The question of how to play the game still needs to be solved. But no matter what, only one person wins for sure. Therefore, the best possible outcome of the game is to pay the winner. This happens with only one form submission. If the participants were allowed to formulate a plan to share the wealth and coordinate their actions with each other – which is not allowed – they would undoubtedly arrange for only one person to submit one form. The maximum number of games is almost a replica. You want all but one person to volunteer, not to submit the form. Ideally, a kBC company lottery would be used to determine who submits the form, but unfortunately, this requires explicit communication between participants, which is not allowed by the rules.
This game is a kind of pure betrayal, and there is no way to cooperate.
There are indeed collectively rational ways to play the maximum number of games. This method is a mixed strategy: use a random method (such as rolling dice) to determine whether a person and each person should be a volunteer. Each person can roll their dice, utterly independent of the others, without any communication. For example, in the case of thirty-six participants, each person rolls two dice, and whoever rolls two submits a form. That sounds good. But people will not accept this arrangement deep down.
That’s not the case in the lottery, and it’s hard to imagine any living group, breathing human being acting that way.
It is difficult to resist the temptation to deceive. You roll two dice, and it comes up as a two—oh no one of them hits the edge, flips over, and it’s a three! You cannot submit it. But who knows if you’ll be playing Lady Luck yourself and flipping the dice back to two? Nobody knows! Everyone rolls the dice in his house and acts alone; who knows when you roll the dice? No one knows! Therefore, this rational arrangement did not change the result at all. If everyone tries to get away with it and everyone submits, the prize money is still for naught. It’s the same as the, if not worse. Everyone is punished according to the proportion of defectors in the contestants.
In complete rounds, even one defector can ruin everyone’s chances.
So what should you do? Do you still roll dice and count points (someone has written as many nines as possible on a card, knowing most other people won’t)? Or do you fill the cards with nines yourself? The only reasonable conclusion is that the maximum number of games could be better.
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