Even under these conditions, you have very slim chances of winning.
I would assume more than 5% of people are greedy, even among rational thinkers there will greedy a-holes who would act against their rational thinking, and send the telegraph with their name.
Waste of time either way. You dont enter, you lose. You enter, you almost certainly lose as there will more than one person who sent the telegraph.
Like I said, a waste of my time.
Even in the most communist societies you will find that more than 5% of the population is greedy and would take free stuff without hesitation.
If you send these letters to the very top one percenters, you might change the odds, but with the general population there is no way to win this game, so why bother playing.
Indeed your are correct no one will win the contest. But that is not because the game cannot be won but because rational action alone is insufficient to win it. Human beings have great difficulty with superrationality. We need to be selfless which is not something humans are very good at.
It is rational to always try and claim the prize. The cost is near zero and the potential reward very large. You need to be something better then rational to win this particular game. Indeed in the long run for games of this nature everyone must be superrational if you want to win. As you said playing the game with traditional rational actors make cooperation impossible. You can't win.
So how do we make someone superrational where there is always short term profits from defection. The only way I am aware of is to truly and totally ground oneself in the infinite. Anonymint stated it well which is why I quoted his comments on the matter.
We can instead choose to believe in superrational God that loves us and emulate that ideal, thus applying superrational sacrifice to our motivation and decisions. IOW, that everything is motivated by what is best for the other person, not for ourselves. In that case, there are no Prisoners dilemmas. The key is recognizing that only selflessness is compatible with unconditional love. And that the choice of a belief (and love) in the unfalsifiable God is a choice that one makes because our existence is but an illusion of our choice in the multiverse. Consciousness is but what we choose it to be. Nihilism will illogically reject this as unfounded, and instead choose no foundation at all, no purpose, no life. Love in the form of selflessness is the only form of life. That is what Jesus came to exemplify. All those who claim that such unfounded belief makes people vulnerable to insane collective actions (e.g. the Inquisition) fail to understand that was a reversion from unconditional love to animalism, Nihilism and Prisoners dilemmas, i.e. that was not true Christianity.
Superrationality itself is just a formalization of Immanuel Kant's categorical imperative: Act only according to that maxim whereby you can, at the same time, will that it should become a universal law. The categorical imperative is in turn is a valiant but incomplete attempt to codify much older wisdom into a logical framework. Matthew 7:12 "So in everything, do to others what you would have them do to you, for this sums up the Law and the Prophets." Easy to say very difficult to live by. See:
Superrationality and the Infinite for more.
You want to win the game you have to change and not just yourself but eventually everyone because winning requires everyone playing to be better then simple rational actors.
You dont need external God to be a good person.
Learn biology, human psychology, do not cause harm to sentient life, eventually you will become selfless.