In this post, I demonstrate that if we treat acausal trade as being analogous to ordinary trade (in the sense that it obeys supply and demand), the final result is the Nash bargaining solution.

Supply and demand of utility

Alice is an AI that wants to maximize apples. Bob is an AI that wants to maximize oranges. One day, they decide to trade.

Let's say Alice releases that by producing 3 fewer apples, she could produce 4 oranges. And Bob can produce 4 apples by forgoing 3 oranges. So Alice and Bob can achieve a better outcome if they both do this. Notice that this is, in essence, just ordinary trade. (Note that in the acausal context where Alice only believes Bob has a 50% chance of existing, the apples he produces only count for 50%. For simplicity, we will treat them as having physically met and both having a 100% chance of existing.)

How much more should Alice and Bob trade? We can imagine there being a supply and demand curve for apples, priced in oranges. Alice demands apples (by paying a price in oranges) and Bob supplies apples (by being paid a price in oranges). Where these curves meet is the optimal trade. In particular, the point consists of a price $p$ and a number of apples $x$. The number of oranges in the trade will be $y=px$.

The two properties of utility trading

This trade has the following two properties:

1. The "books balance": the amount of oranges provided by Alice are equal in value to the apples provide by Bob when compared with the price $p$.
2. The solution maximizes value: the trade maximizes the value when we treat an orange as being $p$ times as valuable as an apple. In particular, the trade is on the Pareto front. Note that it does not need be the only solution that maximizes the value. The reason the trade maximizes value is as follows: If there is a more valuable solution with more apples, than Bob's supply curve was too low (he could have offered more apples at the price $p$). If there is a superior solution with more oranges, the price was too high (Alice could have offered more oranges, thus lowering the price).

In practice, acausal trade is more than just trading physical objects. But I still think these two criteria are natural criteria for a satisfactory trade. If a solution gives one agent twice as many utils as the other, they should be willing to give up one of their utils to give their trading partner two utils (note that each agents utils should actually be thought of as different units in the sense of dimensional analysis). So we regard an acausal trade as resulting in an outcome such that there exists a price p that satisfies the two criteria. The price is not actually part of the outcome; we just require that such a price exists.

Proof that Nash bargaining is the unique solution satisfying these two properties

Theorem: A solution has some price p that satisfies the two criteria iff it is the Nash bargaining solution.

Proof: $(⟹)$ Let $x$ and $y$ be the utility gains to Alice and Bob respectively in some outcome. By condition 1, the price is $p=\frac{y}{x}$. The value is therefore $\frac{y}{x}x+y$.  Let's say that the Nash bargaining solution has utility gains $x+\Delta x$ and $y+\Delta y$. Utilizing mixed strategies, we also get a spectrum of solutions with utility gains $x+k\Delta x$ and $y+k\Delta y$ where $k$ ranges over $[0,1]$. If we take the derivative of $(x+k\Delta x)(y+k\Delta y)$ with respect to $k$ the result is $y\Delta x+x\Delta y+2k\Delta y\Delta x$. Since the Nash bargaining solution ($k=1$) maximizes the product, we get that $y\Delta x+x\Delta y+2\Delta y\Delta x\ge 0$.

$y\Delta x+x\Delta y+2\Delta y\Delta x\ge 0$

$y\Delta x+x\Delta y\ge -2\Delta y\Delta x$

Assuming $(x,y)$ and $(x+\Delta x,y+\Delta y)$ are different, $y\Delta x+x\Delta y>0$ (since both solutions are pareto efficient, $\Delta x$ and $\Delta y$ must have opposite signs).

$y(x+\Delta x)+x(y+\Delta y)>yx+xy$

$\frac{y}{x}(x+\Delta x)+(y+\Delta y)>\frac{y}{x}x+y$

And thus if the $(x,y)$ solution wasn't the Nash bargaining solution, the Nash bargaining solution is better according to the price p. Therefore, $(x,y)$ only satisfies the two criteria if it is the Nash bargaining solution. $\square$

$(⟸)$ Let's say the Nash bargaining solution has utility gains $x$ and $y$. Let $p=\frac{y}{x}$ (this satisfies condition 1). The value is therefore $\frac{y}{x}x+y$. Consider a different solution with utility gains $x+\Delta x$ and $y+\Delta y$. Utilizing mixed strategies, we also get a spectrum of solutions with utility gains $x+k\Delta x$ and $y+k\Delta y$ where $k$ ranges over $[0,1]$. If we take the derivative of $(x+k\Delta x)(y+k\Delta y)$ with respect to $k$ the result is $y\Delta x+x\Delta y+2k\Delta y\Delta x$. Since the Nash bargaining solution ($k=0$) maximizes the product, we get that $y\Delta x+x\Delta y\le 0$.

$y\Delta x+x\Delta y\le 0$

$y(x+\Delta x)+x(y+\Delta y)\le yx+xy$

$\frac{y}{x}(x+\Delta x)+(y+\Delta y)\le \frac{y}{x}x+y$

Thus the Nash bargaining solution is at least as good (when judged using price $p=\frac{y}{x}$) as any other solution, and thus meets criteria 2. $\square$

So there you have it. That's how you calculate a trade of utility!

Open questions

1. To determine the utility gains, you need a way to determine the "no trade" outcome. How do you do this? My first thought is to use the Nash equilibrium, but that gives counter-intuitive results in the ultimatum game, and it is also not unique. If you try minimax, you get a way to calculate acausal blackmail instead of acausal trade.

2. How does this generalize to more than two participants? I'm thinking that you will need to involve Shapely Values somehow when determining how much each agent should credit the others. See Shapley values: Better than counterfactuals.