Sure, but non-adversarial cases (really, any cases where u is determined independently of strategies chosen) can just choose R as a fixed part of the strategy, rather than a random shared component determined later.
Nope. Random choice gives a specific value for R each game. The outcome for that iteration is IDENTICAL to the outcome if that R was chosen intentionally. Randomness only has game value as a mechanism to keep information from an adversarial actor.
To be clear, by “worst-case guarantee” I mean “the expected utility is guaranteed to be pretty good regardless of uy”, which is unattainable without shared randomness (claim 1).
I think you are either misunderstanding or disagreeing with a lot of the terminology on randomized algorithms and worst-case guarantees that are commonly used in CS and statistics. This article is a decent introduction to this topic.
Sure, but non-adversarial cases (really, any cases where u is determined independently of strategies chosen) can just choose R as a fixed part of the strategy, rather than a random shared component determined later.
That’s right, but getting the worst-case guarantee requires this initial choice to be random.
Nope. Random choice gives a specific value for R each game. The outcome for that iteration is IDENTICAL to the outcome if that R was chosen intentionally. Randomness only has game value as a mechanism to keep information from an adversarial actor.
To be clear, by “worst-case guarantee” I mean “the expected utility is guaranteed to be pretty good regardless of uy”, which is unattainable without shared randomness (claim 1).
I think you are either misunderstanding or disagreeing with a lot of the terminology on randomized algorithms and worst-case guarantees that are commonly used in CS and statistics. This article is a decent introduction to this topic.