Given agents a1,a2 and numbers p1,p2 such that p1+p2=1, there is an aggregate agent called p1a1+p2a2 which means “agents a1 and a2 acting together as a group, in which the relative power of a1 versus a2 is the ratio of p1 to p2”. The group does not make decisions by combining their utility functions, but instead by negotiating or fighting or something.
Aggregation should be associative, so 13a1+23(12a2+12a3)=13a1+13a2+13a3=23(12a1+12a2)+13a3.
If you spell out all the associativity relations, you’ll find that aggregation of agents is an algebra over the operad of topological simplices. (See Example 2 https://arxiv.org/abs/2107.09581.)
Of course we still have the old VNM-rational utility-maximizing agents. But now we also have aggregates of such agents, which are “less Law-aspiring” than their parts.
In order to specify the behavior of an aggregate, we might need more data than the component agents ai and their relative power pi. In that case we’d use some other operad.
Ah, great! To fill in some of the details:
Given agents a1,a2 and numbers p1,p2 such that p1+p2=1, there is an aggregate agent called p1a1+p2a2 which means “agents a1 and a2 acting together as a group, in which the relative power of a1 versus a2 is the ratio of p1 to p2”. The group does not make decisions by combining their utility functions, but instead by negotiating or fighting or something.
Aggregation should be associative, so 13a1+23(12a2+12a3)=13a1+13a2+13a3=23(12a1+12a2)+13a3.
If you spell out all the associativity relations, you’ll find that aggregation of agents is an algebra over the operad of topological simplices. (See Example 2 https://arxiv.org/abs/2107.09581.)
Of course we still have the old VNM-rational utility-maximizing agents. But now we also have aggregates of such agents, which are “less Law-aspiring” than their parts.
In order to specify the behavior of an aggregate, we might need more data than the component agents ai and their relative power pi. In that case we’d use some other operad.