Let $U_1,U_2,\ldots,U_n$ be $n\geq 2$ mutually independent Bernoulli random variables. There are two cases ofofaccor interest:
$$R(n,(p,q))-R^*(n,(\mathbf{p},\mathbf{q}))=\left(\frac{1}{2}+\frac{1}{2} \sum_{k=0}^t\binom{n}{k} \left[(1-q)^k q^{n-k}+p^k(1-p)^{n-k}\right]\right)-\left(\frac{1}{2}\left(\sum_{\{u_1,\ldots,u_n\}\in \mathcal{S}_0^*}\prod_{u_k=0}q_k \prod_{u_k=1}(1-q_k)+\sum_{\{u_1,\ldots,u_n\}\in \mathcal{S}_1^*}\prod_{u_k=0}(1-p_k)\prod_{u_k=1}p_k \right)\right)$$
$$R(n,(p,q))-R^*(n,(\mathbf{p},\mathbf{q}))=\left(\frac{1}{2}+\frac{1}{2} \sum_{k=0}^t\binom{n}{k} \left[(1-q)^k q^{n-k}+p^k(1-p)^{n-k}\right]\right)-\left(\frac{1}{2}\left(\sum_{\{u_1,\ldots,u_n\}\in \mathcal{S}_0^*}\prod_{u_k=0}q_k \prod_{u_k=1}(1-q_k)+\sum_{\{u_1,\ldots,u_n\}\in \mathcal{S}_1^*}\prod_{u_k=0}(1-p_k)\prod_{u_k=1}p_k \right)\right)$$
This function needs to be maximized over all convex continuous curves for the minimizing $(p,q)$ and $(\mathbf{p},\mathbf{q})$.
This function needs to be maximized over all convex continuous curves for the minimizing $(p,q)$ and $(\mathbf{p},\mathbf{q})$.
