Very helpful. Here's what I write in my paper.

   ' Wittgenstein writes:
But if someone wished to say: "There is something common to all these constructions--namely the disjunction of all their common properties"-- I should  reply: Now you are only playing with words.  You might as well say: "Something runs through the whole thread-- namely the continuous overlapping of those fibres".

But a game, on our account, is typically played for the recreation of participants or spectators or to sharpen skills. Isn't this the sort of disjunctive definition Wittgenstein explicitly rejects? Well, no. First, all games must share the feature of being rule-defined activities involving a state that counts as performing the activity successfully because an arbitrary rule so defines it. Plainly Wittgenstein has in mind a more radical disjunction, where there is no important commonality and games share only the disjunction of different properties. Second, the disjuncts themselves have something in common, namely, they reflect the fact that success in games is created by an arbitrary rule. We do not typically play a game because we consider its success state intrinsically valuable, nor do we play it because its success state has pre-existing causal connections to other states we value. We create a rule-defined success state and pursue it, not because it has intrinsic value or pre-existing instrumental value, but because we value the pursuit.'
  .'
I do think the definition I give that is most plausible is this:

In sum: A game is a rule-defined activity involving a state which counts as performing the activity successfully because it is so defined by an arbitrary rule, an activity typically performed not because we consider its success state intrinsically valuable or because it has pre-existing instrumental value, but because we value the pursuit.'

But I also think the disjunction of motives matters ( played for the recreation of participants or spectators or to sharpen skills). The two definitions are meant to
operate in tandem. I think the bomber pilots you mention are involved in a kind of recreation--that's a plausible motive, given your description. Same for the executioners.

I wrote this:

'Then why not omit the disjunctive condition from the definition? If we lived in a world where legal suits were settled by chess, believing that God would allow only the innocent party to checkmate, and this was the only venue for chess, we would not consider chess a game nor would we consider such endeavours play. In fact, games are not essentially games, a feature our theory preserves.'

We might imagine that this means of settling suits, the rules and so on, were given to Moses on Mt. Sinai. Of course we would say that chess is LIKE a game. I'm after checkmate,
not because I value the pursuit, but because it proves my innocence. Note though that success is defined by God's arbitrary rule (he could just as well have made it taking
all your opponents pieces except his king).

Thanks, Jim