UDT is “updateless” about its utility function

Updateless decision theory (UDT) (or some variant thereof) seems to be widely accepted as the best current solution to decision theory by MIRI researchers and LessWrong users. In this short post, I outline one potential implication of being completely updateless. My intention is not to refute UDT, but to show that:

  1. It is not clear how updateless one might want to be, as this could have unforeseen consequences.
  2. If one endorses UDT, one should also endorse superrational cooperation on a very deep level.

My argument is simple, and draws on the idea of multiverse-wide superrational cooperation (MSR), which is a form of acausal trade between agents with correlated decision algorithms. Thinking about MSR instead of general acausal trade has the advantage that it seems conceptually easier, while the conclusions gained should hold in the general case as well. Nevertheless, I am very uncertain and expect the reality of acausal cooperation between AIs to look different from the picture I draw in this post.

Suppose humans have created a friendly AI with a CEV utility function and UDT as its decision theory. This version of UDT has solved the problem of logical counterfactuals and algorithmic correlation, and can readily spot any correlated agent in the world. Such an AI will be inclined to trade acausally with other agents—agents in parts of the world it does not have causal access to. This is, for instance, to achieve gains from comparative advantages given empirical circumstances, and to exploit diminishing marginal returns of pursuing any single value system at once.

For the trade implied by MSR, the AI does not have to simulate other agents and engage in some kind of Löbian bargain with them. Instead, the AI has to find out whether the agents’ decision algorithms are functionally equivalent to the AI’s decision algorithm, it has to find out about the agents’ utility functions, and it has to make sure the agents are in an empirical situation such that trade benefits both parties in expectation. (Of course, to do this, the AI might also have to perform a simulation.) The easiest trading step seems to be the one with all other agents using updateless decision theory and the same prior. In this context, it is possible to neglect many of the usual obstacles to acausal trade. These agents share everything except their utility function, so there will be little if any “friction”—as long as the compromise takes differences between utility functions into account, the correlation between the agents will be perfect. It would get more complicated if the versions of UDT diverged a bit, and if the priors were slightly different. (More on this later.) I assume here that the agents can find out about the other agents’ utility functions. Although these are logically determined by the prior, the agents might be logically uncertain, and calculating the distribution of utility functions of UDT agents might be computationally expensive. I will ignore this consideration here.

A possible approach to this trade is to effectively choose policies based on a weighted sum of the utility functions of all UDT agents in all the possible worlds contained in the AI’s prior (see Oesterheld 2017, section 2.8 for further details). Here, the weights will be assigned such that in expectation, all agents will have an incentive to pursue this sum of utility functions. It is not exactly clear how such weights will be calculated, but it is likely that all agents will adopt the same weights, and it seems clear that once this weighting is done based on the prior, it won’t change after finding out which of the possible worlds from the prior is actual (Oesterheld 2017, section 2.8.6). If all agents adopt the policy of always pursuing a sum of their utility functions, the expected marginal additional goal fulfillment for all AIs at any point in the future will be highest. The agents will act according to the “greatest good for the greatest number.” Any individual agent won’t know whether they will benefit in reality, but that is irrelevant from the updateless perspective. This becomes clear if we compare the situation to thought experiments like the Counterfactual Mugging. Even if in the actual world, the AI cannot benefit from engaging in the compromise, then it was still worth it from the prior viewpoint, since (given sufficient weight in the sum of utility functions) the AI would have stood to gain even more in another, non-actual world.

If the agents are also logically updatelessness, this reduces the information the weights of the agents’ utility functions are based on. There probably are many logical implications that could be drawn from an empirical prior and the utility functions about aspects of the trade—e.g., that the trade will benefit only the most common utility functions, that some values won’t be pursued by anyone in practice, etc.—that might be one logical implication step away from a logical prior. If the AI is logically updateless, it will always perform the action that it would have committed to before it got to know about these implications. Of course, logical updatelessness is an unresolved issue, and its implications for MSR will depend on possible solutions to the problem.

In conclusion, in order to implement the MSR compromise, the AI will start looking for other UDT agents in all possible (and, possibly, impossible) worlds in its prior. It will find out about their utility functions and calculate a weighted sum over all of them. This is what I mean by the statement that UDT is “updateless” about its utility function: no matter what utility function it starts out with, its own function might still have negligible weight in the goals the UDT AI will pursue in practice. At this point, it becomes clear that it really matters what this prior looks like. What is the distribution of the utility functions of all UDT agents given the universal prior? There might be worlds less complex than the world humans live in—for instance, a cellular automaton, such as Rule 110 or Game of Life, with a relatively simple initial state—which still contain UDT agents. Given that these worlds might have a higher prior probability than the human world, they might get a higher weight in the compromise utility function. The AI might end up maximizing the goal functions of the agents in the simplest worlds.

Is updating on your existence a sin?

One of the features of UDT is that it does not even condition the prior on the agent’s own existence—when evaluating policies, UDT also considers their implications in worlds that do not contain an instantiation of the agent, even though by the time the agent thinks its first thought, it can be sure that these worlds do not exist. This might not be a problem if one assigns high weight to a modal realism/Tegmark Level 4 universe anyway. An observation can never distinguish between a world in which all worlds exist, and one in which only the world featuring the current observation exists. So if the measure of all the “single worlds” is small, then updating on existence won’t change much.

Suppose that this is not the case. Then there might be many worlds that can already be excluded as non-actual based on the fact that they don’t contain humans. Nevertheless, they might contain UDT agents with alien goals. This poses a difficult choice: Given UDT’s prior, the AI will still cooperate with agents living in non-actual (and impossible, if the AI is logically updatelessness) worlds. This is because given UDT’s prior, it could have been not humans, but these alien agents, that turned out actual—in which case they could have benefited humans in return. On the other hand, if the AI is allowed to condition on such information, then it loses in a kind of counterfactual prisoner’s dilemma:

  • Counterfactual prisoner’s dilemma: Omega has gained total control over one universe. In the pursuit of philosophy, Omega flips a fair coin to determine which of two agents she should create. If the coin comes up heads, Omega will create a paperclip maximizer. If it comes up tails, she creates a perfectly identical agent, but with one difference: the agent is a staple maximizer. After the creation of these agents, Omega hands either of them total control over the universe and lets them know about this procedure. There are gains from trade: producing both paperclips and staples creates 60% utility for both of the agents, while producing only one of those creates 100% for one of the agents. Hence, both agents would (in expectation) benefit from a joint precommitment to a compromise utility function, even if only one of the agents is actually created. What should the created agent do?

If the agents condition on their existence, then they will not gain as much in expectation as they could otherwise expect to gain before the coin flip (when neither of the agents existed). I have chosen this thought experiment because it is not confounded by the involvement of simulated agents, a factor which could lead to anthropic uncertainty and hence make the agents more updateless than they would otherwise be.

UDT agents with differing priors

What about UDT agents using differing priors? For simplicity, I suppose there are only two agents. I also assume that both agents have equal capacity to create utilons in their universes. (If this is not the case, the weights in the resulting compromise utility function have to be adjusted.) Suppose both agents start out with the same prior, but update it on their own existence—i.e., they both exclude any worlds that don’t contain an instantiation of themselves. This posterior is then used to select policies. Agent B can’t benefit from any cooperative actions by agent A in a world that only exists in agent A’s posterior. Conversely, agent A also can’t benefit from agent B in worlds that agent A doesn’t think could be actual anymore. So the UDT policy will recommend pursuing a compromise function only in worlds lying in the intersection of worlds that exist in both agent’s posteriors. If either agent updates that they are in some of the worlds to which the other agent assigns approximately zero probability, then they won’t cooperate.

More generally, if both agents know which world is actual, and this is a world which they both inhabit, then it doesn’t matter which prior they used to select their policies. (Of course, this world must have nonzero probability in both of their priors; otherwise they wouldn’t ever update that said world is actual.) From the prior perspective, for agent A, every sacrificed utilon in this world is weighted by its prior measure of the world. Every gained utilon from agent B is also weighted by the same prior measure. So there is no friction in this compromise—if both agents decide between action a which gives themselves d utilons, and an action b which gives the other agent c utilons, then any agent will prefer option b iff c divided by this agent’s prior measure of the world is greater than d divided by the same prior measure, so iff c is greater than d. Given that there is a way to normalize both agents’ utility functions, pursuing a sum of those utility functions seems optimal.

We can even expand this to the case wherein the two agents have any differing priors with a nonempty intersection between the corresponding sets of possible worlds. In expectation, the policy that says: “if any world outside the intersection is actual: don’t compromise; if any world from the intersection is actual: do the standard UDT compromise, but use the posterior distribution in which all worlds outside the intersection have zero probability for policy selection” seems best. When evaluating this policy, both agents can weight both utilons sacrificed for others, as well as utilons gained from others, in any of the worlds from the intersection by the measure of the entire intersection in their own respective priors. This again creates a symmetrical situation with a 1:1 trade ratio between utilons sacrificed and gained.

Another case to consider is if the agents also distribute the relative weights between the worlds in the intersection differently. I think that this does not lead to asymmetries (in the sense that conditional on some of the worlds being actual, one agent stands to gain and lose more than the other agent). Suppose agent A has 30% on world S1, and 20% on World S2. Agent B, on the other hand, has 10% on world S1 and 20% on world S2. If both agents follow the policy of pursuing the sum of utility functions, given that they find themselves in either of the two shared worlds, then, ceteris paribus, both will in expectation benefit to an equal degree. For instance, let c1 (c2) be the amount of utilons either agent can create for the other agent in world S1 (S2), and d1 (d2) the respective amount agents can create for themselves. Then agent A gets either 0.3×c1+0.2×c2 or 0.3×d1+0.2×d2, while B chooses between 0.1×c1+0.2×c2 and 0.1×d1+0.2×d2. Here, it’s not the case that A prefers cooperating iff B prefers cooperating. But assuming that in expectation, c1 = c2 as well as d1 = d2, this leads to a situation where both prefer cooperation iff c1 > d1. It follows that just pursuing a sum of both agents’ utility functions is, in expectation, optimal for both agents.

Lastly, consider a combination of non-identical priors with empirical uncertainty. For UDT, empirical uncertainty between worlds translates into anthropic uncertainty about which of the possible worlds the agent inhabits. In this case, as expected, there is “friction”. For example, suppose agent A assigns p to the intersection of the worlds in both agents’ priors, while agent B assigns p/q. Before they find out whether one of the worlds from the intersection or some other world is actual, the situation is the following: B can benefit from A’s cooperation in only p/q of the worlds. A can benefit in p of the worlds from B, but for everything A does, this will only mean p/q as much to agent B. Now each agent can again either create d utilons for themselves, or perform a cooperative action that gives c utilons to the other agent in the world where the action is performed. Given uncertainty about which world is actual, if both agents choose cooperation, agent A receives c×p utilons in expectation, while agent B receives c×p/q utilons in expectation. Defection gives both agents d utilons. So for cooperation to be worth it, c×p and c×p/q both have to be greater than d. If this is the case, then if p is unequal to p/q, both agents’ gains from trade are still not equal. This appears to be a bargaining problem that doesn’t solve as easily as the examples from above.

Conclusion

I actually endorse the conclusion that humans should cooperate with all correlating agents. Although humans’ decision algorithms might not correlate with as many other agents, and they might not be able to compromise as efficiently as super-human AIs, humans should nevertheless pursue some multiverse-wide sum of values. What I’m uncertain about is how far updatelessness should go. For instance, it is not clear to me which empirical and logical evidence humans should and shouldn’t take into account when selecting policies. If an AI does not start out with the knowledge that humans possess but instead uses the universal prior, then it might perform actions that seem irrational given human knowledge. Even if observations are logically inconsistent with the existence of a fellow cooperation partner (i.e., in the updated distribution, the cooperation partner’s world has zero probability), then UDT might still cooperate with and possibly adopt that partner’s values. I doubt at this point whether everyone still agrees with the hypothesis that UDT always achieves the highest utility.

Acknowledgements

I thank Caspar Oesterheld, Max Daniel, Lukas Gloor, and David Althaus for helpful comments on a draft of this post, and Adrian Rorheim for copy editing.

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