I have no idea what other entertainment facilities (if any) the Hippodrome had historically, but wealthier patrons had better seats and private boxes. The Basileus' private box is described as a multi-room complex that was something of a miniature palace. Since everyone will be playing a Senator, we will all be at least low ranking nobility with access to most luxuries. It seems to me that we can just assume that there's plenty of food or drink to be had at all times in various places, be they in the public stands, Senators' boxes, or the Basileus' box. I say just RP it however you want.
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