Source: In the Vastness of Transmitting Enlightenment
Author: Yao Bin
How can unconventional knowledge help us become better investors? The logic of diversity provides individuals with varied perspectives, heuristics, and interpretations, enabling them to solve problems more effectively than even highly intelligent individuals with limited tools. Research shows that fox-type individuals perform better in long-term forecasting than hedgehog-type individuals. Michael Mauboussin, from an investor’s perspective, offers specific examples of where intellectual diversity can provide useful insights.
01, Diversity: How to Better Solve Problems
Legg Mason Capital Management is recognized as being “more like an academic enclave or workshop” rather than a typical asset management company. They devote time to discussions, reading, and conferences on topics often considered outside the realm of finance and investment. In particular, the Santa Fe Institute, an interdisciplinary research organization focused on studying complex systems, plays a significant role in stimulating their thinking.
All this leads to perhaps the most frequently asked question: how do these topics help them become better investors? Mauboussin attempts to answer this by examining diversity theory, presenting evidence for the conditions necessary for sustained accurate forecasting, and providing examples of how he uses diverse thinking to view common investment issues.
Mauboussin argues that generating excess returns is a difficult problem. Markets are complex; business environments are constantly changing; information is abundant but often ambiguous; facts and conjectures are filtered through enormous human psychological biases. The challenge lies in gaining insight—an advantage that others lack. Clearly, if our input information is identical to everyone else’s, it is unlikely we will gain an edge. Many sources of information—business media, corporate disclosures, and analyst reports—are necessary but insufficient to create an advantage. More importantly, one must interpret information in a way that differs from and surpasses other investors. Gaining an edge requires considerable effort: reading, thinking, and maintaining intellectual independence.
The notion that diversity benefits business has become a cliché, which is unfortunate. Understanding when and why diversity works (and recognizing that it does not always work) is as important as appreciating its advantages. Therefore, we will first briefly discuss how diversity contributes to better problem-solving capabilities. Social scientists have now demonstrated the value of diversity, showing that it is no longer a soft concept but a real and powerful method for solving problems.
02, Winning 'Sum to Fifteen': Greater Likelihood of Success
To see the power of perspective, try playing the “Sum to Fifteen” game. Designed by economist Herbert Simon, the rules are simple. Place nine cards numbered 1 to 9 face up on the table. Two players take turns selecting cards, aiming to hold exactly three cards whose sum equals fifteen. The game is moderately challenging because it requires continuously calculating both your own and your opponent's sums mentally. It also involves balancing offense (getting three cards that sum to fifteen) with defense (preventing your opponent from doing the same). Often, one player wins while the other becomes entangled in numerical confusion.
Now, Mauboussin introduces a magical magic square, offering a perspective that makes the game much easier. Here is the magic square for this game:
8 3 4
1 5 9
6 7 2
Notice that whether viewed vertically, horizontally, or diagonally, the numbers always add up to 15. Suddenly, the game becomes very simple: it transforms into the childhood favorite tic-tac-toe. Once you view the game as tic-tac-toe, winning becomes much easier, with a draw being the worst-case scenario and losing being unforgivable.
In 'The Difference: How the Power of Diversity Creates Better Groups, Firms, Schools, and Societies,' Scott Page uses the game 'Sum to Fifteen' as part of a broader case study to explain why diversity can outperform individual ability under specific conditions. In recent years, diversity has become a popular topic, but most discussions revolve around social identity diversity—gender, race, and ethnicity. Page rigorously demonstrates how diverse perspectives, heuristics, and interpretations contribute to better collective problem-solving and predictive capabilities.
We can consider diversity on two levels. The first is at the group level—teams, units, organizations. In this context, each individual contributes to diversity, and it is crucial to think about what each person brings and how they interact. The second is at the individual level, or the diversity of mental models. This pertains to the number of approaches available for solving problems.
Diversity works because it provides a large toolkit for solving difficult problems, increasing the likelihood that one of the tools (or a combination of tools) will be effective. For any specific problem, we may have the right tool in our minds. However, if the problems we face are challenging and variable, and the number of available tools is limited, finding high-quality solutions becomes unlikely.
While Page provides examples of how diversity functions in the real world, his greatest contribution lies in the logic of diversity. He shows that, under appropriate conditions, diversity is not merely an add-on but essential for finding optimal solutions. As he writes, the theorem that 'diversity trumps ability' is 'not just a metaphor or a cute anecdote that might not hold true a decade later. It is a logical truth.'
The 'Sum to Fifteen' game is an interesting and easy-to-understand way to illustrate a broader point: the more problem-solving methods we possess, the greater the likelihood of success. If our tools are identical to others’—the same business school education, TV programs, Wall Street research reports—it becomes almost impossible to gain unique insights.
03, Diversity: A Toolkit for Solving Complex Problems
Although Page's argument for diversity is logically rigorous, one might ask whether there is empirical evidence that diversity adds value in tasks similar to those faced by investors—predicting outcomes in complex systems. The answer is yes, and it comes from Philip Tetlock’s groundbreaking research summarized in his book 'Expert Political Judgment.' We tend to highly value experts; we watch them on television, seek their advice, and follow their opinions. But how good are expert predictions? Tetlock asked nearly 300 experts to make tens of thousands of predictions over nearly two decades. These were difficult predictions involving political and economic outcomes—similar to the challenges investors face.
The results were unimpressive. Even when expert forecasters improved, their performance was marginal, only slightly better than simple statistical models. When Tetlock pointed out the poor accuracy of their predictions, they defended their views just like anyone else. Tetlock did not describe what happened when expert opinions were aggregated, but his research showed that when problems are difficult, expertise-defined ability does not lead to good forecasting.
After analyzing the data, Tetlock found that while expert predictions overall were poor, some experts performed better than others. What mattered was not who these people were or what they believed, but how they thought. Tetlock categorized experts into hedgehogs and foxes. Hedgehogs are specialists in one big thing and extend its explanatory scope to everything they encounter. Foxes, on the other hand, know a little about many things and do not rely on a single explanation for complex issues.
Two findings from Tetlock's research are relevant. The first is the correlation between media exposure and poor forecasting. Tetlock noted, 'Higher-profile forecasters—those more likely to be sought after by the media—are worse calibrated than their less prominent peers.' This study offers another reason to be cautious of 'talking experts' on television and radio.
Secondly, Tetlock found that fox-like individuals tend to predict better than hedgehog-like individuals. High scorers resemble foxes: they know many small things (the tricks of their trade), are skeptical of grand schemes, view explanations and predictions not as deductive exercises but as flexible 'improvisational' practices that require stitching together different sources of information, and have relatively little confidence in their predictive abilities.
Using Page’s analogy, we can say that hedgehog-like individuals possess a single powerful tool, while fox-like individuals have a toolbox filled with various tools. Hedgehog-like individuals excel at solving certain specific problems — they will certainly gain fleeting fame — but over time, especially when conditions change, their predictive ability is inferior to that of fox-like individuals. Tetlock's research provides academic evidence for the power of diversity.
Leadership studies also offer conclusions supporting the importance of diversity. The best way to predict leadership success is to use a weighted combination of predictors. Among these predictors, learning agility stands out as the best predictor. Learning agility has many definitions, but it generally includes critical thinking, which is the ability to carefully examine issues and establish new connections; a desire to acquire new capabilities to become effective; and the ability to cope with novelty, which is the capacity to perform effectively when encountering something for the first time or under different conditions.
We can now answer the question, 'How do these things help you?' In short, cognitive diversity creates a large set of tools for solving complex problems. If our tools are no different from others, there is no reason to believe we can consistently outperform the market.
Let us now look at several practical examples illustrating how thinking inspired by the Santa Fe Institute has provided insights into investment issues.
04, Complex Adaptive Systems: The Collective Wisdom and Whimsy
Since the late 1960s, the Efficient Market Hypothesis has been one of the cornerstones of financial theory. This hypothesis asserts that security prices reflect all available information, implying that no investor can consistently generate excess returns. Overwhelming evidence collected over decades indeed confirms that, over time, most active fund managers underperform passive indices.
Market efficiency is a very important topic for active investment managers. Without a clear understanding of how and why markets are efficient or inefficient, investors lack the foundation to build an investment strategy. However, few investors carefully consider market efficiency, and most take for granted that markets are inefficient.
There are three basic ways to achieve market efficiency. The first assumes that investors are rational, meaning they correctly update their beliefs when new information arises and make appropriate choices based on expected utility theory. The second relaxes the assumption that all investors are rational, relying instead on a small group of rational investors exploiting arbitrage to eliminate pricing errors. The final approach depends on the interaction and aggregation among many independent investors. This pathway to achieving efficiency, commonly known as 'the wisdom of crowds,' is an example of a complex adaptive system.
Almost all financial models stem from one of the first two approaches. For instance, mean-variance efficiency, where investors linearly weigh risk and return, is based on investor rationality. If someone has ever mentioned alpha or beta in a non-pejorative way, they have employed the rational agent approach. The arbitrage approach forms the core of most option pricing models, including the Black-Scholes model. Rational agent and arbitrage models are the primary tools in financial economists' toolboxes. The wisdom-of-crowds approach has received limited attention and, in some cases, is even completely ignored.
Scientists evaluate the validity of a theory by assessing the plausibility of its assumptions and the accuracy of its predictions. By these criteria, both the rational agent and arbitrage models have faced challenges. In both cases, modelers assume mechanisms—namely, investor rationality—to derive outcomes. If common sense and experience were not enough, psychologists have conclusively demonstrated that investor behavior often deviates significantly from the ideal of rationality.
More troubling is the significant gap between the predictions generated by these models and the outcomes we observe in markets. While these tools have undoubtedly advanced our understanding and offer the advantage of mathematical tractability, they remain severely limited in explaining the real world.
Since its founding, a consistent theme at the Santa Fe Institute has been the study of complex adaptive systems. Scientists at the Santa Fe Institute were early to identify key characteristics of these systems and consider interdisciplinary similarities and differences. Complex adaptive systems typically share common features:
① Composed of individual agents (e.g., investors, ants, neurons) with evolving decision-making rules.
② An aggregation mechanism (e.g., stock exchange, pheromone trails, synaptic connections) that captures interactions among agents.
③ The emergence of a higher-level system with characteristics different from the sum of its parts (e.g., stock market, ant colony, mind).
A key feature of complex adaptive systems is non-additivity: the whole cannot be understood by summing its parts. We can disassemble most mechanical systems, identify the function of each component, and reassemble them. Causality is transparent. This is not true for complex adaptive systems; the system emerges from the interactions of individual agents. Just as interviewing a single ant cannot predict the dynamics of an ant colony, no individual investor can explain how the stock market operates.
For a complex adaptive system to solve problems effectively, certain conditions must be met, including diversity of agents, mechanisms for aggregating information, and appropriate incentives. Note that these conditions align closely with Page’s framework. In market terms, when these conditions hold, markets tend to be efficient—they reflect available information and do not offer opportunities for systematic excess returns.
Conversely, when one or more of these conditions are violated, markets can and do become inefficient. To date, the condition most likely to be violated is diversity. Humans are natural imitators, and investors periodically synchronize their behavior, leading to extreme overreactions. Thus, this approach clearly delineates regimes of efficiency based on specific conditions. Economists have used agent-based models to confirm these findings.
Why is viewing the stock market as a complex adaptive system superior to the other two approaches? First, the foundational assumptions of this framework are far more realistic. Markets exhibit considerable diversity: long-term and short-term investment horizons, fundamental and technical analysis, growth versus value investing preferences. We do not need to assume anyone is rational, yet this approach naturally accommodates rationality.
Secondly, although complex adaptive systems do not make specific predictions, their systemic behaviors are consistent with what we empirically observe in markets. One of the greatest challenges to standard financial theory is explaining the existence of major events (booms and crashes). Applying mean-variance statistics suggests that the 1987 crash was, in fact, impossible. In contrast, the complex adaptive systems approach allows for occasional large-scale fluctuations.
Finally, the complex adaptive systems approach identifies specific conditions or scenarios under which markets may be right or wrong. A reasonable default assumption is that the conditions for collective wisdom hold. However, when diversity collapses, it can create attractive investment opportunities. Yet, capitalizing on these opportunities requires overcoming psychological and organizational barriers, which most investors cannot achieve.
Viewing markets as complex adaptive systems provides an unconventional yet robust perspective. This framework specifies the conditions under which markets operate effectively and those under which they fail. Since complex adaptive systems exist across many domains, we have various contexts from which to gain insights into how they function.
05, Network Theory: Building or Expanding the Circle of Competence
Many value investors, most notably Warren Buffett, have refrained from allocating capital to the technology sector due to its perceived lack of predictability. Buffett has eloquently emphasized the importance of investors recognizing and staying within their circle of competence, undoubtedly sound advice. However, Buffett merely stated that tech investing was outside his circle of competence while acknowledging that some investors might possess insight.
An intriguing characteristic of the technology market is that, although the lifecycle of individual products tends to be short, some companies manage to acquire and maintain extremely high market shares. In many consumer goods markets, leading companies with strong competitive advantages typically hold market shares ranging from 30% to 50%. Consider Coca-Cola, Nike, and Anheuser-Busch. In contrast, in certain technology sectors, the distribution of market share is more uneven; market leaders often command 90% or more of the market share (e.g., Microsoft in operating systems and eBay in auctions). Is there a perspective that can help us understand why market share varies so significantly?
One of the pillars of microeconomics is that competitive forces ensure that firms' return on capital converges toward their cost of capital over time. Researchers have repeatedly documented the phenomenon of diminishing returns. However, increasing returns also exist and have existed historically. Although economists have long recognized the concept of increasing returns (Adam Smith’s pin factory being an early example), this idea was largely overlooked by mainstream economists until recently.
Brian Arthur, who has been involved with the Santa Fe Institute since its early days, is one of the more prominent and outspoken economists emphasizing the importance of increasing returns. Arthur's work spans multiple areas, but his focus on increasing returns driven by network effects has garnered the most attention. Network effects occur when the value of a product or service increases as more people use it. A classic example is the telephone system; the more people who own telephones, the greater the value of the entire network.
When network effects are strong, one network often emerges as dominant. Although multiple networks frequently compete for leadership, positive feedback mechanisms ensure that one prevails. Classic examples include the QWERTY keyboard, VHS videotapes, and Intel microprocessors. Several leading thinkers in network theory, including Duncan Watts, Mark Newman, and Steven Strogatz, have ties to the Santa Fe Institute.
Most investors are aware of network effects but apply the concept too loosely. Specifically, applying network theory to investing involves three core issues. The first is having a clear understanding of network taxonomy, particularly where network effects might exhibit strong vitality. Investors often invoke network effects inappropriately. The second issue is how network effects translate into value creation drivers: sales growth, profit margins, risk, and sustainable competitive advantage (the latter being Buffett’s primary focus). When network effects are at play, these value drivers collectively drive up returns on invested capital and reduce risk. The final issue concerns network formation and diffusion, an area heavily informed by epidemiology and sociology. Understanding network formation enables investors to better anticipate changes in growth rates compared to the market.
A thorough understanding of network theory provides a range of perspectives that can help build or expand one’s circle of competence. Notably, most of the knowledge we acquire in classrooms is based on classical economics centered on the supply and demand of physical goods, where diminishing returns play a dominant role. Moreover, network theory is inherently interdisciplinary, drawing insights from various fields.
06. The Power of Power Laws: Investment Insights Follow
Our final example, power laws, is more speculative but holds promise to become a fascinating line of research and a potential source of insights in the coming years. Power laws effectively represent many relationships in biology (animal mass and metabolic rate), physics (frequency and magnitude of earthquakes), and sociology (city size and rank). Visually, a power law appears as a straight line sloping downward from the upper left to the lower right, with variables plotted logarithmically on both axes. In the case of earthquakes, for instance, power laws imply that we frequently observe small quakes while large ones occur less frequently.
Power laws appear in several domains relevant to investment, including company size and stock price movements. However, unlike power laws in biology, where some causal relationships are better understood, no one knows how power laws emerge in most social systems. We do know that some theoretical mechanisms generating power laws fail empirical tests.
How does understanding power laws assist investors? First, knowing that stock price changes follow a power-law distribution helps recalibrate our understanding of risk. Most financial theories, including risk models, are based on a bell-shaped normal distribution of price changes. Power-law distributions suggest that although infrequent, there will be periodic price swings much larger than standard theories predict. This “fat-tail” phenomenon is critical for portfolio construction, leverage, and insurance.
Second, power laws indicate an underlying order within self-organizing systems. While we may not understand how they form, ample evidence suggests their existence, enabling us to make structural predictions about future distributions. For instance, under reasonable growth projections, we can forecast the distribution and scale of U.S. companies. Unfortunately, we cannot determine where individual companies will ultimately fall within the distribution.
Finally, the science of power laws offers insights into understanding growth. For example, efficiency varies with scale: cells in large mammals do not “work” as hard as those in smaller ones. Specialization also tends to increase with scale, which explains why large cities offer more culinary options than smaller ones. Investors can apply these perspectives as companies progress through their life cycles.
Much like complex adaptive systems, the universality of power laws is remarkable, yet remains poorly understood in many contexts. As scientists develop theories explaining broader power laws, investment insights are likely to follow.
Summary
If diversity is logically and empirically useful, why don’t investors spend more time cultivating diverse perspectives? The first obvious answer is that continuous learning requires significant effort. In a time-constrained world, allocating time to ideas outside business and finance is highly challenging. However, this difficulty is unlikely to be the ultimate answer, given the substantial rewards of success. A more plausible reason lies in belief formation and maintenance. While most investors strive to input relevant information, few are introspective enough to question their own beliefs. Why do I believe what I believe? Can this belief withstand empirical scrutiny? These are uncomfortable, even unnatural questions.
Once a belief—most of which stem from people around us—is established, we are reluctant to change it. Social psychologist Robert Cialdini identifies two deeply rooted reasons for this. First, consistency allows us to stop thinking about the issue—it gives us mental rest. Second, belief consistency enables us to avoid the consequences of rational thought—that is, we must change. The first allows us to stop thinking; the second allows us to avoid action.
Charlie Munger said that you must understand the core ideas in all major disciplines and use them regularly — all of them, not just a few. The logic of diversity requires that if we wish to successfully and continuously solve complex problems, we must constantly develop new tools. Continuous learning and open-mindedness are the best ways to achieve this goal, but they are tedious and usually not innate tendencies. At Legg Mason, they strive to embrace diversity to make their investment process as robust as possible.
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Editor/Rice