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DanielLC comments on A Series of Increasingly Perverse and Destructive Games - Less Wrong Discussion
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I don't think your alternative strategies for GAME1 would work. They might be worth trying if you had a lot more than three people.
That being said, I think there's a way you could improve it. Write a program to randomly pick a program. It then attempts to prove that it won't halt for TREE(3) steps. If it cannot, it runs the program. If it halts in less than TREE(3) steps, it picks a new program. This way, you can get rid of most of the programs that don't halt, and you won't get small numbers.
You can make this strategy more sophisticated by taking several programs that don't seem to halt but don't halt quickly, and waiting until a certain number halt. If you pick a proportion too low, you won't get a sufficiently large number. If you pick a proportion too high, you won't get enough that halt.
Note that the code is being run halting oracle hypercomputer, which simplifies your strategy to strategy number two.
So? I'm not allowed to actually use the oracle. It's just used to make sure my program halts.
No. Strategy number two has an upper bound for how high it can answer, where mine does not. For example, it may be that you reach a program that does not halt before you reach one that takes TREE(3) steps to halt. In fact, I'm pretty sure you will. Second, strategy two is highly likely to fail due to reaching an obviously unhalting program. My version would not do so.
This was supposed to be an improvement on strategy two.