Second-guessing Experts

Second-guessing experts work in the framework Prediction with expert advice. Each expert  announces a continuous function of the Forecaster's predictions. Therefore, the loss of each expert is determined by the Forecasterís prediction as well as by the outcome chosen by Reality. Second-guessing experts are a generalization of experts in the standard Protocol: a standard expert can be interpreted as a constant function. The protocol of the game is as follows. Protocol:
{$\quad$}Initialize {$L_0=0, L_0^k=0, k=1,2,\dots$}
{$\quad$}FOR {$t=1,2,\dots$}:
{$\qquad$}Experts announce {$\xi^k\in\Gamma\to\Gamma, k=1,2,\dots$}
{$\qquad$}Forecaster announces {$\gamma\in\Gamma$}
{$\qquad$}Reality announces {$\omega\in\Omega$}
{$\qquad$}Update cumulative loss {$L_t=L_{t-1} + \lambda(\omega,\gamma)$}, {$L_t(k)=L_{t-1}(k) + \lambda(\omega,\xi^k()), k=1,2,\dots$}

As it is proven by Chernov, 2010, it is possible to modify the Aggregating Algorithm or use directly the defensive forecasting technique to prove the same bounds on the learner's loss as in the standard protocol.