Newcomb problem cannot but be retrocausal, if there is genuine evidential dependence of the predictor’s behaviour on the agent’s choice, from the agent’s point of view.

We do not believe in retrocausality, at least not as an objective feature of the world. Any subjectively apparent retrocausality, we believe, must be some sort of illusion that reduces to genuine, right-side-up causality. Timeless or acausal decision-making resolves the apparent retrocausality by noticing that different "agents" in Newcomblike problems are actually reproductions of the same algorithm, and that they can thus be logically correlated without any direct causal link.

We further prime our intuitions about Newcomb-like problems with the observation that CDT-wielding Newcomb players who bind themselves to a precommitment to one-box before Omega predicts their actions will win the $1,000,000:

At t = 0 you can take a pill that turns you into a “one boxer”. The pill will lead the mad scientist to predict (at t = ½) that you will take one box, and so will cause you to receive £1,000,000 but will also cause you to leave a free £1,000 on the table at t = 1. CDT tells you to take the pill at t = 0: it is obviously the act, among those available at t = 0, that has the best overall causal consequences.

The "paradox", then, lies in how the CDT agent comes to believe that their choice is completely detached from which box contains how much money, when in fact Omega's prediction of their choice was accurate, and directly caused Omega to place money in boxes accordingly, all of this despite no retrocausality occurring. Everything makes perfect sense prior to Omega's prediction.

What, then, goes wrong with CDT? CDT agents will attempt to cheat against Omega: to be predicted as a one-boxer and then actually take both boxes. If given a way to obtain more money by precommitting to one-boxing, they will do so, but will subsequently feel regret over having followed their precommitment and "irrationally" taken only one box when both contained money. They may even begin to complain about the presence or absence of free will, as if this could change the game and enable their strategy to actually work.

When we cease such protestations and accept that CDT behaves irrationally, the real question becomes: which outcomes are genuinely possible in Newcomb's Problem, which outcomes are preferable, and why does CDT fail to locate these?

Plainly if we believe that Omega has a negligible or even null error rate, then in fact only two outcomes are possible:

- Our agent is predicted to take both boxes, and does so, receiving only $1000 since Omega has not filled the opaque box.
- Our agent is predicted to take the opaque box, which Omega fills, and the agent does take the opaque box, receiving $1,000,000.

Plainly, $1 million is a greater sum than $1000, and the former outcome state is thus preferable to the latter. We require an algorithm that can search out and select this outcome based on general principles, in any Newcomblike game rather than based on special-case heuristics.

Whence, then, a causal explanation of what to do? The authors' intuition was sparked by a bit of reading about the famously "spooky" phenomenon of quantum entanglement, also sometimes theorized to involve retrocausality. Two particles interact and become entangled; from then on, their quantum states will remain correlated until measurement collapses the wave-function of one particle or the other. Neither party performing a measurement will ever be able to tell which measurement took place first in time, but both measurements will always yield correlated results. This occurs despite the fact that quantum theory is confirmed to have no hidden variables, and even when general relativity's light-speed limit on the transmission of information prevents the entangled particles from "communicating" any quantum information. A paradox is apparent and most people find it scientifically unaesthetic.

In reality, there is no paradox at all. All that has happened is that the pair of particles are in quantum superposition together: their observables are mutually governed by a single joint probability distribution. The measured observable states do not go from "randomized" to "correlated" as the measurement is made. The measurement only "samples" a single classical outcome governing both particles from the joint probability distribution that is actually there. The joint probability distribution was actually caused by the 100% local and slower-than-light interaction that entangled the two particles in the first place.

Likewise for Newcomb's Problem in decision theory. As the theorists of precommitment had intuited, the outcome is not actually caused when the CDT agent believes itself to be making a decision. Instead, the outcome was caused when Omega measured the agent and predicted its choice ahead of time: the state of the agent at this time causes both Omega's prediction and the agent's eventual action.

We thus develop an intuition that like a pair of particles, the two correlated decision processes behind Omega's prediction and behind the agent's "real" choice are in some sense entangled: correlated due to a causal interaction in their mutual past. All we then require to win at Newcomb's Problem is a rigorous conception of such entanglement and a way of handling it algorithmically to make regret-minimizing decisions when entangled.

## Formalized decision entanglement

Let us begin by assuming that an agent can be defined as a function from a set of Beliefs and a Decision to an Action. There will not be very much actual proof-code given here, and what is given was written in the Coq proof assistant. The proofs, short though they be, were thus mechanically checked before being given here; "do try this at home, kids."

Definition Agent (Beliefs Decision Action: Type) : Type := Beliefs -> Decision -> Action.

We can then broaden and redefine our definition of decision entanglement as saying, essentially, "Two agents are entangled when either one of them would do what the other is doing, were they to trade places and thus beliefs but face equivalent decisions." More simply, if a certain two agents are entangled over a certain two equivalent decisions, any differences in what decisions they actually make arise from differences in beliefs.

Inductive entangled {Beliefs Decision Action} (a1 a2: Agent Beliefs Decision Action) d1 d2 :=

| ent : (forall (b: Beliefs), a1 b d1 = a2 b d2) -> d1 = d2 -> entangled a1 a2 d1 d2.

This kind of entanglement can then, quite quickly, be shown to be an equivalence relation, thus partitioning the set of all logical nodes in a causal graph into Yudkowsky's "groups where every node in a group was the result of the same calculation", with these groups being equivalence classes.

Theorem entangled_reflexive {B D A} : forall (a: Agent B D A) d,

` entangled a a d d.`

Proof.

` intros.`

` constructor.`

` intros. reflexivity. reflexivity.`

Qed.

Theorem entangled_symmetric {B D A}: forall (a1 a2: Agent B D A) d1 d2,

` entangled a1 a2 d1 d2 ->`

` entangled a2 a1 d2 d1.`

Proof.

` intros.`

` constructor;`

` induction H;`

intros; symmetry.

` apply e. apply e0.`

Qed.

Theorem entangled_transitive {B D A}: forall (a1 a2 a3: Agent B D A) d1 d2 d3,

` entangled a1 a2 d1 d2 ->`

` entangled a2 a3 d2 d3 ->`

` entangled a1 a3 d1 d3.`

Proof.

` intros a1 a2 a3 d1 d2 d3 H12 H23.`

` constructor;`

induction H12; induction H23; subst.

` intros b. rewrite e. rewrite e1.`

` reflexivity. reflexivity.`

Qed.

Actually proving that this relation holds simply consists of proving that two agents given equivalent decisions will always decide upon the same action (similar to proving program equilibrium) no matter what set of arbitrary beliefs is given them -- hence the usage of a second-order forall. Proving this does not require actually running the decision function of either agent. Instead, it requires demonstrating that the abstract-syntax trees of the two decision functions can be made to unify, up to the renaming of universally-quantified variables. This is what allows us to prove the entanglement relation's symmetry and transitivity: our assumptions give us rewritings known to hold over the universally-quantified agent functions and decisions, thus letting us employ unification as a proof tool without knowing what specific functions we might be handling.

Thanks to employing the unification of syntax trees rather than the actual running of algorithms, we can conservatively extend Causal Decision Theory with logical nodes and entanglement to adequately handle timeless decision-making, without any recourse to retrocausality nor to the potentially-infinitely loops of Sicilian Reasoning. (Potential applications of timeless decision-making to win at Ro Sham Bo remain an open matter for the imagination.)

Decision-theoretically, since our relation doesn't have to know anything about the given functions other than (forall (b: Beliefs), a1 b d = a2 b d), we can test whether our relationship holds over any two logical/algorithm nodes in an arbitrary causal graph, since all such nodes can be written as functions from their causal inputs to their logical output. We thus do not need a particular conception of what constitutes an "agent" in order to make decisions rigorously: we only need to know what decision we are making, and where in a given causal graph we are making it. From there, we can use simple (though inefficient) pairwise testing to find the equivalence class of all logical nodes in the causal graph equivalent to our decision node, and then select a utility-maximizing output for each of those nodes using the logic of ordinary Causal Decision Theory.

The slogan of a Causal Decision Theory with Entanglement (CDT+E) can then be summed up as, "select the decision which maximizes utility for the equivalence class of nodes to which I belong, with all of us acting and exerting our causal effects in concert, across space and time (but subject to our respective belief structures)."

The performance of CDT with entanglement on common problems

While we have not yet actually programmed a software agent with a CDT+E decision algorithm over Bayesian causal graphs (any readers who can point us to a corpus of preexisting source code for building, testing, and reasoning about decision-theory algorithms will be much appreciated, as we can then replace this wordy section with a formal evaluation), we can provide informal but still somewhat rigorous explanations of what it should do on several popular problems and why.

First, the simplest case: when a CDT+E agent is placed into Newcomb's Problem, provided that the causal graph expresses the "agenty-ness" of whatever code Omega runs to predict our agent's actions, both versions of the agent (the "simulated" and the "real") will look at the causal graph they are given, detect their entanglement with each-other via pairwise checking and proof-searching (which may take large amounts of computational power), and subsequently restrict their decision-making to choose the best outcome over worlds where they both make the same decision. This will lead the CDT+E agent to take only the opaque box (one-boxing) and win $1,000,000. This is the same behavior for the same reasons as is obtained with Timeless Decision Theory, but with less human intervention in the reasoning process.

Provided that the CDT+E agent maintains some model of past events in its causal network, the Parfit’s Hitchhiker Problem trivially falls to the same reasoning as found in the original Newcomb’s Problem.

Furthermore, two CDT+E agents placed into the one-shot Prisoners' Dilemma and given knowledge of each-other's algorithms as embodied logical nodes in the two causal graphs will notice that they are entangled, choose the most preferable action over worlds in which both agents choose identically, and thus choose to cooperate. Should a CDT+E agent playing the one-shot Prisoner's Dilemma against an arbitrary agent with potentially non-identical code fail to prove entanglement with its opponent (fail to prove that its opponent's decisions mirror its own, up to differences in beliefs), it will refuse to trust its opponent and defect. A more optimal agent for the Prisoners' Dilemma would in fact demand from itself a proof that either it is or is not entangled with its opponent, and would be able to reason specifically about worlds in which the decisions made by two nodes cannot be the same. Doing so requires the Principle of the Excluded Middle, an axiom not normally used in the constructive logic of automated theorem-proving systems.

Lastly, different versions of CDT+E yield interestingly different results in the Counterfactual Mugging Problem. Let us assume that the causal graph given to the agent contains three logical nodes: the actual agent making its choice to pay Omega $100, Omega's prediction of what the agent will do in this case, and Omega's imagination of the agent receiving $1,000 had the coin come up the other way. The version of the entanglement relation here quantifies over decisions themselves at the first-order level, and thus the two versions of the agent who are dealing with the prospect of giving Omega $100 will become entangled. Despite being entangled, they will see no situation of any benefit to themselves, and will refuse to pay Omega the money. However, consider the stricter definition of entanglement given below:

Inductive strongly_entangled {Beliefs Decision Action} (a1 a2: Agent Beliefs Decision Action) :=

| ent : (forall (b: Beliefs) (d: Decision), a1 b d = a2 b d) -> entangled a1 a2.

This definition says that two agents are strongly entangled when they yield the same decisions for every possible pair of beliefs and decision problem that can be given to them. This continues to match our original intuition regarding decision entanglement: that we are dealing with the same algorithm (agent), with the same values, being instantiated at multiple locations in time and space. It is somewhat stronger than the reasoning behind Timeless Decision Theory: it can recognize two instantiations of the same agent that face two different decisions, and enable them to reason that they are entangled with each-other.

Under this stronger version of the entanglement relation (whose proofs for being an equivalence relation are somewhat simpler, by the way), a CDT+E agent given the Counterfactual Mugging will recognize itself as entangled not only with the predicted factual version of itself that might give Omega $100, but also with the predicted counterfactual version of itself that receives $1000 on the alternate coin flip. Each instance of the agent then independently computes the same appropriate tuple of output actions to maximize profit across the entire equivalence class (namely: predicted-factual gives $100, real-factual gives $100, predicted-counterfactual receives $1000).

Switching entirely to the stronger version of entanglement would cause a CDT+E agent to lose certain games requiring cooperation with other agents that are even trivially different (for instance, if one agent likes chocolate and the other hates it, they are not strongly entangled). These games remain winnable with the weaker, original form of entanglement.

Future research

Future research could represent the probabilistic possibility of entanglement within a causal graph by writing down multiple parallel logical/algorithm nodes as children of the same parent, each of which exists and acts with a probability conditional on the outcome of the parent node. A proof engine extended with probabilities over logical sentences (which, to the authors' knowledge, is not yet accomplished for second-order constructive logics of the kind used here) could also begin to assign probabilities to entanglement between logical/algorithm nodes. These probabilistic beliefs can then integrate into the action-selection algorithm of Causal Decision Theory just like any other probabilistic beliefs; the case of pure logic and pure proof from axioms merely constitutes assigning a degenerate probability of 1.0 to some belief.

Previous researchers have noted that decision-making over probabilistic acausal entanglement with other agents can be used to represent the notion of "universalizability" from Kantian deontological ethics. We note that entanglements with decision nodes in the past and future of a single given agent actually lead to behavior not unlike a "virtue ethics" (that is, the agent will start trying to enforce desirable properties up and down its own life history). When we begin to employ probabilities on entanglement, the Kantian and virtue-ethical strategies will become more or less decision-theoretically dominant based on the confidence with which CDT+E agents believe they are entangled with other agents or with their past and future selves.

Acausal trade/cooperation with agents other than the given CDT+E agent itself can also be considered, at least under the weaker definition of entanglement. In such cases, seemingly undesirable behaviors such as subjection to acausal versions of Pascal's Mugging could appear. However, entanglements (whether Boolean, constructive, or probabilistically believed-in) occur between logical/decision nodes in the causal graph, which are linked by edges denoting conditional probabilities. Each CDT+E agent will thus weight the other in accordance with their beliefs about the probability mass of causal link from one to the other, making acausal Muggings have the same impact on decision-making as normal ones.

The discovery that games can have different outcomes under different versions of entanglement leads us to believe that our current concept of entanglement between agents and decisions is incomplete. We believe it is possible to build a form of entanglement that will pay Omega in the Counterfactual Mugging without trivially losing at the Prisoners’ Dilemma (as strong entanglement can), but our current attempts to do so sacrifice the transitivity of entanglement. We do not yet know if there are any game-theoretic losses inherent in that sacrifice. Still, we hope that further development of the entanglement concept can lead to a decision theory that will more fully reflect the "timeless" decision-making intuition of retrospectively detecting rational precommitments and acting according to them in the present.

CDT+E opens up room for a fully formal and algorithmic treatment of the "timeless" decision-making processes proposed by Yudkowsky, including acausal "communication" (regarding symmetry or nonsymmetry) and acausal trade in general. However, like the original Timeless Decision Theory, it still does not actually have an algorithmic process for placing the logical/decision nodes into the causal graph -- only for dividing the set of all such nodes into equivalence classes based on decision entanglement. Were such an algorithmic process to be found, it could be used by an agent to locate itself within its model of the world via the stronger definition of entanglement. This could potentially reduce the problem of naturalizing induction to the subproblems of building a causal model that contains logical or algorithmic nodes, locating the node in the present model whose decisions are strongly entangled with those of the agent, and then proceeding to engage in "virtue ethical" planning for near-future probabilistically strongly-entangled versions of the agent's logical node up to the agent's planning horizon.

Acknowledgements

The authors would like to thank Joshua and Benjamin Fox for their enlightening lectures on Updateless Decision Theory, and to additionally thank Benjamin Fox in specific for his abundant knowledge, deep intuition and clear guidance regarding acausal decision-making methods that actually win. Both Benjamin Fox and David Steinberg have our thanks for initial reviewing and help clarifying the text.