 Definition 3.9  Given a game \(\Gamma = (N, S, \boldsymbol{u})\) together with a finite set of possible worlds \(\Omega\), the vector valued function \(f: \Omega \rightarrow S\) is a correlated n-tuple. If \(f(\omega) = (f_1 (\omega), \ldots ,f_n (\omega))\) denotes the components of \(f\) for the agents of \(N\), then agent \(k\)’s recommended strategy at \(\omega\) is \(f_k (\omega).\) \(f\) is an Aumann correlated equilibrium iff   \[ E(u_k \circ f) \ge E(u_k (f_{-k}, g_k)), \]   for each \(k \in N\) and for any function \(g_k\) that is a function of \(f_i\). 