# A short calculation about a Twitter poll

Recently, the following poll was [posted on Twitter](https://twitter.com/lisatomic5/status/1690904441967575040):

> Everyone responding to this poll chooses between a blue pill or red pill.
>    - if > 50% of ppl choose blue pill, everyone lives
>    - if not, red pills live and blue pills die
>
>    Which do you choose?

If we linearize the problem of how much you value the lives of other poll respondents compared to your own, there's actually a way to quantify whether you should vote red or blue depending on your beliefs about how the rest of the population will vote.

Suppose that we normalize the value of choosing the red pill at zero, and let \\( N \\) be the total number of people responding to the poll excluding you. Then, choosing the blue pill only makes a difference in the following two cases:

1. If the number of people who vote blue excluding you, hereafter denoted \\( B \\), is *exactly* \\( \lceil N/2 \rceil \\), then your vote saves the lives of \\( \lceil N/2 \rceil \\) people.
2. If \\( B < \lceil N/2 \rceil \\), then you die and get nothing in return.

If you value the lives of other poll respondents at a constant multiple \\( \nu \\) of your own life, then you should pick blue if

\\[ \mathbb P(B = \lceil N/2 \rceil) \cdot (\nu \lceil N/2 \rceil) > \mathbb P(B < \lceil N/2 \rceil) \\]

and you should pick red if the inequality goes in the other direction. 

A rough heuristic here is that if the distribution of \\( B \\) is unimodal with a maximum to the right of \\( \lceil N/2 \rceil \\), you expect

\\[ \lceil N/2 \rceil \mathbb P(B = \lceil N/2 \rceil) \geq \mathbb P(B < \lceil N/2 \rceil) \\]

so if \\( \nu = 1 \\) (meaning you value the lives of other poll respondents equally to your own) and you expect a blue majority in the precise sense defined above, you should always take the blue pill yourself. The exact cutoff on \\( \nu \\) for you to take the blue pill is

\\[ \nu \geq \frac{\mathbb P(B < \lceil N/2 \rceil)}{\mathbb P(B = \lceil N/2 \rceil) \lceil N/2 \rceil} \\]

A general heuristic is that if we think a red victory is a distinct possibility, so that \\( \mathbb P(B < \lceil N/2 \rceil) = O(1) \\), and the distribution doesn't decay too sharply around \\( N/2 \\), in general, we'll have \\( \mathbb P(B = \lceil N/2 \rceil) = O(1/N) \\) so that the cutoff ends up being \\( \nu \geq O(1) \\). In other words, you pick blue in realistic circumstances if you value other people's lives similarly to your own: \\( \nu \\) can't be too far away from \\( 1 \\) if picking blue is to make sense as an individually altruistic decision, ignoring social pressures to pick blue *et cetera*.

Approximating \\( B/N \\) by a beta distribution for large \\( N \\) gives the following rough results:

| Probability of a red victory | Expected share of blue votes | Cutoff value of \\( 1/\nu \\), i.e. minimum value of your life in units of others' lives to choose red |
|------------------------------|------------------------------|--------------------------------------------------------------------------------------------------------|
| \\( 10 \% \\)                | \\( 55 \% \\)                | 22.4                                                                                                   |
| \\( 10 \% \\)                | \\( 65 \% \\)                | 7.1                                                                                                    |
| \\( 10 \% \\)                | \\( 75 \% \\)                | 3.7                                                                                                    |
| \\( 20 \% \\)                | \\( 55 \% \\)                | 11.6                                                                                                   |
| \\( 20 \% \\)                | \\( 65 \% \\)                | 3.5                                                                                                    |
| \\( 20 \% \\)                | \\( 75 \% \\)                | 1.4                                                                                                    |

I think the degree of altruism implied by choosing blue in this context is pretty strong for plausible distributions over \\( B/N \\). For that reason, I think picking red is easily defensible even from an individually altruistic point of view (would you sacrifice your life to save the life of five strangers?)

There are higher-order considerations that are relevant beyond individual altruism, of course: society might have a set of norms to impose penalties on people who choose to take the red pill. However, the possible cost of not taking the red pill is losing your life, which suggests that such penalties would have to be quite steep to change these basic calculations as long as there is a non-negligible probability of a red victory that survives.

I suspect that if this were a real situation most people would choose to take the red pill in a situation where monitoring costs are high (e.g. which pill you take is a secret decision unless red wins) and social punishments are therefore difficult to enforce.