Games had an important ceremonial and religious role, particularly in regards to the imperial cult. So, theoretically, the emperor was in charge of all games, even if he wasn’t in the city.

On a practical level, the urban prefect ran the show in the imperial era. This office grew in importance over time, particularly when the imperial seat was removed from Rome. He was basically the top official over the city and its immediate surroundings – kind of like a super mayor – and he would host the games as the direct representative of the emperor.

Some of the games had a very strong connection with various pagan rites, which caused problems as Christianity spread. However, the office of urban prefect for a long period was held by a pagan, even when Christianity became dominant. In this way problems with hosting the games were avoided.

The Colosseum is an interesting case, because later in the Empire individual senators appear to have become occasionally responsible for the games there. The urban prefect was more concerned with the spectacles in the Circus. But still the senators would undoubtedly have worked under the acquiescence of the urban prefect and the games would have been held as representatives of the emperor.

After the office of western emperor died out there are indications that senators hosted and presided over games in the Colosseum in their own names. Some even had their names carved on seats, claiming areas for themselves.

Source: Chenault, Rome Without Emperors: The Revival of a Senatorial City in the Fourth Century CE