Let's approach this for a purely Game-theory perspective (which means, everyone simply wants to get the biggest possible payout, to live, and no one does actions like shooting someone out of revenge for shooting at them, etc.)
We can deduce two things logically from the beginning. The first is that since Juan is such an excellent marksman, at the end of the first "round" (where everyone has had a chance to shoot at least once, assuming they haven't died), the game will always end with either one winner left alive, or a "two-man standoff".
The second thing we can deduce is that since everyone has an infinite amount of bullets (though, only three would actually be required per person until the game ends), if two people are left standing, the best thing they can do is keep shooting at each other until they or the other person is dead.
If you enjoy graphs, I have created a little ASCII diagram which illustrates how the first three turns can play out:
[Jorge]---(Shoot José)--- 0.3 HIT ---[DEAD]-------------------------------[Juan]--(Shoot Jorge)-- # Juan WINS
\ `-- 0.7 MISS ---[José]---(Shoot Jorge)--- 0.7 HIT --[Juan]--(Shoot José )-- # Juan WINS
\ \ `-- 0.3 MISS --[Juan]--(Shoot Jorge)-- # José vs. Juan two-man standoff
\ \ `-(Shoot José )-- # Jorge vs. Juan two-man standoff
\ `--(Shoot Juan )--- 0.7 HIT --[DEAD]----------------- # Jorge vs. José two-man standoff
\ `-- 0.3 MISS --[Juan]--(Shoot Jorge)-- # José vs. Juan two-man standoff
\ `-(Shoot José )-- # Jorge vs. Juan two-man standoff
`--(Shoot Juan)--- 0.3 HIT ---[José]---(Shoot Jorge)--- 0.7 HIT --[DEAD]----------------- # José WINS
\ `-- 0.3 MISS --[DEAD]----------------- # Jorge vs. José two-man standoff
`-- 0.7 MISS ---[José]---(Shoot Jorge)--- 0.7 HIT --[Juan]--(Shoot José )-- # Juan WINS
\ `-- 0.3 MISS --[Juan]--(Shoot Jorge)-- # José vs. Juan two-man standoff
\ `-(Shoot José )-- # Jorge vs. Juan two-man standoff
`--(Shoot Juan )--- 0.7 HIT --[DEAD]----------------- # Jorge vs. José two-man standoff
`-- 0.3 MISS --[Juan]--(Shoot Jorge)-- # José vs. Juan two-man standoff
`-(Shoot José )-- # Jorge vs. Juan two-man standoff
Two-man standoffs
There are three possible two-man standoffs at the end of the first round. Note that since everyone has had a chance to fire once, the person who is the poorest shot will be allowed to shoot first.
Jorge vs. Juan
Jorge is first to fire:
> If he hits Juan (30%
chance), he wins.
> If he misses (70%
chance), Juan fires back, and being an excellent marksman, is guaranteed to win the standoff.
José vs. Juan
Jorge is first to fire:
> If he hits Juan (70%
chance), he wins.
> If he misses (30%
chance), same as above, Juan guarantees a win.
Jorge vs. José
If Jorge faces off against José, since no one can hit the other person with 100%
accuracy, there is an infinitesimally small probability that the standoff will go on forever with each person missing their shot until they both die of old age.
It's possible to calculate an "exact" probability out of this game using integrals, however, we are going to stick to high-school algebra and basic logic to solve this puzzle in order to make it understandable to as many people as possible. Let's just say since Jorge shoots first, he has a greater than 30%
chance of winning, and José has a less than %70
chance of winning. This is actually all the information we need to get an answer.
Summary
Let's summarize the two-man standoffs. Instead of using percentages, we will be using values from 0
to 1
to make the math we will be doing later easier. The following values represent the probabilities of each person surviving the encounter:
Jorge vs. Juan: { Jorge: 0.3, José: 0.0, Juan: 0.7 }
José vs. Juan: { Jorge: 0.0, José: 0.7, Juan: 0.3 }
Jorge vs. José: { Jorge: >0.3, José: <0.7, Juan: 0.0 }
Juan
Let's start with Juan. If only one person remains after José and Jorge had their turns, he will obviously shoot the remaining person, hitting the target perfectly, and will win the game.
If the other two have gotten their allowed shots, and Juan is left standing, he has two options:
A. Shoot José
B. Shoot Jorge
Since he will always hit his opponent, the person he shoots will not make it to the next "round". That means, he will be left in a two-man standoff (see Deduction #2) with the person he didn't shoot.
Looking at our previous summary, we know that he would rather be in a two-man standoff with the worst shooter (Jorge), so he will always shoot José to avoid a two-man standoff with him.
We can now update our graph with this information:
[Jorge]---(Shoot José)--- 0.3 HIT ---[DEAD]------------------------------- # Juan WINS
\ `-- 0.7 MISS ---[José]---(Shoot Jorge)--- 0.7 HIT -- # Juan WINS
\ \ `-- 0.3 MISS -- # Jorge vs. Juan
\ `--(Shoot Juan )--- 0.7 HIT -- # Jorge vs. José
\ `-- 0.3 MISS -- # Jorge vs. Juan
`--(Shoot Juan)--- 0.3 HIT ---[José]---(Shoot Jorge)--- 0.7 HIT -- # José WINS
\ `-- 0.3 MISS -- # Jorge vs. José
`-- 0.7 MISS ---[José]---(Shoot Jorge)--- 0.7 HIT -- # Juan WINS
\ `-- 0.3 MISS -- # Jorge vs. Juan
`--(Shoot Juan )--- 0.7 HIT -- # Jorge vs. José
`-- 0.3 MISS -- # Jorge vs. Juan
José
Now it's José's turn. Obviously, if only one person remains, José will shoot whoever is left! But what if three people remain?
We will update the graph to include the payoffs, which will see who José is better off shooting:
[José]---(Shoot Jorge)--- 0.7 * { Jorge: 0.0, José: 0.0, Juan: 1.0 }
\ `-- 0.3 * { Jorge: 0.3, José: 0.0, Juan: 0.7 }
`--(Shoot Juan )--- 0.7 * { Jorge: >0.3, José: <0.7, Juan: 0.0 }
`-- 0.3 * { Jorge: 0.3, José: 0.0, Juan: 0.7 }
If we multiply the probability of José hitting the target by the probability that José will win after that round, and then get the average of the two, we get the following results:
[José]---(Shoot Jorge)--- { Jorge: 0.09, José: 0.00, Juan: 0.91 }
`--(Shoot Juan )--- { Jorge: >0.30, José: <0.49, Juan: 0.21 }
We can now clearly see that if there are three people remaining when José is ready to shoot, if he shoots Jorge he has absolutely 0%
chance of surviving. On the other hand, if he shoots Juan, he has a bit less than 49%
chance of winning. Not being an idiot, he will choose the latter!
Our tree now looks like this:
[Jorge]---(Shoot José)--- 0.3 HIT ---[DEAD]------------------- { Jorge: 0.00, José: 0.00, Juan: 1.00 }
\ `-- 0.7 MISS ---[José]---(Shoot Juan )--- { Jorge: >0.30, José: <0.49, Juan: 0.21 }
`--(Shoot Juan)--- 0.3 HIT ---[José]---(Shoot Jorge)--- { Jorge: >0.09, José: <0.91, Juan: 0.00 }
`-- 0.7 MISS ---[José]---(Shoot Juan )--- { Jorge: >0.30, José: <0.49, Juan: 0.21 }
Jorge
Finally, it's Jorge's turn (well, actually, he shot first, but thinking backwards logically, we finally arrive at him).
Let's take a final look at that tree, collapsing the tree down so we see what Jorge's probabilities of winning are (hiding José's move):
[Jorge]---(Shoot José)--- 0.3 * { Jorge: 0.00, José: 0.00, Juan: 1.00 }
\ `-- 0.7 * { Jorge: >0.30, José: <0.49, Juan: 0.21 }
`--(Shoot Juan)--- 0.3 * { Jorge: >0.09, José: <0.91, Juan: 0.00 }
`-- 0.7 * { Jorge: >0.30, José: <0.49, Juan: 0.21 }
And if we multiply the payoffs by the odds of them happening:
[Jorge]---(Shoot José)--- { Jorge: >0.210, José: <0.343, Juan: 0.447 }
`--(Shoot Juan)--- { Jorge: >0.237, José: <0.616, Juan: 0.147 }
It's a close tie! Jose has a slightly larger than 21.0%
chance of winning if he shoots Jorge, but a very slight lead of slightly larger than 23.7%
chance of surviving if he shoots Juan.
There isn't a very large chance that he will survive anyway, but Jorge hopes for the best, and aims his sights at Juan.
Results
Assuming everyone plays rationally, here is how the game will go:
[Jorge]---(Shoot Juan)--- 0.3 HIT ---[José]---(Shoot Jorge)--- 0.7 HIT --[DEAD]--------------------- { Jorge: 0.0, José: 1.0, Juan: 0.0 }
\ `-- 0.3 MISS --[DEAD]---[Jorge vs. José]-- { Jorge: >0.3, José: <0.7, Juan: 0.0 }
`-- 0.7 MISS ---[José]---(Shoot Juan )--- 0.7 HIT --[DEAD]---[Jorge vs. José]-- { Jorge: >0.3, José: <0.7, Juan: 0.0 }
`-- 0.3 MISS --[Juan]---(Shoot José)------ { Jorge: 0.3, José: 0.0, Juan: 0.7 }
In the end, these are the combined chances that the respective marksmen will win:
- Jorge: >23.7%
- José: <61.6%
- Juan: 14.7%
So surprisingly, despite being a semi-poor shot, assuming Jorge acts rationally, the odds are in José's favor! Yet, one can never guarantee anything with randomness.
Addendum
There is a "bonus answer" for Jorge getting a better payoff if you are willing to "think outside the box", but I'll leave that one for you to decipher. ;)