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I want to ask what's the usage of " 's ". For instance, Alice's cat is cute. In this case, " 's " represent the possession. So when we want to express the possession, we can use " 's ". So how about this "Did you see Tony's game yesterday?" here what does " 's " in that case represent? We all know that we use adjectives to modify the nouns, and also we can use nouns to modify the nouns, too. Therefore, does " 's " have any relationships with modifiers? Please explain it to me, THanks.
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Possessives are determiners, so they are a subclass of noun modifiers.

(Alice's cat and Tony's game are the same structure: the wording of your post seems to suggest that they are different.)
Comments  
As you've realised, the " 's " can be used in a wider sense than just to denote possession of an article. It is often used as a shorthand way of expressing association of some kind. In your example, "Tony's game" is a convenient shorthand way of saying "the game in which Tony was playing" (i.e. the game with which Tony was associated). Other examples would be:

John's journey to London was eventful. (i.e. the journey made by John)

John's punishment was well deserved. (i.e. the punishment inflicted upon John)
 Mister Micawber's reply was promoted to an answer.