when i reading about this paper, in the
Setup of SIM section 5.2, it says that "According to the definition of PFS game", i wonder what the definition of PFS game is?
PFS, or perfect forward secrecy, is a desirable (in many cases) property for cryptographic protocols. It says that even if your long-term secret (or static key in the paper) is revealed at some point, messages sent in the past should still be kept secret.
Right after that line you quote, they define their PFS game. I'll reword it here to make it more generic.
The game is between three parties. Alice, Bob, and Trudy. Alice and Bob are able to exchange messages using the protocol for some period of time (the test session in the paper). During this period, Trudy can only watch the communications (i.e., encrypted messages). After that period of time, Trudy can, whenever she chooses, force Alice or Bob or both to reveal their long-term secrets (could be their corresponding private keys, or it could be a long-term shared symmetric key). At this point, Trudy's goal is to reveal the plaintext of one of the prior messages. If she can do this, the protocol does not have the PFS property.