Stuart_Armstrong comments on Acausal trade: double decrease

Stuart_Armstrong 19 May 2017 11:57 UTC
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AF
I think it does do the double decrease for the known smaller network.

Take three agent $A_{1}$ , $A_{2}$ , and $A_{3}$ , with utilities $u_{1}$ , $u_{2}$ , and $u_{3}$ . Assume the indexes $i$ , $j$ , and $k$ are always distinct.

For each $A_{i}$ , they can boost $u_{j}$ at the cost described above in terms of $u_{i}$ .

What I haven’t really specified is the three-way synergy—can $A_{i}$ boost $u_{j} + u_{k}$ more efficiently that simply boosting $u_{j}$ and $u_{k}$ independently? In general yes (the two utilities $u_{j}$ and $u_{k}$ are synergistic with each other, after all), but let’s first assume there is zero three-way synergy.

Then each agent $A_{i}$ will sacrifice $1 / 2 + 1 / 2 = 1$ in $u_{i}$ to boost $u_{j}$ and $u_{k}$ each by $1$ . Overall, each utility function goes up by $1 + 1 - 1 = 1$ . This scales linearly with the size of the trade network each agent sees (excluding themselves): if there were two agents total, each utility would go up by $1 / 2$ , as in the top post example. And if there were $n + 1$ agents, each utility would go up by $n / 2$ .

However, if there are any three-way, four-way,..., or $n$ -way synergies, then the trade network is more efficient than that. So there is a double decrease (or double increase, from the other perspective), as long as there are higher-order synergies between the utilities.