Alice purchases a lottery ticket with odds of only 1/1 quadrillion to win. She discovers that she has the winning ticket. Upon going to collect her prize, she learns that this lottery company has a policy of hiring a sniper to follow around every ticket purchaser, and kill them if they ever discover that their ticket is not the winning one.
Should Alice infer via the anthropic principle that there were a huge number of people playing in this lottery?
Resolves once I'm convinced of the correct answer.
1,000
3.00The answer is no if you use a reasonable theory of anthropics (in particular, one that isn't Dutch-bookable), and depends on the details otherwise. Consider a version with 100 possible tickets, and each major theory of anthropics in turn (SIA, non-minimum reference class SSA, minimum reference class SSA):
- Suppose that before finding out that she won, Alice is 50:50 between only her playing (and buying ticket 1), and 100 people playing (and the other players buying tickets 2,...,100). She thinks that the winning ticket is equally likely to be each of the numbers 1,...,100.
- Now, there are 200 possible worlds, of equal ex ante (before making anthropic updates) probability: 100 in which Alice is the only player, in 99 of which she loses and is shot, and in 1 of which she wins, and 100 in whcih Alice is one of 100 players, in 99 of which she loses and is shot, and in 1 of which she wins.
- So before making anthropic updates, the the two worlds in which Alice wins (the one with many players, and the one with few players) are equally likely. Let's now turn to each theory of anthropics in turn:
- SIA: SIA splits its credence evenly between expected copies of Alice with her current observations (so it updates in favour of worlds in proportion to the number of copies of Alice with her current observations in that world). It therefore assigns 0 credence to worlds in which Alice loses, and splits its credence evenly between the two possible worlds where Alice wins (since both just have one winning Alice in them) -> No update
- minimum reference class SSA, or generalised double halfing (GDH) (taking the terminology from this paper - https://www.andrew.cmu.edu/user/coesterh/DeSeVsExAnte.pdf): This only updates against worlds with no copies of Alice with her current observations, and otherwise divides credence according to the prior. So it is again 50:50 between the two worlds where Alice wins -> No update
- Finally, non-minimum reference class SSA, or generalised single halfing (GSH), sometimes just called SSA: This updates against worlds in proportion to the proportion of agents in the agent's reference class that aren't observing the agent's current observations. For instance, in the Doomsday argument, if you are a priori 50:50 between large world and small world, it is a large update towards small world, after you observe that you are early. If the other agents aren't in Alice's reference class, this is the same as minimum reference class SSA. What if Alice's reference class is just everyone who enters the lottery? Then in the world where Alice is the only one to enter, and wins, 100% of agents in Alice's reference class are observing what she observes, whereas in the world where Alice is 1/100, the other agents observe other things (not winning, not being Alice, dying), so this is actually an update away from the world where many people enter. What if the other agents still exist in the other world, but don't enter the lottery? Then if we remove the sniper, there would be no update, since the same proportion of Alice's reference class is observing her observations in either case (1/100*T, where T is the number of time steps). But with the sniper, there can be an update in favour of the world where many agents enter: since the other agents have less total observations when they die, and more when they don't enter and live, a higher proportion of Alice's reference class is then observing her observations in the worlds where she is one of many entrants (since more people are dead in that world). For instance, if the 100 agents are the whole population of the world, and they live half as long when they are shot, this is about a twofold update in favour of the world where they do enter. If they live zero time when they are shot, this is instead a 100-fold update. But it isn't really important whether they enter for this, GSH would make this same update if they didn't enter the lottery but a crazed gunman killed the other 99 agents for fun. Finally, if there are many other agents in the world, the update is much smaller -- e.g., if there are 1,000,000 agents in the world who never enter the lottery, and they are also in Alice's anthropic reference class, then the fraction of Alice's reference class that observes what she observes is not very sensitive to whether 100 agents are shot or not, and so she makes only a small update.
So GSH/non-minimum reference class SSA can update in favour of the world with more entrants, depending on the details. However, this update is essentially just towards worlds where there are fewer other people alive. Moreover, I would argue it is not a very good theory of anthropics. For instance, it is Dutch-bookable when combined with CDT or EDT in a setting in which those decision theories are not Dutch-bookable when combined with better theories of anthropics. (Cf. https://www.andrew.cmu.edu/user/coesterh/DeSeVsExAnte.pdf)
Depending on the information Alice has, the anthropic principle could lead Alice to infer that probably a lot of other people bought the ticket just from the fact that she bought it. But the snipers and the fact that she won add no extra probability to this - they will not update her probability even a tiny bit. The is true no matter what version of the anthropic principle (SIA vs SSA) you use.
This question was inspired by this article, for reference.
I think it's a terrible article. The inverse gambler's fallacy is not analogous to the fine-tuning argument, nor is the modified version with a sniper. The actually analogy to fine-tuning would be, "Betty would not exist at all unless a specific set of numbers came up in some Bingo machine tonight. Given that Betty exists and knows this, she infers that there must be lots of people playing Bingo tonight." This is clearly a valid inference, it's just a standard application of Bayesian reasoning.
What makes the article even worse than the fallacious argument, though, is that it presents the argument as if it's a decisive refutation that has been mathematically proven or is agreed upon by probability theorists. This is 100% false - anthropics is contentious, but it is certainly not true that, "experts in the mathematics of probability have identified the inference from fine-tuning to a multiverse as an instance of fallacious reasoning." Not even close.
Partial, incomplete thoughts: I think the answer is almost certainly no. Before Alice buys a ticket, do she and her eventual sniper have different beliefs about the number of participants? Say no potential participants know about the snipers. Then they certainly must agree (assuming the same priors and info outside of the snipers thing.) For everybody who might potentially buy a ticket, Alice and her sniper have the same beliefs about that person's thought process, so they must estimate the entire sequence of conditional probabilities the same.
After Alice buys the ticket but before she checks if it's winning, do her beliefs and those of her sniper diverge? No, for the same reason as above.
After learning that Alice has won, do their beliefs diverge? No, they now have access to the same information.
Finally, does the sniper update their belief about the number of participants based on Alice winning? No, because none of their beliefs about the potential participants have changed, and the sniper is not themselves subject to any sort of anthropic reasoning here.
Anthropic Principle only seems to be effective when updating about the probability of an event that would wipe out all possible observers.
Since the goal here is to convince you, can you explain what you think the snipers contribute to the thought experiment?
To me, they look like a way to make the stakes of losing be non-existence, which then is an attempt to make the anthropic principle more relevant.
However, I think the principle is just as relevant even if there are no snipers and she would survive losing, because she can still consider the chances of there being other players, given that she won (and imothat's all the existential stakes of the anthropic principle are for, to emphasise that this is a conditional probability question, not an absolute probability question).
So imo the snipers muddy the waters by raising questions of how it is that she would have missed all the dead people if losers die. That question seems to not be relevant to what you want to ask, so I think introducing the snipers just confuses things for no benefit.
I might be mistaken here though, so if you explain what you think the snipers meaningfully change abotu the situation then that would help either side try to convince you.
@MatthewLeong Hmm, yeah. Even without the snipers, the chance of anyone winning is very low if there are few players, so Alice could infer that since someone has won (which happened to be her), there were lots of players.
That doesn't seem right though...
@IsaacKing "Alice could infer that since someone has won (which happened to be her), there were lots of players."
I don't think that is valid, because of the way information is arrivd up to update Alice's knowledge.
If someone else won, would she know? The question doesn't say she would learn this information.
So "I won" is quite different to "an unknown person won".
If an unknown person wins, that could be evidence of a huge pool of gamblers that made that win more likely, in order for you to be likely to witness a victory.
However, if a single person who you already knew bought a ticket wins (such as yourself), then I don't think you need to justify their win with more people to pad out the ticket sales.
The difference here is pretty subtle, so if you want more explanation I'll try to provide it.
I think it is similar to (but not exatly the same) to the Monty Hall Problem, in that the way in which information is presented to you matters.
Commenters here really need to learn how to interact with thought experiments in a productive manner.
https://www.lesswrong.com/posts/neQ7eXuaXpiYw7SBy/the-least-convenient-possible-world
@IsaacKing In the particular case of anthropic puzzles, you can't assume that you are in the least convenient possible world because the question of what world you are in is part of the puzzle.
The odds actually don't even matter due to survivor bias. If the odds were 1 in 2 or 1 in 10^100, it's the same.
It's the 1 in x chances that counts bc Alice is alive. If the odds were 1 in 2, would this change anything? Assuming this lottery can have multiple winners.
If Alice is the only person to play Power Ball or 300 million people play Power Ball, the odds for Alice winning remain constant.
I think the only way she can conclude that lots of people lost, would be to know that other people are stupid enough to try this gamble.
You might try assuming that a similar proportion of people play the lottery (some quick googling suggests maybe 1/3rd of people do).
But lotteries are a bad deal even with 1/millions win chance, and if you made the game almost ~1billion times harder to win, I think you'd get fewer buyers.
Like, a lottery game kind feels fair, becaue all you need to do is guess like 5 or 6 numbers or whatever.
With those sorts of chances, its almost like you need to win the normal lottery twice back-to-back (like winning the lottery puts you into the draw to attempt another lottery, and only if you win both rounds do you win overall).
And this is all in a realm of behavioral economics, and far beyond the 'pure logic' realm that I think the spirit of your thought experiment was in (since you tried to invoke the anthropic principle).
Alice simply doesn't know if other people would also buy lottery tickets. If she starts seeing huge numbers of people dying from sniper shots, then she'll have her answer, but imo people won't buy such an over-rigged lottery in the first place.
Alice should infer she is the only player of this lottery because the odds are so small. Assuming normal societal conditions with no sniper epidemics, Alice should infer she's the only player due to the absense of dead lottery losers, with single gun shot wounds to the head, littering the streets. There would be piles of bodies at any lottery terminals where people checked their tickets if Alice weren't the only player (and winner).
Its the Fermi Paradox in a microcosm. It ceases to be a paradox and is solvable at this scale. Dead bodies from snipers = aliens. No dead bodies, therefore no other lottery players. It's not like we wouldn't know the dead lotto losers existed or not like the current situation we find ourselves in with alien life. This situation would be akin to having access to explore the entirety of the universe for alien life like we would have access to explore all the earth for all the dead bodies from all the snipers.
I think the snipers are muddying the water here. Without them I would say that yes, you should use anthropic principle to deduce that there were a lot of players. But with them either Alice knows about a lot of people being killed by snipers and this affects the probability, or she doesn't know, which is very implausible and affects the probability in the opposite direction.
I don't think the anthropic principle applies in the case where you have agency in taking the extreme chance.
It is usually used to describe some selection bias in the creation of the universe.
By deploying the snipers you've artificially made this lottery a matter of life&death, but crucially, only people who chose to play it, played.
Even though the players didn't know all the rules, they did know something about the chance at play (1/trillion to win, with the unknown kicker of death).
When it comes to the universe, we didn't choose to gamble on the fundemental physical constants of the universe and that's a crucial part of the anthropic principle, since us existing requires us to have 'won' the comic lottery before our species even evolved.
Alice, on the other hand, already existed, and gets to buy her independent ticket with a known 1/quadrillion chance of victory.
Aiming snipers at the players doesn't change things, since those snipers only kill people in the present, not retroactively make the universe the losers came from never exist in the first place.
For practical purposes, Alice's odds of winning were the same whether she bought a ticket or not. Who knows why she bought the ticket, for fun maybe, but unlikely she's now going to become fascinated by the "anthropic principle" and try to infer anything at all. She'll just go on, take the money and run.
The sniper could also be outside the story: "Alice is a fictional character. But nobody reads boring stories. So stories where Alice wins unlikely lotteries are more likely to be instantiated into stories. The odds are closer to 70:30 in favor of her winning the lottery from the Fictional Anthropic Principle."
@Mira Yeah, I think she infers that she is in a thought experiment and probably the simulating philosopher isn't clever enough to simulate more than a few players in any depth.