**Why Tzameti 13 Is A
Positive Sum Game.**

**This is a positive-sum game because at its end one player
wins all, while the other loses everything (including his life). In a zero-sum
game nobody wins. In a zero-sum game if we
add together expected probabilities of the opponents they sum to zero, thus the
name. Nuclear war is a good example in
reality. If both parties annihilate each
neither party wins. **

**In positive sum games somebody wins. Most games are of
this type: soccer, football, basketball, chess, etc. We can see it is a positive-sum game
(sometimes called a win-win game) when we remember that the two shooters
are not the only players. In fact, they
are proxy players for the investors providing the money to bet on this
game. When one player wins, his backer
does also, and a win-win outcome obtains for this team of players. We can illustrate this with an example from Tzameti
13.**

**Consider the payoff matrix T of the players Number 13 (X _{1)}
and Number 6 (Y_{1)} pay-off outcomes below:**

** ****Y _{1}
Y_{2}**

**T= X _{1}
**

** X _{2}
**

**In this example, X and Y have ˝ probabilities for each of
the following payoffs:**

**(Row) X _{1}Y_{1}=850,000 Fr **

**(Column) X _{1}Y_{2}=-850,000 Fr**

**(Row) X _{2}Y_{1}=-850,000 Fr **

**(Column) X _{2}Y_{2}=850,000 Fr.**

**The expected average payoff probability E(P) will be **

**E(P)=1/2(X _{1}Y_{1})+
1/2(X_{1}Y_{2}) + 1/2(X_{2}Y_{1}) + 1/2(X_{2}Y_{2})
=0**

**As shown above if this game could be played repeatedly,
then the outcome probabilities would sum to zero. This game is played until there is a winner
and this means until a player dies, so the choice of the players to fire first
or not doesn’t matter. Whoever has the
good fortune to fire and kill the other will win. Waiting for the other player to fire or
firing first doesn’t help, since the probability is ˝. This game can never be a zero-sum game, and
must always be a positive-sum game regardless of each player’s strategy. In the case where both players fire and kill
each other it’s a draw as previously stated.
This can be a zero-sum for the players’ lives but not for the proxy
betting players. That is to say both players can kill each other and that would
be zero-sum for them. The possibility of each chambering a round with no bullet
and firing will only restart the game. If they both kill each presumably the
game restarts with the last living game players. Remember there can be no draw.**

**In the case where the game is played until the death of
one player, then we must reformulate the payoff matrix to be: **

**Y _{1} Y_{2}**

**X _{1 }│1 0 **

**= 850,000 Fr**

**X _{2 }│ 0 1 **

**This matrix resolves to 2 solutions:**

**1) ****X _{1}Y_{1
}(1) + 0 = 850,000 Fr Row wins (number 13) and Column loses (number 6
dies)**

**2) **** X _{2}Y_{1 }(0)+ (1 *X_{2}Y_{2})
= 850,000 Fr Column wins (number 6) and Row loses (number 13 dies)**

**Nash Equilibrium Strategy**

**Nash equilibrium is an algorithm that quantifies opposing
player strategies with the following rule:**

*Given two or more
players in a game, their individual strategies could not be unilaterally
changed and yield an advantage*. It
is named after the famous game theorist John Nash that first described the
idea. It is alternately described as *no individual player in a game having an
incentive to change his/her strategies to yield a positive outcome.* It means that if we have a payoff matrix for
any number of players, one combination is best compared to all others. That is, it yields an advantage compared to
all others. Let’s revisit the above
matrix and apply Nash Equilibrium. The elements of the Nash Equilibrium matrix
will be a 3X3 matrix. By the way, this strategy can apply to more than human
players in a game, as I’m sure any reader realizes.

**Elements are:**

**Row/Column X _{1}Y_{1} =[1,0]**

**Row/Column X _{2}Y_{1} =[0,1]**

**Row/Column X _{3}Y_{3} =[0,0]**

**And the matrix is:**

** Y _{1 }Y_{2 }Y_{3} **

**X _{1} **

**X _{2} **

**X _{3} **

**Here we have the first row/column combination X _{1}Y_{1,
}X_{2}Y_{1,} X_{3}Y_{3}
is the only winning strategy for row is X_{1}Y_{1}. The only winning strategy for column is X_{2}Y_{2.
} All other combinations result in
no wins. Thus X_{1}Y_{1}
=[1,0] is a Nash Equilibrium for Row (number 6) and X_{2}Y_{1}
=[0,1] is a Nash Equilibrium for Column (number 13). What this means is that for any choice of
other than the 2 described above when one opponent chooses a move, the other
opponent can’t do better than choosing the option given. Whenever X_{x} is 1 then the opponent
Y_{x} must be 0, i.e., he dies. And vice versa whenever Y_{x}
is 1, then X_{x} is 0. Thus, the optimal strategy is as described
above. **

**Of course, the problem with applying Nash Equilibrium to
this game is even though the players have this knowledge they had no choice in
the matter. They both knew their guns
were loaded with 5 of 6 bullets and they’re chances were 50%, but couldn’t opt
out of playing the duelist round. Thus,
this movie really doesn’t portray true Nash Equilibrium. But, the outcome estimates above show the
possible Nash Equilibrium.**

**The key point to take away from this analysis is there is
no draw in the duelist portion of Tzameti 13, somebody has to die and if both die, the monstrous game would have to
be restarted by its own rules! Not a game many would want to play, huh?**

**9/6/09 Ken Wais: Updated 4/6/14**