The Good, the Bad and the Ugly
Imagine a triel in the wild west (a duel between three people). Without loss of generality (as if there were generalities in triel in the old west), we will call the people involved Good, Bad and Ugly.
In this triel, whoever was hit is considered eliminated from the dispute. The firing order is well defined and respected among those involved who are still in the dispute (that is, who have not been hit). Thus, each one has the right to fire a shot, starting with the Ugly, then the Bad, then the Good, then the cycle starts again with those who are still in the dispute. But each of them has a different accuracy of accuracy:
Ugly: hits the target 1/3 of the time;
Bad: hits the target 1/2 of the time;
Good: hit your target every time.
We have to consider that there is a clear order of threats among those involved:
Good) If the Good needs to shoot, and the Ugly and the Bad are in triel, the Good will choose to shoot the Bad, because the chance of the Bad to hit him is greater than the chance of the Ugly to hit him (the Bad is greater threat);
Bad) If the Bad needs to shoot, and the Ugly and the Good are in contention, the Bad will try to shoot the Good, as he knows that the Good considers him a greater threat and will eliminate him in the next action.
The question remains, what is the best strategy for the Ugly?
To help this analysis, we will expand the possibilities of destination through the choices of the Ugly:
A) The Ugly shoots the Bad, but misses, the Bad shoots the Good, but misses (Miss), the Good shoots the Bad, and hits, eliminating him. The Ugly and the Good are left in the triangle.
B) The Ugly shoots the Bad, but misses, the Bad shoots the Good, and hits, eliminating him. The ugly and the bad remain in the triel.
C) The Ugly shoots the Bad, and hits, eliminating him, the Good shoots the Ugly, and hits, eliminating him. Only the Good is left in the triangle.
D) The Ugly shoots the Good, but misses, the Bad shoots the Good, but misses, the Good shoots the Bad, and hits, eliminating him. The Ugly and the Good are left in the triangle.
E) The Ugly shoots the Good, but misses, the Bad shoots the Good, and hits, eliminating it. The Ugly and the Bad remain in the triel.
F) The Ugly shoots the Good, and hits, eliminating him, the Bad shoots the Ugly, but misses. The Ugly and the Bad remain in the triumph.
G) The Ugly shoots the Good, and hits, eliminating him, the Bad shoots the Ugly, and hits, eliminating him. Only the Bad is left in the triel.
Analyzing these possibilities presented:
1. If Ugly shoots Bad, he has a 66% chance of staying in the contest until the next round;
2. If Ugly shoots Good, he has an 83% chance of staying in contention until the next round.
It seems that the solution is to shoot Good… but is there a better option? After all, in all expanded cases, the worst results involved the Ugly hitting the target. So, the whole problem is linked to the Ugly hitting whoever shot him. That way, what if he misses on purpose?
This ensures that the Bad will try to hit the Good, and the Good if it survives, will hit the Bad. Thus, the Ugly has a 100% chance of surviving this first round, and only the Bad or the Good will be in contention, as one will try to eliminate the other in his opportunities to shoot.