Reviews: Dying Experts: Efficient Algorithms with Optimal Regret Bounds

The dying expert setting is interesting, it would be appreciated to give more examples. The overall writing is good and easy to follow. I have a simple question on the performance measure, ranking regret. In the definition of (1), authors claim \sigma t(\pi) is the first alive expert of ordering \pi in round t. So why do we need to specify the "first" alive expert, rather than the alive expert with the optimal performance?