never sending it, we dene the following value:

$$\Delta^{T}(t_{0},i,\,\beta)\equiv E\left[\sum_{t=0}^{T-t_{0}}R^{t_{0}+t}\left|\Sigma_{i}^{t_{0}}=\sigma_{G},T_{G}=t_{0},K_{G}=i,\,b_{\bullet\bullet}^{t_{0}}\Sigma=\beta\right]\right.$$ $$\left.-\,E\left[\sum_{t=0}^{T-t_{0}}R^{t_{0}+t}\left|\Sigma_{i}^{t_{0}}=\text{null},\,T_{G}=t_{0},K_{G}=i,\,b_{\bullet\bullet}^{t_{0}}\Sigma=\beta\right]\right.\tag{13}$$

We assume that, for all times other than TG, the agents follow some 
ommuni
ation poli
y,
, that never spe
ies G. Thus, T measures the dieren
e in expe
ted reward that hinges on agent i's spe i de ision to send or not send G at time t0. Given this denition, it is lo ally optimal for agent i to send the spe ial message, G, at time t0, if and only if T  0. We dene the ommuni ation poli y, + , as the 
ommuni
ation poli
y following  for all agents at all times, ex
ept for agent i under belief state , when agent i sends message . With this denition, +G , is the poli
y under whi
h agent i ommuni ates the a hievement of G, and 

+null is the poli
y under whi
h it does not.

Therefore, we an alternatively des ribe agent i's de ision riterion as hoosing +G

over +null if and only if T  0.

Unfortunately, while Equation 13 identies an exa t riterion for lo ally optimal ommuni ation, this riterion is not yet operational. In other words, we an not dire tly implement it as a ommuni ation poli y for the agents. Furthermore, Equation 13 hides the underlying omplexity of the omputation involved, whi h is one of the key goals of our analysis.

Therefore, we use the COM-MTDP model to derive an operational expression of T  0.

For simpli ity, we dene notational shorthand for various sequen es and ombinations of values. We dene a partial sequen e of random variables, X<t
, to be the sequen e of random variables for all times before t: X0
, X1
, : : : , Xt1
. We make similar denitions for the other relational operators (i.e., X>t
, Xt
, et
.). The expression, (S)

T

, denotes the ross produ t over states of the world, QT
t=0 S, as distinguished from the time-indexed random variable, ST
, whi h denotes the value of the state at time T . The notation, st0
[t℄, spe ies the element in slot t within the ve tor st0
. We dene the fun tion, , as shorthand within our probability expressions. It allows us to ompa tly represent a parti ular subsequen e of world and agent belief states o urring, onditioned on the urrent situation, as follows:

$$\Pr\left(\Upsilon\left(\left\langle t,t^{\prime}\right\rangle,s,\mathbf{\beta_{\Phi^{\Sigma}}}\right)\right)\equiv\Pr(S^{\geq t,\leq t^{\prime}}=s,\mathbf{b_{0\Phi^{\Sigma}}}^{\geq t,\leq t^{\prime}}=\mathbf{\beta_{\Phi^{\Sigma}}}\left|T_{G}-t_{0},K_{G}-i,b_{1\Phi^{\Sigma}}^{t_{\Phi}}=\beta\right.)\tag{14}$$

$$\left(\,15\right)$$

Informally,  (ht; t0i ; s; ) represents the event that the world and belief states from time t through t 0 orrespond to the spe ied sequen es, s and , respe
tively, 
onditioned on agent i being the rst to know of G's a hievement at time t0 with a belief state, . We dene the fun tion, , to map a preommuni
ation belief state into the postommuni
ation belief state that arises from a ommuni ation poli y:

$$\beta_{\Sigma\bullet}(\beta_{\bullet\Sigma},\pi_{\alpha\Sigma})\equiv S E_{\alpha\Sigma\bullet}(\beta_{\bullet\Sigma},\pi_{\alpha\Sigma}(\beta_{\bullet\Sigma}))$$
(; )  SE(; ()) (15)
This denition of  is a well-dened fun
tion be
ause of the deterministi nature of the poli y, , and state-estimator fun
tion, SE.