I have considered the following decision rule:

1. Calculate the vector of "estimated expected loss" (denote it by $\widehat{el}(\mathbf{L},\hat p_0)$) consisting of elements corresponding to classification decisions $j=1,\dots,J$ (assigning class $j$ to $\hat y$) as $\widehat{el}(\mathbf{L},\hat p_0):=\mathbf{L}^\top \hat p_0$.
2. Find the minimal element of the vector and select the corresponding predicted class.

I wonder if tis is optimal.

I have considered the following decision rule:

1. Calculate the vector of "estimated expected loss" (denote it by $\widehat{el}(\mathbf{L},\hat p_0)$) consisting of elements corresponding to classification decisions $j=1,\dots,J$ (assigning class $j$ to $\hat y$) as $\widehat{el}(\mathbf{L},\hat p_0):=\mathbf{L}^\top \hat p_0$.
2. Find the minimal element of the vector and select the corresponding predicted class.

I wonder if this is optimal.