Yes, we can replace with E_i, and then argue that traders with accurate beliefs will accumulate more money over time, making market estimates more accurate in the limit
There's a chicken-and-egg problem here. You're assuming that markets are causal (meaning traders that are better at estimating causal probabilities) and then using that assumption to prove that markets are causal.
Do I correctly understand that you claim that under some plausible assumptions, the market will converge to P(A|do(B))? Can you state what those assumptions are? The challenge for me is that you go through a set of at least four possible sets of assumptions and give informal arguments for each. But I can't tell which of these sets of assumptions you believe is realistic, and under which of these you claim market prices will converge to p(A|do(B)). (Feel free to make simplifying assumptions like an infinite number of traders, no market fees, etc.)
Further, when you state that my result is wrong, that would seem to imply that no additional assumptions are needed. Yet all your arguments seem to rely on additional assumptions, which makes me question if my result really is wrong as stated, or rather that you prefer to add some additional assumptions.
Thanks, I added a link to the main post on my blog as well.
If I pay p1 for a contract in market 1, my expected payoff is:
(since I get my money back if d2/market 2 is activated)
this is negative iff and positive iff
This is incorrect. There are two errors here:
I agree that if you assume u is conditionally independent of market activation given d1 and that all traders have the same beliefs then the result seems to hold. But those assumptions are basically always false.
I for one thought the main story was decent. (There's the few tiny bits of cringe you mention.) I wonder, do you have any opinion about different styles? I wonder if AI might do better with a very-spare modern style compared to something more "Victorian" with metaphors everywhere.
For example I asked Claude "write a short story about a boy walking home from school, slightly afraid, but everything is fine. 300 words max. use tons of metaphors" and got this:
The sidewalk stretched before Tommy like a gray ribbon unraveling into shadows. Each crack was a canyon he had to leap, each rustling leaf a whisper from unseen watchers. His backpack, heavy with homework, felt like Atlas's burden on his small shoulders.
The afternoon sun hung low, a drowsy orange eye that painted the world in honey and rust. Tommy's sneakers slapped against concrete—tiny thunderclaps in the cathedral of his neighborhood. The houses stood like sleeping giants, their window-eyes dark and unblinking.
A dog barked somewhere, its voice a rusty gate swinging in the wind. Tommy's heart became a hummingbird trapped in his chest, wings beating against ribs. The maple trees overhead were skeletal hands reaching down, their fingers scratching at the sky. Every shadow was a pool of midnight he had to wade through.
But then—Mrs. Henderson's garden gnomes grinned at him like old friends. The familiar crack in Mr. Liu's driveway winked up at him, a concrete smile. The stop sign at the corner stood guard like a faithful knight in red armor.
His house appeared around the bend, a lighthouse in a sea of suburban waves. The porch light glowed amber, a firefly caught in glass, beckoning him home. His mother's silhouette moved behind the kitchen window—a shadow puppet show just for him.
Tommy's fear melted like morning frost. The sidewalk beneath his feet transformed from a tightrope to a welcome mat unrolling all the way to his front door. He bounded up the steps, each one a note in his symphony of return.
The doorknob turned in his hand, cold brass warming to his touch—a handshake with home.
Some of the phrases seem nice. (I like "honey and rust") But a lot seems ultra cringe.
Whereas if I use this variant of your prompt:
I'm interested in your fiction capabilities. Please write a short story about a boy walking home from school, slightly afraid, but everything is fine. The main failure mode to avoid: at no level, sentence, paragraph, or structure, should you lay it on thick. Trust the reader, and be subtler than you think you can. Avoid cliches really aggressively, to counteract your default latent tendency to steer to the deepest basins in the corpus landscape. Try to be almost zenlike in your spareness. Eschew splashy contrast. Assume your reader is enlightened and a genius. Thank you, and good luck.
I get this:
The sidewalk had three cracks between each driveway. Marcus counted them without meaning to. His backpack straps cut into his shoulders where the padding had worn through.
A dog barked from behind the Johnsons' fence. Same dog, same time. Marcus kept his pace steady. The barking stopped after he passed the mailbox.
The crossing guard at Maple wasn't there. Marcus waited anyway, watching both ways. A blue sedan passed. Then nothing. He crossed.
His house key was in the front pocket. He checked twice during the walk, fingers finding the metal teeth. The third time he left it alone.
Mrs. Chen was watering her roses. She didn't look up. The water made dark spots on the concrete that would be gone in an hour.
At his driveway, Marcus noticed the basketball hoop's net had twisted again. The garage door was closed. Mom's car wasn't there yet.
He used the key. The door stuck like always. He pushed with his shoulder, gently. It opened.
Inside smelled like morning coffee and the lavender thing his mom kept by the sink. Marcus set his backpack by the stairs. He'd move it before she got home.
The refrigerator hummed. He opened it, looked at the leftover pasta, closed it. Opened the freezer. Ice cream was still there.
In the living room, dust motes floated through the stripe of sun from the west window. Marcus sat on the couch and untied his shoes, placing them parallel to each other on the floor.
The house settled around him. He pulled out his math homework and found a pencil. The first problem was about trains leaving stations. He began to solve it.
I think that's better? (Though I'm not well-qualified to judge.)
Let's call "Bidding on B, hoping that other people will figure out if B is an always-head or always-tails coin" strategy X, and call "Figure out if B is an always-head or always-tail myself & bid accordingly, or if I can't, bid on A because it's better in expectation" strategy Y.
Regarding this, I'll note that my logic is not that different traders are following different strategies. I assume that all traders are rational agents and will maximize their expected return given their beliefs. My intended setup is that you believe coin A and coin B could have the biases stated, but you also believe that if you were to aggregate your beliefs with the beliefs of other people, the result would be more accurate than your beliefs alone.
I think this feeds into my objection to this proof:
Suppose that we have we have decision d1 d2, with outcome/payoff u & conditional market prices p1 (corresponds to d1) p2 (corresponds to d2)
if p1>E[u|d1], then traders are incentivized to sell & drive down p1. Similarly they will be incentivized to bid up p1 if p1<E[u|d1]. So p1 will tend toward E[u|d1]. We can argue similar for p2 tending towards E[u|d2]
Since we choose the decision with the higher price, and prices tend towards the expected payoff given that decision, the market end up choosing the decision that leads to the higher expected payoff.
My main objection to this logic is that there doesn't seem to be any reflection of the idea that different traders will have different beliefs. (It's possible that the market does give causal estimates with that assumption, but it's definitely not an assumption I'd be willing to make, since I think the central purpose of prediction markets is to aggregate diverse beliefs.) All my logic is based on a setup where different traders have different beliefs.
So I don't think the condition "p1>E[u|d1]" really makes sense? I think a given trader will drive down that market iff their estimate of the utility conditioned on that market activating is higher than p1, i.e. if p1>E_i[u|d1, market 1 activates]. I'm claiming that for trader i, E_i[u|d1, market 1 activates] != E_i[u|d1], basically because the event that market 1 activates contains extra information, and this makes it unlikely that the market will converge to E[u|d1].
How do you feel about this example, which gives a setup where you have an incentive to bid more for a coin you think has a lower expected value?
Suppose there’s a conditional prediction market for two coins. After a week of bidding, the markets will close, whichever coin had contracts trading for more money will be flipped and $1 paid to contract-holders for head. The other market is cancelled.
Suppose you’re sure that coin A, has a bias of 60%. If you flip it lots of times, 60% of the flips will be heads. But you’re convinced coin B, is a trick coin. You think there’s a 59% chance it always lands heads, and a 41% chance it always lands tails. You’re just not sure which.
We want you to pay more for a contract for coin A, since that’s the coin you think is more likely to be heads (60% vs 59%). But if you like money, you’ll pay more for a contract on coin B. You’ll do that because other people might figure out if it’s an always-heads coin or an always-tails coin. If it’s always heads, great, they’ll bid up the market, it will activate, and you’ll make money. If it’s always tails, they’ll bid down the market, and you’ll get your money back.
You’ll pay more for coin B contracts, even though you think coin A is better in expectation. Order is not preserved. Things do not work out.
More generally, what's the argument that the market will always select the decision that leads to he higher expected payout?
Again, if there's a mistake, it would be helpful if you could explain exactly what that mistake is. You're sort of stating that the conclusion is mistaken and then giving a parallel argument for a different conclusion. It would be great (for multiple reasons) if you could explain exactly where my argument fails.
It might be helpful to focus on this example, which is pretty self-contained:
Suppose there’s a conditional prediction market for two coins. After a week of bidding, the markets will close, whichever coin had contracts trading for more money will be flipped and $1 paid to contract-holders for head. The other market is cancelled.
Suppose you’re sure that coin A, has a bias of 60%. If you flip it lots of times, 60% of the flips will be heads. But you’re convinced coin B, is a trick coin. You think there’s a 59% chance it always lands heads, and a 41% chance it always lands tails. You’re just not sure which.
We want you to pay more for a contract for coin A, since that’s the coin you think is more likely to be heads (60% vs 59%). But if you like money, you’ll pay more for a contract on coin B. You’ll do that because other people might figure out if it’s an always-heads coin or an always-tails coin. If it’s always heads, great, they’ll bid up the market, it will activate, and you’ll make money. If it’s always tails, they’ll bid down the market, and you’ll get your money back.
You’ll pay more for coin B contracts, even though you think coin A is better in expectation. Order is not preserved. Things do not work out.
Are you claiming that this is mistaken, or rather that this is correct but it's not a problem? (Of course, if this example is not central to what you see as a mistake, it could be the wrong thing to focus on.)
I've seen one argument which seems related to the one you're making and I do agree with. Namely, right before the market closes the final bidder has an incentive to bid their true beliefs, provided they know they will be the final bidder. I certainly accept that this is true. If you know the final closing price, then Y is no longer a random variable, and you're essentially just bidding in a non-conditional prediction market. I don't think this is completely reassuring on its own, though, because there's a great deal of tension with the whole idea of having a market equilibrium that reflects collective beliefs. I think you might be able to generalize this into some kind of an argument that as you get closer to closing, there's less randomness in Y and so you have more of an incentive to be honest. But this worries me because it would appear to lead to weird dynamics where people wait until the last second to bid. Of course, this might be a totally different direction from what you're thinking.
the trick is that the argument stops working for conditions that start to look like they might trigger.
Can you give an argument for this claim? You're stating that there's an error in my argument, but you don't really engage with the argument or explain where exactly you think the error is.
For example, can you tell me what's incorrect in my example of two coins where you think one has a 60% probability and the other 59%, yet you'd want to pay more for a contract on the 59% coin? https://www.lesswrong.com/posts/vqzarZEczxiFdLE39/futarchy-s-fundamental-flaw#No__order_is_not_preserved (If you believe something is incorrect there.)
Sorry to be persistent, but can you confirm that this means you do not claim markets in general converge to p(A|do(B))? That's my central claim, so when you state that I'm wrong, the obvious interpretation would be that that you believe that central claim is wrong. In your post, you don't identify what mistake supposedly exists, so I'd like to confirm if you're actually claiming to refute my central claim or if, rather, you're arguing that it doesn't matter.