Actually, the 2nd is not the usual native utterance, which is: A: Let me know who wins. B: I don't know/know when he arrives.
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jooneyA: Let me know who wins.The real question is how the meaning changes, if at all, when certain parameters are varied.
B: I don't know/know when he will arrive.
jooneyHow do you formulate rules on this?Here are some possible factors to consider when researching this.
jooneyBut some verbs like hope and bet offer a choice between the simple present and will: I hope we (will) win. I bet you (will) win."Apparently, the strength of the general rule is diluted by exceptions, which is not at all unusual.