“If you can completely see what a program will do, what’s the point of running it? … To make the program worth actually running, there have to be parts of its behaviour you can’t foresee… Once there’s anything you can’t foresee, there tends to be a lot you can’t foresee. In other words, the system will tend to be full of computational irreducibility.” Stephen Wolfram, July 2026.
This quote comes from yet another great note from the incomparable Stephen Wolfram. Stephen, for those not aware, has developed a philosophy and science around what he terms “ruliology”, which is the computational universe, the experimental study of how complexity emerges from simple rules. This note makes the point that even the simplest of computational models will, eventually, surprise us with what he dubs as “bugs”.
Towards a Theory of Bugs: The Ruliology of the Unexpected—Stephen Wolfram Writings
“So let’s say we’re running a program that we think will operate in a certain way. But it doesn’t. And instead it has what we can consider a bug… in effect we’ve discovered something we didn’t expect. And indeed we can think of much of the progress of ruliology as being precisely about the discovery of a long sequence of “bugs” – or at least bugs relative to our normal intuition about how things work. But now, with what we learn from ruliology, we can turn this around and start to understand some of the foundations of the ubiquitous and fundamental phenomenon of bugs, wherever they may be found.”
What he is talking about here is really “the ubiquitous and foundational nature” of emergent properties. Things that arise from complexity that we cannot foresee, until going through the actual process. We cannot cut short time in the running of our computations and comfortably believe that there would be no “bugs” in the unexplored future.
As a practical example of this we show below what, in the investment world, is commonly known, though widely ignored, as ‘volatility drag’. This is the wildly popular Korean listed chip maker, SK Hynix, and the Hong Kong listed 2x Leveraged ETF on SK Hynix. We have normalised the performance over the last three months.
Figure 1: SK Hynix 000660 KS (white) vs SK Hynix 2x SK Hynix ETF 7709 HK (blue). May 2026 – July 2026. Normalised
Source: Bloomberg, Convex Strategies
Over the last three months, we have seen decent moves up, decent moves down, and a fair bit of volatility for SK Hynix. In the end, it has gained 19.47% over that period. The 2x leveraged ETF, however, has lost 27.37%. The point here is that the leveraged version is on daily performance, not performance over the entire period, and exposes holders to the inevitable “bug” of volatility drag. From the beginning of the period, the 2x ETF topped out at a gain of 220.39%, then had a drawdown of 86.49% from the peak. Even after a 67.67% one-day recovery on the last day of the period, it is still down that 27.37% for the full three-month period. Arithmetic returns are meaningless. Geometric returns are what matters.
If you run the process long enough, leverage and volatility will ultimately guide such products towards zero. As Mr. Wolfram’s ruliology would show, the “bug” is inevitable.
This got us thinking of learning in general, of how complacency sets in for the simplistic model / back-test that hasn’t run long enough or searched far enough to find the inevitable emergent nature. That led us to the world of anthropology and work done by Alan R. Rogers, a professor of anthropology and biology at the highly esteemed University of Utah. Mr. Rogers wrote this wonderful paper, “Does Biology Constrain Culture?”, which led to the formulation of what has come to be known as the Rogers’ Paradox.
Rogers discusses how a society learns and breaks that down into two mechanisms of learning.
“I distinguish individual learning (i.e., learning directly from the environment) from social learning (i.e., learning from others)” Alan Rogers, 1988.
Individual learning is difficult, involving trial and error, experimentation, gathering information about the environment, adapting to changing circumstances, and the requisite accountability that comes along with failures. Social learning, on the other hand, is easier and more reliable as it is simply adopting the behaviour that is seen as being successfully implemented by other members of society, what he dubs as ‘cultural parents’.
The paradox comes from the impact on overall societal fitness as the share of learning coming from social learning becomes more or less dominant in the overall. He provides this simple visual representation.
Figure 2: The Rogers’ Paradox. Individual Learning (IL) and Social Learning (SL) by Fitness vs Frequency of Social Learning
Source: Rogers-AA-90-819.pdf
Rogers describes his simple visual very clearly in the paper.
- The fitness effect of individual learning depends on its cost and benefits, but not on what others are doing. Thus, its fitness is a horizontal line.
- If social learning is rare, nearly all cultural parents will be individual learners. Thus, social learners will acquire relatively recent information about the environment. They will acquire, that is, behaviors that were appropriate in the immediately preceding generation. Their fitness will therefore exceed that of individual learners, provided that social learning is sufficiently cheap and environmental change sufficiently slow.
- When all learning is social, no one is monitoring the environment, and the information acquired will eventually be many generations old. Social learning will then have lower fitness than individual learning because, by assumption, information is worthless if sufficiently out of date.
A society that becomes overly reliant on social learning risks losing the fitness to adapt to a changing environment. It is easy to see how this becomes particularly dangerous in complex systems with emergent properties. Models that are developed, looking backwards, on what did work are likely to not be prepared for the “bugs” that have yet to be discovered. It is this acquiescence towards social learning, as periods of stasis make it look like the easy road to learning, that then leads to the fragility that generates punctuated equilibrium jumps in the realm of self-organizing complex systems. Per Bak, the father of self-organized criticality, lays it out beautifully in his must-read book on the topic – “How Nature Works”.
“… the apparent equilibrium is only a period of tranquillity, or stasis, between intermittent bursts of activity and volatility in which many species become extinct and new ones emerge… This phenomenon is called punctuated equilibrium. Punctuated equilibrium turns out to be at the heart of the dynamics of complex systems. Large intermittent bursts have no place in equilibrium systems, but are ubiquitous in history, biology, and economics.” Per Bak, “How Nature Works”, 1996.
Rogers’ Paradox is telling us exactly this. If there is insufficient individual learning, thus too much reliance on accepted learning from what worked in previous environments, a culture risks losing its fitness to adapt to unforeseen changes in circumstances. Long periods of stasis (suppressed volatility), lead to growing general acceptance of existing models, which leads to fragility to unforeseen emergent properties.
An obvious place where we should be concerned with just this issue of an undue shift towards social learning is the accelerating adoption of Artificial Intelligence. We discussed a closely related dynamic in our May 2026 Update – “The Erosion of Trust” Convex Strategies | Risk Update: May 2026 – “The Erosion of Trust”. We discussed in this note precisely the risk of foregone learning as more and more hand over thinking to their trusted LLMs. We quoted from this paper, thusly.
“Desirable difficulties are purposeful challenges at study time (e.g. retrieval, spacing, generation) that lower short-term fluency but improve long-term retention… By potentially eliminating these beneficial difficulties, AI tools might optimize for immediate task completion while undermining the deeper learning processes necessary for durable knowledge construction.” Andrew Barcaui, 2025.
After all, LLMs are just that, a searchable collection of the past learning of cultural parents, with a statistical engine to give us the aggregate average of what has worked within its available data set. AI is one big amalgamation of existing learning for dissemination to ease the process of social learning.
This brings us to this very interesting paper by Katherine Collins, Umang Bhatt and Ilia Sucholutsky, “Revisiting Rogers’ Paradox in the Context of Human-AI Interaction”. They ask precisely the question that is on our mind – does AI risk cheapening the cost and easing the access to social learning such that we risk getting (further?) on the wrong side of Rogers’ Paradox?
[2501.10476] Revisiting Rogers’ Paradox in the Context of Human-AI Interaction
“Humans learn about the world, and how to act in the world, in many ways: from individually conducting experiments to observing and reproducing others’ behaviour. Different learning strategies come with different costs and likelihoods of successfully learning more about the world. The choice that any one individual makes of how to learn can have an impact on the collective understanding of a whole population if people learn from each other. Alan Rogers developed simulations of a population of agents to study these network phenomena where agents could individually or socially learn amidst a dynamic, uncertain world – and uncovered a confusing result: the availability of cheap social learning yielded no benefit to population fitness over individual learning. This paradox (Rogers’ Paradox) spawned decades of work trying to understand this equilibrium and uncover factors that foster the relative benefit of social learning that centuries of human behavior suggest exists. But what happens in such network models now that humans can socially learn from AI systems that are themselves socially learning from us?” Collins, Bhatt, Sucholutsky, January 2025.
The authors go on to develop a model that incorporates the human-AI interaction and social learning feedback loop. Not surprisingly, they end up much like our discussion above in our “Erosion of Trust” Update, noting that “learning from AI may impact our own ability to learn about the world”. Some of their, what we would call, common sensical findings:
“This suggests a novel form of Rogers’ Paradox for the AI age: the widespread availability of cheap AI systems trained on all human data in the world may not, on its own in the long-term, improve our collective world model.”
“… it remains important that the population engages in some form of critical appraisal on whether or not to override the output of the AI system.” (For an interesting discussion around this issue, see this note from Dan Davies on discerning good from bad output – (3) the legend of john henry’s cerebellum)
“… if you can always socially learn for cheap and therefore ‘avoid’ individual learning, your future ability to re-engage with individual learning may be substantially weakened.” (Again, see our comments in our “Erosion of Trust” Update linked above.)
“However, it is possible – and one may argue, even the current state of society – where changes from the AI system change the rate of change in the environment.”
That last one, a nod to complex systems, has obvious feedback loops to the genesis of the Rogers’ Paradox. The more the system is changing/evolving, the more important individual learning is to maintaining social fitness and, likewise, the more damaging a high share of social learning is to that social fitness. Bit of a mind bender when you start going around in those circles.
As if by telepathy, our friend Pippa Malmgren found her way to write on a very similar theme and, as always, she finds a way to put her unique optimistic spin on the topic. In her wonderful recent Substack article, “FIFA, Footballs, and Firms: Aligning Humans in an AI-Led World”, Pippa emphasizes the power of AI as a tool for individual learning, not simply as a crutch of social learning to fall back on. She sees AI as a powerful tool arming individual learners with the capacity to break free of the sclerotic institutions of our past, the accumulated social learning resisting adaptation to the very changing environment that AI brings.
FIFA, Footballs, and Firms: Aligning Humans in an AI-Led World
The note, as ever, is a wonderful rollick through a number of weaving themes, including touching on interactions between her father, Harald Malmgren, his PhD supervisor at Oxford, Sir John Hicks, and modern-era luminary in what is known in economics circle as “Theory of the Firm”, Sir Ronald Coase. She sees AI as an enabling tool for the most critical of individual learners to go forth and adapt to the evolving world.
“The most important quality in this new world is our capacity to imagine. Our powers of creation now allow us to build what was previously unimaginable. But who actually does the imagining? It is the person who used to be seen as a source of friction inside the firm. The person who asked the uncomfortable questions. The person who saw things… Conformity is no longer the solution. It is the problem.” Pippa Malmgren, July 2026.
We love the concept of being a source of friction. We proudly hold ourselves up as a multi-decade source of friction against the conformity of Sharpe World practices in the realms of economics and finance.
“Questioning threatens the old forms of alignment. It undermines the old consensus. We have an entire class of managers, executives, journalists, writers, and economists whose status, identity and wealth are tied to being the gatekeepers of the ‘old truths’”. Pippa Malmgren, July 2026.
We feel like this is a great lens through which to view the out-in-the-open struggles of Kevin Warsh and his efforts to come into the Federal Reserve and orchestrate some evolution away from the accumulated buildup of historical consensus social learning. Our friend Marvin Barth put out this fantastic piece on just what it is, at its core, that Mr. Warsh is up against, aka the New Keynesian Synthesis (NKS).
(4) Pink slips for the New Keynesian Synthesis
This is a really fun note to read, and a wonderful quick tool to get some background history on the dominant economic philosophy, the overwhelming driver of social learning dominance, in central banking.
“It’s easy to see why economists would fall in love with the NKS framework. It provides a mathematically elegant – economists love math! – description of the world that claims to be predictive and makes economists into all-powerful managers of our expectations using variables that they construct (the output gap and neutral interest rates)” Marvin Barth, July 2026.
The accepted consensus of these models, the reliance on social learning and near total lack of any sort of accountability (something readers know we harp on about relentlessly), has moved central banks, the world over, into Rogers extreme bottom right quadrants of a devastating lack of fitness. Every failure of their accepted form of copying from the past was met with more of the same. Always accepting knowledge from cultural parents of the past, heedless to changing environments. Openly mocking those would be friction generators of individual learning, those willing to take the risk of being held accountable.
We can think of several things that have changed that might necessitate some level of adaptability in the population of central banking elites and their acolytes. Taking the particularly eyepopping circumstances of China as an example of what is universally an issue across major global economies, we will suggest that demographic circumstances have changed.
Figure 3: China Working Age Population Adjusted Forecasts for 2019, 2022, 2024 (left chart). China % of Young Adults Who Desire No Children (right chart)
Source: Peter Berezin, BCA. UN Population Division. The Rise of Zero Fertility Desire in China.
Amazing work that we were alerted to by Peter Berezin of BCA. We have long been noting the 2022 UN numbers indicating that China’s working age population was set to decline by 600 million people over the course of this century. Just two years later, to 2024, and the UN has updated those numbers to decline by circa 700 million people. Remember, to a great extent, these are not per se ‘forecasts’, they are already carved in stone in the numbers of current young children and the last several generations of potential parents, including the generations that had to endure the ‘one-child policy’. For any who think this is likely to start improving, the chart on the right (a new study out of Brown University), showing the stated desires of young adults to have children, ought to pour some cold water on those hopes.
Does anybody really think that economic analysis and policy that may have seemed useful/successful, from a social learning perspective, during the years of massive growth in working age populations are still going to be worthwhile in the dramatically changing environment of the collapse of this population? It seems rather obvious that some individual learning, adaptability, is going to be needed going forward.
Here is another example that we noted back in our June 2024 Update – “Hunger Games II” Convex Strategies | Risk Update: June 2024 – “Hunger Games II” with this all-world quote from the true High Priestess of the NKS faction of Sharpe World, then Treasury Secretary Janet Yellen:
“The interest burden of the debt is at what I would call normal historical levels.” Janet Yellen. June 2024.
Figure 4: US Govt Interest Expenditures
Source: Bloomberg
Copying/social learning off past behaviour that was deemed to have worked, may not be the right adaptable skill set for the environment of current evolving initial conditions. This picture is a simple representation of what is referred to as fiscal dominance. There are plenty of voices out there, besides ours, pointing this out. One of our favourites is former FOMC member, and notorious dissenter back when dissenting was a big deal (friction!), Thomas Hoenig.
Fortunately for all, Tom continues to be very active in publicly speaking and writing on these topics. Below we link a note from his very active FinRegRag website and a recent podcast with Kathleen Hays on her Central Bank Central program.
The Fed’s Balance Sheet Girth: A Symptom, Not the Problem
“If the Fed and regulators don’t accommodate Treasury debt growth, interest rates will rise until something breaks. If they continue to accommodate Treasury debt growth, interest rates will be subdued until inflation forces everyone’s hand… Only Congress can solve the problem.” Thomas Hoenig, June 2026.
Hoenig: Fed Must Stop Monetizing U.S. Debt, Start Reducing Balance Sheet Now
“So you’re really not solving the problem. The problem is the United States is spending more than it’s taking in every year by dramatic amounts. And the consequence of that has been to lower the real growth rate, give us an implicit inflation rate that is far beyond price stability and far beyond 2%. And those are adverse consequences. So switching out who’s going to grow their balance sheet doesn’t solve the fundamental problem.” Thomas Hoenig, July 2026.
At least verbally, the things that Tom is advocating appear to be things that Mr. Warsh wants to address. In practice, many (us included) are pointing out that nothing is happening yet.
Another worthwhile voice in this realm is Hanno Lustig. His recent Substack note, “The United States Capital Structure”, is an ode to Stein’s Law with some great visuals.
The United States Capital Structure – by Hanno Lustig
Figure 5: US Spending as % of GDP: Discretionary (blue), Mandatory (red), Net Interest Expense (yellow)
Source: Hanno Lustig The United States Capital Structure – by Hanno Lustig
The challenge is quite clear. The differing circumstances to the past ought to be obvious. We, as always, would throw in the foundational amplifier to this is, being that, for every year going forward, there will be fewer taxpayers. See the China population pictures in figure 3.
Another note from Hanno Lustig, “Marked to Model”, aligns exactly with what we are trying to portray in the tie to the Rogers’ Paradox. Hanno is shouting for more individual learning, for trying something different, for breaking free from the old models (ridiculously high frequency of social learning) and calling out past such speeches from none other than Kevin Warsh.
Marked to Model – by Hanno Lustig – The Two Cents
“In August 2016, on the eve of the Fed’s annual Jackson Hole conference, Kevin Warsh published an opinion piece in the Wall Street Journal under the headline “The Federal Reserve Needs New Thinking.” Warsh had served as a Fed governor from 2006 to 2011, so this was not an outsider lobbing grenades. His diagnosis: the conduct of monetary policy had been deeply flawed, and the deeper problem was intellectual. He identified groupthink within what he called the academic economics guild as a key culprit. And rather than confronting its forecasting record, “the guild tightens its grip when it should open its mind” — to new data, new analytics, and especially new economic models.” Hanno Lustig, July 2026.
Time will tell if Mr. Warsh can introduce some fresh individual learning into not just the Fed but the world of social-learning-dominated central bankers and the existing NKS establishment. It will no doubt entail two major challenges: 1) massive resistance from the entrenched pack of the socially learned (in Pippa’s terms the keepers of old truths), and 2) the accumulated lack of fitness in the existing system leaving it inordinately fragile to the risk of a punctuated equilibrium shock.
On the practitioner side of this is the monolith of the financial industry and its inhabitants, Rational Accounting Man, the tool of implementation for the central planners in their efforts of financial repression and economic/market manipulation. The part that we swim in, fiduciary investment management, is going through their own evolution. This has most prominently taken the form of what has been dubbed Total Portfolio Approach (TPA). The principles behind TPA are simply common sense, i.e. investment/risk decisions should be made based upon their contributions/impact on a whole of portfolio perspective.
Like with NKS in economic-think inside academia and central banking, Sharpe World social learning has come to dominate the world of financial risk taking. Exactly per Mr. Wolfram’s examples and the Rogers’ Paradox circumstances, given enough time bugs and poor fitness will appear. There has been enough of that, particularly as the foregone efficacy of bonds as a portfolio diversifying benefit have dispelled the efficacy of social learning from past cultural parents in a realm that no longer exists. TPA is the attempt at moving towards an environment of much greater individual learning, in hopes of restoring some fitness.
We have been long term advocates of the principles that are now grouped under the TPA efforts. One of the earliest major names to officially adopt and espouse TPA was the Singapore Sovereign Wealth Fund, GIC (Government Investment Corporation). They are out with their annual report so we thought we should just give a very quick update to follow on from our more extensive covering said report last year that we wrote up in our July 2025 Risk Update – “Preservation” Convex Strategies | Risk Update: July2025 – “Preservation”
GIC Report on the Management of the Government’s Portfolio for the Year 2025/26
Anybody knowledgeable about market performance over the last 5, 10, 20 years will likely spot why there might be some pressure on GIC to rethink their adoption/weighting of socially learned skills.
Figure 6: Singapore’s GIC Annual Returns and Volatility.
Source: GIC Report on the Management of the Government’s Portfolio for the Year 2025/26
In our more recent February 2026 Update – “Step #1: Free Up Capital” Convex Strategies | Risk Update: February 2026 – “Step #1: Free Up Capital”, we went through the full nature of GIC’s own research on diversifying strategies (done in conjunction with JPM AM). In that research they came to the conclusion that bonds could indeed be beneficially replaced with alternative hedge fund strategies. It would appear, based on this year’s announcement, that they are finally taking at least some of that research to heart, announcing an increased allocation to hedge funds of $30 billion, as noted in this Reuters article.
Singapore’s GIC to invest an additional $30 billion in hedge funds | Reuters
“Singapore sovereign wealth fund GIC plans to deploy an additional $30 billion into hedge funds and is spreading its bets across artificial intelligence, top executive said, while reporting its weakest long-run earnings since 2020… Group Chief Investment Officer Bryan Yeo said the money would be allocated over three years and that the fund’s top executives see opportunities in global macro, quantitative and multi-strategy funds, which span multiple assets and can adjust quickly when conditions change.” Reuters on GIC, July 2026.
We say that they are only taking to heart some of the conclusions of their own research because, while that research did indeed conclude that replacing bonds with hedge funds could improve risk-adjusted returns, it also explicitly concluded that 100% of the allocation should go to “Loss Mitigation” strategies. Our revised version of their research, as discussed in the above linked February 2026 Update, which included Long Volatility actively managed hedge fund strategies (which had inexplicably been left out of the GIC/JPMAM universe of funds), which we dubbed “Explicit Loss Mitigation”, even more conclusively took up 100% of the bond-replacement allocation.
With GIC’s latest 20 year returns out, we can revise our comparison of hypothetical compounding paths between a Proxy Portfolio that approximates the announced returns from GIC (48% MSCI World/52% Global Bond Aggregate), the Reference Portfolio that GIC uses to define the risk appetite as mandated to them by Parliament (65% MSCI World/35% Global Bond Aggregate) (worth noting that, for the first time since they started this sort of annual reporting, they no longer included the performance numbers of the Reference Portfolio in the report), the Barbell Portfolio that is the conclusion reached in our adapted version of the GIC/JPMAM research (80% MSCI World/20% Long Vol), and something we dubbed the Preservation Portfolio as a theoretical sample of what aligns with GIC’s own internal mantra of “Preserve and Enhance” (50% Nasdaq/25% Gold/25% Long Vol). In all cases we perform annual rebalancing of these simple worked examples.
Figure 7: 20yr Hypothetical Compounding Paths vs Target Return. Reference Portfolio (light blue). Proxy Portfolio (dark blue). Barbell Portfolio (gold). Preservation Portfolio (Fuchsia). 2005-2025
Source: Bloomberg, Convex Strategies
The Preservation Portfolio is a hypothetical example of a class of strategies that we refer to as “Aggressively Defensive”. It has a full 50% of its capital allocated to what we would consider defensive strategies, gold and Long Volatility, very explicitly intended to preserve capital. The other half, now that you have the preserve-case locked up, is the enhance portfolio. As a proxy for growth and innovation we have used Nasdaq, a sophisticated investor may just park capital there while they go out and look for something even better. Going back to Rogers’ Paradox, this is just an example of where we might opine that some individual learning, trial and error, experimenting, might aid in better overall fitness, as opposed to sticking with social learning that is no longer relevant to the environment we now inhabit.
Risk, to the future retiree, isn’t simply about negative drawdowns, it is also about foregone opportunities. It is about not achieving the compounded terminal wealth needed in retirement. Underperformance in the wings, both the painful left wing as well as the forgone right wing, is what will impede that compounding path. Simply driving slowly in the multiple lap Grand Prix investment race appears a sub-optimal way to undertake the challenge of managing wealth for someone’s retirement.
For more on our thoughts about why the point of a strong defence is so that you can be aggressive in attack, we would refer everybody to our recently released “Convex Goalkeeper Theorem” here https://convex-strategies.com/research/. Quoting our own research that tracks every game from the recent World Cup:
“Efficient defensive tail management reduces capital required for loss-prevention, releasing it for offensive right-tail participation. The goalkeeper enables the striker. The hedge enables the enhanced risk positioning. Defence liberates capital, rather than consuming it.” Convex Strategies, July 2026.
What it is NOT is a traditional prediction model that tries to tell who is going to win a game or a tournament. It is a model that tells you how to wingames and tournaments.
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