The description I gave here about how to balance and negotiate between different utility functions, is a bit incomplete, as this post will show. Here I'll give more details on possible decision algorithms the agents could run.

Here, two agents $A_{1}$ and $A_{2}$ , who maximise $u_{1}$ and $u_{2}$ , exist with probabilities $q_{1}$ and $q_{2}$ , not necessarily independent. The joint probability that both agents exist is $r_{12}$ .

Waiting souls

In this perspective, we imagine that each utility function is represented by a disembodied entity, that negotiate the terms of acausal trade before the universe begins and before any agents exist or not. This is the Rawlsian veil of ignorance, that I said we'd ignore in the original post, with justifications presented here.

Now, however, we need to consider it, as a gateway to the case we want. How would the agents balance each other's utilities?

One possibility is that the agents assign equal weight to both utilities. In that case they will be both maximising $u_{1} + u_{2}$ . But this poses a continuity problem as the probability of any agent declines towards $0$ . So the best option seems to be to have the agents agree to maximise $q_{1} u_{1} + q_{2} u_{2}$ .

Then, in the situation presented in the previous post, both agents would increase the other utility until the marginal cost of doing so increased to either $q_{2} / q_{1}$ (for agent $A_{1}$ ) or $q_{1} / q_{2}$ (for agent $A_{2}$ ).

There is no "double decrease" in this situation.

Existing souls

Here we restrict ourselves to agents that actually exist. So, if for instance $r_{12} = 0$ (the two agents cannot both exist) then agent $A_{1}$ , should they exist, will have no compunction about not maximising $u_{2}$ in any way.

One way of modelling this is to go back to the "waiting souls" formalism, but replace $u_{i}$ with $u_{i}^{'} = u_{i} I_{i}$ where $I_{i}$ is the indicator variable on whether agent $A_{i}$ existed at any point in the universe. Thus all utilities depend on the existence of the agent that prefers them, in order to be maximised by anyone.

There is not longer a continuity issue with $u_{1}^{'} + u_{2}^{'}$ when the probabilities $q_{i}$ tend to zero, since low $q_{i}$ mean that changes to $u_{i}^{'}$ become smaller and smaller in expectation.

So, when maximising $u_{1}^{'} + u_{2}^{'}$ , the agent $A_{1}$ will consider that increases to $u_{2}^{'}$ have an effect that is $r_{12} / q_{1}$ times as large as increases to $u_{2}$ (while increases to $u_{1}$ and $u_{1}^{'}$ are identical from its perspective, since the agent exists). Thus it will increase $u_{2}$ until the marginal cost of further in

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Acausal Trade