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Slider comments on The Number Choosing Game: Against the existence of perfect theoretical rationality - Less Wrong
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Define "dominant decision" as an action that no other option would result in bigger utility.
Then we could define an agent to be perfect if it chooses the dominant decision out of its options whenever it exists.
We could also define a dominant agent whos choice is always the dominant decision.
a dominant agent can't play the number naming game whereas a perfect agent isn't constrained to pick a unique one.
You might be assuming that when options have utility values that are not equal then there is a dominant decision. For finite option palettes this migth be the case.
Define a "sucker" option to be a an option with a lower utility value than a some other possible choice.
A dominant decision is never a sucker option but a perfect agent migth end up choosing a sucker option. In the number naming game every option is a sucker option.
Thus "winning" is different from "not losing".
I would argue that a perfect agent can never choose a "sucker" option (edit:) and still be a perfect agent. It follows straight from my definition. Of course, if you use a different definition, you'll obtain a different result.
Thus why the dominant agent can't play the number naming game as it can't choose any of the options.