I don’t quite understand what you’re saying. All I am saying is that this argument is invalid:
Even if (3) is true, something more than just (1) and (2) is required to justify it. Either (1) and (2) are red herrings or there’s at least one missing premise.
We are not just cogs in the machine of a culture. But neither are we solipsistic islands. Belief is not free-floating. What we understand and believe is a product of our interactions with others in the unique circumstances of a cultural environment with its shared language.Even when we disagree with each other, that disagreement is made possible by a partially shared overarching understanding of what we are talking about.
Two points follow from this. First, there is a relative stability to cultural values, even as they slowly evolve. Second, when an individual departs from those conventions, there is no way of telling whether that will
incline them more toward blue than red. We could then imagine 20% of voters firmly entrenched within a cultural value system which commits them to blue, 20% equally entrenched toward red, and an undecided middle ground.
I think every approach does and should have an ethical pull for the community ensconced within that value system. Value systems exquisitely reflect the unique circumstances and needs of a cultural environment. MAGA’s values reflect a traditional way of life found in rural environments. I am not a part of that environment, so my ‘oughts’ can not be foisted onto them. I will vote based on my own brand of cultural and ethical relativism, but would never assume this ‘should’ be theirs, or that my perspective has any pull except among those who are already predisposed to resonate with it.
It is not unsupported, because it’s right there in the math, but I could have done a better job of highlighting it, perhaps. Here’s my elaboration:
If x \in S, and if you can apply the Principle of Indifference, then the probability that x=s is \frac{1}{|S|}, whereas the chance that x \ne s is \frac{|S|-1}{|S|}, which displays an extreme imbalance that quickly gets a lot worse as |S| grows. If x\in [a,b], then P(x=s) =0, quite simply. The probability P_r may truly exist in the interval [0,1], or perhaps the world is discretized even for probabilities, in which case P_r \in S, with |S| likely being a very large number.
Now, we probably do not have the grounds to apply the Principle of Indifference, but the above sets a baseline, because if you quantify your “bias towards 0.5”, you can start seeing just how extreme that bias needs to be to make P_r = 0.5 even remotely likely.
Let’s quantify the bias in a relative way, where we say for all p\ne 0.5, P(P_r = 0.5) = bP(P_r = p), where b is the bias multiplier. If you set b = 100, you say that P_r is a hundred times more likely to be 0.5 than any other probability (of course, in a real situation, the distribution would be more complicated, but this should suffice for now).
So, if P_r \in S, we can model the bias multiplier simply as converting S into a multi-set that only has unique elements, except for 0.5, which now has 100 instances. Then, we see that P(P_r = 0.5) = \frac{100}{|S|+99}. Yeah, if |S| is a big number (which I’d imagine it is, given that it contains all real-life, possible probabilities from 0 to 1), that pretty extreme bias of 100 does little. Scale b however you like, the probability will always just be \frac{b}{|S| + b - 1}, nothing to write home about unless b starts becoming sizeable fractions of |S|…
But P_r does not need to be exactly 0.5 for you to have a point! There is a whole transition window around P_r = 0.5 with some slightly different behavior than the rather boring piecewise function I showed in my first post. And this relates to my simplification above, since we could say it’s not just P_r = 0.5 that would have a bias, but perhaps multiple values around P_r = 0.5.
I point out that there is such a transition window in my first post:
If you look at the mathematics, you can calculate that transition window, choosing whatever cut-off parameter you want, where the parameter decides how close P_r needs to be to 1 or 0 before you just round it.
The reason I left this as an exercise to the reader is because the cut-off parameter is because I thought it didn’t really matter to get into the details here. But I was slightly wrong.
In my last post I talked about the case in which P_r = 0.5…
What follows in my post shows that your ability to affect the outcome in the best-case scenario that P_r = 0.5 is limited to a probability of 10^{-5} that your vote matters. For any P_r value even remotely removed from 0.5, that tiny probability drops to unimaginably low numbers.
You are right, we should quantify things. In my first post, I made the mistake of saying that even at P_r = 0.5, the expected value of redding is higher than that of blueing. This was a careless mistake on my part. It is false, and I correct this in the section titled Extreme 2.
I agree, and I say so in my post:
I only included sections 2.2 and 2.3 for the sake of completeness.
Whether you participate after or before the threshold, it does not matter as far as the outcome is concerned. But we’re talking about ethics here, and some people consider it metaphysically important whether you contribute to a change in the probability of P(\text{bluers die}) or not. If the “determined” condition (section 2.3) is met, you actually cannot affect the ontic probability P(\text{bluers die}), and that matters to some people. I found it prudent to highlight this.
You are right that, since you don’t know if P_r=0.5, to calculate your actual probability impact, you would need to integrate over all possible P_r-values. But this would not be adding probability to the 10^{-5} probability… In fact, it would greatly subtract from it. To understand why, consider the following overly gracious simplification:
So now, consider you’re in that room, and you have no idea what P_r is, but you want to calculate your impact on P(\text{bluers die}). Well, using the above simplification, you’d see that the average probability across all situations is 10^{-5}, and since they’re all the same in this simplified model, if you add some bias for P_r-values close to 0.5, it does not change this resultant probability impact.
Now, in actuality, for most P_r-values, \Delta P(\text{bluers die}) \ll 10^{-5}, so taking them into account, even with an extreme bias for P_r-values close to 0.5, it would drastically reduce \Delta P(\text{bluers die}). I really was showing the best-case scenario in which you magically know that P_r = 0.5, and thus you’re free to not integrate over those other, far smaller probabilities.
Your misstep was thinking you could just add those other probabilities on for free, but the fact is, whether you’re taking a normal or weighted average (with some bias for central values), you’re not just adding, you’re also dividing… (integration has a dx too, remember)
You don’t add the other probabilities to the 10^{-5}, you dilute the 10^{-5} with the other probabilities.
Now, if you’re curious, if we apply the Principle of Indifference (incorrect, but simpler) to P_r-values, and we integrate over all possible P_r-values, from 0 to 1, we can see what the resultant difference in P(\text{bluers die}) that we can cause. Firstly, we make \Delta P(\text{bluers die}) a function of P_r like this: \Delta P(\text{bluers die})[P_r] = \text{Diff}(P_r). Then, the derivation looks like this:
Setting \epsilon = P_r - \tfrac{1}{2} and using the local CLT, the impact function is approximately Gaussian in P_r:
It is centered at P_r = \tfrac{1}{2} with standard deviation \sigma_{P_r} = 1/(2\sqrt{m}) — for m = 8 \times 10^9, an absurdly narrow spike of width \approx 5.6 \times 10^{-6}. Under a uniform prior, the spike is so concentrated we can integrate over \mathbb{R} without meaningful error:
Plugging in m = 8 \times 10^9:
This gives us that our choice in that room comes with \Delta P(\text{bluers die}) \approx 10^{-10}. Now, as I’ve said before, I don’t think the Principle of Indifference holds for P_r here, but if we impose a bias towards P_r-values close to 0.5, the \Delta-value would only be able to approach an upper bound of 10^{-5}.
So, your concerns for the rigor were valid, but they do not improve \Delta… That’s why I didn’t include any of this, because I didn’t want to complicate it when the upper-bound of 10^{-5} was already tiny.
But it turns out it wasn’t so tiny after all…
I sloppily assumed the expected value for picking red was still best even when P_r =0.5, which is why I didn’t elaborate on the actual \Delta P(\text{bluers die}). Because it turns out, it does matter!
Let’s take a look at the expected values of picking blue / red as a function of P_r. We assume all lives equal, and since m = 8 billion, the number of bluers will be effectively identical to P_bm = (1 - P_r)m.
And:
We have two extremes we can substitute \text{Diff}(P_r) with: the value 10^{-10} and the value 10^{-5}.
Checking the former, we see these expected values:
Now, looking at the EV of picking blue:
The square brackets denote Iverson brackets.
Conclusion 1:
Comparing the two, we can see that the expected value of redding is always greater than that of blueing given \text{Diff}(P_r) = 10^{-10}. See here:
Let’s look at the situation where you have the most impact on the probability, which is when P_r = 0.5, which means \text{Diff}(P_r) = \text{Diff}(0.5) = 10^{-5}.
Let’s again assume all lives are equal, with a value of 1. Also, since m = 8 billion, we know that the number of bluers will approximately be mP_b. Also, this all is made simpler by the fact that P_b = P_r = 0.5. We thus have the expected values here:
Now, looking at the EV of picking blue:
Conclusion 2:
Clearly, 1 - 4\cdot 10^{4} \ll 4\cdot 10^{4} - 0.5, which means that in this case, \text{EV}[\text{picking red}] \ll \text{EV}[\text{picking blue}].
You seem to think there’s a real chance that P_r is close to 0.5. Well, I think so too, but here the absolutely essential question is this: how close?
The function \text{Diff}(P_r) is not even remotely a linear function that goes from 10^{-10} up to 10^{-5} and then back down again as P_r varies from 0 to 0.5 to 1. Instead, it looks far more like the Dirac \delta-function:
The difference is that \text{Diff}(P_r) is centered on P_r = 0.5 and the max output is roughly 10^{-5} as opposed to 1, and its spike isn’t literally infinitely thin, but it is practically that thin. Now, the EV of blueing as a function of \text{Diff}(P_r) is thusly also practically infinitely thin, though slightly asymmetric around the center P_r =\tfrac 12.
And I’ll prove it.
Let’s find the interval [a,b] such that:
I call this the bluey interval, the interval in which blueing doesn’t have a lower expected value than redding.
To get this, we set the two EVs equal and find the two roots r_0 and r_1, since that gives us the bluey interval as [r_0, r_1].
Outside a thin transition window of width \sigma_{P_r} = 1/(2\sqrt{m}) around P_r = \tfrac{1}{2}, the probability P(\text{you die}\mid\text{you picked blue}) \simeq [P_r > \tfrac{1}{2}] to extreme precision; we’ll assume this and verify afterward that both roots fall outside that window.
Setting the EVs equal and using the Gaussian form \text{Diff}(P_r) = \frac{1}{\sqrt{\pi m/2}}\exp\!\left(-2m(P_r - \tfrac{1}{2})^2\right), we have:
Letting u = P_r - \tfrac{1}{2} and using 1-P_r \to \tfrac{1}{2} to leading order (valid since |u| \ll 1):
Taking logs:
For m = 8\times 10^9 we have \sqrt{2m/\pi} \approx 7.14\times 10^4, giving:
Both lie roughly 4.6\,\sigma_{P_r} outside the transition window of width \sigma_{P_r} \approx 5.6\times 10^{-6}, confirming the Iverson approximation we’ve been using.
And therefore, the bluey interval is:
That is a window of total width \approx 5.20\times 10^{-5}. As you can see, it’s tiny, and the chance that P_r actually falls inside it under any non-degenerate prior is so small you can pretty much disregard it. That means redding has the highest expected value in essentially every realistic scenario.
You say this: “Picking blue contributes to your death only: …”
If you picked blue and died, your death is the death of a bluer. So, your choice caused the death of a bluer… That is simple logic. You are not applying the Copernican principle here.
I must thank you for prompting me to make my argument more rigorous. I did not highlight the fact of just how tiny the bluey interval is, but I should have.
Now that I have shown that, I hope you understand that the expected utility of redding is almost always higher than the expected utility of blueing, given that all lives are equal.
Yes and that’s what I was saying, you assume it’s uniform so 0.5 is extremely unlikely but that’s probably not true.
I am not sure how convincing that is given I can say the same thing about my coin and yet I am very tempted to believe my coin is fair.
You are right. Point taken.
The expected value calculations are sloppy.
In your extreme 1, where you talk about the “aggregate” or P_r, I don’t think it makes sense to do the calculations for all P_r at once.
The problem is putting P(survive) at 1. Like I said before, it’s not 1 because there are times where you survive despite picking blue. So red only adds you value when picking blue would have led to your death. So for P_r = 0.5 for example, it’s only (almost) half the time (when red is already a majority basically). The actual expected value of red (blue is just the opposite) is, for one value of P_r:
P(more than m/2 red votes) - \frac{m}{2} P(exactly m/2 red votes)
For the aggregate (which isn’t useful in my opinion), it’s the integral. What happens is that the expected value of blue is greater when P_r \leq 0.5 + \epsilon and the expected value of red is greater otherwise. On the red side, the maximum expected value you can get is 1 (when P_r = 1) and this drops to 0 at P_r = 0.5 + \epsilon. And with blue, the maximum expected value you can get is 50462.15 (around 4 billion times the 10^{-5} but more precise) when P_r = 0.5 and it drops towards 0 when we get close to P_r = 0.5 + \epsilon or to P_r = 0.
Not really.
I don’t think so. The expected value of blue is greater (but very small) even when P_r < \tfrac{1}{2} - 2.64\times 10^{-5}. I checked that with a program.
Not tiny anymore.
Sure, there is contributing to the genocide as in being one victim of the genocide and contributing as in being perpetrator. I don’t think contributing is usually used in the first sense but it’s just words anyway.
I would have hoped it clear from my previous responses, prior to your joining us, that that’s just not the approach I adopted. Those who are familiar with my writing elsewhere will be aware of my fondness for logic.
@AlveK presumes that the best result is the survival of the individual, and then give a very long-winded explanation. That is tangential to the Blue argument, which is to seek the best outcome overall. His premise is the survival of the self - that is how he frame the whole discussion.
So here:
See the discussion of Foot’s article above. This framing is erroneous. Picking red does do harm.
Here again the presumption is the survival of the individual. Yes, your life is valuable. “If”.
It’s difficult to see how such a conclusion can be maintained.
What these quotes show is that your presumption that the morally relevant consideration is the direct, probabilistic impact of one’s individual action on the aggregate outcome, measured in terms of individual survival. That is what blue rejects.
So back to my question to @AlveK. You say you have shown that the rational response is red, yet the evidence shows a large majority will pick blue. Again, how can you account for this discrepancy?
Either most folk are unable to follow your mathematics, and they are mistaken. Or they have not accepted that yours is the correct approach.
I put it to you that we might account for the majority vote by pointing out that folk will cut through the calculation by seeing that if they vote blue, and trust others to do likewise, the result will be the best possible one of everyone surviving.
So will we say that most folk are irrational, or will we say that most folk use a different form of reasoning to your calculation? That they reject the mathematical, individual-survival-maximizing framework as the only legitimate approach?
Take care not to fall to the simplistic error @Hanover makes, of thinking the argument is “Most vote blue, so I should too”.
And it’s worth adding that I do not expect you to agree, and perhaps not even to recognise the point being made. One problem with making long calculations is that it takes commitment, and so one ends up “nailing one’s flag to the mast”. Over-intellectualising the problem brings its own blindness. My suggestion is that having the mathematical tools at hand, the temptation to use them leads to a sort of entrapment, to only seeing the problem in a way that is amenable to the one solution.
I mean I think the point is quite clear.
In a rational society, this wouldn’t even require a second of thought. Nearly everybody votes blue, everybody survives.
That we live in a world where the results of this poll are 50/50, is an indictment of the intellectual culture.
It assumes that a large portion of the population are such cynics that they think they ought to let others die in a genocide.
No doubt questions like these arise from the mistaken assumption of game theory…
Yep.
The way that @Michael and @AlveK see the scenario is mistaken.
And so they reach the wrong conclusion.
Like the rest of your post, there’s not much for me to object to here, but there’s more at stake than what you have sufficient warrant to assume. People learn from each other. They even learn things that they are resistant to learning but are capable of learning. There aren’t all that many teenagers you’d call ‘predisposed’ to learn calculus, but there are a considerable number that are capable of it. You meet them where they are and try to bring them to calculus as you bring calculus to them.
And you can see the same thing happen with moral issues: a person’s thinking can change based on what another says to them, or something they read, or something they see. I’d agree that we would be wrong to expect this, to assume that it will happen, but it is clearly possible.
And that’s one reason we talk to each other, as we do here.
Most responses seem to fall into three buckets:
People who perceive this thing as a sort of puzzle to be solved work out quickly that choosing Red means you survive no matter what, and that makes Red the right answer. The morally charged language used in the puzzle statement counts as a sort of misdirection. The puzzle has a right answer and you just have to work out the logic without being misled, and you win.
People who perceive it as a moral question mostly still work through the same steps as the puzzle-solvers, but they take the moral language seriously, even if they don’t all reach the same conclusion.
And finally there are people who see it primarily as a matter of guessing how a given population will vote, based on whatever—empirical research, some gut instinct about what humans are like and how they behave, a lot of options here.
I think we are all capable of seeing the problem in each of these ways (and in others as well), but for reasons that are certainly unclear to me people are drawn particularly to one framing or another (as both you and @Moliere , among others, have noted).
Given that we each have some level of commitment to one framing or another, you’re right that we can’t assume anyone will see things the same way we do, and if we demand that they do we’re likely to be disappointed.
But not only is it still worthwhile, there isn’t only one sort of thing happening here. There’s a real difference between me engaging you as a moral agent, urging a moral viewpoint on you, and engaging you as a rational agent, telling you how to solve a puzzle. Life is complicated enough that we often have to juggle all three buckets—understand the lay of the land empirically, work out a rational response to it, guided by our values. It’s probably not so much that these three types of behavior are constantly entangled, but that they are three aspects of a unity, and we’re fluent at picking out one aspect or another, either holding the others fixed or disregarding them altogether—as circumstances or consensus allow.
And we’re prone to forget that’s what we’re doing, which leads to the sorts of impasse we’ve seen here, where there’s lots of incredulity at what the other side is leaving out.
I don’t have a fix for that, but I think we all agree that part of the point of doing philosophy at all is learning to see your perspective as a perspective, to see this framing as a framing, and the usual way to achieve that is seeing something from another perspective or framed differently. What happens then, I don’t know, but I think we want something more than “well it depends on how you look at it.” You should know more when you’ve seen something from more than one perspective. It’s why we bother to learn how other people see things, and why we tell them how we see things.
Jeez man. What’s going to be the next contentious topic?
Press red to torture others babies, blue to torture only yours? I don’t know.
A neat analysis. My main misgiving is calling the first option rational. It might encourage the naive to think the other two options are therefore irrational.
What about calling it the “Bookkeeping” response?
I get that. Maybe “rationalistic”.
I don’t really want to wade into the game theory, but since in some ways this is a variation on Prisoner’s Dilemma, it might be worth pointing out that the point of Prisoner’s Dilemma is that it’s a paradox, on its own terms. You don’t get a prize for figuring out there’s a dominant strategy; the whole point is that the dominant strategy leads to a suboptimal outcome. Sometimes game theory is a trap.
In this case the thought experiment seems to have gone out of its way to make Red the dominant strategy, to the point of—unlike in standard PD—giving it no apparent, measurable downside at all. (The deaths of others would count as, ahem, a “disutility” to most people, but there’s no measurable value for that.) And then the moral or social cues all run the other way.
So which is it? Are the social cues misleading? Or is the dominance of Red misleading? I’m genuinely not sure, and maybe there just was no particular purpose to the way it’s set up. I spent some time digging around in r/polls and there are buttons galore. This one happened to land on something that has traction no one seems to really understand.
Or perhaps they have chosen the correct approach for them, and there is no such thing as THE correct approach. Why not celebrate your blue vote and avoid proclaiming what others ought to do? Thar sounds like a nice cooperative gesture to me.
Very much so. But this can happen in two ways.
Perhaps we sometimes use game theory in order to comfort ourselves that we can always rationally find an optimal action, but of course that’s not how things are.
Or perhaps the clarity of the mathematics of game theory deludes us into an unwarranted certainty. Any formalisation must abstract away from reality, taking some things as important and others as irrelevant.
Certainly when one puts substantial effort into a calculation, it becomes harder to see that effort as misguided.
Nevertheless we must choose. That’s the underpinning existential state that we find ourselves in.
So in choosing, will we consider only our own survival, or will we consider others?
It seems to me to be pretty much that simple.
Will you consider that many red voters would say you’re putting words into their mouths, or do you prefer to proclaim on their behalf how they think?
There’s really no question about it.
One thing that’s interesting to me about this puzzle is that it really pushes you to confront genuine uncertainty. It’s a popular strategy to rewrite all uncertainty as manageable risk, all probabilities and expected utility. But here the stakes are so high they swamp the probabilities and the expected utility values are all maximal. Not only is there really nothing to calculate with—no way of knowing how the vote would turn out—the calculation wouldn’t do you much good.
So I think a lot of this is about real uncertainty, that black hole of ignorance we can do nothing about, and how we act accepting that we don’t know shit.
This is the focus of the Anscombe-Foot discussion. Ethics is not algorithmic.
Indeed, very little is algorithmic, including maths and logic.
So some—many—when confronted with the “trolley” problem presume that there is a deducible correct answer and argue accordingly.
Others—the better thinkers, I will hold—will see the Tram problem; that there is no optimal solution, yet nevertheless we must act.
The red voters see the trolley problem. The blue voters might be seeing the tram problem.
Here’s another way to think about it. Suppose we really were put in the position described, perhaps by a victorious Kim Jong Un or some such.
The simplest way to give him the finger would be to refuse participation. But if that is not an option, vote blue so as to maintain the status quo and so undermine the very problem as offered.
Voting blue as recognising the framing and rejecting the tyranny.
@Moliere ?
to the whole thing?
Damnit, I’ve spent too long on this silly thread.
Me too.
But, to be fair, I’ve been having a hard time saying something at all, so it brought me out even if just to complain and gripe 
That’s the way I’m inclined just because it seems like the only way to answer is to bring your presuppositions to it.
What you’ve said of bluepushers fits my ethical intuitions. It’s not necessarily a reasonable proposition, but it is at the same time quite reasonable, given what we know about human beings.
I’m inclined to throw the finger mostly because it looks like a party question which is fun, but then we can spend too much time on the thread (as I have too)
