Path Dependence and the Non-Ergodic Nature of Markets
Most of the analytical machinery used in mainstream finance rests on an assumption so foundational that it is rarely examined: the assumption of ergodicity. It is worth examining. The ergodic assumption is not a neutral technical choice. It is a claim about the nature of the system being studied, and in financial markets, it is wrong in ways that have direct consequences for how risk is measured, how positions are sized, and how long-run geometric compounding actually works.
Ergodic and Non-Ergodic Systems
An ergodic system is one that, given sufficient time, visits all conceivable states. Consider a gas released into a sealed glass box. Given enough time, the molecules distribute uniformly throughout the space. The distribution across time aligns with the distribution across space. This equivalence is what makes the ergodic hypothesis useful: it allows complex dynamic descriptions to be simplified into probabilistic models, sidelining the role of time entirely. The expectation value, calculated across the ensemble of possible states, equals the time-average experienced by any individual trajectory through the system. History does not matter. Where the system has been does not constrain where it will go.
Non-ergodic systems are structurally different. They do not visit every conceivable state. Their trajectories are shaped and constrained by their history. Where the system has been determines, at least in part, where it can go next. The evolution of life on Earth is the most vivid example: not every imaginable life form will emerge in our universe, because evolution does not explore all possibilities. It follows a path shaped by what came before, by the specific sequence of selection pressures, environmental conditions, and heritable variations that produced the present state. The past is not irrelevant. It is constitutive.
Financial markets are non-ergodic. The ergodic assumption has been embedded in economics since foundational ideas about risk and randomness were developed in the 17th century, predating the introduction of ergodicity in 19th-century physics by a considerable margin. The concepts were borrowed into economics before their limitations were fully understood, and prevailing theories of expected utility and portfolio optimisation still carry that assumption forward. In a system that is genuinely non-ergodic, this produces systematic errors that compound over time.
The Ole Peters Gambling Scenario
Ole Peters’ work on non-ergodic systems in economics makes the problem concrete.¹ Consider a simple game: start with a bankroll of $100. On each flip of a coin, heads increases the bankroll by 50% and tails reduces it by 40%. Calculated across the ensemble of possible outcomes, the game appears to have a small positive expected value. Under ergodic thinking, this is a game worth playing.
Peters’ simulation produced a different result. Run the game forward in time across many individual trajectories, and a consistent pattern emerges: regardless of early gains, every wealth trajectory eventually trends toward ruin. The ensemble average climbs, pulled upward by a small number of lucky trajectories, while the median outcome deteriorates steadily (Figure 1).
Figure 1: The Ole Peters Gambling Scenario, wealth trajectories over 1,000 gamble rounds
The reason is path dependence. The game is not a static calculation of expected value. It is an iterative process in which each outcome is applied to the current state of the bankroll, not to some fixed reference point. Starting from $100, a win produces $150. A subsequent loss does not deduct a fixed amount: it removes 40% of $150, leaving $90. The sequence of outcomes matters. A win followed by a loss produces a different result from a loss followed by a win, even though the ergodic model treats both as equivalent. The system has memory. The path taken shapes the outcome in ways that expectation values calculated across the ensemble cannot capture.
This is the non-ergodic problem in its simplest form. Applying ergodic tools to a non-ergodic system produces the illusion of a positive edge where the reality, experienced along a single path through time, is eventual ruin. Financial markets operate on the same principle. The question this raises is direct: are the analytical tools used to evaluate market strategies ergodic or non-ergodic? Are they computing averages across an ensemble of possible worlds, or are they tracking the actual path-dependent evolution of wealth through time?
Strange Attractors and Path-Dependent Uncertainty
The path-dependent properties of non-ergodic systems become visible in a different way when complex adaptive systems are studied through the lens of chaos theory. When the trajectories of non-linear systems are plotted in state space over time, a striking structure emerges. Rather than exploring all possible states, the trajectories are drawn toward a constrained geometry: an attractor.
Edward Lorenz discovered this property while studying non-linear differential equations in the context of atmospheric dynamics. The mathematics involves non-linear systems in which inputs and outputs do not change proportionately. Financial markets, with their numerous interconnected dependencies and feedback loops, share this non-linear character. The emotional response of market participants illustrates the intuition: a $10 million gain does not produce ten times the behavioural response of a $1 million gain. The relationship between stimulus and response is non-linear throughout.
Plotting trajectories of such non-linear systems produces a structure that constrains the paths available to the system without fully determining them. The trajectories exhibit periods of relative predictability, orbiting one lobe of the attractor, before making abrupt transitions to a different trajectory path with its own periodicity. Critically, the trajectory never returns to its starting point. This is the signature of a non-ergodic system: the path through state space is unique, history-dependent, and non-repeating. As Tim Palmer illustrates in The Primacy of Doubt, the attractor structure emerges progressively as the simulation extends, beginning with the early formation of the dual-lobed geometry (Figure 2) and unfolding toward the full structure as trajectories accumulate (Figure 3).²
Figure 2: The Lorenz Attractor in early formation (Palmer, The Primacy of Doubt, 2022)
Figure 3: Unfolding of the Lorenz Attractor (Palmer, The Primacy of Doubt, 2022)
Regardless of starting point, trajectories eventually align with this geometry, but never repeat within it. The full butterfly shape, one of the defining images of chaos theory, captures this property precisely (Figure 4): the attractor both constrains and defies prediction simultaneously.²
Figure 4: The Lorenz Attractor in all its Glory (Palmer, The Primacy of Doubt, 2022)
From Predictability to Chaos
Palmer’s analysis of predictability in complex systems extends directly to the question of where uncertainty originates in a non-ergodic environment.² The uncertainty of a trajectory is not fixed. It depends on where the trajectory is located on the attractor at any given moment, which is itself a function of the path taken to arrive there (Figure 5).²
Figure 5: From Predictability to Chaos, three uncertainty scanarios (Palmer, The Primacy of Doubt, 2022)
In the top left scenario, an initial ring of uncertainty remains compact as the system evolves. The trajectory is in a region of the attractor where predictability is relatively high. In the top right scenario, the ring deforms into a banana or boomerang shape, indicating that the system is approaching a transition between the two lobes. Uncertainty about which state the system will occupy has increased significantly, though the shape of the distribution still carries information about the likely transition. In the bottom scenario, the ring has dispersed entirely. The trajectory has entered a region of maximum uncertainty, and predicting the future state from the current one is no longer tractable.
The practical implication is precise: the degree of predictability available at any given moment is not a property of the system in the abstract. It is a property of where the system is on its attractor, which is determined by the path it has taken to get there. Ergodic tools that treat predictability as a stable feature of the system’s statistical properties will produce systematically wrong estimates of uncertainty whenever the trajectory is approaching or crossing a lobe transition.
What This Means for the Outlier Hunter
The non-ergodic, path-dependent nature of financial markets has direct implications for how a systematic trend follower thinks about position sizing, drawdown management, and long-run geometric compounding.
Variance drain is the mechanism through which non-ergodicity destroys wealth in ways that ensemble averages conceal. A portfolio that gains 50% and then loses 40% ends at 90% of its starting value, despite a positive arithmetic average return. The geometric return, which reflects the actual path-dependent experience of wealth through time, is negative. Strategies that appear attractive on the basis of ergodic expectation calculations can produce systematic losses along the actual path through time.
The Cut Back Rule addresses this directly. By systematically reducing position size as drawdown geometry worsens, it reduces exposure during the periods when the system’s trajectory is most likely to be in a high-uncertainty region of the attractor, and preserves capital for redeployment when the trajectory has stabilised. It is a rule designed for a path-dependent world, not an ergodic one.
The Outlier Hunter’s approach to diversification reflects the same logic. Because trajectories through non-ergodic systems are unique and non-repeating, concentration in any single market or strategy is a bet on a specific path. Wide diversification across markets, systems, and timeframes distributes the portfolio across multiple trajectories simultaneously, reducing the variance drain that any single path-dependent sequence of outcomes would impose.
Expectations are a tool for ergodic systems. Markets are non-ergodic. The path matters, the sequence matters, and the geometry of the drawdown matters in ways that no ensemble average can capture. Building a process that respects this is not a refinement of conventional portfolio management. It is a departure from its foundational assumptions.
Footnotes
- Ole Peters, “The Ergodicity Problem in Economics,” Nature Physics, published 2 December 2019.
- Tim Palmer, The Primacy of Doubt, Oxford University Press, 2022.