Notation

S_n,t is the state of agent n at time t.
S is the state of all agents across all time.
G(S) is the world utility.
Subworlds are sets of agents that make up an exhaustive partition of all agents.
- For each subworld w, all agents in it have the same subworld utility functions g_w(S)
- At one extreme each agent could be its own subworld.
A constraint-aligned system is one in which any change to the state of the agents in subworld w at time 0 will have no effect on the state of agents outside of w at times later than 0. (rarely happens).
A subworld-factored is one where a change at time 0 to the agents in w results in an increased vale for g_w(S) if and only if it results in an increased value for G(S).
The wonderful life utility (WL) for all agents in w is defined as G(S) - G(CL_w(S)) where CL_w(S) is the vector S modified by clamping the states of all agents in w, across all time, to an arbitrary fixed value (0).
- We can view it as the change in world utility has w not existed.
- It is purely mathematical. No assumption is being made that CL_w(S) could ever happen.
A constraint-aligned system with wonderful life subworld utilities is subworld-factored and, the authors have shown, will approach near-optimal values of the world utility!
Of course, constraint-aligned systems are very rare, so experiments are needed to determine how well WL performs under a not-so-aligned system.