Complexity-minded antitrust

Complexity science permeates the policy spectrum but not antitrust. This is unfortunate. Complexity science provides a high-resolution screen on the empirical realities of markets. And it enables a rich understanding of competition, beyond the reductionist descriptions of markets and firms proposed by neoclassical models and their contemporary neo-Brandeisian critique. New insights arise from the key teachings of complexity science, like feedback loops and the role of uncertainty. The present article lays down the building blocks of a complexity-minded antitrust method.


Introduction
Antitrust law is in the spotlight. Rising concentration levels in the economy fuel demands for antitrust to do more. However, the neoclassical economic methods applied in practice constrain application of antitrust law to a limited number of straightforward cases like cartels and mergers to monopoly. Other antitrust cases are excessively costly to bring and fail to deliver appropriate and timely responses to business conduct and transactions that harm competition. Besides, the practice of antitrust law is governed by a set of protocols, rules, and tools that makes limitative assumptions about the economy. The methods of antitrust law were good enough for a simple agrarian economy Thibault Schrepel and Nicolas Petit contributed equally to this paper.
* Thibault Schrepel t.schrepel@vu.nl with decreasing returns, fungible production factors, and technological maturity. The sophistication brought by price theory and microeconomics also worked well in the context of the 20th-century industrial economy, with the growth of large, multi-product firms and global markets. But today, the method of antitrust law appears ill-suited to a complex economy with unprecedented types and levels of increasing returns, feedback loops, and technological dynamism. Unless neoclassical antitrust makes a 'complexity' leap, it is doomed to irrelevance, and a diminishing role in the "broader gestalt of technology and industrial policies." (Teece 2018).

Problems of methods in antitrust
The task of antitrust is to maintain a competitive economy. 1 Thus, the central question is whether the economy is competitive. To find out, antitrust relies on analytical methods derived from neoclassical economics (2.1). 2 Neoclassical methods are reaching their limits in an ever-complexifying economy (2.2). Unfortunately, contemporary debates amongst antitrust experts have not (yet) invited a reexamination, but have tended to correlate with ideological predispositions, and an overly stylized view of the economy (2.3).

Neoclassical antitrust
One main activity in antitrust is defining whether there is competition. The search for information about the existence and degree of competition in the economy is constrained by application of methods drawn from a modernized version of neoclassical economics. The main question of interest under a neoclassical economics method is whether positions of control over output result from monopoly or efficiency. A related question of interest consists in studying the conditions of minimization of probability of errors and costs of decision incurred by antitrust and regulatory institutions. 3 In addressing these questions, a legitimate concern for exactitude and practicality has justified reliance on a method of competitive analysis that works on a set of limitative assumptions, units of analysis, and focal points (Kauper 2008). These limitations concern the business environment (i), markets (ii), and firms (iii).
(i) The business environment is stable, so investment is a quantifiable tradeoff between risks and returns (experts talk of expected value). 4 Limited government and application of the rule of law are deemed the norm. Under these constraints, markets move towards predictable equilibrium positions. The role of antitrust and regulation is to move market structures and business conduct closer to the idealized yet unattainable competitive equilibrium conditions described in static representations of the economy. 1 We talk of antitrust in the broad sense, including market power regulation. 2 We leave aside the institutional question, which is about evidence-based fact-finding processes (e.g., litigation). 3 For example, the Courts have held that no elaborate market inquiry was required in areas where there is "sufficiently reliable and robust experience for the view" that business conduct injures competition, see CJEU, Budapest Bank, ECLI:EU:C:2020:265, para 76. 4 Works in economic theory on uncertainty, although numerous, have had little policy impact.

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Complexity-minded antitrust (ii) Antitrust and regulation focus on individual "markets," an approach known as "partial equilibrium" optimization. The assumption is that competition in individual markets averages well over the economy. In the same averaging spirit, neoclassical antitrust overlooks asymmetries and heterogeneity amongst agents. Instead, neoclassical antitrust assumes that all agents pursue utility maximization.
In addition, not all markets matter to a neoclassical eye. Markets where firms compete for production factors like labor and capital are neglected. Product market competition is the focal point. Further, not all products exchanged in a market are accounted for as competition. Product rivalry is only deemed to exist under conditions of substitutability. The term of art is that antitrust and regulation look at competition in a "relevant market." (iii) Monopoly or efficiency are considered at the firm's level. Rivalry with decreasing returns amongst profit-maximizing firms determines performance, prices, costs, and output levels. Gains in welfare can logically be achieved by increasing the number of firms or by decreasing opportunities for interfirm cooperation. Limited attention is given to divisional, organizational or managerial competition within the firm.
Some further issues are given short shrift in neoclassical methods. An assumption of fungible labor, for example, implies that the development of firm-level competitive advantages from organization or human resources is not a central issue in antitrust and regulation fact-finding. Relatedly, the role of technological competition in market outcomes has been considered a footnote. Until the 1970s, innovation was treated as an exogenous force in mainstream economics (Solow 1956;Swan 1956). The learning changed in economics with the recognition that innovation is endogenous. But antitrust and regulation methods remain wedded to methods that leave a nominal role to innovation in practice. Last, neoclassical antitrust also tends to correlate competition (the whole) with the sum of its parts (companies active in the market). Neoclassical antitrust thus assumes away the non-linearity of competition. By assuming that interactions between agents are additive, neoclassical antitrust fails to capture that the whole is different from the sum of the parts. 5

The knowledge-economy
As the economy is complexifying, the limitations of neoclassical methods come in broad daylight. Antitrust and regulation methods' inaptitude to answer fundamental questions motivates a methodological reexamination.

Complexification of the economy
The economy is increasingly complex, meaning there is a global increase in the number of activities, and interactions between them. 6 One critical channel of economic 1 3 complexification is the transition from an industrial to a knowledge-economy. Since the early 1970s and the Intel microprocessor announcement in Santa Clara, the economic system has experienced tremendous growth. The decoupling of information from matter has expanded the production possibility frontier. 7 New markets, industries, and economic sectors have relentlessly emerged fueled by entrepreneurial effort and public investments in technological infrastructure. Today, a universe of economic opportunities is on the horizon with the advent of new value creation propositions (like microcontent, influencing, or the metaverse), new forms of value capture (like targeted-advertisement or non-fungible tokens), and new modes of economic transaction (like the sharing economy or blockchain).
The nature of competition is also changing. The years following the wave of economic destruction of the 1999 Dot-Com crash witnessed the formation of dozens of large digital firms in the US (and in China) and hundreds of smaller ones (Goodnight and Green 2010). Google, Apple, Facebook, Amazon, and Microsoft are household names. But there are many more big and small firms in this group than just a few consumer-facing companies.
Digital firms challenge our understanding of competition in ways that the good old big oil, big aluminum, and big tobacco monopoly suppliers did not. First, large digital firms sit at the center of interdependent ecosystems that connect their own services ("platforms") 8 and multiple agents with whom they entertain cooperative and/ or competitive relationships. Second, increasing returns on the supply and demand side due to economies of scale and network effects create powerful incentives for efficient growth and diversification amongst all ecosystems agents. 9 Third, compared to the industrial economy, competitive advantage -and business survival -appear dependent on the firm's environment (e.g., dynamics at the industry, not just market level) and technological resources.

Problems in policy
Neoclassical antitrust methods would be tolerable if they predicted the existence or absence of competition with reasonable certainty. However, neoclassical methods have supplied confounding answers in the context of a complex knowledge-economy. Some examples of illogicalities, oddities, and sometimes absurdities drive the point home.
The application of neoclassical methods led the European Commission to consider that Apple and Google are not competitors in an antitrust-relevant market. 10 Does this imply that the EC would treat a merger between Apple and Google as presumptively lawful, in line with the standard approach towards mergers between noncompetitors? The Competition and Markets Authority in the UK considered Giphy a close competitor to Facebook. But a decade before, it refused to consider Instagram a potential competitor to Facebook. Why? 1 3

Complexity-minded antitrust
In a complaint, the US government has alleged that Google's payment to Apple in exchange for exclusive pre-installation of its search engine constituted exclusionary monopolization. But without any serious competitor in search, Google has no one to exclude. By contrast, it defies understanding that the US government never asks whether Apple extracting billions of dollars for preferential access to iPhones constitutes evidence of monopoly power in use.
More than 20 years after the case, the jury is still out on whether the antitrust litigation in US v Microsoft promoted, reduced, or was irrelevant to competition and innovation in digital industries. To paraphrase Frank Easterbrook, we are approaching a point where we have as many ex-post rationalizations about the Microsoft case as there were positions on what dragons looked like 600 years ago (Easterbrook 1981).

Complexity denialism
Different reactions have emerged from the antitrust field in reaction to the creation, expansion, and maturation of the knowledge-economy. The Neo-Brandesians have been first to propose a complete antitrust analysis of the knowledge-economy. Neo-Brandeisians recognize complexity to the extent that they consider that large digital firms hold a specific form of structural power over markets. 11 According to Neo-Brandeisians, "gauging real competition in the twenty-first century marketplace-especially in the case of online platforms-requires analyzing the underlying structure and dynamics of markets (…) a company's power and the potential anticompetitive nature of that power cannot be fully understood without looking to the structure of a business and the structural role it plays in markets" (Khan 2017).
Failing to translate that methodological observation into practice, however, Neo-Brandesians paint a one-size-fits-all 'big is bad' picture of large digital firms. 12 The solutions proposed are the same as those advocated by Justice Brandeis in the early twentieth century. Against the background of an assumption of decreasing returns, neo-Brandesians defend break-ups and want to classify platforms as "public utilities." There is little interest in empirical facts, except those denoting corporate size, dominant shares, and conglomeration. The industry environment and technological resources are ignored.
The Neo-Chicagoan reply has been equally ideological in analyzing the knowledge-economy. Neo-Chicagoans rely on observations of rising output in the digital sector to draw a general inference of economic efficiency and justify a blanket laissez-faire approach. Emergent issues like privacy loss or excessive ad load are glossed over. According to Neo-Chicagoans, rational users freely barter their privacy in exchange for free goods and services. The ever-increasing targeted advertisement supplied by large digital firms is treated as an informational improvement. Where Neo Brandesians see bigness everywhere, Neo Chicagoans witness affluence of gains in consumer surplus.

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In last analysis, Neo-Chicagoans double down on the methods developed by the Chicago school. 13 They model the economy as an equilibrium system in which firms have equal initial access to technology, capital, and information, and compete for advantage through product and cost-efficiency. Monopoly rents knowledge-economy's cannot exist unless firms enjoy government protection. Neo-Chicagoans' disinterest in technology allows them to turn a blind eye to specific properties of the knowledge-economy. 14 Increasing returns to adoption, lock-in by historical events, and path dependence 15 which play a determinant role in selecting market outcomes, and raise complexity, are looked down upon as fables (Liebowitz and Margolis 1990). A hidden assumption appears that if a market is locked into an inferior technology, the costs of improving this outcome through government intervention will exceed its benefits.
In contrast to Neo-Brandeisians and Neo-Chicagoans, Neo-Schumpeterians draw richer methodological implications from economic complexification. 16 Neo-Schumpeterians also consider markets as the main focus of interest. But markets are envisioned as a "selective device" amongst different firms (Dosi 1982b). Firms struggle to survive, and the reason why they win or fail matters. From this Darwinian predicate, Neo-Schumpeterians look at competition through a whole array of perspectives. A wide variety of analytical paradigms exist in the literature, like dynamic capabilities (Teece et al. 1997), organizational routines, and the resource-based theory (Nelson and Winter 1982). In their inquiries, Neo-Schumpeterians place heavy emphasis on technology and dynamism (Arthur 1989;Dosi 1982aDosi , 1988. But their work predates the knowledge-economy and has not yet been updated to account for new empirics (Pyka and Nelson 2018; Dosi 1984). Moreover, Neo-Schumpeterians overlook the role of the public sector in the economy in general and in relation to technological capability in particular. A related implication is that Neo-Schumpeterian hardly ever consider the role of antitrust policies in their discussion of competitive outcomes. 17 In the 1990s, Ken Arrow called upon his profession to develop a new approach to the theory of oligopoly in response to the rise of the knowledge economy (Arrow 1996). What have antitrust experts done? In hindsight, all three schools have circled back on ideology. 18 This background is what motivates taking a different direction in response to the advent of the knowledge-economy, and avoiding knee-jerk denialism of complexity. If established antitrust ideologies have limited answers to propose, a promising place to start a methodological reexamination might lie in complexity science, as Ken Arrow had conjectured. 19 13 Wright (2012) (underlining the lack of differences between the Neo-Chicago School and the original Chicago School). 14 Horton (2012) (arguing that Chicago School and its modern ramifications "completely ignores the risks of lost variation, diversity, and complexity"). 15 Araujo and Harrison (2022) ("path dependence is associated with two types of event sequences: selfreinforcing and reactive sequences (…) self-reinforcing sequences are dominated by structural mechanisms, often remote in terms of their spatial and temporal origins, which keep events moving along a particular track. Reactive sequences are characterized by consequential, path shaping actions that often rearticulate existing structures and carve new trajectories"). 16 For an overview of the Neo-Schumpeterian movement, see Freeman (1994), Hanusch and Pyka (2006

Perspectives from complexity science
Complexity science is widely used across disciplines (3.1) but not in antitrust. We fill this gap showing that it allows for a better understanding of what competition is (3.2), and we explore the resulting antitrust framework (3.3).

Definition, history, and applications
Complexity science studies how "micro-level interactions lead to the emergence of macro-level patterns of behavior" and how these patterns influence back micro-level interactions (Whitt and Schultze 2009b). Another conventional description of complexity science stresses its focus on systems and how they adaptively change through the backpropagation of the context they create. 20 In the nineteenth century, Darwin pioneered works on complexity by studying the relationship between species, varieties, and their environment. 21 Though not phrased in such terms, Darwin laid down the foundations of what would become systems thinking, multilevel analysis, and evolutionary theory. In the following century, complexity science irrigated various fields, including biology (Gordon 2011), political economy (Axelrod 1986), physics (Holovatch et al. 2017, game theory (Lindgren 1991), archeology (Kohler 2011), finance (Marimon et al. 1990), sociology (Castellani and Hafferty 2009) In so far as economics is concerned, complexity science has also gained momentum. 22 Since the 1980s, an increasing number of studies have considered the economic system as a living organism instead of a machine. The economy is looked at as a set of systems made of components that combine and recombine. A complexity perspective considers that economic systems and their elements grow, shrink, and change. The ambition is to provide insights into the determinants of evolutionary processes in the economy. In complexity economics, the focus of analysis is on: 20 For an introduction, see Holland (2014a), Thurner et al. (2018), Waldrop (1992, Mitchell (2009a). 21 Dopfer and Nelson (2018) (underlining that "the proposition that cultural, social, political, and economic structures and modes of operation should be understood as evolving predates Darwin," see "Mandeville (1714) regarding the evolution of ship design. Hume's description, in 1762, of how the British social structure and culture of his day came to be clearly is evolutionary in spirit, as is Smith's (1776) analysis of what is going on in the economy"). 22 Arthur 2021; Dosi and Nelson (1994) ("Nelson and Winter (1982) has been followed by several other works also exploring evolutionary theory in economics (among others, Dosi et al. (1988), Saviotti and Metcalfe (1991), Anderson, Arrow and Pines (1989), Day and Eliasson (1986), Winter (1984) and(1987), Witt (1992), DeBresson (1988), Langlois and Everett (1992), Metcalfe (1992), Stiglitz (1992)"); Kirman (1989); Frenken (2006). Please note, however, that economists have been questioning economic assumptions for a long time, see Veblen (1898), Sargent (1993), Arthur (2014a).
1. Organizational characteristics of the firm, e.g., resources, capabilities, management, ownership, etc. 2. Business strategy, e.g., products and services sold; transactional relations with suppliers, customers, and consumers; learning from experience (Nelson 1996;Maital et al. 1994), routines (Whitt and Schultze 2009c), etc. 3. Competitive environment, e.g., industrial, institutional, and technological forces. 4. Interaction between the above-mentioned variables. 23 Mainstream economics has a troubled relationship with complexity science. The idea of incommensurability of complex phenomena has been a hard pill to swallow in a field that reifies measurement and quantitative analysis. Friedrich Hayek explained the problem 24 : "Unlike the position that exists in the physical sciences, in economics and other disciplines that deal with essentially complex phenomena, the aspects of the events to be accounted for about which we can get quantitative data are necessarily limited and may not include the important ones. While in the physical sciences it is generally assumed, probably with good reason, that any important factor which determines the observed events will itself be directly observable and measurable, in the study of such complex phenomena as the market, which depend on the actions of many individuals, all the circumstances which will determine the outcome of a process, for reasons which I shall explain later, will hardly ever be fully known or measurable" (von Hayek 1974).
This predicament has long ostracized complexity science in subfields like Austrian economics, evolutionary economics, or institutional economics. But a wind of change can be felt. Progress in techniques -essentially computational -like agent-based and patternoriented modeling are allowing improved observation, estimation, and prediction (Gallagher et al. 2021). Fruitful applications of complexity theory arise in fields such as the economics of technological change (Constant 1980;Basalla 1988;Mokyr 1990Mokyr , 2009Arthur 2009 Of course, computational techniques capture at best a fraction of economic complexity. Even in advanced computational models, many aspects of economic systems are ignored. But computational techniques walk in the right direction. They highlight the necessity (and difficulty) of considering more dimensions of economic systems. They also stress the relevance of change, dynamism, and processes. In some important fields, like financial economics, experts are today opening their eyes to the relevance of 1 3 Complexity-minded antitrust complex, evolutionary, and multi-level dynamics (Soros 2013a;Lo 2017;Bookstaber 2017). The same new perspectives can inform a reexamination of antitrust methods.

Complexity in antitrust: channels of relevance
Complexity science was never deployed in an antitrust context. 25 A claim that an introduction of complexity science in antitrust can improve the state of affairs is necessarily fragile. But the same holds true for a claim that complexity science holds limited potential to ameliorate antitrust short of empirical testing in real-life cases.
Two reasons allow a belief that complexity science holds relevant insights for antitrust. First, complexity science provides the multilevel lens on competition that has been long called for by antitrust scholars and practitioners. Second, complexity science supplies an understanding of competition as uncertainty richer than contemporary associations with rivalry.

Multilevel analysis
Complexity science develops abstract frameworks that consider interdependent multilevel systems. How can this help antitrust? An understanding of competition as a multilevel system provides a broad lens on competitive and anticompetitive forces discarded in a single-level neoclassical antitrust evaluation. What are the levels at which one can attempt to observe competition dynamics with a complexity mindset?
At this stage of elaboration, one can draw a line between the macro, meso, and micro-levels of a competitive system. 26 These levels are not selected arbitrarily. They correspond to established understandings of competition developed by scholars outside of neoclassical economics.
The macro-level of a competitive system is the industry. Within an industry, firms of different sizes, positions, and countries engage in "related," not just rival activity. The focus of firms within an industry is on short-term growth and long-term survival (Farjoun 2007). Firms within an industry seek competitive parity. Firms compete in the sense that they try to maintain financial, managerial, and technological capabilities comparable to their peers. 27 An example is the ICT and software industry.
The meso-level of a competitive system is the market. 28 Firms within a market supply (or purchase) substitutable products or services. The focus of firms within a market is on profit maximization. Firms within a market compete for share. An 25 There are a few articles discussing the usefulness of complexity science in antitrust analysis, but they are not deploying a new and related antitrust method, see Gundlach (2006), Horton (2006), Farrell (2006, Gundlach and Foer (2006). 26 For an illustration, see here. 27 A complexity lens would discuss this level in terms of the "environment" of the competitive system. Other possible framings are innovation competition, broad spectrum competition, Schumpeterian competition, long-term competition. 28 Or buy input, with exception of labor and capital. example is the market for online advertising services where Facebook and Google compete to attract advertisers. 29 The micro-level of a competitive system is the firm. Inside a firm, agents compete and/or cooperate to maximize individual payoffs. Corporate governance aligns agents' incentives with the firm's goals. A firm will select a mix of cooperation and competition called "co-opetition" between individuals, units, and divisions. And it will enforce co-opetition by exercise of hierarchical control over employees, managers, and contractors. A striking example of co-opetition is Meta, where What-sApp and Messenger compete and cooperate in developing messaging services. 30 An understanding of competition as a multilevel system is already progress. Neoclassical antitrust nominally acknowledges both rivalry beyond markets and organizational arrangements within firms. For analytical convenience, however, neoclassical antitrust treats practically both forces as irrelevant variables in an evaluation of competition.
But an even more relevant insight can be gained by considering interconnections between the various levels. What does complexity economics teach? That competitive selection at the meso-level might be a dependent variable of competitive changes at the macro and micro competitive levels, not just market rivalry. The history of Netflix in the 2000s provides a good anecdotal illustration of the importance of multilevel analysis. 31 Let us look at the sequence of events: • In 2004, Netflix's main competitors are Blockbuster and Walmart in DVD rental; 32 • In 2005, Netflix realizes that Internet delivery of content to the home will "surpass DVD" by virtue of increasingly competitive high-speed Internet and cloud services offerings; 33 • In 2007, Netflix introduces a streaming service that competes with legacy DVD operations, and incumbent ISPs with eyes on content; • In 2008, Netflix starts using Amazon AWS for cloud services, pivoting away from its own logistical operations; 34 • In 2009, Netflix develops an internal prize program to improve algorithmic recommendation accuracy in a context of increased competition with cable content providers; 35 29 A complexity lens would discuss this level in terms of competition between "species." Michael Porter talked of "extended rivalry," see Porter (1980). 30 A complexity lens would discuss this level in terms of the "organism" of the competitive system. Another possible framing is co-opetition, see Brandenburger and Nalebuff (1996). 31 One could also look at Spotify, see Thompson (2022). 32 In 2006, Netflix was still involved in a price war with Blockbuster. 33 As well as Amazon SimpleDB, S3, and Cassandra for file storage, see Berman et al. (2012). 34 Cockroft and Orzell (2011) ("Netflix manages to build its Internet video delivery service with little infrastructure of its own"). 35 Netflix offered a $1 million prize to anyone who could improve the accuracy of its algorithm recommendation by 10% and awarded it to BellKor Pragmatic Chaos in 2009, see Van Buskirk (2009). The program helped Netflix better identify an "addressable audience;" Koren (2009). The prize helped Netflix improve the baseline predictors and Restricted Boltzmann Machines, but also to address temporal dynamics.

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Complexity-minded antitrust • In 2011, Netflix becomes the single largest source of Internet traffic in the US despite the lack of significant in-house infrastructures (Adhikari et al. 2012;Schonfeld 2011); • In 2013, Netflix enters content production. Award-winning TV shows and movies (like House of Cards) now compete with major film studios; 36 • In 2016, Netflix introduces a download-and-go feature that allows users to watch content offline and changes the way content is encoded to enable users to stream content on a smartphone easily. 37 • Today, Netflix faces direct competition from Amazon Prime, Apple TV, HBO Max, Disney + , and Hulu. 38 And it recently entered the gaming industry. 39 In stylized terms, growth of Netflix's industry peers at the macro-level prompted a reconfiguration of co-opetition arrangements at the micro-level, and led to the selection of new rivals at the meso-level.
Admittedly, it would be bad scholarship to derive a general rule from a simple anecdote. But complexity-minded scholars have made many similar observations in other industries. Together, their works suggest that firms and markets respond to broader competitive forces than just product rivalry at the meso-level. 40 This pattern is relevant to competition analysis.

Uncertainty
In interconnected multilevel systems, firms face a challenge: making sense of complexity. Depending on the concrete properties of the competitive system, macro, meso, and micro interconnections give rise to unexpected, nonlinear (i.e., non-additive), and multidimensional changes. 41 For industries, businesses, and managers, the challenge associated with predicting the future in a complex economy leads to uncertainty. 42  Arthur, Eric D. Beinhocker, Allison Stanger (SFI Press, 2020) (describes the economy as "a constantly developing set of technological innovations, institutions, and arrangements that draw forth further innovations, institutions, and arrangements" that manifests itself at the meso level); Kirman (2010) ("the behaviour of the aggregate can be assimilated to that of an individual"). 41 Knight (1921) (making a distinction between risk and uncertainty. Risk occurs when the probabilities of different future states are known. On the contrary, uncertainty occurs when the probabilities of future states are not known). 42 As Hayek put it, with social sciences, "it's the subject that's much more complicated [than in physical sciences]" because it changes behaviors depending on others, see Friedrich von Hayek & Leo Rosten Part III, youtuBe (1978); also, Soros (2013b) (explaining that in natural sciences, observation does not impact phenomenon. In social sciences, it does because the subject "thinks," He concludes that "[t]he resulting uncertainty hinders the social sciences in producing laws similar to Newton's physics").

Complexity < -> Uncertainty < -> Competition
Firms respond to uncertainty in distinct ways. Some firms hustle under uncertainty. 43 Firms in this category 'cognize.' They do not make perfectly informed decisions. Firms that hustle under uncertainty diversify, explore, and innovate. Other firms hustle against uncertainty. To paraphrase George Stigler, they jump out of the uncertainty "frying pan" (Stigler 1975). Firms that hustle against uncertainty collude, lock in users and/or trading partners, and seek to maintain the status quo. 44 The competitive pressure bearing on complex systems highlights another objective for antitrust. In addition to rivalry, antitrust might maintain or promote uncertainty. Granted, rivalry is a powerful adjuvant of uncertainty. And antitrust laws, to a certain extent, already embody an orientation towards uncertainty at the meso-level. Cartel laws raise the cost of uncertainty-reducing communications between competitors. 45 Monopolization laws increase uncertainty by maintaining opportunities for contestability to new and potential entrants. Merger control systems prevent markets from falling under unified control and allow firms to combine and recombine, thereby promoting uncertainty.
With uncertainty as a function of antitrust, new policy targets emerge. Uncertainty can be promoted at the macro and micro-level through antitrust intervention at the meso-level. For example, antitrust might impose access duties, line of business restrictions, and M&A bans on firms exposed to insufficient levels of uncertainty. Such policies foster competitive responses when they encourage market power firms to direct their efforts towards uncertain related products and services at the macrolevel. 46 Another example of antitrust intervention might consist in raising the share of internal activities competing with core products and services at the micro-level. For example, antitrust might impose strong Chinese walls on a firm with competing divisions to safeguard healthy internal rivalry. Antitrust might also promote innovation. A last possibility for antitrust is to orient towards the promotion of innovation. 47 As Joseph Schumpeter has emphatically advocated, innovation is the strongest force of uncertainty in the long term. What optimal antitrust intervention towards innovation concretely entails, however, is a matter of considerate disagreement amongst economists. 43 Clark (1985) ("uncertainty is more than a precondition for evolution, it is also a determinant of its pattern (…) The pattern of innovation (…), the kinds of changes introduced, the timing of particular changes and so forth, will depend in part on the pattern of uncertainty, and the way in which new understanding is developed"). 44 As Joliet noted, "cartels tend to preserve the status quo and keep less efficient business units in existence, thereby enabling the more efficient firms to make comfortable profits," Joliet (1970). 45 This theory was at the heart of the UK agricultural tractors case, CJEU, C-7/95, John Deere Limited v Commission, ECLI:EU:C:1998:256, para 88-90. 46 To illustrate, Facebook's Metaverse might be a reaction to rising regulation of its market power position in social media. 47 Note also that innovation can lead to mixed welfare outcomes. For example, innovation in advertising markets produces ambiguous effects on consumer welfare. In some cases, innovation reduces welfare outcomes. Financial innovation is a case in point.

Complexity-minded antitrust
An uncertainty mindset also helps when multi-level systems decomplexify. In a context of increasing returns, complex systems can lead to lock-in. Classic lock-in examples are QWERTY's domination over simplified Dvorak Simplified keyboards (David 1985); VHS video cassette recorder standard over Betamax (Arthur 1990), and light-water reactors over gas-cooled reactors (Arthur 1989). More recently, the prevalence of cable-chargers over wireless chargers, the constant domination of USB-A (1996) over faster and more versatile solutions such as USB-C (2014), and the use of Bluetooth (1998) over Low-Power WiFi solutions show how increasing returns arising from coordination externalities can result in lock-ins. Until now, neoclassical antitrust has not addressed the problem of lockin. Lock-in can be beneficial. This is the case when the selected system is the superior one. However, a test allowing to sort inferior from superior systems does not exist. Antitrust has been understandably reluctant to address head on the issue of lock-in. Under an uncertainty mindset, not an efficiency one, antitrust qualms fade away. With uncertainty as the rationale for intervention, addressing lock-in becomes legitimate to the extent that it suppresses complexity.

Complexity in antitrust: a framework
Compared to a neoclassical approach, what does a complexity-minded antitrust entail? A full treatment of the issue is beyond the ambition of this paper. Some building blocks can be identified.

Positive feedback loops
An understanding of feedback loops is critical to developing complexity-minded antitrust. Feedback loops (also called "recursive loops" or just "loops") are intuitively easy to grasp, but their effects are hard to predict (Newman et al. 2006). A feedback loop occurs when a variation of output ('O1') ignites another change in output ('O2'), and so on, and so forth (On). 48 When competition takes place with positive feedback loops, an attraction force pulls the firms away from their initial equilibrium, leading to extreme outcomes.
Consider the example of PayPal. Several strategic decisions affecting PayPal's customer base (O1) contributed to further increases in its customer base (O2). In 1999, PayPal triggered a self-reinforcing network effect by offering new users $10 for joining (Thiel 2001). With exponential growth of its customer base, Pay-Pal could sell to eBay and become its subsidiary in 2002. The acquisition helped democratize online payment and thus contributed to changing the online sales environment. In 2014, PayPal went back to being a separate publicly traded company and soon acquired Xoom Corporation (2015), iZettle (2018), Honey (2019), thus further expanding online payment solutions and use.
With competition under negative feedback loops, an attraction force straps firms to the initial equilibrium. No leader emerges. Negative feedback loops in marketplace competition lead to balanced equilibrium, and oligopoly is the norm.
Neoclassical antitrust tends to favor negative feedback loops associated with rivalrous structures. But positive feedback loops associated with monopoly can also be good for competition. The uncertainty arising from future feedback loops produces competitive pressure, non-inertial behavior, and business dynamism. Positive feedback loops are associated with technological change, the development of new products and services, and adoption of innovative business models. That said, positive feedback loops also reduce uncertainty when they create winner-takes-all advantages, market tipping, and path dependence. 49 Positive feedback loops are associated with the problems of lock-in into inferior technologies, products, and services.
An understanding of positive feedback loops highlights a more sensible path for antitrust. Rivalry is limited when a positive feedback loop locks a system into a rigid developmental path (regardless of whether the option is superior or inferior). Neoclassical antitrust does not understand this. A concrete example underscores the point. Today, output-increasing monopolies are dealt with in a binary manner. On the one hand, some agencies and courts equate large size with monopoly. This is the approach followed by the EU in digital markets. Gatekeeping firms are deemed structurally anticompetitive, regardless of output growth at firm and industry levels. On the one hand, US antitrust courts associate output growth with competition. In the Amex case of 2017, reported increases in payment card transactions at firm and industry levels allowed the Supreme Court to dismiss direct evidence of exercize of monopoly power by the defendant.
In both the Google and Amex cases, a more relevant question might have consisted in wondering whether a lock-in situation existed so that growth benefited defendant firms disproportionately relative to other industry participants. The reason why this question is relevant is not that lock-in reduces rivalry. The point is that lock-in might, or not, diminish change, limit uncertainty, and blunt competitive pressure at the various levels of a competitive system. The question is ultimately an empirical one.

Random events
What forces set positive feedback loops in motion? Complexity theory teaches that in increasing returns contexts, "random" events trigger positive feedback loops. Random events are not to be confused with events that happen by chance. Random events are often known facts. But they are facts whose possible outcomes are all equally likely. 50 1 3

Complexity-minded antitrust
Their effects can be big or small, substantial or insignificant. 51 Complexity literature refers to the idea of contingency in outcomes by talking of "historical" events. 52 As Brian Arthur explains, random events by nature render costly to predict competitive or monopoly consequences with any degree of certainty (Arthur 1989).
Diverse types of random events contribute to positive feedback loops. Some are technological. For example, Apple's anticipation of the success of devices smaller than computers triggered massive expansions of output in industries like advertising, content creation, and finance. Others are economic. For example, business model innovations like Google's introduction of search advertising inspired the development of new modes of personalized advertising, marketing, and pricing across digital markets. Last, political and regulatory events also cause positive feedback loops. For example, Section 230 of the US 1996 Communications Decency Act has spurred the growth of online intermediaries by expressly shielding them from liability for publication, moderation, or censorship of content (including that posted by third parties).
Given this, neoclassical antitrust understandably responds to random events by discounting their role as drivers of positive feedback loops. 53 Micro-level developments like firms' endogenous research and development efforts, monetization experimentation, or organizational and managerial changes are never really considered relevant) in competition law analysis. The same is true of macro-level developments like exogenous government subsidies (i.e., concessionary finance to state-owned firms) or the introduction of general-purpose technologies (for example, the adoption of new communications protocols) in upstream industries. Neoclassical antitrust rationalizes outcomes as if they were known from the start.
But neoclassical antitrust does not neglect all random events. Antitrust intervention is regarded as a random event with power to trigger positive feedback loops. Most celebrated accounts of the US v Microsoft case implicitly develop a positive feedback loop argument to rationalize government intervention as the engine of innovation in the 2000s. 54 51 In other words, not all random events generate positive feedback loops. Only the random events that impact the core routing of a system generate positive feedback loops, see Holland (2014c). 52 Arthur (1989) ("historical 'small events' are not averaged away and 'forgotten' by the dynamics -they may decide the outcome.") 53 The low predictability of random events' outcomes complicates the evaluation of business conduct in the future, but also in the past. History is contingent. Assigning causality to established facts is hard. Besides, the low observability of random events undermines the restorative function of antitrust remedies. When antitrust agencies and courts seek to introduce a negative feedback loop to restore the previous procompetitive state, they do this ignoring possible events at the micro and macro-levels that prevent a competitive reset. 54 European Commission, 24 March 2004, Case AT.37792, Microsoft, para. 459 ("In conclusion, the 'positive feedback loop' protects Microsoft's high market shares in the client PC operating system market from effective competition from a potential new entrant. The term 'applications barrier to entry' has been coined to describe this phenomenon."). So, how should antitrust agencies and courts approach random events? Attempting to predict their impact on feedback loops appears illusory. Standard algebraic computation does not work. In the words of Hayek, "[a] theory of essentially complex phenomena must refer to a large number of particular facts; and to derive a prediction from it, or to test it, we have to ascertain all these particular facts" (von Hayek 1974). By contrast, a more humble research direction consists of enriching our empirical understanding of feedback loops to identify patterns (Holland 2014b). Will they be of the pro-competitive kind, raising uncertainty? Or will they be of the anticompetitive kind, locking in users into rigid technological trajectories? And after how much time can an antitrust institution consider that change is long overdue? By studying cases, patterns will emerge, which will inform antitrust rulemaking and adjudication. 55

Increasing returns
Increasing returns are the economic property-driving force-that allows random events to amplify and turn into positive feedback loops. In an economy with increasing returns, an increase in supply causes a reduction in the cost of production; and an increase in demand causes a rise in the benefits from consumption. This contrasts with a decreasing returns economy, where supply and/or demand growth correlates with diseconomies and/or disutility.
The idea of increasing returns was introduced in the economics literature in 1926, when Pietro Sraffa mentioned the existence of a functional connection between cost and quantity, and said that in a large dimension firm, a greater division of labor is possible, leading to increases in output. 56 For long, the concept of increasing returns was confined to policy applications in industries where natural monopolies were the efficient market structure, subject to regulation of pricing and access conditions like in transport and communication, or in relation to economies of scale in manufacturing and equipment (Shapiro and Varian 1998). The works of Brian Arthur in the 1980s underlined three novel dimensions of increasing returns. First, economists started to pay more attention to increasing returns on the demand side. Second, increasing returns on the demand side appeared more widespread in an economy working on the basis of networks, like communications systems. Third, and with the caveat of non-pecuniary externalities, the coexistence of increasing returns on the supply and demand side suggested the possibility of efficient, non-output reducing monopoly. 57 55 Ibid (exploring the concept of "pattern predictions" as "predictions of some of the general attributes of the structures that will form themselves, but not containing specific statements about the individual elements of which the structures will be made up"). 56 Sraffa had also observed that individual firms benefited from external economies as a result of the growth of the industry in the aggregate. 57 When there are increasing returns on both sides, a firm might enjoy considerate pricing power due to the progressive marginalization of other competing firms in the market. But output grows, and there is no deadweight loss.
Increasing or decreasing returns determine industry structure and growth. In industries with increasing returns, firms have efficiency incentives for efficient concentration. 58 And consumers experience rising benefits from marginal increases in consumption. By contrast, in industries with decreasing returns, firms achieve limited efficiency gains from concentration. Similarly, consumers experience economic disutility from marginal consumption.
Feedback loops, i.e., dramatic increases in output to the benefit of one firm, cannot exist absent increasing returns on the supply and/or demand side. A good understanding of whether an industry empirically displays increasing or decreasing returns to scale is therefore key to antitrust law and policymaking. Antitrust laws that strike unilateral conduct and mergers to monopoly make sense in the decreasing returns case, but do not in the increasing returns one. By contrast, antitrust laws that strike price fixing make sense in the increasing returns case because cartels prevent the competitive process of selecting monopoly winners and stifle demand growth. But they do not in the decreasing returns case, because cartels maintain supply side plurality and limit consumption.
Unfortunately, antitrust laws formulate grievances toward monopolies and cartels independent of the nature of the returns involved. The inconsistency might be because policymakers want to protect consumers above all. This entails protecting consumer choice, by a strict prohibition of monopolization. And protecting consumer wealth by a strict prohibition of price-fixing cartels.
A modern antitrust law for an increasing returns economy would look very different. It would not prohibit abuse of dominance but behaviors that slow the competitive process of monopoly formation. In addition, the antitrust system would embed a clause allowing intervention when a positive feedback loop locks in users to an inefficient monopoly system, compared to possible alternatives. 59 Antitrust would not wait for superior technologies, products, and services to reveal themselves.
The test would consist in showing that the monopoly in place has exhausted all increasing returns to scale (and falls in a situation of decreasing returns to scale), while a competitive system demonstrates an ability to further deliver increasing returns to scale. And it might have provisions allowing for price control in the extreme case where a monopolist with increasing returns exploits user lock-in by re-contracting with them on unfair terms without triggering a self-collapsing negative feedback loop or an outside positive feedback loop towards competitors. 58 Arthur (2014b) ("complex technologies often display increasing returns to adoption in that the more they are adopted, the more experience is gained with them, and the more they are improved"). Rosenberg calls this "Learning by Using," see Rosenberg (1983). See also Atkinson and Stiglitz (1969). 59 Arthur (2014c) (Arthur talks about non-ergodicity (or path-dependence), i.e., "historical 'small events' are not averaged away and "forgotten" by the dynamics-they may decide the outcome). Several improvements to neoclassical antitrust emerge from the above discussion. 60 First, complexity highlights a distinct mental model for competition agencies and courts. Compared to the "physicist" mindset of neoclassical antitrust institutions, a complexity-minded antitrust institution will rethink its role in terms of what Brian Arthur has called a "park-ranger" spirit. The difference in mental models is that physicists seek to reach static and predictable outcomes (moving a monopoly towards competition), while park rangers seek to maintain dynamic and unpredictable processes (moving a monopoly towards competition or towards a new monopoly). 61

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Complexity-minded antitrust Second, complexity highlights an additional function for competition law: promoting uncertainty. 62 Compared to the rivalry improvement function of neoclassical antitrust, a complexity-minded antitrust considers that increases in uncertainty can, in some circumstances, deliver important incentives to competitive effort, noninertial behavior, and innovation. 63 True, rivalry often correlates with uncertainty. But rivalry is neither a necessary, nor a sufficient condition for uncertainty. 64 Firms that compete under uncertainty are motivated by threats not limited to rival products, including the reconfiguration of supply and demand conditions by imperfect substitutes, complements, or new combinations (Petit 2020). Now, the reason why antitrust is a legitimate instrument to raise uncertainty levels owes to its mode of enforcement. Antitrust intervention certainly raises (legal) uncertainty by undermining free contract and property. But it does so in a limited, narrow, and facts-specific way so that incentives effects from rising levels of legal uncertainty are very limited.
Third, complexity highlights an additional method for competition law: that of "unfreezing" markets. Compared to the deconcentration method of neoclassical antitrust, a complexity-minded antitrust enables competition by complexification. Adding noise at one or more of the various levels of a frozen competitive system "destabiliz[es] rational speculation," (de Long et al. 1990) leading to new opportunities for natural selection (Whitt and Schultze 2009a).
Fourth, complexity points to a specific remedy to administer an uncertaintyincreasing approach. Compared to the negative feedback loop approach of neoclassical antitrust, a complexity-minded antitrust considers that positive feedback loops that grow output help prevent monopolies from living what Sir John Hicks called the "quiet life." 65 By adding a positive feedback loop, antitrust can 'shake' markets. In the monopoly case, adding a positive feedback loop can be done by imposing a duty to deal on the dominant firm, so its inputs ('I') are shared with rivals in support of a new composition of output (new outputs, 'O2'). In the lock-in case, one can impose a line of business restriction on the output of the winning system (current outputs, 'O1'), so its input ('I') cannot be leveraged in support of diversification, leaving external options an opportunity to grow ('O2'). In both cases, the idea is to allow feedback loops to fuel other, competing systems. Note that a remedy might create a negative feedback loop, depending on the circumstances. In the duty to deal case, if beneficiaries align their innovation trajectories on the winning system, the market will move back to the competitive equilibrium. 66 In the line of business restriction case, market partitioning, and monopoly power for the winning system and new entrants, is also possible outcome. 62 We say "additional" because, once more, complexity-minded antitrust builds on top of neoclassical antitrust as opposed to replacing neoclassical antitrust entirely. 63 We give competition law a broad meaning that includes the enforcement activities of competition authorities, but also ex-ante tools such as the Digital Markets Act. 64 When several firms compete but one is clearly superior to others, rivalry does not create uncertainty. 65 In both cases of lost rivalry due to either output reducing monopoly or output increasing lock-in, market uncertainty is reduced. 66 Access regulation in Western European telecom markets led new entrants to develop copper networks, modeling incumbents' technological choices. This retarded innovation into fiber.
Fifth, complexity highlights actionable targets for competition law. A firm has incentives to exploit market power where it is less costly. This is why monopolists' market power strategies target predominantly core, adjacent, and related markets where uncertainty is limited. Anticompetitive leveraging is a case in point. Given the above, one way to promote competition is to inject cost-raising complexity in markets worth monopolizing and incentivize monopoly firms to compete in distant, long-term, and unrelated markets where uncertainty is higher. 67 Sixth, complexity highlights a new division of labor between competition law and regulation. Not all competition, change, or growth arising from positive feedback loops is welfare-enhancing. Unrestrained competitive innovation in the financial sector paved the way to the subprime crisis. Growth of digital advertisement coincides with unprecedented levels of privacy extraction, and free trade policies have raised the carbon footprint of the economy at possibly unsustainable levels.
These few examples suggest that while a complexity-minded antitrust should work towards positive feedback loops, a complexity-minded regulation should seek to address their negative externalities. The clear division of labor between antitrust institutions 'unfreezing' markets, and regulators 'cooling off' changes in markets will allocate decision-making to those with the proper expertise rather than tasking enforcers to arbitrate between different objectives and opening the door to capture. It will also clarify the role of regulators when tackling competition issues.
Complexity is a mindset, not a rigid code of conduct. Competition law systems can select diverse insights from complexity. Some will emphasize the short-term, others the long-term. Some will focus on the macro-level, others on the micro-level. Several complexity-minded antitrust policies are thus possible. But their common feature is to work with a clear objective: ensuring uncertainty. 68

Test and illustration
A complexity-minded test of legality is whether business conduct freezes or shakes the market. Some basic examples show how the test roughly works in concrete antitrust cases. position in social networks to extract excessive data from users across other business segments. A test of freezing versus shaking would have asked other questions: did Facebook's data combination practice shake the market, by leading to a positive feedback loop, a change of output, and raised levels of uncertainty; or did Facebook's data combination trigger a negative feedback loop by further consolidating its dominant position? Reports of (i) intense macro-level competition from innovative players like TikTok, and (ii) stagnation of Facebook's ability to capture new users at the meso-level cast doubt on the freezing hypothesis. 70 • Several technology firms prohibit blockchain advertisements. The argument behind the ban is based on disputable security concerns. 71 Beyond this, however, the relevant test of legality for antitrust is whether technology firms' refusal to deal in blockchain advertisement is likely to freeze the market around existing technologies. This, in turn, depends on considerations like the boycotting firms' position over advertisement at the meso-level and whether other industries or governments at the macro-level sponsor blockchain technology. A random event likely to shake the market is Facebook's reversal of its ban on crypto-advertising right after the company presented its metaverse and crypto-compatible products. 72 • On several occasions, Apple has denied compatibility to songs bought outside of iTunes by modifying encryption methods. Music platforms like RealNetworks and Harmony have occasionally tried to reintroduce interoperability. But Apple denied interoperability every time (Schrepel 2018), thus froze the market around its technology. Interoperability denials triggered a negative feedback loop. A complexity-minded antitrust remedy would have prohibited Apple from introducing innovation to remove interoperability developed by external parties on the basis of publicly available information. • In the browser wars, Microsoft focused on acquiring and eliminating complementors of its ecosystem, hampering evolution and innovation of multiple software capabilities. Microsoft took Netscape's threat of disruption so literally that it preferred to "freeze" the technology around its Operating System, rather than trying to surf the "tidal wave" of new Internet applications (Teece 2013

Complexity leeway in antitrust doctrine
Is a complexity-minded antitrust realistic? To think about this, one can ask if the antitrust case law provides leeway to interpret statutory instruments in light of complexity science. To be clear, the issue is not whether the case law implements complexity science. If this is the question, the answer is a straight no. The reason is obvious. Courts have not possibly expounded a theory that, even to this day, has remained incomplete. Besides, lawyers are well placed to know that judicial language is treacherous. If a court writes "complexity" in an opinion, it does not mean that complexity science played a role in the case. 74 The best way to assess complexity leeway in the case law is this. Can we see analogies to complexity thinking in judicial reasoning? The answer is a sure yes.
To start, we can witness sympathy towards an antitrust norm of uncertainty in the case law. In Socony Vacuum, the Court held that price fixing to be per se unlawful, even when prices remain at reasonable levels. 75 To close the door to any discussion of price levels in price fixing cases, the Court said this. The problem with price fixing is that prices are stabilized. This is not in line with the desirable state of affairs in the free market case, where "prices (have) no constancy, due to the dynamic quality of business facts underlying price structures." 76 Similarly, the case law shows openness to multilevel analysis. The Lorain Journal opinion of 1951 supplies an illustration. 77 Concerned about the growing competition of radio channels over advertising budgets, a local monopoly newspaper had refused to deal with advertisers who were planning to place ads on a neighboring broadcasting network. To affirm liability, a multilevel analysis, even in crude form, was necessary. The Court, in our view correctly, understood that newspapers and radio broadcasters competed at the macro level for advertisement money, even if both businesses were not in the same product market at the meso level.
A concern towards protecting positive feedback loops also appears in the cases. The story behind the Otter Tail opinion of 1973 stands out. 78 Otter Tail, an incumbent local power supplier with expiring retail franchises had used various 74 For comparison, antitrust journals are replete with articles which discuss whether nominal references to "innovation" in cases really meant innovation. 75 US v Socony-Vacuum Oil Co 310 US 150 (1940). Most commentators underline that Socony remains good law, yet they stress that the Supreme Court has practically brought derogations by permitting defendants to raise rule of reason type arguments. See, e.g., Gellhorn and Kovacic (1994). 76 A norm of uncertainty also appears in the Sugar Institute case. Here, defendants had coordinated over advance price announcements. In affirming liability under the per se prohibition rule, the Court held that: "The unreasonable restraints which defendants imposed lay not in advance announcements, but in the steps taken to secure adherence, without deviation, to prices and terms thus announced. It was that concerted undertaking which cut off opportunities for variation in the course of competition however fair and appropriate they might be," Sugar Institute v. United States, 297 U.S. 553 (1936). 77 Lorain Journal Co. v United States 342 U.S. 143 (1951). 78 Otter Tail Power Co. v. United States, 410 U.S. 366 (1973).

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Complexity-minded antitrust tactics (including litigation) to dissuade towns from establishing their own power systems. A clear judicial intent of protection of outside feedback loops undergirds the opinion. Out of 465 towns potentially open to competition, the antitrust case against Otter Tail focused on just 4 towns. This finding could have sufficed to dismiss the case on grounds of insubstantial effects. And yet, as the dissent of Justice Stewart stresses, the majority considered "that Otter Tail's actions … resulted in … maintenance of monopoly control by hindering the emergence of municipal power companies." The defendant conceded the feedback loop point, arguing that "without the weapons which it used, more and more municipalities will turn to public power and Otter Tail will go downhill." Brown Shoe exemplifies an antitrust doctrine sensitive to random events. Considered by many experts the worst opinion in antitrust history, Brown Shoe mutters complexity reasoning. 79 Here, the US Government was trying to block a merger between the nation's 3rd and 8th largest shoe suppliers. Large in size, the merged entity was small in share. The evidence suggested a post-merger share comprised between 5 to 10% of the market. The Court nonetheless approved the Government case. It held that "if a merger achieving 5% were now approved, we might be required to approve future merger's efforts by Brown's competitors seeking similar market share." 80 The Brown Shoe opinion deserves criticism to the extent that if each of the 20 firms in an industry with a 5% market share merged with another firm, a market of 10 firms with a 10% market share would likely remain competitive. In the words of Chief Justice Warren, the Court expressed an ideological preference for "retaining 'local control' over industry and the protection of small businesses." 81 But the Brown Shoe Court merits approval when it considers that the signal sent by a judicial opinion is a small event with wide implications. The Court correctly understood that the joint action of stare decisis and of equal treatment principles could unleash a concentrative feedback loop. If defendants in merger cases argue that a precedent allows mergers that incrementally raise the parties' share of output by no more than 5%, a risk of "cumulative mergers" with accretion of monopoly power is in the cards. Limiting principles like a market share cap can prevent a judicial opinion from locking in a market into a path-dependent trajectory of monopoly consolidation. The Court, erroneously in Brown Shoe, might have set the cap too low. 82 The focus on dynamics in other opinions of the same period also betrays a complexity-mindset. The Von's Grocery Court opposition to a merger leading to a 7.5% market share in the Los Angeles groceries market breathes 79 One of such individuals include Bork (1978). 80 Brown Shoe Co., Inc. v. United States, 370 U.S. 294 (1962). 81 Ibid. 82 A few years later, in US v Philadelphia National Bank, the Court will correct course by holding that a 30% market share allows a presumptive inference that a merger substantially lessens competition. United States v. Philadelphia Nat'l Bank, 374 U.S. 321 (1963) holding: "Without attempting to specify the smallest market share which would still be considered to threaten undue concentration, we are clear that 30% presents that threat.". complexity science when it warns against a "trend towards fewer and fewer" competitors and suggests a precautionary approach if concentration is "gaining momentum in a market." 83 The parallels with complexity science of Brown Shoe, Von's Grocery, and other contemporary opinions are, however, imperfect. Opinions like Pabst Brewing held that a merger ought to be prohibited on account of a "trend" towards concentration, regardless of its "causes." 84 And in International Salt, the Court found "immaterial that the tendency [towards concentration] is a creeping one rather than one that proceeds at full gallop." 85 These statements are not reconcilable with a complexity mindset. A complexity-minded antitrust court would have asked if a trend towards concentration locks markets towards a superior or inferior technology. The causes of feedback loops, economization or monopolization, remain relevant in a complexityminded antitrust analysis. Similarly, the antitrust concern with feedback loops is that they work on a fast clock. The idea that it is irrelevant whether the trend is slow or rapid does not sit well with complexity science.
To close, increasing returns are perhaps the element of complexity science best understood in case law. The economic theory of increasing returns predicts a possibility of irreversible lock-in of markets into inferior products, services, or technology. From there, the guideline for policymakers is crystal clear: two birds in the bush is better than one in the hand. In cases of increasing returns, policymakers must safeguard alternative products, services, and technology, even if this requires imposing a tax or granting a subsidy to maintain second best options. In FTC v Staples, the District Court (reluctantly) supplied an antitrust translation of the increasing returns guideline. The District Court observed that the post-merger entity would benefit from unmatchable economies of scale in advertising and distribution, preventing future entry of new office superstores. 86 In spite of a better product, the Court refused to let the merger go through to safeguard alternative options: "Despite the Court's sympathy toward the plight of the defendants in this case, the Court finds that the Commission has shown a "reasonable probability" that the proposed merger between Staples and Office Depot may substantially impair competition." FTC v Staples shows that complexity minded antitrust requires difficult calls from antitrust courts and agencies. The Court expressed its discomfort as follows: "In light of the undeniable benefits that Staples and Office Depot have brought to consumers, it is with regret that the Court reaches the decision that it must in this case. This decision will most likely kill the merger. The Court feels, to some extent, that the defendants are being punished for their own successes and for the benefits that they have brought to consumers. In effect, they have been hoisted with their own petards." Now, unlike the Brown Shoe Court, the FTC v Staples Court took great care to emphasize the specificity of its holding. The Court closed with a statement stressing that its opinion was strictly based on the facts in an "extremely complex matter," and not be construed as a recognition of a new general antitrust doctrine.
The above cases supply an inference that there is no judicial obstacle to an interpretation of statutory antitrust law aligned with complexity science. Many opinions suggest judicial proximity, perhaps even affinity, to fundamental aspects of complexity science like the norm of uncertainty. Besides, several of complexity science's elemental concepts like feedback loops and increasing returns live in the case law, even though not under the terminology employed in complexity literature.
One last point. A complexity-minded antitrust law is not a recipe for less-enforcement. Most, but not all, the above cases correlate with periods of intense antitrust activism. 87 But a complexity-minded antitrust law is not either a mandate for aggressive interventionism. FTC v Staples exemplifies how a court leaning towards the protection of incentives to invest, business acumen, and competition on the merits can nonetheless understands the need to avoid irreversible path-dependent outcomes in the particular case. A complexity-minded antitrust, in reality, correlates with hard policy choices for antitrust. In brief, a complexity minded antitrust corresponds to what European lawyers have in mind when they say "dura lex, sed lex." 88

Research agenda
The neoclassical paradigm of antitrust policy is bounded. The limit is not ideology, but methodology. Antitrust policy relies on reductionist assumptions, frameworks, and units of analysis on competition and innovation that harness just a fraction of economic complexity. Because antitrust policymaking operates under practical constraints, neoclassical economics has remained the best game in town absent better actionable concepts, methods, and tools. 87 For instance, in Brooke Group, the fact featured an incumbent trying to repress an outside feedback loop. See Brooke Group Ltd. v. Brown & Williamson Tobacco Corp., 509 U.S. 209 (1993) ("Liggett took an unusual step to revive its prospects: It developed a line of black and white generic cigarettes. When introduced in 1980, black and whites were offered to consumers at a list price roughly 30% lower than the list price of full-priced, branded cigarettes." "They were also promoted at the wholesale level by means of rebates that increased with the volume of cigarettes ordered. Black and white cigarettes thus represented a new marketing category. The category's principal competitive characteristic was low price. Liggett's black and whites were an immediate and considerable success, growing from a fraction of a percent of the market at their introduction to over 4% of the total cigarette market by early 1984"). The Court, however, vacated the case. 88 "The law is harsh, but it is the law.".
With this background, this article introduces complexity theory and outlines the first steps toward a more complexity-minded antitrust. Multilevel analysis, feedback loops, and uncertainty-increasing intervention constitute the starting points for an intellectual renovation of neoclassical antitrust method.
Much work remains needed. Three directions of a future research agenda can be outlined. First, a complexity-minded antitrust requires a good understanding of when and why markets develop. Antitrust scholars should conduct historical work on the emergence and growth of markets, firms, and technologies. A key focus of analysis should be placed on the nature of feedback loops, their properties, duration, intensity, context dependence, and impacts on welfare.
Second, a complexity-minded antitrust must inevitably be a provisional system of market control. Short of predictive power, application of complexity theory in an antitrust context is bound to happen on the spot. Antitrust scholars should thus develop tools and methods that allow antitrust intervention to be more adaptive, but not abusive; timely, but not discretionary. In this context, a key question is how to adjust antitrust intervention and regulation in light of real-time data documenting feedback loops (Schrepel 2021a).
Third, a complexity-minded antitrust demands affinity for methods that allow an understanding of uncertainty. Research on agent-based modeling is a fruitful area for progress. Computerized simulations of institutions and agents with individual characteristics will contribute to a better understanding of nonlinear behavior. They will document the potential impact of antitrust policies on complex ecosystems, and, perhaps more importantly, the limits of what we imagine we can design. 89 Now, we want to conclude with takeaways for legal research beyond antitrust. Complexity requires an innovative perspective on governance and regulatory processes. The suggestion that provisional enforcement might be a way forward for antitrust law echoes with legal scholarship that insists on the necessary development of soft law for the regulation of new technologies (Hagemann et al. 2018); the acceptance that technological innovation is better supported by rules providing less legal certainty 90 ; and advocacy towards experimental techniques like regulatory sandboxes or proactive adjudication that increase information levels. 91 But this is not all. Our work is, at heart, a call for regulatory humility. Genuine uncertainty creates "wicked problems" for law and policymakers. Often, a mix of substandard solutions will be the best that can be achieved in a complex environment (Marchant 2020). 89 In agent-based modeling, agents interact through prescribed rules, see Farmer and Foley (2009). Simulations often show that changing these rules the slightest impacts the entire ecosystem in hard-to-predict dimensions. 90 Van Utytsel (2021). Adaptive regulations provide less legal certainty than fixed regulations, but adaptivity avoids rigidity, see Pentland and Mahari (2022). 91 Sherkow (2022). Such mechanisms of proactive adjudication include progress in prediction markets such as illustrated by Augur, reputation systems, blockchain dispute resolution, crowdsourced or peer-topeer dispute resolution such as Aragon, and decentralized arbitration. These dispute resolution mechanisms facilitate legal outcomes and thus free resources for companies to make sense of business uncertainty. For more on the subject, see Guillaume and Riva (2022), and Abramowicz (2016).

Conclusion
Under neoclassical antitrust, uncertainty is often a pretext for discretion. When data are absent, ambiguous, or incomplete, decision-making is difficult. Neoclassical antitrust institutions must in turn make a binary choice between intervention and nonintervention. Such choices are often based on experience, ideology, and/or opportunism. In a policymaking discipline committed to empirics, the drivers of these choices are faulty because they are disconnected from current facts. This is the advantage of a complexity-minded antitrust. Computing difficulties do not paralyze application of the law. Complexity-minded antitrust is action-oriented in the face of uncertainty.

Declarations
Human and animal rights This research does not involve human or animal participants.

Conflict of interest The author does not have a conflict of interest.
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