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Forum Games and Roleplaying / Re: Systematically Choose Who Wins: Day 3
« on: August 06, 2010, 03:10:38 pm »
Non-Partisan Analysis By A Dude With A Basic Understanding of Mathematics/Probability And Too Much Time On His Hands
Criptfiend's plan is flawed. For one thing, he has to keep recruiting cabal members in order to get a sure majority.
FIRST TURN
There are 14 players in the game, eight in the cabal.
Six will win.
Eight players vote for Criptfiend, assuring his victory.
SECOND TURN
There are 13 players in the game, seven in the cabal.
Five more will win.
Seven players vote for some cabal member, assuring his victory (the random roll, by the way, is not secure. Criptfiend can easily, easily fake them. Here's how: suppose Bill is his friend, and Bill is #3. He rolls a 1d7 repeatedly until he gets a 3. Then he posts that roll. Easy as pie. One reliable way is to say "Tomorrow, I will take the opening price of the NYSE, take the last two digits (the cents, which are essentially random given how much it wavers), and take that modulus the number of players, and that will be the person we vote for. It is completely transparent, anyone can verify it, and it cannot be changed.
THIRD TURN
There are twelve players in the game, six in the cabal. Oh no! He doesn't have enough for victory! He has to recruit one more player; there are seven in the cabal now.
Four more will win. (Note that the "crude" chance of winning has declined to 4/7 or 57%. I will calculate the "true" chance of winning later)
Seven players vote for the cabal member, assuring victory.
FOURTH TURN
There are 11 players in the game, 6 cabal.
3 more will win.
6 vote for the chosen cabal member, assuring victory.
FIFTH TURN
There are 10 players in the game, 5 cabal. One cabal member is recruited, now 6 cabal.
2 more will win. (Crude chance of cabal winning has declined to 33%)
6 vote for the chosen, assuring victory.
SIXTH TURN
There are 9 players in the game, 5 cabal.
1 more will win. (20% win chance)
Five people vote for the chosen, who wins.
Now that is the point that Pandarsenic illustrated - even if it works at the start, it will surely collapse later, when there are fewer cabal members. To keep a majority you will have to get more and more members, few of whom will win. Now we'll calculate the "true" win chance.
Criptfiend touts his cabal idea on the first turn. You figure, hey, why not? I want to win, right? That turn, you have a 0 chance of winning, since Cript will win.
The second turn, you have a 1/7 chance of winning.
The third turn, you have a 1/7 chance of winning, since a new member had to be recruited.
The fourth turn, you have a 1/6 chance of winning.
The fifth turn, you have a 1/6 chance of winning.
The sixth turn, you have a 1/5 chance of winning.
The total chance of winning is: (chance that you'll win the first round) + (chance that you'll lose the first round but win the second) + (chance that you'll win the first two rounds but win the third)... etc.
It's not them all added up, because obviously you can't win if you've already won, so you have to lose the first to win the second. Otherwise you get dumb stuff like "I have a 1/2 chance of getting heads, so if I flip twice in a row my chance of getting heads is 50%+50%=100%!" No, two tails is unlikely but possible.
So. The total chance is 0+(1/7)+(6/7)(1/7)+(6/7)(6/7)(1/6)+(6/7)(6/7)(5/6)(1/6)+(6/7)(6/7)(5/6)(5/6)(1/5) = 59.2%. Just over half; far from the glorious 85% he promised (which, as Pandar said, doesn't make any sense anyways. 85% = 17/20. So does that mean there are 20 cabal members and 17 winners? Huh?)
And the chance of winning if you join later on is even more abysmal; if you join on the third turn, the chance of winning is (1/7)+(6/7)(1/6)+(6/7)(5/6)(1/6)+(6/7)(5/6)(5/6)(1/5) = 52.4%. A few percentage points over half. That's definitely not the almost sure win he promised.
If you join on the fifth turn, the chance is (1/6)+(5/6)(1/5) = 33%. Ouch. (Actually, due to an oversight on my part, the cabal will have to recruit even more members than I thought, since one can't vote for oneself - for example, on the second turn, there will be 6 non-cabal, and seven cabal, and since you can't vote for yourself, only six cabal can vote for a given person, which could result in a tie; thus you would need to recruit already by the second turn)
"But wait!" you cry. "If we don't join the Cabal, we won't win at all!" That may seem so. But it remains, firstly, that Criptfiend has blatantly lied to you (I'd say that he was merely incorrect, but judging by the false statistics he's put up, with no calculations behind them, I doubt it), and another thing...
On the sixth turn, five people will vote for the chosen cabal member, and he will win... or will they?
Imagine you're a cabal member. On the last turn, you aren't chosen. Well, that's to be expected, you had only a 20% of being chosen that turn. Wait! You still have a small chance of winning if you defect and try to gain favor! So you do so. The cabal fails, and the chosen is no better than any other man.
Now, why have you joined the cabal in the first place? The cabal's lure is thus: "Vote for who we want you to, and we'll vote for you later on." Does this hold true?
On the fifth turn, you might follow the rules like always. But this would be unwise. Even if you vote for the chosen person, this will confer you no special advantage on the sixth turn; all will descend into chaos by then and you will have no supporters. Thus, on the fifth turn, if you are not chosen, you should defect and try to get people to vote for yourself.
On the fourth turn, the same thing. On the third turn, and so on.
I would like to point to a game called Prisoner's Dilemma. In one game, the best strategy is to defect, as this is profitable. In a series of games of unknown length, a very good strategy is to play "Tit-For-Tat"; when the other player defects, defect, and when the other player cooperates, cooperate (this strategy attempts to enforce cooperation and thus better both players). But in a series of games where the length is known, well, you're both going to defect on the last round, because you've got nothing to lose. And knowing this, you're going to defect on the penultimate round, because the other guy's not going to cooperate even if you do cooperate... and so on. This is the situation we have here.
The cabal is not stable. Criptfeind lied, deceived you, played you for fools. Every man for himself.
War!
Chaos!
BLOOD FOR THE BLOOD GOD!
Criptfiend's plan is flawed. For one thing, he has to keep recruiting cabal members in order to get a sure majority.
FIRST TURN
There are 14 players in the game, eight in the cabal.
Six will win.
Eight players vote for Criptfiend, assuring his victory.
SECOND TURN
There are 13 players in the game, seven in the cabal.
Five more will win.
Seven players vote for some cabal member, assuring his victory (the random roll, by the way, is not secure. Criptfiend can easily, easily fake them. Here's how: suppose Bill is his friend, and Bill is #3. He rolls a 1d7 repeatedly until he gets a 3. Then he posts that roll. Easy as pie. One reliable way is to say "Tomorrow, I will take the opening price of the NYSE, take the last two digits (the cents, which are essentially random given how much it wavers), and take that modulus the number of players, and that will be the person we vote for. It is completely transparent, anyone can verify it, and it cannot be changed.
THIRD TURN
There are twelve players in the game, six in the cabal. Oh no! He doesn't have enough for victory! He has to recruit one more player; there are seven in the cabal now.
Four more will win. (Note that the "crude" chance of winning has declined to 4/7 or 57%. I will calculate the "true" chance of winning later)
Seven players vote for the cabal member, assuring victory.
FOURTH TURN
There are 11 players in the game, 6 cabal.
3 more will win.
6 vote for the chosen cabal member, assuring victory.
FIFTH TURN
There are 10 players in the game, 5 cabal. One cabal member is recruited, now 6 cabal.
2 more will win. (Crude chance of cabal winning has declined to 33%)
6 vote for the chosen, assuring victory.
SIXTH TURN
There are 9 players in the game, 5 cabal.
1 more will win. (20% win chance)
Five people vote for the chosen, who wins.
Now that is the point that Pandarsenic illustrated - even if it works at the start, it will surely collapse later, when there are fewer cabal members. To keep a majority you will have to get more and more members, few of whom will win. Now we'll calculate the "true" win chance.
Criptfiend touts his cabal idea on the first turn. You figure, hey, why not? I want to win, right? That turn, you have a 0 chance of winning, since Cript will win.
The second turn, you have a 1/7 chance of winning.
The third turn, you have a 1/7 chance of winning, since a new member had to be recruited.
The fourth turn, you have a 1/6 chance of winning.
The fifth turn, you have a 1/6 chance of winning.
The sixth turn, you have a 1/5 chance of winning.
The total chance of winning is: (chance that you'll win the first round) + (chance that you'll lose the first round but win the second) + (chance that you'll win the first two rounds but win the third)... etc.
It's not them all added up, because obviously you can't win if you've already won, so you have to lose the first to win the second. Otherwise you get dumb stuff like "I have a 1/2 chance of getting heads, so if I flip twice in a row my chance of getting heads is 50%+50%=100%!" No, two tails is unlikely but possible.
So. The total chance is 0+(1/7)+(6/7)(1/7)+(6/7)(6/7)(1/6)+(6/7)(6/7)(5/6)(1/6)+(6/7)(6/7)(5/6)(5/6)(1/5) = 59.2%. Just over half; far from the glorious 85% he promised (which, as Pandar said, doesn't make any sense anyways. 85% = 17/20. So does that mean there are 20 cabal members and 17 winners? Huh?)
And the chance of winning if you join later on is even more abysmal; if you join on the third turn, the chance of winning is (1/7)+(6/7)(1/6)+(6/7)(5/6)(1/6)+(6/7)(5/6)(5/6)(1/5) = 52.4%. A few percentage points over half. That's definitely not the almost sure win he promised.
If you join on the fifth turn, the chance is (1/6)+(5/6)(1/5) = 33%. Ouch. (Actually, due to an oversight on my part, the cabal will have to recruit even more members than I thought, since one can't vote for oneself - for example, on the second turn, there will be 6 non-cabal, and seven cabal, and since you can't vote for yourself, only six cabal can vote for a given person, which could result in a tie; thus you would need to recruit already by the second turn)
"But wait!" you cry. "If we don't join the Cabal, we won't win at all!" That may seem so. But it remains, firstly, that Criptfiend has blatantly lied to you (I'd say that he was merely incorrect, but judging by the false statistics he's put up, with no calculations behind them, I doubt it), and another thing...
On the sixth turn, five people will vote for the chosen cabal member, and he will win... or will they?
Imagine you're a cabal member. On the last turn, you aren't chosen. Well, that's to be expected, you had only a 20% of being chosen that turn. Wait! You still have a small chance of winning if you defect and try to gain favor! So you do so. The cabal fails, and the chosen is no better than any other man.
Now, why have you joined the cabal in the first place? The cabal's lure is thus: "Vote for who we want you to, and we'll vote for you later on." Does this hold true?
On the fifth turn, you might follow the rules like always. But this would be unwise. Even if you vote for the chosen person, this will confer you no special advantage on the sixth turn; all will descend into chaos by then and you will have no supporters. Thus, on the fifth turn, if you are not chosen, you should defect and try to get people to vote for yourself.
On the fourth turn, the same thing. On the third turn, and so on.
I would like to point to a game called Prisoner's Dilemma. In one game, the best strategy is to defect, as this is profitable. In a series of games of unknown length, a very good strategy is to play "Tit-For-Tat"; when the other player defects, defect, and when the other player cooperates, cooperate (this strategy attempts to enforce cooperation and thus better both players). But in a series of games where the length is known, well, you're both going to defect on the last round, because you've got nothing to lose. And knowing this, you're going to defect on the penultimate round, because the other guy's not going to cooperate even if you do cooperate... and so on. This is the situation we have here.
The cabal is not stable. Criptfeind lied, deceived you, played you for fools. Every man for himself.
War!
Chaos!
BLOOD FOR THE BLOOD GOD!
