The author looks credible:
https://philipmaymin.com/about-philip
Thank you for sharing this on HN.
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To the mods: The title needs to be edited to replace the equal sign with not-equal.
But, both free markets and supply/demand are useful enough concepts to talk loosely about processes to understand the interest that I'll enjoy digging into this.
“=“ <> “!=“Most filters are to avoid sensational titles, AFAIK.
Markets are efficient if and only if P = NP https://arxiv.org/abs/1002.2284
:)
"Markets are competitive if and only if P != NP"
Seems that HN's auto-headline rewriting in this case has made a critical error :)
>Artificial intelligence, by expanding firms' computational capabilities, is pushing markets from the competitive regime toward the collusive regime, explaining the empirical emergence of algorithmic collusion without explicit coordination.
I have to dig more into the paper but I don't see how this follows, except in the most straightforward way. Basically, if everyone uses the same methods to derive price, of course there will be "collusion", or in other words, everyone will have the same price. But this doesn't seem like a result of compute per se, but simply better communication networks and information flows. You could have gotten the same result in medieval England by having everyone post their selling prices on the town square board.
Again, I haven't dug into the paper yet, but it seems like what really matters for firms is "compute"/$ (if the "compute" is an LLM or an assistant that has to go walk the 10 minutes down to the square makes little difference)
Edit: Isn't another implication of this, that increased compute -> collusion imply that increased compute -> communism becomes feasible?
I think this goes to my point above though, the primary problem preventing fully automated luxury communism isn't compute per se, but actually observing the information flows to make it possible. Capitalism famously solves this information problem through the pricing mechanism. So in effect, he's arguing that extra compute makes information gathering more efficient, and at the limit you get perfect information. Which, yeah, I guess so. Assuming everything can be perfectly measured, even theoretically.
The term ‘perfect information’ is a bit of a mirage, and has been shown to be impossible in physics (uncertainty principle).
What really matters is information advantage: Does your inexact expected value function consistently beat others’ calculations in the market. Here, the true value - value really is just a word and is dependent on people - is irrelevant.
If P!=NP then it is arbitrarily smaller, for the same reason that e^x > Cx^N for any constants C and N, as long as x grows big enough. There is no epsilon in that can overcome that, no matter how big you make it, because x will eventually dominate the equation.
There are a lot of cases where pragmatically x remains small enough that it doesn't matter, and a P algorithm will give you an answer more quickly. (For the same reason I only ever write bubble sorts: I would only write my own at all if I knew that the list would never be bigger than 10. Even then it's only when using the library is too much trouble for some reason.)
But we care about P and NP when the number can potentially be very, very large.
So I'm not talking about the number of steps needed to prove optimality with a correct P algorithm versus an exponential one.
I'm only talking about how this applies to the efficient market hypothesis.
NP problems gets solved with heuristics every day.
NP-Completeness is the norm, not the exception. Any system that's complex enough is almost surely NP-Complete. For similar reasons, Turing Machine Equivalence is also the norm, not the exception.
These results are interesting but not unexpected. A more interesting question is under what conditions is the problem difficult to find solutions for. Many NP-Complete instance ensembles turn out to effectively have polynomial time solutions (3-SAT w/ uniform clause variable choice, Hamilton Cycles in Erdos-Renyi random graphs), so proving NP-Completeness is not a death knell for approximation.
Really interesting conclusion, but I can't help but feel this is overly reductive, as stated. Surely market efficiency is a sliding scale and so is market competitiveness.
Okay, so a perfectly competitive market cannot also be a perfectly efficient market. Interesting! But I'm confused about how this may work when efficiency and competitiveness are a sliding scale. Should we think of this as one axis (with a spectrum from efficiency to competitiveness) or as two separate axes that just happen to have an exclusive relationship between their extremes?
No expensive analysis is needed. Beating them just requires (1) having disproportionate capital to others, and (2) having disproportionate control (and therefore know in advance) whatever the markets are pricing.
It helps that there are enough believers (in the religious/cult sense) that markets are 100% efficient, that they will deny that any of that is happening - even while they lose their shirts to those who are doing the market manipulation.
Now seriously, I wonder if AI collusion/use in investments would add to the market inefficiency and create opportunities for observing investors.
The annoying part is that, as the same Adam Smith says, regulating industries would end up enforcing such assemblies, reinforcing the problem... after all, industries can share information via the market itself...
And proposed solutions end up being controversial: employees ownership, open source, paying taxes over stocks ownership... or just hoping that colluders will be broken by a randomly ocurring incumbent...
In a certain sense I guess this is why every anti-trust suit fundamentally comes down to defining the market bubble more than anything else.
Want to buy/sell a stock?
Humans need to manually submit in the system.
First, this definition has always been circular: what’s the most accurate price? The one the market comes up with. More market, more accuracy!
Second, there is never any reconciliation of the costs society is saddled with in order to chase arbitrarily more accurate prices, the most obvious of which is the massive quantity of fat skimmed off by the financial services sector.
Third, as an index investor, I more or less couldn’t care less. This hyperfixation on accuracy only really matters to people who are actively trading, which is already a fool’s game.
This "should" is doing a lot of work here. The paper is mainly about a game-theoretic model allegedly corresponding to real markets, but establishing what regulators ought to do requires far more rationale than mere math. It requires a bridge from "is" to "ought." It reminds me of Hume's warning about this kind of non-sequitur:
"In every system of morality, which I have hitherto met with, I have always remarked, that the author proceeds for some time in the ordinary ways of reasoning, ... ; when all of a sudden I am surprised to find, that instead of the usual copulations of propositions, is, and is not, I meet with no proposition that is not connected with an ought, or an ought not. This change is imperceptible; but is however, of the last consequence."
> Combined with Maymin (2011), who proved that market efficiency requires P = NP, this yields a fundamental impossibility: markets can be informationally efficient or competitive, but not both.
(Note that Maymin is the author of both papers.)
Wow... this is quite fascinating. It has been theorized for a little while that widespread AI could form accidental trusts due to optimization around one-another. This seems to be taking it a step further and arguing that if P!=NP, then markets are certain to trend towards collusive.
And yet we’ve clearly observed stable price fixing cartels. Maybe the word “unstable” means too much or the game theory model used doesn’t describe the real world accurately. When theory is contradicted by the evidence, it would be wise to consider the theory is flawed.
The behavioral economics/Freakonomics thing was like "Hey, here's this thing that might if you squint real hard fall outside of efficient market theory" and then for a decade people took that to mean that that the base concepts were worthless, which was a severe overcorrection from people that didn't understand economics.
BASIC[1] came out in 1964, and Pascal[2] came out in 1970.
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[1] https://en.wikipedia.org/wiki/Dartmouth_BASIC
[2] https://en.wikipedia.org/wiki/Pascal_(programming_language)
YieldStar was technically an “AI” product, but I don’t really think the computational abilities were what enabled the collusion. RealPage’s employees (according to the DoJ[0]) would actively monitor whether companies were following their pricing recommendations and call up companies that defected. And the software itself used dark patterns to make it easier to simply follow the YieldStar pricing suggestions, rather than set a lower rental rate and be more competitive. The algorithmic pricing I think did allow people to launder their own judgement and simple “trust the process” in a way that in the past would have required knowing complicity with the cartel, but I don’t think it required substantial compute capacity.
(This isn’t a comment on the paper by the way, which I glanced at but did not have the background knowledge to fully comprehend)
[0] See the section labeled “RealPage Uses Multiple Mechanisms To Increase Compliance With Price Recommendations” https://www.federalregister.gov/documents/2026/01/21/2026-01...
You could imagine the exact same scheme without the use of a computer.
“Markets are competitive if and only if P ≠ NP”
It’s 2026, people, you don't have to use crude ASCII approximations of mathematical symbols any more.
> If P != NP, the collusion detection problem is computationally infeasible for markets satisfying a natural instance-hardness condition on their demand structure, rendering punishment threats non-credible and collusion unstable.
...and then from the paper:
> Stigler (1964) famously argued that the “chief difficulty” of collusion is detecting “secret price-cutting.”
The thing is that Stigler's insight is far from proven, and indeed, the primary difficulty in collusion is often not the detection of defection. Firms know they're being undercut all the time. The problem is that very often, there is nothing they can do about it. Markets are specifically structured as firm-to-firm transactions, where competing firms have no leverage over what your firm can do or what sort of transactions you can conduct, and as long as this condition holds it doesn't matter if you know that a competitor is fucking you over, you can't do anything about it.
I'd argue that the increase in collusion and anticompetitive behavior lately is because these conditions increasingly don't hold. When you intersperse another party in the transaction, eg. a regulatory agency, permitting body, or exclusive distribution deal, you introduce a leverage point for incumbents to punish competitors who choose to undercut them.
I'm eagerly awaiting the day someone proves that P != NP and HN edits the title of the announcement post in this exact same way.
A more capable actor can anticipate the actions of other participants to a greater degree. Imagine that all sellers are such actors. Consider that collusion happens (and is bad) because it enables sellers to extract much higher prices than the market would otherwise set. When all sellers are such "hyper rational" actors they can act cooperatively to maximize their profits without the need to explicitly coordinate in secret. The same end result without the illegal step sure feels like an end run around the spirit of the law.
> Edit: Isn't another implication of this, that increased compute -> collusion imply that increased compute -> communism becomes feasible?
That depends on what you think the problem that communism faced was. AI increases our ability to centrally plan but it probably doesn't do much in and of itself to combat various forms of corruption. Human greed is an invariant; by glorifying and directly making use of it capitalism is hardened against a number of otherwise pathological behaviors.
A problem in P can be solved in polynomial time - the computation required grows relatively slowly as the input size increases. Like sorting a list of numbers.
A problem in NP requires exponential time or greater, but a proposed solution can be verified quickly. For example, checking a completed Sudoku puzzle.
It is believed but unproven that all problems in NP are NOT in P.
Every single problem in P is in NP. What is believed but unproven is that some problems in NP are not in P.
I feel like I have not yet found the proper explanation. Or I'm just too dense to get it.
- P means polynomial - NP means nondeterministic polynomial
Roughly polynomial (P) represents the upper bound of the cost to verify, and the polynomial characterization says that given a problem with a certain input (e.g. the input can be the number of training examples in ML training set or the number of constraints/conditions in a general problem— e.g. all the places you want to visit on your next trip, given many "wants" in a group)
When the cost is polynomial relative to the input size it means it can only be finitely larger than the input - that's a characteristic of the polynomial which is just a finite sum of powers of x (the input).
When the cost is nondeterministic polynomial, one way to think about it in what is called a nondeterministic Turing machine. The nondeterministic part refers to the "states" that the machines can transition to from any current state. When the transition can happen to more than one state, we say it's non-deterministic— and can imagine it's determined by some probability.
The general assumption is that polynomial (P) is easier than nondeterministic polnomial (NP). This isn't necessarily the case as there can be arbitrarily large finite numbers (making P solutions intractible)
The P vs NP problem is one of the main open problems and generally considered a crank magnet and general confusion. For a good (likely the best) resource see https://scottaaronson.blog/
This video explains in detail: https://www.youtube.com/watch?v=YX40hbAHx3s
In short, P means Polynomial time (i.e. markets can solve computation problems efficiently) and NP means Non-Deterministic Polynomial time (i.e. markets can verify solutions of computation problems efficiently but solutions are found by luck).
If P != NP, it means luck CANNOT be engineered and markets are competitive.
P complexity class
https://en.wikipedia.org/wiki/P_(complexity)
NP complexity class
https://en.wikipedia.org/wiki/NP_(complexity)
P vs NP question
one such class is the Polynomial class, or P, where the time to solution is some fixed exponent of N (like N^2, or 3).
the next big step is NP, which require a polynomial number of nondeterministic steps, whose solution can only be verified in polynomial time. usually solutions to NP problems are exponential in cost with respect to N (like 2^N), but thats not part of their definition.
problems in NP are generally identified by mapping them into a well known problem known to be in NP, where the mapping has to occur in polynomial time.
its an open question as to whether NP as a class can actually be solved in P time, but most people doubt that that is really the case.
Nothing to do with ideology, but with the nature of the field. Take epidemiological research in the areas of food and medicine: incredibly hard and expensive to get right and even then with often tenuous results. Now try doing that with ridiculously heterogeneous nations influenced by potentially almost everything on the planet.
It's a small miracle that economists manage to get some useful insights out of the data, but we should definitely be aware of how weakly most of them are supported (don't start talking about "error bars" with economists).
Tbf, most americans just sort of skate along on vibes and don't have many concrete values or material understanding of our society at all, regardless of how they identify.
Cf any argument over rent control: people either want it out of some sort of justice for inhumanely priced rents, or people bought into some kind of idea that it doesn't work because some think tank they put faith in pushed that messaging. Few people actually study the literally thousands of ways that almost every other society manages to house their population more effectively. It's all just vibes.
I mention calculus specifically because it does feel like more of the progressive movements in economics are focused on rate of change rather than point in time. I personally often tend to get into electrical circuit analogies that current/resistance/impedance are more useful than steady state voltage. I find I especially bring these metaphors up a lot when discussing cryptocurrencies (and why I distrust them: inflation/deflation is the wrong axis, in my opinion, some inflation is a useful sign of "impedance" in the circuit, the trick to a healthy economy is not "deflationary" [most of real world history states that the deflationary economies are the worst to live in] but carefully managing the rate of change of inflation).
That progressive science of economics is happening, slowly (this paper seems relevant), but so much of conservative discussions get trapped in (misreads of) Adam Smith still. From a progressive point of view it does seem easy to dismiss conservative views of economics as bunk pseudoscience if they begin and nearly always end with Adam Smith, ignore some of Adam Smith's own warnings (for instance, relevant to above conversation: Adam Smith was also often the first to admit that the "free market" is an ideal/a model and unlikely to ever be a reality because humans are messy and ultimately irrational as individuals), and also ignore centuries worth of work since then, especially a lot of the "pre-calculus" stuff.
> Combined with Maymin (2011), who proved that market efficiency requires P = NP, this yields a fundamental impossibility: markets can be informationally efficient or competitive, but not both.
Market makers and HFT don't determine price: price is usually purely determined by the net inflows and outflows as decided by humans. MMs just smooth it out over time so everyone gets good pricing at the time and in the size they want it.
> Second, there is never any reconciliation of the costs society is saddled with in order to chase arbitrarily more accurate prices
By definition market makers are earning a fraction of the price improvement they provide, ergo the costs to society have to be less that the benefits for better pricing for the companies to stay in business!
> Third, as an index investor, I more or less couldn’t care less
As an index investor you should absolutely care! How do you think you are able to buy into the fund at a reasonable price? And then how do you think the fund is able to rebalance without transaction costs destroying performance long-term?
"as decided by humans", more than 90% of trades are led by bots, so no, price is not being determined by humans.
In fact, we have seen time and again how some weird companies get a huge inflow of purchases simply because of bots misinterpreting news articles.
Both papers seem to be jokes about it, based on complete caricatures of competitiveness and efficiency. It's kinda like a recent paper that was posted here proving "general intelligence" impossible while ignoring that humans exist.
Nuclear has been in maintenance mode for so long that there are doubts about if anyone could right now detonate one without shitting their pants on account if it would even go off.
For whatever reason, the OS documentation lacks a list of allowed compose key sequences. But they are intuitive enough that you can find many of them through experimentation. For example:
Musical sharp ("♯"): compose + "#" + "#".
Interrobang ("‽"): compose + "!" + "?".
Letter "ñ" as in "jalapeño": compose + "n" + "~".
Copyright ("ⓒ"): compose + "(" + c + ")".
However, the reason all NP algos are part of the same category is because you can solve any problem in NP by switching the problem into another problem in the same category and solving that. For example, you can turn sudoku into a graph coloring problem, which is also NP. You can turn sorting (P) into something like balancing a tree, which is also P.
The major question is "is there any algorithm that would allow us to change some NP problems into P problems, solve it, then use it for the original problem". E.g. could we take graph coloring and turn it into sorting a list of numbers?
So basically, if there is any way to bridge the two, then it might mean every NP problem is actually solvable by a P algorithm, under some transformation. This would be immense because it would completely change the way we solve those algorithms and greatly reduce compute costs.
While this seems far-fetched, realize that there are some problems that seem extremely expensive if done the naiive way, but are actually solvable in P. For example, you _could_ write an exponential sorting algo (try every element in every position), but clever people found a way to make it efficient (P). So its possible we just need the right algo to completely change the landscape of computing.
However, as you say, its almost self-evidently true that P != NP, but has never been proven so (to do so, we need to prove that no such algorithm can exist). But clearly, solving an exponentially complex problem using a O(log n) algo would be remarkable.
To take a concrete example, currently the best algos to exhaustively check a board game like chess or go are exponential (NP). Its easy to verify the winner, but its exponential to enumerate every possible move (e.g. 80^turns states). If we found a polynomial way to solve this (even by converting to something simplified), then it would mean we could exhaustively search chess polynomial to the number of moves (e.g. turns^100). This changes it from "cannot be done in the lifespan of universe" to "its possible with a powerful computer in measurable time". We already use heuristics and estimates to explore the exponential space in efficient time, so if we had a polynomial algo chess, markets, optimization, and other NP problems would be extremely efficient to solve.
Examples of the former: cutting prices yourself; increasing product quality; differentiating yourself; spending more on advertising to get the word out about your product.
Examples of the latter: crafting exclusive deals with your distributors to prevent your competitors from getting shelf space; politically influencing regulatory bodies to declare your competitors' existence illegal; making direct agreements with the leadership of opposing firms to not drop prices or hike wages; assassinating, extorting, or kidnapping rival business leaders.
Basically it comes down to "control yourself, because you cannot control others". In a functioning market, you have no control over what rival firms offer. Your only legal reaction to competition is to improve your own offering until it is the best it can be. In pathological markets where the assumption is (as in the paper) that you can punish rivals for not colluding, you actively make your competitor's offering worse.
Those pathological markets exist today, but if you're analyzing markets economically, your root assumption should not be that pathology is normal and only the lack of information keeps it in check, it should be that information is abundant and it is the lack of ability that keeps it in check.