by invitation only
my opponent cast by invitation only and I was not given an option to respond to it when he picked a number. I was under the assumption that all parameters of a spell had to be decided when cast and not on resolution. It cost me the game as when he chose the number the spell just went to resolution without giving me a chance to respond to it.
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[Deleted User] commented
Only very specific things are chosen on cast: modes, the value of X, targets, distribution of stuff (like damage or counters) among targets, optional additional costs, how to pay for special mana symbols (phyrexian, hybrid) , etc. (rule 601.2b-d). The choice of number for By Invitation Only is not among them, so it gets chosen on resolution. You want to respond, you have to do so without knowing what number your opponent is going to pick. You can't wait until the number is chosen, because that is part of the spell's effect, and followed by the rest of the spell's effects during resolution, which you cannot interrupt.