Why Do Small Early Chances Matter So Much Later? Comparing Theories with the Same Structure as Cumulative Advantage across Economics, Political Science, Statistical Physics, Chemistry, Biology, Developmental Research, and Linguistics

Is the explanation that early chance matters later the same across fields?

Do Those Who Have Receive More? Mechanisms, Causal Identification, and Limits of Cumulative Advantage and the Matthew Effect in the Academic Literature examined, through the sociology of science, education, and network research, the process by which those who were initially advantaged keep accumulating advantages (cumulative advantage). Is the First Advantage Luck? Option Luck and Brute Luck, Ex Ante and Ex Post, and the Evidence on Initial Advantages and Fairness Judgments in the Academic Literature asked whether that initial advantage is luck and, if so, what kind of luck. Both notes stayed within the world of people and organizations.

The explanation that a small early chance becomes a large difference over time, however, appears repeatedly in other disciplines. Economics has explained the layout of the typewriter keyboard in this way, political science the formation of institutions, chemistry the fact that biological molecules are built from only one mirror-image form, and biology the history of life. Krugman (1991) compared the division of regions into a core and a periphery with the structure of ice that crystallizes as water is cooled. If such comparisons seem to work across fields, is a single mechanism at work?

Some cases that look like one mechanism are not. Chaos, which showed the limits of weather forecasting, the fact that the genetic code does not change, and the lifelong effect of early childhood experience can all be summarized as “the beginning decides what follows.” The reasons why an early difference becomes large, however, differ from case to case. This note compares the theories of each field by where chance enters, what makes the difference grow, and what remains at the end. The answer that emerges is that only theories in which three conditions are present together, namely chance, self-reinforcement that grows with the current state, and a process that does not forget its past, have the same structure as cumulative advantage, and that many theories have only one or two of these conditions.

Scope and method

• Coverage: original statements, representative tests and refutations, reviews, and conceptual clarifications of theories which hold that small, chance differences early on come to have large effects over time. Theories that look similar but differ in mechanism (sensitive dependence on initial conditions, proportional growth, hysteresis, sensitive periods, genetic drift, and others) were included for comparison. No field was excluded; sources came from economics, political science and historical sociology, statistical physics and network science, physics and cosmology, chemistry, biology, developmental research and the economics of education, linguistics and cultural evolution, and philosophy.
• Exclusions: empirical studies of cumulative advantage in sociology (covered in the two existing notes). Theories stated only in general books (such as Gould’s Wonderful Life) were treated only to the extent that peer-reviewed papers summarized them.
• Collection: 90 candidates were gathered across academic databases (Crossref, Semantic Scholar, OpenAlex, PMC, arXiv, NBER, and RePEc). After one out-of-scope item was removed, 89 remained, from which the 43 needed for the comparison were selected. No retracted papers or predatory journals were found.
• Verification: for 21 of the 43 sources, the full text (journal version, PMC, arXiv, author version, or working paper) was obtained, and numbers and claims were checked against the relevant passages. For sources read in a working paper version (Arthur 1989, Davis and Weinstein 2002, Bleakley and Lin 2012, Blanchard and Summers 1986, and Cunha and Heckman 2007), only claims that do not conflict with the abstract of the journal version were used. The two papers by Liebowitz and Margolis were read in the versions posted on the authors’ site (a 2019 copy). Of the 22 sources whose full text could not be obtained, 19 were treated as the authors’ claims within the scope of their abstracts. For the remaining three, Frank (1953), Soai et al. (1995), and Crick (1968), neither the full text nor the abstract could be obtained, and their content was taken from reviews that cite them (Blackmond 2010; Koonin and Novozhilov 2009).
• Presentation: for each theory, the point at which chance enters, the mechanism that enlarges the difference, the outcome, and the counterevidence are described separately.

What must be present for two theories to share the same structure?

The comparison uses as its standard the properties that the economist Arthur (1989) demonstrated for competition between technologies. Arthur considered a model in which users, whose order of arrival is not known in advance, choose one at a time between two technologies (for example, two standards)¹. He compared how the outcome changes depending on whether a technology’s payoff falls (diminishing returns), stays the same (constant returns), or rises (increasing returns) as more people choose it. All four properties appeared together only under increasing returns (under constant returns, only inflexibility appeared). The aggregate outcome can be inefficient even when individual choices are rational (potential inefficiency). Market shares become locked in and cannot be shifted by small policy measures (inflexibility, or lock-in). Knowledge of supply and demand conditions is not enough to predict the final market shares (non-predictability). Small historical events are not averaged away but determine the path of market shares (non-ergodicity). Under diminishing and constant returns, the outcome was predictable, and different sequences of events led, with probability one, to the same outcome (ergodicity).

In this model, “chance” is defined as events outside the knowledge of an observer. Arthur granted that infinitely detailed knowledge would make the outcome determinable, and he called the events that the observer’s model cannot distinguish “historical small events.” Because chance is placed in the limits of the observer rather than in the nature of the world, the question shifts from whether chance exists to whether differences that the observer cannot distinguish disappear or remain later. In Arthur’s words, under increasing returns small events are magnified by positive feedbacks, and their cumulation decides the outcome.

The Pólya urn of probability theory shows clearly how the strength of self-reinforcement changes the outcome. When one ball is drawn from an urn and returned together with an added ball of the same color, and this is repeated, the color drawn becomes more likely to be drawn next. According to the review by Pemantle (2007), in the basic Pólya urn, where the probability of being chosen is proportional to the current proportion, the proportion converges to a limit, but that limit is itself determined by chance. If, through economies of scale, adoption rates are proportional to a power greater than 1 of the current market share, the proportion converges to either 0 or 1, and which of the two occurs depends on chance fluctuations near the beginning. This qualitative behavior persists even if one technology has a small intrinsic advantage, as long as the form of the self-reinforcement stays the same. On the side of diminishing returns, as Arthur (1989) showed, chance is averaged away.

Three conditions for the comparison follow from this. The first is that there are chance events that the observer cannot distinguish. The second is that the current state determines the probability or speed of the next change, with an effect that is proportional to the state or stronger (self-reinforcement). The third is that the process does not forget its past, so that the effects of early events remain later (non-ergodicity). This note calls theories in which all three are present “the same structure” and treats theories that have only some of them as theories that “look similar but differ in mechanism.” The strength of self-reinforcement also determines the kind of outcome: convergence to a chance-determined proportion when it is linear, lock-in to one side when it increases faster than proportionally, and the disappearance of chance when it diminishes.

In economics, what produces lock-in, and what has undermined it?

Research on path dependence in economics grew around Arthur’s (1989) model and the history of the QWERTY keyboard. Arthur’s working paper gave the QWERTY keyboard as an example of a technology that is manifestly inefficient for modern use and yet locked in.

Liebowitz and Margolis (1990) disputed this example. They obtained the report of the 1944 Navy experiment on which claims of the superiority of the Dvorak layout rested, and they cited Strong’s report that the experiment had been conducted by Dvorak himself, who held the patent on the layout. In a controlled experiment by Strong for the General Services Administration in 1956, ten typists who were retrained on the Dvorak layout needed well over 25 days of four-hour daily training to return to their former speed. In the training that followed, the typists retrained on Dvorak progressed more slowly than typists who continued training on QWERTY. Liebowitz and Margolis concluded that the continued use of QWERTY was efficient given the current understanding of keyboard design, and they called the Dvorak example a fable of market failure that does not withstand examination of the historical record.

Liebowitz and Margolis (1995) organized this debate by dividing path dependence into three degrees. The first is sensitivity to starting points without any implied inefficiency (first-degree). The second is a case in which the inferiority of a chosen path was unknowable at the time of choice, but in retrospect another path would have been better; the outcome is regrettable and costly to change, but not inefficient in any meaningful sense, given the limits on knowledge (second-degree). The third is a case in which a feasible arrangement for recognizing and achieving a better outcome existed or exists, yet that outcome is not obtained (third-degree). They wrote that only the third degree conflicts with the neoclassical model of rational behavior. By this distinction, the “potential inefficiency” shown in Arthur’s model is a third-degree claim and needs to be tested separately from the claim that early chance determines the outcome (the first and second degrees).

In economic geography, the same structure appears as the differentiation of cities and regions. Krugman (1991) argued that when an index combining transportation costs, economies of scale, and the share of expenditure on nonagricultural goods crosses a critical value, population begins to concentrate, regions diverge, and the process feeds on itself. Which region gains population depends sensitively on initial conditions: the region that has slightly more population at the moment when transportation costs fall below the critical level gains population at the expense of the others. Krugman compared this to a “random broken symmetry,” like ice crystallizing as water is cooled, and wrote that the detailed structure may depend on small accidents of early history. In this account, chance matters around the time when the critical value is crossed, not equally at every time.

Two studies tested this prediction with large shocks. Davis and Weinstein (2002) used the Allied bombing of Japanese cities in World War II as a shock to relative city sizes. Most cities returned to their former position in the distribution of city sizes within about 15 years. The coefficient on population growth in 1940–47 in explaining growth in 1947–60 was −1.0, and the typical city had recovered from the shock by 1960. Hiroshima and Nagasaki also returned to their prewar growth trends, although this took somewhat longer than in other cities. Davis and Weinstein read this result as striking most strongly at the random growth theory, which predicts that shocks persist without reversion, and wrote that for the increasing returns theory, however appealing spatial catastrophes may be in theory, the actual distribution of city sizes is robust to large temporary shocks. The abstract of the journal version states that the results support a hybrid theory in which locational fundamentals establish the pattern of relative regional densities and increasing returns help to determine the degree of differentiation.

Bleakley and Lin (2012) provided evidence pointing in the other direction. Many North American cities formed at portage sites, where water navigation was interrupted and goods had to be carried overland; commerce gathered there, and the water power of the falls attracted early manufacturing. Examining portage sites in the U.S. South, Mid-Atlantic, and Midwest, they documented the continuing importance of these sites long after their original advantages had become obsolete. They interpreted this as path dependence and contrasted explanations that combine sunk costs with diminishing returns and with increasing returns to scale. Taken together, the two studies show cases in which a temporary shock disappears (wartime bombing) and cases in which a location persists after its original reason has disappeared (portage). Both studies deal with geography as an initial condition and with later shocks, and they need to be read as distinct from studies that directly measure chance early differences.

Hysteresis in unemployment shares the feature that the past persists, but its mechanism is different. Blanchard and Summers (1986) regarded the steady rise of European unemployment over fifteen years as a persistence of shocks that standard theories cannot explain. In their hypothesis, wages are set in bargaining between employed insiders and firms, and outsiders are excluded from bargaining. A shock that reduces employment changes the number of insiders and thereby the subsequent equilibrium wage, so that the equilibrium unemployment rate follows the actual unemployment rate. In this process, the effect of a shock persists, but an early difference does not reinforce itself and widen. Moreover, what persists is not limited to early shocks; a shock at any time will do.

What did political science and historical sociology take from economics, and what did they add?

Political science took over Arthur’s properties directly. The abstract of Pierson (2000) conceptualizes path dependence as a social process grounded in a dynamic of “increasing returns” and argues that such processes are likely to be prevalent in politics. The same abstract summarizes the claims of historical institutionalism in five points. Specific patterns of timing and sequence matter; a wide range of social outcomes may be possible; large consequences may result from relatively small or contingent events; particular courses of action, once introduced, can be almost impossible to reverse; and political development is punctuated by critical moments or junctures that shape the basic contours of social life. The first four correspond to Arthur’s four properties.

The fifth, the “critical moments,” is the element that political science added. The abstract of Capoccia and Kelemen (2007) reconstructs the concept of a critical juncture as a period in which choices close off alternative options and lead to institutions that generate self-reinforcing path-dependent processes. They identified contingency as the key characteristic of critical junctures and wrote that counterfactual reasoning and narrative methods are necessary to analyze it. In this framework, chance matters through the choices made at critical junctures, after which the self-reinforcing process protects the path. As with the moment around Krugman’s critical value, the point at which chance enters is limited in time.

Management and organization studies questioned how path dependence is tested. The abstract of Vergne and Durand (2010) points out that the literature has conflated path dependence as a process, in which history unfolds in a self-reinforcing manner, with path dependence as an outcome, a persisting state called lock-in. They redefined the necessary conditions for path dependence, distinguished it from other “history matters” mechanisms, and recommended moving from case studies to more controlled designs such as simulations, experiments, and counterfactual investigation. The abstract of Sydow, Schreyögg, and Koch (2009) describes how organizations become path dependent in three stages centered on self-reinforcing mechanisms and also discusses breakouts from lock-in.

On the philosophical side, the abstract of Ben-Menahem (1997) redefined necessity and contingency in the historical context. In this definition, contingency increases in direct proportion to sensitivity to initial conditions, and necessity in reverse proportion. The abstract lists as merits of the definition that it avoids conflating causality with necessity and contingency with chance. Under this definition, the question for historical explanation is not whether chance was present but how sensitive the outcome was to small differences in initial conditions.

In distributions and markets, how have the forms of reinforcement been distinguished?

Statistical physics and network science have tried to read self-reinforcing processes from the shape of distributions. According to the review by Newman (2005), in the Yule process new objects appear from time to time, and the new elements added between appearances are allocated in proportion to the number that each object already has (Yule devised this process to explain the number of species per genus of plants). Simon called this kind of “rich-get-richer” process the Gibrat principle, and the same process has also been called the Matthew effect, cumulative advantage, and preferential attachment. The process generates a distribution with a power-law tail, whose exponent is determined by the number with which new objects appear, a proportionality constant, and the number of elements added between the appearances of new objects.

Barabási and Albert (1999) showed that two ingredients, the continuous addition of nodes and the preferential attachment of new nodes to nodes that already have many connections, produce networks whose numbers of connections follow a power law. In this model, the number of connections of node i grows in proportion to the 0.5 power of the time elapsed since it was added at time t_i, so that older nodes increase their connections at the expense of younger ones. Here the early advantage is less a chance event than early arrival itself. Bianconi and Barabási (2001) added a fitness that differs from node to node (the ability to attract links) and showed that fitter nodes can overtake nodes that are more connected but less fit. When differences in intrinsic quality are added to self-reinforcement, the advantage of early arrival can be overturned.

“Proportional” growth of a different kind, Gibrat’s law of proportionate effect, produces a different outcome. According to the review by Mitzenmacher (2004), in a multiplicative process, size grows or shrinks by a random factor at each step. The logarithm of size then equals the logarithm of the initial size plus the sum of the logarithms of the factors at each step, and by the central limit theorem the distribution approaches the lognormal. As the form of the equation shows, early and late shocks are added with the same weight on the log scale, and none of them disappears. Mitzenmacher also showed that adding a single reflecting lower barrier makes the same multiplicative process produce a power-law distribution. This random proportional growth is the theory that Davis and Weinstein (2002) rejected as predicting that wartime shocks would not be reversed.

That a process cannot be inferred from the shape of a distribution has also been confirmed with real data. Broido and Clauset (2019) tested 928 network data sets from biological, technological, social, transportation, and information domains and reported that only 4% (Strongest) and 10% (Strong) met the criteria for strong scale-free (power-law) structure, while for most networks lognormal distributions fit the data as well as or better than power laws. On the cultural side, the abstract of Bentley, Hahn, and Shennan (2004) states that the frequency distributions of first names, archaeological pottery, and technology patent applications follow power laws that can be explained by a simple model of random copying (drift). Heavy-tailed distributions can arise from self-reinforcement, from multiplicative processes, and from drift without any preference.

To observe the process itself rather than the distribution, there are experiments that rerun history many times from the same conditions. Salganik, Dodds, and Watts (2006) created an artificial music market in which 14,341 participants listened to and downloaded unknown songs, and they assigned participants to eight “worlds” in which others’ download counts were visible or to one world in which they were not. Increasing the strength of social influence increased both the inequality and the unpredictability of success. Success was not determined by quality alone, but the best songs rarely did poorly and the worst songs rarely did well. That outcomes differed among eight worlds that started from the same songs indicates that early chance choices shaped later success. That outcomes converged at the top and bottom of quality indicates that intrinsic quality remains as a counterweight to self-reinforcement.

Informational cascades in economics explain the working of reinforcement through people’s reasoning. Bikhchandani, Hirshleifer, and Welch (1992) defined an informational cascade as a situation in which it is optimal for an individual, having observed the actions of those ahead, to follow the preceding individual without regard to his or her own information. In a cascade, people converge rapidly on one action on the basis of very little information, and if even a little new information arrives, or if people merely suspect that circumstances have changed, the social equilibrium may shift radically. They wrote that cascades make society systematically land, on little information, close to the borderline between alternatives, so that mass behavior is fragile. The lock-in here is fragile, unlike the lock-in under Arthur’s increasing returns. Although in both cases the choices of a few early individuals decide what follows, the firmness of the locked-in state differs depending on whether the source of reinforcement is payoffs or information.

In chemistry and physics, how do small imbalances grow, and when do they not?

That biological amino acids and sugars are built from only one mirror-image form (left-handed or right-handed), or homochirality, is a classic question in chemistry. According to the review by Blackmond (2010), fluctuations in the numbers of molecules create left-right imbalances by chance, but unless some process sustains and amplifies such an imbalance, it averages out to the state of equal amounts. Whether the imbalance arose by chance or came from space, approaching homochirality requires an amplification mechanism.

Frank (1953) showed such an amplification mechanism mathematically. In Blackmond’s (2010) account, a molecule in Frank’s model acts as a catalyst for its own production (autocatalysis) and, at the same time, when it meets the opposite mirror image, both lose their activity (mutual antagonism). Starting from three left-handed and two right-handed molecules, when one pair meets and is deactivated and the rest reproduce themselves, the ratio becomes 4 to 2; repeating this gives 6 to 2 and then 10 to 2, widening the gap. The smaller the initial imbalance, the more molecules are lost before the imbalance becomes appreciable, but if the substrate pool is large enough, the production of one form eventually dominates. Soai et al. (1995) reported an asymmetric autocatalytic reaction in which the product accelerates its own formation and showed that a highly imbalanced product can be obtained from a slight imbalance in the initial catalyst. According to Blackmond, later work reported an imbalance of 85% obtained from an initiator with an imbalance of 0.1%, and the kinetic model of Blackmond and colleagues showed that this reaction has the two features of Frank’s model. The same review calculates that an autocatalyst with a 1% imbalance approaches homochirality after about 5000 catalytic cycles, even with weakly selective combinations.

The same amplification has been found in a physical process. According to the abstract of Kondepudi, Kaufman, and Singh (1990), when sodium chlorate was crystallized from aqueous solution without stirring, left-handed and right-handed crystals were found in statistically equal numbers, but when the solution was stirred, almost all of the crystals in a sample (99.7%) had the same handedness, which could be either left or right. In Blackmond’s (2010) account, stirring breaks the first crystal that forms into many small crystals of the same handedness, which then rapidly deplete the molecules in solution. That the handedness fell by chance to the left or the right in repeated experiments indicates that a single early event decided the outcome and that self-reinforcement spread it throughout the system. The same review, however, also describes a process near equilibrium in which crystals repeatedly dissolve and re-form and “forget” their former handedness (chiral amnesia). Whether an imbalance stays locked in depends on conditions.

Chaos in physics also enlarges small differences, but by a different mechanism. The abstract of Lorenz (1963) states that in systems of deterministic nonlinear differential equations representing dissipative flow with bounded solutions, nonperiodic solutions are unstable with respect to small modifications, so that slightly differing initial states can evolve into considerably different states, and it examines the feasibility of very long-range weather prediction in light of this result. Here what widens the difference is the instability of solutions, not reinforcement that grows with the state. Because the instability described in the abstract applies not only to the initial state but to any state along a solution, it can be read as implying that initial differences do not matter in any special way and that the system is not locked into any one state. In light of Ben-Menahem’s (1997) definition of contingency as sensitivity to initial conditions, chaos is among the most contingent systems, yet it lacks the structure of cumulative advantage.

Cosmic structure formation combines chance seeds with deterministic growth. According to the review by Springel, Frenk, and White (2006), at the beginning of the universe quantum fluctuations in the inflaton field were blown up to macroscopic scales and became weak ripples in the cosmic energy density. Over 14 billion years these ripples were amplified by gravity into concentrations of dark matter, within which galaxies formed. Gravitational growth resembles self-reinforcement in that denser regions attract more matter. However, the review writes that simple models of inflation predict the statistical properties of the initial fluctuations and that the cosmic web is a nonlinear gravitational “sharpening” of the pattern already present in the Gaussian random field of initial fluctuations. Where galaxies form is decided by the chance of the seeds, but the statistics of the structure are predictable, which differs from Arthur’s non-predictability.

In biology, does contingent history or convergence prevail?

Biology has several theories about chance early events, each with a different mechanism. The founder effect concerns the chance of sampling when a few individuals settle in a new place. According to the abstract of Templeton (1980), in Mayr’s original model a founder event causes inbreeding and an imbalance in gene combinations, which alters how selection acts and can cause a rapid shift to a new adaptive peak (a genetic revolution). The abstract of the review by Barton and Charlesworth (1984) argues that founder effects may cause speciation only under sufficiently stringent conditions and are only one extreme of a continuous range of possibilities. Here the early chance enters as a sampling bias in a small population and remains through subsequent drift and selection.

The frozen accident is fixation through the cost of change. According to the review by Koonin and Novozhilov (2009), Crick’s (1968) frozen accident hypothesis holds that the standard genetic code might have no special properties but was fixed simply because all extant life forms share a common ancestor, with subsequent changes to the code mostly precluded by the deleterious effect of codon reassignment. The review writes that mathematical analysis shows the standard code to be highly robust to translational misreading, but that there are numerous more robust codes, and that much of the code’s evolution could be explained by a combination of frozen accident and selection for error minimization. What is fixed here is a single assignment, and the reason it remains is the harm that a change would cause.

Historical contingency was posed as the question of whether replaying the history of life would produce the same result. Blount, Borland, and Lenski (2008) summarized Gould’s argument that evolutionary outcomes depend on the details of long chains of antecedent states, small changes in which may have enormous long-term repercussions, so that replaying the “tape of life” would produce a world far different from the present one. They set against it Conway Morris’s counterargument that natural selection constrains organisms to relatively few adaptive options, so that the routes are many but the destinations are limited. They then studied 12 populations of E. coli founded in 1988 from the same ancestor. A variant able to use citrate as a carbon source under oxic conditions did not appear in any population for more than 30,000 generations and evolved in only one population by 31,500 generations. When evolution was replayed from different points in that population’s history, no citrate-using mutants arose among 8.4×10^12 ancestral cells or among 9×10^12 cells from 60 clones sampled in the first 15,000 generations, but later clones were more likely to produce them. The authors concluded that a mutation that made the later change more likely (a potentiating mutation) had arisen by 20,000 generations. Chance in this case acted as a condition that made later events possible.

The same line of research has also produced evidence of convergence. According to the abstract of Travisano et al. (1995), bacterial populations grown in identical environments achieved similar fitness regardless of prior history or subsequent chance events, whereas cell size, a trait only weakly correlated with fitness, was more strongly influenced by history and chance. According to the abstract of Losos et al. (1998), the same set of habitat specialists occurs among the Anolis lizards of the four islands of the Greater Antilles, and phylogenetic analysis indicates that they originated independently on each island. The abstract of the review by Blount, Lenski, and Losos (2018) summarizes that replay experiments often show parallel changes, especially in overall performance, and that evolution is surprisingly repeatable among closely related lineages, but that disparate outcomes become more likely as the footprint of history grows deeper. Whether chance matters later depends on which trait is examined (fitness or a trait weakly linked to it) and on how distantly related the compared lineages are.

Priority effects in community ecology have the structure closest to cumulative advantage within biology. According to the abstract of the review by Fukami (2015), the order and timing of species immigration during community assembly affect species abundances at multiple spatial scales, and these priority effects cause historical contingency in the structure and function of communities, resulting in alternative stable states, alternative transient states, or compositional cycles. The mechanisms fall into two categories: early-arriving species occupy available resources (niche preemption) or change the environment (niche modification). For historical contingency to occur, two requirements must be met: the regional species pool must contain species that can cause priority effects, and local dynamics must be fast enough for early-arriving species to preempt or modify niches before other species arrive. The abstract of Chase (2010) reports that in a long-term pond experiment, stochastic community assembly became stronger with higher productivity, and that this result is most consistent with the hypothesis that more intense priority effects lead to multiple stable equilibria, which may explain why species composition differs more among sites where productivity is high. The three conditions are met here, as the chance of arrival order is locked in by the self-reinforcement of resource preemption.

Runaway sexual selection has the same structure. Lande (1981) modeled the joint evolution of female mating preferences and male secondary sexual characters. According to the abstract, some types of preferences can create a runaway process even under stabilizing natural selection on males, and the outcome of phenotypic evolution depends critically on the genetic variation parameters and the initial conditions of the population. Self-reinforcement is present in that the linkage between preference and trait strengthens both.

In development and language, how are timing effects and self-reinforcement distinguished?

Developmental research first explains the persistence of early experience as a property of timing. The abstract of Knudsen (2004) defines a sensitive period as a limited period in development during which the effect of experience on the brain is particularly strong, and a critical period as one in which experience provides information essential for normal development and alters performance permanently. The hypothesis is that experience during a sensitive period modifies the architecture of a circuit in fundamental ways, making certain patterns of connectivity highly stable, and that plasticity after the sensitive period alters connectivity within the architectural constraints established during that period. In their review of the developmental origins of health and disease (DOHaD), Gluckman et al. (2008) point to developmental plasticity and its epigenetic mechanisms and write that the effects of developmental changes may not manifest until later in life. A body pattern induced by a poor intrauterine environment may favor survival in a poor environment, but if the later environment is nutritionally rich, the mismatch between pattern and environment increases the risk of metabolic disease. In both accounts, early events act strongly because of high plasticity at that time, not because early differences reinforce themselves.

Cunha and Heckman (2007), in the economics of education, separated the two within a single model. In their model of skill formation, a stage that is especially productive for a given skill is a sensitive period, and if only one stage is effective, it is a critical period. Separately, skills acquired at one stage raise the skills attained at later stages (self-productivity), and skills acquired at one stage raise the productivity of investment at later stages (dynamic complementarity). They wrote that the two together produce multiplier effects through which “skills beget skills.” Complementarity also implies that early investment must be followed by later investment for the early investment to be productive. Sensitive periods concern timing, whereas self-productivity and complementarity are self-reinforcement, and claims that early differences widen in education (such as the Matthew effect in reading, discussed in Do Those Who Have Receive More? Mechanisms, Causal Identification, and Limits of Cumulative Advantage and the Matthew Effect in the Academic Literature) belong to the latter.

In linguistics, the formation of creole languages has been explained by analogy with the founder effect of population genetics. The abstract of Mufwene (1996) compares a language to a population rather than to an organism and a linguistic feature to a gene, and views features as competing for selection into the systems of creoles in situations of language contact. It then argues that the founder populations of early colonies played a greater role than previously thought in determining which features gained a selective advantage (the founder principle). On the propagation of language change, the abstract of Blythe and Croft (2012) states that the widely observed S-shaped trajectory of change can be captured only if the mechanisms include selection that weights the competing variants differently, except under highly specialized circumstances. That neutral drift alone rarely produces an S-curve, set beside Bentley et al.’s (2004) explanation of cultural frequency distributions by drift, shows that the shape of a distribution and the trajectory over time can lead to different inferences about the mechanism at work.

Comparison by where chance enters, the mechanism of reinforcement, the outcome, firmness, and counterevidence

The column “Three conditions” shows how many of the three conditions (chance, self-reinforcement, and a process that does not forget its past) are present.

The rows marked “All three” have all three conditions, even though they differ in where chance enters and in how firm the lock-in is. In the rows marked “Two” and “One” early events persist, but the reason they persist is not self-reinforcement.

What shares the structure of cumulative advantage, and what does not?

Read through the three conditions, theories with the same structure as cumulative advantage were found in the natural sciences, the social sciences, and the humanities alike. They are increasing returns and lock-in in economics, the core and periphery in economic geography, path dependence and critical junctures in political science, preferential attachment in network science, markets with social influence and informational cascades, asymmetric autocatalysis and stirred crystallization in chemistry, priority effects in ecology, runaway sexual selection in evolutionary biology, and the self-productivity of skills in the economics of education. In these theories, an early chance is carried by self-reinforcement that grows with the current state, and because the process does not forget it, it becomes a large difference later. Given that stirred crystallization in chemistry and economic geography share the same skeleton, a single early chance event and self-reinforcement that spreads it, Krugman’s (1991) comparison of urban differentiation with crystallization is more than a metaphor.

Within the same structure, the outcome depends on the strength of self-reinforcement. When reinforcement is linear, as in the Pólya urn, the proportion settles at a value set by chance. When reinforcement increases faster than proportionally, as under increasing returns or Frank’s mutual antagonism, the result is lock-in to one side. When it diminishes, chance disappears. The firmness of the locked-in state also differs: lock-in under increasing returns is not moved by small policies, informational cascades collapse with a little new information, and crystals near equilibrium can forget their handedness.

Theories that look similar but differ in mechanism lack one or more of the three conditions. Sensitive dependence on initial conditions has sensitivity to chance but neither reinforcement nor lock-in. Proportional growth preserves shocks but does not weight early shocks more heavily than late ones. Hysteresis makes shocks persist but does not widen differences. The frozen accident involves fixation, but its reason is the cost of change. Sensitive periods and DOHaD give early events more weight, but the reason is the plasticity of that period. Cosmic structure formation grows chance seeds, but its statistics are predictable. Historical contingency showed that chance can become a condition that opens later possibilities, but it is not a process in which differences reinforce themselves and widen. Calling these theories “the same” as cumulative advantage makes it easy to draw wrong conclusions about why early intervention works (plasticity or self-reinforcement) and whether a state can be changed later (cost or lock-in).

Theories with the same structure also come with counterevidence and counterweights. Japanese cities hit by wartime bombing returned to their former relative sizes within about 15 years (Davis and Weinstein 2002). The evidence for the inefficiency of QWERTY was weak (Liebowitz and Margolis 1990). Fitter nodes can overtake earlier ones (Bianconi and Barabási 2001). In the music market, outcomes converged at the top and bottom of quality (Salganik et al. 2006). Bacterial fitness converged to similar values regardless of history (Travisano et al. 1995). What counteracts chance are determining factors outside it, such as locational fundamentals, intrinsic quality, fitness, and selection. The qualification by Merton (1988) discussed in Do Those Who Have Receive More? Mechanisms, Causal Identification, and Limits of Cumulative Advantage and the Matthew Effect in the Academic Literature, that differences widen until dampened by countervailing processes, applies equally to the theories of other fields.

Finally, the methods for testing whether chance matters later are also similar across fields. Experiments that rerun history many times from the same conditions (the eight worlds of the music market, the 12 E. coli populations and their replays, repeated crystallizations, and experiments that vary the order of community assembly) measure the effect of chance most directly. Natural experiments (wartime bombing, portage) distinguish self-reinforcement from fundamentals by whether a temporary shock disappears or persists. Inference from the shape of distributions is the weakest, because self-reinforcement, multiplicative processes, and drift can produce the same shape. Whether a small early chance matters a great deal later depends on four factors: when chance enters, how strong the self-reinforcement is, how firm the locked-in state is, and how far determining factors outside chance counteract it. Which of the four is at work can be established only through replay experiments and responses to shocks.

Gaps (unresolved issues)

• Estimating the strength of self-reinforcement: the outcome of theories with the same structure depends on whether reinforcement is linear or increasing, but within the scope of this note there were few studies that estimated the exponent of reinforcement for social processes from real data and predicted the kind of outcome.
• Comparing the firmness of locked-in states: the firmness of lock-in differs across theories, as with lock-in under increasing returns, informational cascades, and the forgetting of crystal handedness near equilibrium, but no study comparing them within one framework was found.
• Tests on the humanities side: within the scope of the abstracts, contingency in history and the founder principle in linguistics have not been tested down to the mechanism of self-reinforcement. For canon formation and counterfactual history, no peer-reviewed papers treating the mechanism were identified, and they are not covered.
• The initial advantage and luck: what kind of luck the initial advantage from which accumulation starts is was discussed in Is the First Advantage Luck? Option Luck and Brute Luck, Ex Ante and Ex Post, and the Evidence on Initial Advantages and Fairness Judgments in the Academic Literature.

Unverified items

• No main claim remains unverified.
• Nineteen sources were handled within the scope of their abstracts because their full texts could not be obtained (Pierson 2000; Capoccia and Kelemen 2007; Vergne and Durand 2010; Sydow et al. 2009; Ben-Menahem 1997; Bentley et al. 2004; Lorenz 1963; Kondepudi et al. 1990; Templeton 1980; Barton and Charlesworth 1984; Travisano et al. 1995; Losos et al. 1998; Blount et al. 2018; Fukami 2015; Chase 2010; Lande 1981; Knudsen 2004; Mufwene 1996; Blythe and Croft 2012). In the text, all of them are attributed to the authors within the scope of what the abstracts state.
• For Frank (1953), Soai et al. (1995), and Crick (1968), neither the full text nor the abstract could be obtained, and their content was taken from reviews (Blackmond 2010; Koonin and Novozhilov 2009).
• For Arthur (1989), the full text of the journal version could not be obtained, and claims were checked against the working paper presenting the same model (IIASA WP-83-090, 1983).

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