Other posts in the series: Introduction, Double decrease, Preexistence deals, Full decision algorithms, Breaking acausal trade, Trade in different types of utility functions, Being unusual, and Summary.
A reminder that we won’t be looking at any “utilities I might have had before I knew who I was” scenarios.
This post is for illustrating a point about acausal trade: weakening the acausal trade network(s) in any way tends to reduce acausal trade more than linearly, as the traders cut back further on their trading. And the converse for strengthening the acausal trade network(s).
How to weaken the network
How could the acausal trade network be weakened? In potentially many ways. Greater uncertainty about the existence or the utilities of other agents, for instance. More agents who might defect from the trade, not have right utility function, or with who you can’t reach a deal because of negotiation breakdown.
Basically anything that lowers the expected number of agents acausally trading with you  and also causes those agents to similarly have a lower expectation on the number of agents trading with you.
Illustration
Take the case where \(N=2\), so there are only two possible agents, you (\(A_1\)) and one other (\(A_2\)), with utilities \(u_1\) and \(u_2\) respectively. Both agents are sure to exist, so \(q_1=q_2=1\).
Trade can’t happen unless there is some gain from trade  if it costs you more (in terms of \(u_1\)) to increase \(u_2\), than the gain in \(u_1\) that the other agent is capable of giving you in exchange, then there is no trade that can happen.
So suppose you can increase \(u_2\) quite easily initially, but it gets harder and harder as you increase it more. Specifically, if you’ve already increased \(u_2\) by \(x\), then it costs you, marginally, \(x\) to increase \(u_2\) further.
So the marginal cost is linear in \(x\); cost, here, always refers to the decrease in \(u_1\) needed to pay for the increase in \(u_2\).
Assume the other agent is in exactly the same situation, mirrored.
Then, since we’re assuming that the negotiations divide the gains from trade equally, then if you and the other agent have full knowledge of these facts and use a functional decision theory.
Then the rational acausal decision is for both of you to increase the utility of the other agent by \(1\), paying \(\int_0^1 x dx = 1/2\) utility each, and hence each gaining \(1/2\) utility total.
But now imagine that the probability of each agent existing is \(q=q_1=q_2\), and that \(q\) is not necessarily \(1\). You know you yourself exist, so put the probability of the other agent existing at \(q\) (note that this argument is robust to different types of anthropic reasoning, as it’s the change that happens when \(q\) varies that’s important).
Then the rational thing for both of you is to increase the other utility until the marginal cost of doing so reaches \(q\). Thus each agent increases the utility by \(q\), at a cost of \(\int_0^q xdx = q^2/2\). With probability \(q\), the other agent exists and will thus give you \(q\) utility. Thus the expected gain for each of you is \(q(q)q^2/2=q^2/2\).
The fact that this is quadratic in \(q\) rather than linear is the “double decrease” effect: as the expected size of the network goes down, the expected return for participation goes down as well, causing those in it to decrease their own participation, until an equilibrium is reached at a lower level.
