Another (related?) advantage is that the incentives to manipulate and catch manipulation are much better balanced with the negative (“you’re out”) version. Consider:
Perfectly cheating in the positive version improves your chances of winning by (n-1)/n, and stopping you from doing so improves each other person’s chances by 1/n.
Perfectly cheating in a round of the negative version improves your chances of winning by 1/(k(k-1)), where k is the number of people in to start the round. Stopping you from doing so improves each other person’s chances by the same amount.
The total (summed) incentive to manipulate in the negative version is (n-1)/n, the same as in the positive case.
Perfectly cheating in a round of the negative version improves your chances of winning by 1/(k(k-1)), where k is the number of people in to start the round. Stopping you from doing so improves each other person’s chances by the same amount.
I think 1/(k(k-1)) is the improvement in each other person’s chance of getting into the next round, not the improvement in chance of winning the whole thing. The point still holds though since the absolute numbers are so much smaller in any single round.
Oh, you’re right. The net incentive to catch cheaters is actually… 1/(k(k-1)^2), then? The relative incentive story is worse, though still better in total than the positive version, and better still if you assume a constant-size disincentive to be caught cheating.
Another (related?) advantage is that the incentives to manipulate and catch manipulation are much better balanced with the negative (“you’re out”) version. Consider:
Perfectly cheating in the positive version improves your chances of winning by (n-1)/n, and stopping you from doing so improves each other person’s chances by 1/n.
Perfectly cheating in a round of the negative version improves your chances of winning by 1/(k(k-1)), where k is the number of people in to start the round. Stopping you from doing so improves each other person’s chances by the same amount.
The total (summed) incentive to manipulate in the negative version is (n-1)/n, the same as in the positive case.
I think 1/(k(k-1)) is the improvement in each other person’s chance of getting into the next round, not the improvement in chance of winning the whole thing. The point still holds though since the absolute numbers are so much smaller in any single round.
Oh, you’re right. The net incentive to catch cheaters is actually… 1/(k(k-1)^2), then? The relative incentive story is worse, though still better in total than the positive version, and better still if you assume a constant-size disincentive to be caught cheating.