# Probabilistic Implications of Tzameti 13

I began to consider the probability aspect of this game, and found it even more stimulating.

If we take into account the probability of having a bullet aligned with the barrel, as odd as it seems, the spin of the cylinder does not affect the probability. If we have 3 bullets out of 6 chambers in the gun cylinder, the probability is 3/6th=1/2 of any of the 3 bullets landing in alignment with the gun barrel. If we spin the cylinder several times that probability remains the same EVEN IF we align a bullet with the gun barrel. The requirement of spinning the cylinder only added to the perceived fear of the player, not the probability of aligning a killing bullet. Spinning the barrel never affected probability. If we had a game in which we are asked to choose one white or black ball out of a hat, then probabilities would change after each round. Say we had 3 black balls and 2 white out of 5. You choose one white the probabilities become 3/4th for black and 1/4th for white. In this game the probability of a bullet aligning with the firing cylinder increases as more bullets are added. This probability must remain the same for all players. When a player was eliminated by being shot in the head, and a new round began, the guns were all reloaded with the same number of bullets, thus the players that did not kill another player would not gain a distinct advantage: having more bullets in their guns than the ones that killed someone. If the number of bullets each player had in his gun didn't stay the same after each round, this would become an unfair game in the game theory sense of that phrase. Let's look at the game theory side of Tzameti 13.

# Game Theory Implications of Tzameti 13

This game is a positive sum contest with unbiased conditions. That's what it would be called in game theory. Its positive because when we added up the losses (minuses, the dead ones) and gains (pluses, the remaining living), somebody eventually wins. Unbiased, because nobody ever has an advantage. Normally, advantage translates to knowledge in game theory. Here it would be one guy having more bullets than anyone else. It might seem weird that this game would be classified a positive sum game. To see why it's classified as a positive-sum game, see the analysis here: Why Tzameti 13 is a positive-sum game.

Some game theoretic features of this game might surprise viewers. Is this a categorical game? Categorical games allow no draws. Games that allow no draws are in game theory unfair. Unfair meaning if a certain player follows a precise strategy he will always win. So, as I indicated above, if players that killed another player were not allowed to replace their spent bullet, the players that added bullets after not firing one, would gain a distinct advantage, and following this strategy would assure one of those players would win. This game is clearly not unfair, it's brutal as hell, but not unfair. Thus it can't be classified as a categorical game. But, we must have a contradiction. We know this game ended in a winner (number 13 that is), so it must be categorical. But, there is the mistake. This game does allow a draw, though it's so rare no one would ever think it would occur. It is possible that the two final players could both shoot each other in the head and die, thus no one would win. I can't imagine how anybody would declare a winner, if they both shot each in the head and died. As illustrative of its brutality however, by the rules of the duelist portion of the game, at least one player must die to obtain a win, and two would have to die to obtain a draw.

# Postscriptum

A real categorical game existed at the end of World War II between the US and Japan. The US was assured it could crush Japan after dropping a nuclear weapon on them, no matter how hard the Japanese fought back. As you know, the Japanese realized this and surrendered. The US could have followed a strategy that made it certain they would win, regardless of the bravery of the Japanese military machine. Thus, the US military had a distinct advantage in this case, and an unfair game would have been played. It should be obvious why nobody wants to play a categorical game. Tell me who wants to be assured that they will lose in a game? Unfortunately categorical games are played on the international economic stage. There are many international trading situations where categorical games are being played today. For instance, the US and European nations subside their agricultural producers and thus make it virtually impossible for 3rd world farmers to compete with them on worldwide export markets like coffee trading corn and wheat trade, etc. Can you think of any other categorical games that are played today? I am thinking of money transactions....Pawnbrokerage? Payday loans?

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Robleh Wais 4/7/08