I'm reading a paper Code-Based Game-Playing Proofs and the Security of Triple Encryption, and I encounter a question in section 4.3, which designs the following games:
Lets just focus on the medium two pairs of games, $R_1/C_1$ and $R_2/C_T$. $R_1$ and $C_1$ differents only in $R_1$ can query 2 more oracles $T$ and $T^{-1}$, and $C_T$ takes a parameter input $T$ rather than sample it. Then the author says choose the $S$ from the all possible $T$ which can maximise the Adv($A_T^{C_T},A_T^{R_2}$), we can have Adv($A^{C_1},A^{R_1}$) $\le$ Adv($A_S^{C_S},A_S^{R_2}$). Why does this inequality hold？ It seems that the author omits to bound the advantage between the ($R_1,C_1$) and ($R_2,C_T$). So how to bound this or how to explain the inequality Adv($A^{C_1},A^{R_1}$) $\le$ Adv($A_S^{C_S},A_S^{R_2}$)?
