What is a quantifier in game-based definitions?

I’d like to know what are quantifiers in game-based security definitions (in cryptography)

Some relevant examples and sources are also much appreciated.

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Edit: for example, in a formal definition, we say for any security parameters, any function $F$, etc. My question is what should be included in these quantifiers (shall we include everything picked deterministically by the experiment?)is there any clear rule we need to follow?