Because it's always better when you're in control.

Let's say that it's in the first half of the game, before the sleeper agent phase. It's a 5-player game and you think that everyone is human. Someone else is President. Why would you want to spend the action and cards necessary to make yourself President instead? Because one of you might become a Cylon.

Let's say all players are going to get one more loyalty card. So there are five cards, two of which are Cylon cards. The probability that one (or more) of you turns Cylon is $1 - P(\text{neither Cylon})$. $P(\text{neither Cylon}) = \frac{3}{5}\cdot\frac{2}{4}$. That is, one of you gets a human card ($3/5$), and then the other also gets one of the remaining human cards ($2/4$). Thus,

\begin{align}

P(\text{one or more of you become Cylon}) &= 1 - \frac{3}{5}\cdot\frac{2}{4} \\

&= 1 - \frac{3}{10} \\

&= 0.7.

\end{align}

Don't forget about the case where both of you end up as Cylons. Then resources are "wasted", and maybe you look suspicious. But maybe that's okay, because it's a good cover for depleted resources and allows you to stay hidden. But let's take this out, just assume that it's not valuable to switch here.

\begin{align}

\begin{split}

P(\text{one of you become Cylon}) &= P(\text{one or more of you become Cylon}) \\

& \phantom{= } - P(\text{both of you become Cylon})

\end{split}\\

\end{align}

\begin{align}

P(\text{one of you become Cylon}) &= 0.7 - \frac{2}{5}\cdot\frac{1}{4} \\

&= 0.7 - 0.1 \\

&= 0.6

\end{align}

Thus, it is more likely than not that you and the current President are going to end up on opposite teams, so you'd be better off having more power. The remaining question is whether or not you think the benefit is worth the cost.

Let's assign this a shorter variable, $p_1$, for use later on.

\begin{align}

p_0 &= P(\text{one of you become Cylon}) \\

&= 0.6.

\end{align}

I'll be honest in that I used to think it was inefficient to squabble about the Presidency unless you knew the President was a Cylon. But after a forum post about the importance of control in the game got me thinking about this case. It's interesting to me how the game is structured in this way that encourages loyal humans to do selfish things even without explicit personal goals (a la Dead of Winter). This is an interesting emergent property of the rules.

We've considered the case when there aren't any Cylons out, now let's look at having one Cylon out before the sleeper agent phase. There are three sub-cases: you're the Cylon, the President is the Cylon, someone else is the Cylon. We've already tacitly assumed above that if you or the President is or becomes a Cylon, it's worth it to take the Presidency. So those cases are taken care of. If someone else is a Cylon, then we're similar to the above analysis, but with a smaller probability that one of you becomes a Cylon in the sleeper agent phase. In this case there is only one Cylon card out of 4 left.

\begin{align}

P(\text{one of you become a Cylon}) &= 1 - P(\text{both of you stay human}) \\

&= 1 - \frac{4}{5}\cdot\frac{3}{4}\\

&= 1 - \frac{3}{4} \\

&= \frac{1}{4} = 0.25

\end{align}

Here the probability of one of you becoming a Cylon is much lower. One interesting case to consider here is if the existing Cylon gets the second Cylon loyalty card. In this case, the Cylon can then give this card to one of the humans. This is another opportunity for you or the current President to become a Cylon, and thus on opposite teams. Assuming this happens, and not knowing anything about the Cylon's strategy in selecting a target, there is a probability of $2/4 = 0.5$ that the Cylon selects you or the President out of the four humans. However, the President is arguably a higher-value target because of the powers of the office. For the rest of the analysis, we'll assume that a Cylon with two loyalty cards does give the second card away, with equal probability to all players. This increases the probability that one of you becomes a Cylon, $\Delta p = \Delta P(\text{one of you becomes a Cylon})$, is as follows.

\begin{align}

\Delta p &= P(\text{Cylon with two cards}) \cdot P(\text{Cylon gives card to one of you}) \\

&= \frac{1}{5} \cdot \frac{2}{4} \\

&= \frac{1}{10} = 0.1

\end{align}

Thus, our corrected probability for one of you becoming a Cylon is,

\begin{align}

P(\text{one of you become a Cylon}) &= 0.25 + 0.1 \\

&= 0.35.

\end{align}

As before, let's assign this a shorter variable, $p_1$, for use later on.

\begin{align}

p_1 &= P(\text{one of you become a Cylon}) \\

&= 0.35.

\end{align}

Now, let's consider the case when two Cylons are out, which I'd argue doesn't depend on when in the game it occurs. (I choose my words carefully here. I would argue this if pressed, but I'm not going to because I'm lazy.) There are four sub-cases: you are a Cylon, the President is a Cylon, both of you are Cylons, neither of you are Cylons. In the first two cases, it makes sense to get the Presidency, as you and the President are on opposite teams. If neither of you are Cylons, it doesn't make sense (unless the President isn't using the office well). If both of you are Cylons, I'd say in general it doesn't make sense, unless the President is already suspected and you can keep the office on your team by taking it.

What is the confidence (i.e. assessed probability) you must have that a third party is a Cylon to make it not worth while to go for the presidency? As a break-even point we could assign a probability of 0.5. A better approach would be to have values assigned for the benefit of taking the presidency from the opposing team as well as the cost of moving the presidency. Determining those values is beyond the scope of this analysis, so we'll stick with a probability threshold of 0.5, meaning we're looking for the point at which it's more likely than not that you have incentive to take the presidency. This critical point occurs according the following equation.

\begin{align}

0.5 &= p_0 \cdot P(\text{no Cylons}) + p_1 \cdot P(\text{one Cylon}) + 0 \cdot P(\text{two Cylons})\\

&= 0.6 \cdot P(\text{no Cylons}) + 0.35 \cdot P(\text{one Cylon})

\end{align}

I included the two Cylon case to point out that the no and one Cylon cases are not the only possibilities. There's no $1-p$ type tricks here in general. Now, if we do assume that there is at most one other Cylon, then we condition becomes easier to conceptualize. Then it simplifies as follows.

\begin{align}

0.5 &= 0.6 \cdot (1 - P(\text{one Cylon})) + 0.35 \cdot P(\text{one Cylon}) \\

&= 0.6 + (-0.6 + 0.35) \cdot P(\text{one Cylon}) \\

&= 0.6 + -0.25 \cdot P(\text{one Cylon}) \\

P(\text{one Cylon}) &= \frac{0.6-0.5}{0.25} \\

&= 0.4

\end{align}

Here, as long as we believe the probability of one Cylon is 0.4 or less (and the probability of two is zero), then we're more likely than not to end up and opposite teams as the President, even though we are both presently human, and thus have reason to take it.

As a reference it may be useful to know the probability of each of those cases (and the previous cases we've considered). If you have a human card, and you're sure that the President is also human, you know there are two Cylon cards and six human cards left. Three of those cards have been dealt, while five remain for later.

\begin{align}

P(\text{all humans}) &= \frac{6}{8} \cdot \frac{5}{7} \cdot \frac{4}{6} \approx 0.357 \\

P(\text{one cylon}) &= \frac{6}{8} \cdot \frac{5}{7} \cdot \frac{2}{6} \cdot\underbrace{3}_{{\text{3 ways to assign Cylon}}} \approx 0.536 \\

P(\text{two cylons}) &= \frac{6}{8} \cdot \frac{2}{7} \cdot \frac{1}{6} \cdot (\text{3 choose 2 ways to assign cylons) }\\

& = \frac{6}{8} \cdot \frac{2}{7} \cdot \frac{1}{6} \cdot \underbrace{3}_{= \frac{3!}{2! \cdot 1!}} \approx 0.107

\end{align}

Check, does that add up to one? Yes. We can combine this to find the probability that you and the President will end up on opposite teams, given that you start off as human (without any assumptions about the other players.

\begin{align}

p &= p_0 \cdot P(\text{no Cylons}) + p_1 \cdot P(\text{one Cylon}) + 0 \cdot P(\text{two Cylons})\\

&\approx 0.6 \cdot 0.357 + 0.35 \cdot 0.536 + 0 \cdot 0.107 \\

p&\approx0.402

\end{align}

Thus, without any information pertaining to the loyalty of the other players, you're more likely than not to stay on the same team as the President. However, perhaps counter-intuitively, if you feel you can trust everyone, you're more incentivized to go for the presidency yourself.

And what if you're not certain that the President is human? What if all you know is that you're human. Given that, there's a probability of $P(HH|H) = 7/9$ that the President is human with a probability of $P(CH|H) = 7/9$ that is a Cylon with probability $2/9$. Here $P(HH|H)$ refers to the probability that both the President and you are human given that you are human. Similarly, $P(CH|H)$ is the probability that the President is Cylon (now) and you are human (now) given that you are human (now). We can express the probability that you and the President end up on opposite teams given that you are human, $P(O|H)$ as follows. Here $O$ refers to the event that you and the President end up on opposite teams. We're still focusing on opposite teams, and ignoring possible benefits to the Cylons if resources are expended moving the presidency.

\begin{align}

P(O | H) = P(O | HH) \cdot P(HH | H) + P(O | CH) \cdot P(CH | H)

\end{align}

We previously computed $P(O|HH)$, which is the $p$ immediately above.

Now let's find $P(O|CH)$, the probability that you and the President end up on opposite teams given that the President is a Cylon before the sleeper agent phase while you are human. There are similar cases to before. There are two ways that the two of you can remain on opposite teams. First, another player can receive the remaining Cylon card. Since there are five players and five loyalty card, only one of which is Cylon, everyone has the same probability of receiving the Cylon card. Thus, another player receives the Cylon card with probability $3/5$. Second, the President can receive the Cylon card (probability $1/5$) and give it to another player aside from you. Assuming the President gives it out randomly, the probability that this happens and you remain human is $\frac{1}{5}\cdot\frac{3}{4}$.

\begin{align}

P(O|CH) &= \frac{3}{5} + \frac{1}{5}\cdot\frac{3}{4} \\

&=\frac{12+3}{20}\\

&=0.75

\end{align}

Pulling everything together, we get the following.

\begin{align}

P(O | H) &\approx 0.402 \cdot \frac{7}{9} + 0.75 \cdot \frac{2}{9}\\

&\approx 0.479

\end{align}

This is much closer to 0.5, but still less. So given that you're human before the sleeper agent phase, you're just slightly more likely than not to end up on the same side as the president, given all of the assumptions that we've made. However, once you get some information about how the other people are playing, you can throw out most of the probabilities regarding the current state of the game, and only pay attention to those that affect the future of the game. Individual behavior can be much more revealing than statistics.

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