Theorem 7  If we assume that, upon achievement of G, no communication other than σ G
is possible, then the condition ∆ T (to,i,β) ≥ 0 holds if and only if:

Pr(Y((0, to), s ≤ to, β ≤ to )) ′ Pr ( Y ( to, T ), s ≥ to, β ≥ to )  Σ to = σ G, Y ( (0, to), s ≤ to, β ≤ to )) s=to[t], παλ (βΣ• (βαΣ ≥to[t], παΣ+σσ ))) Pr ( Y ( t 0 , T ), s ≥ t 0 , β ≥ t 0 )  ∑ t 0  = null, Y ( ( 0 , t 0 ), s ≤ t 0 , β ≤ t 0 ) s≥ to [t], παλ (βΣ - (βαΣ≥ to [t], παΣ+ null) ) ) Pr (Y({to, to), s, β)) R Σ(s, σ G ) (16) EG PEB
Proof: The complete proof of the following theorem appears in Online Appendix 1.

The definition of A T in Equation 13 is the difference between two expectations, where each expectation is a sum over the possible trajectories of the agent team. Ea trajectory must includes a sequence of possible world states, since the agents' reward at each point in time depends on the particular state of the world at that time. The agents' reward also depends on their actions (both domain- and communication-level). These actions are deterministic, given the agents' policies, παλ and πε, and their belief states.  Thus, in addition to summing over the possible states of the world, we must also sum over the possible states of the agents