by Stuart Armstrong 399 days ago | link | parent 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.

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