• No results found

An Ordoliberal Theory of Algorithmic Exploitation

N/A
N/A
Protected

Academic year: 2023

Share "An Ordoliberal Theory of Algorithmic Exploitation"

Copied!
52
0
0

Bezig met laden.... (Bekijk nu de volledige tekst)

Hele tekst

(1)

An Ordoliberal Theory of Algorithmic Exploitation -

How High-Frequency Pricing Facilitates Tacit Collusion and Deprives Consumers of Their Sovereignty

Johann Henke

johann.henke@student.uva.nl Student Number: 13935607

European Competition Law & Regulation Supervisor: Dr. Giacomo Tagiuri

Date of Submission: 28.06.2022 Word Count: 12.968

(2)

Table of Contents

Abstract ... 4

1. Introduction ... 5

2. Setting the Scene – The Economic and Legal Background of Tacit Collusion ... 6

2.1. The Economics of Tacit Collusion – Oligopoly Theory ... 6

2.1.1. The Prisoner’s Dilemma and Oligopolistic Interdependence ... 6

2.1.2. The Abstract Conditions for Collusion ... 10

2.2. The Legal Approach to Tacit Collusion under EU Competition Law – A Legislative Gap? ... 12

2.2.1. Art.101 TFEU ... 12

2.2.2. Art.102 TFEU ... 15

2.2.3. EUMR ... 16

2.3. Conclusion ... 16

3. Algorithmic Tacit Collusion ... 17

3.1. Basics of Pricing Algorithms ... 17

3.2. Scenarios Leading to Algorithmic Tacit Collusion ... 18

3.2.1. Predictable Agent ... 18

3.2.2. Self-Learning Algorithms ... 18

3.2.3. Conclusion ... 19

3.3. Empirical Evidence ... 19

3.4. Differences from Traditional Tacit Collusion ... 20

4. An Ordoliberal Doctrine of Algorithmic Exploitation ... 22

4.1. Introduction – Ordoliberalism ... 23

4.2. The Ordoliberal Understanding of Competition & Consumer Sovereignty ... 24

4.3. Restriction of Consumer Sovereignty through High-Frequency Pricing ... 26

4.3.1. Decision Making Framework – Price as an Indicator ... 26

4.3.2. Asymmetric Information and Rationality ... 29

4.3.3. Conclusion ... 31

4.4. Operationalization - Algorithmic Exploitation under Art.102 TFEU ... 32

4.4.1. Establishment of Collective Dominance ... 32

4.4.2. Quantitative Assessment of “Unfair Prices” ... 32

4.4.3. Proposal for a Qualitative Assessment ... 34

4.5. Efficiency Defense & The Core of Competition ... 35

(3)

4.5.1. The Ordoliberal Form-Based Approach ... 35

4.5.2. Efficiencies in Current EU Law ... 38

4.5.3. Justifying Algorithmic Exploitation – The Transparency Fallacy ... 39

4.6. Remedies ... 41

5. Enforcement in Practice ... 42

6. Conclusion ... 44

7. Bibliography ... 46

(4)

Abstract

This thesis aims to contribute to the discussion around algorithmic tacit collusion by approaching the subject from an ordoliberal perspective. This builds on the rationale that the apparent legal gap of algorithmic tacit collusion turns out to be only an enforcement gap that, contrary to the prevailing opinion in scholarly literature, can be closed using the current competition law toolbox, more specifically Art.102 TFEU.

After assessing the potential impact of algorithms on the economic conditions of tacit collusion, the thesis will take a more philosophical view on the emerging topic of algorithmic tacit collusion. It will explore the abstract implications of algorithmic high-frequency pricing on the ordoliberal construct of competition as a demand-driven, self-regulating discovery process with the central role of the consumer as a sovereign. The market price mechanism, which is of paramount importance as an information parameter for consumers, is severely harmed by algorithmic dynamic pricing, which leads to irrational decisions and a restriction of consumer sovereignty.

Proposing a theory of harm based on price controls under Art.102(2)(a) TFEU, the thesis will introduce an innovative method to assess the unfairness of prices within that provision, thereby showing that an ordoliberal approach makes it possible to mitigate the potential risks of algorithmic pricing without ignoring the efficiencies that may arise from the use of this technology.

(5)

1. Introduction

Algorithms are prevalent in pretty much all areas of life these days. Especially pricing algorithms are powerful instruments in the digital economy since they enable undertakings to

“outsource” pricing decisions to efficient and reliable technologies. But not only businesses may profit from these technologies. They are also praised for benefitting consumers by enhancing price transparency.

However, the increasing use of algorithms for dynamic pricing has also led to a wide variety of academic literature on how this technology may harm consumer welfare by facilitating collusion between undertakings and substantially impacting market structures. Especially the subject of algorithmic tacit collusion, collusion achieved without any communication or concertation but merely through the interaction of pricing algorithms, has received a lot of attention amongst scholars.

In view of these dangers and risks, an ambivalence can be discerned in the literature.

While some dismiss the mentioned risks as “science fiction”1, others argue for changes to or the application of the law and see competition law agencies as not sufficiently equipped for enforcement against algorithmic tacit collusion.2

The aim of this thesis is to offer an innovative approach to the problem of striking a balance between the concerns and potential benefits of algorithmic pricing by carrying out critical research. Focusing on ordoliberal concepts of consumer sovereignty and protection of the competitive process as such, I will take a hitherto underrepresented standpoint in the discussion around algorithmic pricing, detached from a mere welfare-focused approach. By doing so, I will conceptualize a theory of harm for algorithmic tacit collusion based on current EU competition law to answer the question if and how the current provisions are fit to address the risks of pricing algorithms.

1 Petit (2017), 361; Schwalbe (2019), 600 (“legal sci-fi”).

2 E.g. Ezrachi/Stucke (2017b), 34.

(6)

The thesis is structured as follows:

Section 2 will explore the economic and legal background of tacit collusion to demonstrate why this phenomenon is considered legal under current EU competition law rules.

Section 3 looks at how pricing algorithms affect the economic preconditions for tacit collusion scenarios.

Section 4 will first introduce the reader to the ordoliberal school of thought, particularly its perception of the competitive process. It will then outline how this process is harmed by algorithmic high-frequency pricing. Next, these findings are embedded into a theory of harm based on Art.102(2)(a) TFEU. Finally, the role of efficiencies in this theory of harm is addressed and possible remedies that competition authorities could impose on undertakings after finding an abuse are discussed.

Section 5 deals with practical enforcement challenges.

2. Setting the Scene – The Economic and Legal Background of Tacit Collusion 2.1. THE ECONOMICS OF TACIT COLLUSION –OLIGOPOLY THEORY

Collusion in the economic sense can be defined as “reward-punishment scheme” with the objective of generating profits “above some competitive benchmark”.3 The agreement on such a scheme may take place explicitly or tacitly, resulting in explicit or tacit collusion respectively.

The following section will explore the basic economics of a tacit collusion scenario.

2.1.1. THE PRISONERS DILEMMA AND OLIGOPOLISTIC INTERDEPENDENCE

To understand how oligopolistic markets may end up in a supra-competitive, non-cooperative equilibrium, game theory can be of help.

The prisoner’s dilemma is a model of game theory where two suspects who committed a crime together are interrogated separately.4 The option to confess will decrease the own sentence

3 Calvano et al. (2019), 168; Harrington Jr. (2019), 335.

4 Steward-Moreno (2020), 52; Beneke/Mackenrodt (2018), 115.

(7)

while increasing the others.5 If both confess, the sentence is worse for both than if they had cooperated.6 If we transfer this model to a fictional market with two firms and perfectly homogenous products, they have two possible options – collude and charge a high price or cheat by charging a low price.

High Price/Collude Low Price/Cheat

High Price 300 300 0 500

Low Price 500 0 100 100

Figure 1. Profits arising from different pricing policies

Both firms would prefer to collude and charge the high price for the highest profits. However, if firm A charges a high price while firm B cheats by charging a low price, it is assumed that firm B will serve the entire market, depriving firm A of all his profits. The potential reward from charging a low price is thus higher (500) than the potential reward from charging a high price (300).

A rational player will choose the option with the greater reward, resulting in a cheat-cheat scenario, which is the “Nash equilibrium”, defined as the situation where no player has an incentive to change his conduct considering their opponents likely choice (cheating).7 The dominant strategy in this one-shot scenario is thus to cheat.8

However, the jointly favorable scenario is to cooperate and charge high prices. To achieve this and thereby collectively deviate from the Nash-Equilibrium, the players have two options.

They could agree to collude through communication or concertation, rendering the oligopoly

“non-competitive, cooperative”9. This scenario would constitute explicit collusion and is thus not relevant for this thesis.

I rather focus on non-cooperative game theory, where the players cannot communicate with each other before being interrogated.10

5 Steward-Moreno (2020), 52.

6 Ibid.

7 Steward-Moreno (2020), 52; Beneke/Mackenrodt (2018), 116; Gautier et al. (2020), 416.

8 Steward-Moreno (2020), 52; Schuchmann (2017), 33.

9 Blanco (2011), 176.

10 Beneke/Mackenrodt (2018), 115.

(8)

In oligopolies, the collusive scenario could also be reached merely based on oligopolistic interdependence, leading to a “non-competitive, non-cooperative”11 oligopoly. This is the tacit collusion scenario.

To understand why oligopolists may be able to coordinate their behavior without explicit cooperation, the concept of oligopolistic interdependence needs to be examined.

Oligopolies are markets with few players.12 Hence, rivals “are acutely aware of each other’s presence and are bound to match one another’s marketing strategy”.13 This means that an undertaking, when making commercial decisions, considers the likely reaction by its competitors.14

While every market bears some level of interdependence since every rational business decision considers the possible decisions of competitors, the degree of interdependence in oligopolies is much higher.15

Furthermore, in oligopolies, the same suppliers of substitute products meet repeatedly for an infinite number of times, constituting a “Supergame”.16 The decisions on how to behave on the market then factor in the game’s history.17 This fact combined with the high level of interdependence may, as first explored by Chamberlin in 1929, lead to the elimination of price competition.18 This process can be demonstrated as follows:

As in the scenario above, firm B considers cheating by charging a low price. However, in this Supergame, firm B factors in the likely reaction by firm A in the subsequent period. Firm A would likely react by charging a low price as well. The one-shot Nash equilibrium at the cheat-

11 Blanco (2011), 176.

12 Ibid, 174, “reduced number of competitors”.

13 Whish/Bailey (2021), 590; See also OECD (2017), 19; Ivaldi et al. (2007), 217.

14 Petit (2012), 3; Yao/DeSanti (1993), 116; Blanco (2011), 174.

15 Blanco (2011), 174.

16 Friedman (1971), 1; Kaplow/Shapiro (2007), 1104; Kupcik (2020), 535.

17 Yao/DeSanti (1993), 123.

18 Chamberlin (1929), 85; See Petit (2012), for an overview over economic theory on tacit collusion.

(9)

cheat scenario would thus be re-established in the next period, leading to lower profits for both undertakings.19

A potential price cut decision must therefore be based on the comparison between the potential short-term gains and the long-term profit losses through retaliation.20 The possible retaliation in subsequent rounds of the game keeps firms from favoring the short-term interest over the long-term one. According to Chamberlin’s theory, the incentive for price cuts then gets eliminated in the first place.21 As quoted in the beginning, the supra-competitive equilibrium is here based on threats and rewards.22 If the threat of retaliation is sufficient, each firm will independently conclude that colluding is its best strategy, considering the fact that the other competitors make the same calculation.

Realizing that, the competitors can rise prices to a supra-competitive equilibrium level similar to the monopolistic one.23

There are thus multiple equilibria. The competitive one and an infinite number of collusive equilibria over marginal costs.24 As long as the retaliation threat works, any of those collusive equilibria can constitute a subgame-perfect Nash equilibrium in the sense that in each subgame, all oligopolists charge the same supra-competitive price repeatedly.25 However, problems in finding a commonly accepted specific collusive price level may occur. This will be addressed at a later stage of this thesis26, since it could be solved easily by using algorithms.

19 Steward-Moreno (2020), 53.

20 Chamberlin (1929), 90, 91; Petit (2012), 7; Ivaldi et al. (2007), 5 f.; Blanco (2011), 175.

21 Chamberlin (1929), 85.

22 Gautier et al. (2020), 416.

23 Chamberlin (1929), 89.

24 Yao/DeSanti (1993), 132, “plethora of equilibria”; Kaplow/Shapiro (2007), 1104. While in games played a finite number of times the Nash equilibrium involves each firm setting the price P equal to costs C, in infinitely played games P may be set at different levels above C since every reduction of P may lead to retaliation in subsequent periods consisting of all other competitors setting P equal C or even below C. Retaliation thus means reversion to the one-shot Nash equilibrium.

25 Kaplow/Shapiro (2007), 1105. There is thus a multitude of equilibria (As proven by the “Folk theorem”), see Werden (2004), 729; Wagner-von Papp (2012), 8.

26 See below, section 3.4.

(10)

In “one-shot games”27, contrarily, where players do not interact repeatedly and the level of interdependence is lower, the undertaking does not face a possible threat of retaliation and will thus choose the dominant defect strategy.28 A common policy to cooperate is then unrealistic without communication. 29

2.1.2. THE ABSTRACT CONDITIONS FOR COLLUSION

As the discussion above illustrates, four cumulative conditions are necessary for the supra- competitive equilibrium to occur and be stable.30 Absent an explicit agreement, these conditions usually appear in oligopolies31 so that the phenomenon of tacit collusion is labelled the

“oligopoly problem”.32

First, there needs to be a common perception regarding the level of the collusive equilibrium price (C1). Otherwise, a price raise to different levels would constitute price competition.33 In explicit collusion, such perception is reached through an agreement or a concerted practice.

In multi-round games, as stated above, there are several possible collusive equilibria, resulting in uncertainty as regards the commonly preferred collusive price level. This uncertainty may be reduced if there is a “natural focal point”, which is the same for all rational players, resulting from the price of a similar good or from a market leader.34

However, according to some scholars, explicit coordination is necessary to agree on which of the equilibria to play.35

27 Note that Supergames only constitute one-shot games repeated indefinitely, see Ten Kate (2017), 6.

28 Petit (2012), 6.

29 Steward-Moreno (2020), 52.

30 According to Petit (2012), 7.

31 Kupcik (2020), 536.

32 Posner (1968), 1563.

33 Petit (2012), 7; Wagner-von Papp (2012), 8.

34 AdlC/BKartA (2019), 20; For price leadership see Schuchmann (2017), 55 f.; The uncertainty of equilibrium selection could result in choosing a “safe” equilibrium level below joint payoff maximization to cap losses in case of erroneous deviations, so-called “trembling hand perfection”, see Wagner-von Papp (2012), 9.

35 Lee (2018), 36; See Kaplow/Shapiro (2007), 1106 f. on the role of communication in oligopolistic Supergames;

See also Page (2012), 190 f.

(11)

Second, a credible punishment mechanism must be present to remove the incentive to deviate (C2).36 This retaliation must be “sufficiently likely and costly to outweigh the benefits from cheating”37. It can consist of competitive pricing or even short-term below-cost pricing, which would make it even more unattractive to deviate from the common conduct.38 As game theory shows, the threat of retaliation keeps firms from pursuing their short-term interest in cheating.

Third, monitoring of the competitors’ prices must be possible to allow for a detection of deviations (C3). Unnoticed deviations cannot trigger the punishment mechanism.39 Effective monitoring requires a sufficient degree of price transparency.40

Fourth, the supra-competitive equilibrium can only be sustainable in the absence of external competitive restraints (C4).41 For instance, low barriers to entry the market may destabilize a collusive equilibrium because of potential entries of maverick firms with aggressively low pricing.

The likelihood of these conditions to materialize is influenced by endogenous market features, such as market concentration, frequency of interactions, product homogeneity, capacity constrains and rate of innovation as well as by human limitations and biases developed by behavioral economics.42 It is thus important to mention that the economic theory based on the prisoner’s dilemma cannot hold true for all real-life oligopolistic markets and rather constitutes a very simplified model.

However, as will be outlined in section 3.4, the here mentioned economic conditions and market factors may be facilitated and influenced by pricing algorithms.

36 Gautier et al. (2020), 416; Harrington Jr. (2019), 336.

37 Ivaldi et al. (2007), 218.

38 Blanco (2011), 178; Wagner-von Papp (2012), 11; Harrington Jr. (2019), 335.

39 Blanco (2011), 178; Stigler (1964), 46.

40 See Stigler (1964).

41 Petit (2012), 7, focusing on discouraging entry of external firms.

42 See Ivaldi et al. (2007), 228, for a mathematical illustration of the impact of some endogenous market features on the occurrence and stability of collusion.

(12)

2.2. THE LEGAL APPROACH TO TACIT COLLUSION UNDER EU COMPETITION LAW – A LEGISLATIVE GAP?

EU competition law does not prohibit tacit collusion as such. This could be ascribed to the fact that law, unlike economics, is concerned with the means leading to the collusive market outcome.43 A reason for that is the requirement of predictability of law, which makes law a

“guide for human behavior”.44 Another possible explanation is that collusion itself, meaning the reward-punishment scheme, cannot be proven by competition authorities, because the pricing strategy resembling this scheme is “inside the managers’ heads”.45 For instance, the threat of retaliation does not have to be explicitly expressed to have its deterrent effect.

2.2.1. ART.101TFEU

Coordination in form of communication through an agreement and concertation is illegal under Art.101 TFEU.

The notion of agreement reflects “the existence of a concurrence of wills between at least two parties, the form in which it is manifested being unimportant so long as it constitutes the faithful expression of the parties’ intention”.46 Thus, proving an agreement requires a manifestation of the common will.47

A concerted practice constitutes “a form of coordination between undertakings which, without having reached the stage where an agreement properly so-called has been concluded, knowingly substitutes practical co-operation between them for the risks of competition”48.

In Suiker Unie, the Court clarified this concept by stating that a concerted practice requires “any direct or indirect contact between such operators, the object or effect whereof is either to influence the conduct on the market of an actual or potential competitor or to disclose to such

43 Kupcik (2020), 534; Motta (2004) 138.

44 Kupcik (2020), 534.

45 Harrington Jr. (2019), 349.

46 General Court in Case T-41/96, Bayer AG v Commission, para. 69.

47 OECD (2017), 37.

48 Court in Case 48/69, Imperial Chemical Industries Ltd. v Commission, para. 64.

(13)

a competitor the course of conduct which they themselves have decided to adopt or contemplate adopting on the market”49.

The rationale behind these notions is that effective competition requires autonomous actors carrying out their economic freedoms by acting in uncertainty about the reaction of competitors to the exercise of a competitive parameter.50

If this uncertainty is reduced by means of communication or concertation, this conduct is prohibited as anti-competitive under Art.101 TFEU.

However, in a tacit collusion scenario, the uncertainty about the competitors’ future behavior is reduced by reason of the market structure, not by co-operation of competitors.51

The Court recognized that and held in Wood Pulp that Art.101 TFEU does “not deprive economic operators of the right to adapt themselves intelligently to the existing and anticipated conduct of their competitors”52 and that “parallel conduct cannot be regarded as furnishing proof of concertation unless concertation constitutes the only plausible explanation for such conduct”53. In the case at hand, the oligopolistic market structure provided such a plausible explanation.54 Hence, oligopolists could invoke an “oligopoly defense”55 against the allegation that their parallel behavior results from prior concertation.

Thus, mere oligopolistic interdependency cannot be deemed a concertation.56 If there is actual evidence of such prior concertation, however, the oligopoly defense cannot act as a free ticket to collude.

This finding in Wood Pulp was an important rectification of the Courts prior judgment in Dyestuffs, where it stated that “although parallel behavior may not by itself be identified with a concerted practice, it may however amount to strong evidence of such a practice if it leads to

49 Court in Joined Cases 40-73 etc, Coöperatieve Vereniging “Suiker Unie” UA and others v Commission, para.

174.

50 Kupcik (2020), 541.

51 Ibid, 538.

52 Court in Case C-89/85 etc, A Ahlström Osakeyhtiö v Commission, para. 71.

53 Ibid.

54 Ibid, para. 126.

55 Petit (2012), 30.

56 Beneke/Mackenrodt (2018), 114.

(14)

conditions of competition which do not correspond to the normal conditions of the market, having regard to the nature of the products, the size and number of the undertakings, and the volume of the said market. This is especially the case if the parallel conduct is such as to enable those concerned to attempt to stabilize prices at a level different from that to which competition would have led, and to consolidate established positions to the detriment of effective freedom of movement of the products in the common market and of the freedom of consumers to choose their suppliers”.57

Summarized, when there has not been any contact between the competitors, the parallel conduct in oligopolistic markets does neither constitute a concerted practice nor an agreement. While some scholars proposed different approaches in earlier phases of EU competition law development,58 this constitutes settled-case law today.

Art.101 TFEU has instead been relied on to target so-called facilitating practices. That is, practices which facilitate the likelihood or stability of collusion. For instance, information exchange agreements may facilitate transparency.59 This, in turn, facilitates the monitoring and retaliation mechanisms.

This is recognized by the Commission in its Guidelines on horizontal co-operation agreements, in which the artificial increase of market transparency through information exchange is deemed potentially anti-competitive due to its possible effects on the common understanding on the terms of coordination, the mentioned facilitation of deviation-monitoring and retaliation mechanisms and monitoring potential market entries, which could have impact in terms of market foreclosure.60

57 Court in Case 48/69, Imperial Chemical Industries Ltd. v Commission, paras. 66 f.

58 See, e.g., Posner (1968), 1576, who interprets the parallel behavior as a “meeting of the minds”, whereby the first action constitutes the offer to align market conduct and the subsequent corresponding reaction constitutes the acceptance.

59 Schuchmann (2017), 43.

60 Commission Communication C 11/1, Guidelines on the applicability of Article 101 of the Treaty on the Functioning of the European Union to horizontal co-operation agreements, paras. 65-68.

(15)

2.2.2. ART.102TFEU

Art.102 TFEU applies to abuses of dominance “by one or more undertakings of a dominant position”.

As regards the notion of “dominant position”, the General Court held that independent economic entities may hold a collective dominant position “vis-à-vis the other operators on the same market” when there are “economic links” between the undertakings.61 These economic links may include the interdependence between oligopolists “in a market with the appropriate characteristics, in particular in terms of market concentration, transparency and product homogeneity”.62

The conditions for a finding of collective dominance were ultimately defined in Airtours.

According to the General Court, each member must be able to monitor the adoption of the common policy, there must be the threat of retaliation as a long-term incentive to stick to the common policy and the foreseeable reaction of current and future competitors must not jeopardize the results of the common policy.63 These conditions basically resemble the above- mentioned economic conditions for tacit collusion. Moreover, they essentially restrict the concept of collective dominance to oligopolies since monitoring of the whole market is implausible in markets with a larger number of participants.64

While the case-law was developed in merger control cases, the General Court repeated these conditions subsequently, also with regards to Art.102 TFEU.65 However, as of 2012, the Commission has never applied Art.102 TFEU to oligopolistic tacit collusion scenarios.66 Considering this case-law, the finding of a collective dominant position may thus be manageable in situations of tacit collusion, even if this does not constitute the current state of enforcement.

61 General Court in Joined Cases T-68/89 etc, Societa Italiana Vetro SpA and PPG Vernante Pennitalia SpA v Commission, para. 358.

62 General Court in Case T-102/96, Gencor Ltd v Commission, para. 276.

63 General Court in Case T-342/99, Airtours plc v Commission, para. 62.

64 Kupcik (2020), 540.

65 General Court in Case T-193/02, Laurent Piau v Commission, para. 111.

66 Petit (2012), 47. A possible reason is the difficulty of proving an abuse in oligopolistic market structures, see below.

(16)

However, it is difficult to establish an abuse of the collective dominant position. The argument that prices are above the competitive level because of parallel pricing is not sufficient, since the economic concept of tacit collusion is already used to establish the collective dominance, which in itself is not reproachable.67 Conceivable would be an abuse based on the excessive level of prices, whereby price controls under Art.102 TFEU are rarely conducted.

The theory of harm put forward in this thesis will be based on an exploitative abuse of collective dominance.

2.2.3. EUMR

In practice, the phenomenon of tacit collusion is mainly approached through merger control aimed at preventing the creation of oligopolies. This approach is concerned about preserving non-oligopolistic market structures not prone to tacit collusion.68 However, as an ex-ante instrument, merger control has limited utility to approach already emerged tacit collusion scenarios.

Since the theory of harm developed in this thesis will focus on an abuse of dominance, the merger control regime will not be further investigated.

2.3. CONCLUSION

The economic insights show us that the aligned conduct in a tacit collusion scenario derives from rational and independent decisions of the market players. The collusion is the outcome of the repeated game.69 The oligopolists intelligently adapt to the market conditions.

The law divides conduct in lawful and unlawful. Unlawful conduct decreases uncertainty on the market. However, if the market itself provides enough information for competitors, this uncertainty is naturally reduced. So far as the competitors do not further decrease uncertainty through an agreement or a concerted practice, the use of this information in the process of

67 Whish/Bailey (2021), 608.

68 Petit (2012), 44.

69 Gautier et al. (2020), 417.

(17)

making rational business decisions then does not constitute prohibited conduct under Art.101 TFEU.

Art.102 TFEU is also of limited use since the requirements for proof of an abuse represent a hurdle that is not easy to overcome.

The merger control regime may provide a useful ex-ante instrument but at the same time is of little help for coping with an already emerged tacit collusion scenario.

Concluded, if collusion is achieved tacitly rather than explicitly, this is deemed legal under current EU competition law rules. Considering the undesirable economic implications, the

“oligopoly problem” constitutes a “legislative gap”70 and arguably “the single most important imperfection of modern capitalism”71.

3. Algorithmic Tacit Collusion

After having presented the background of non-algorithmic tacit collusion, I will outline in this section how the use of pricing algorithms may lead to tacit collusion scenarios and how these scenarios differ from non-algorithmic tacit collusion.

3.1. BASICS OF PRICING ALGORITHMS

Algorithms can solve pre-formulated problems by specified sequences of steps.72

Pricing algorithms are here understood as digital tools to monitor market data and adapt pricing strategies by reacting to changes in real-time, or even being able to act anticipatory. 73

70 Lee (2018), 27.

71 Blanco (2011), 73.

72 Beneke/Mackenrodt (2018), 110.

73 Steward-Moreno (2020), 55; See Harrington Jr. (2019), 341, for a more detailed assessment on how pricing and learning algorithms work.

(18)

3.2. SCENARIOS LEADING TO ALGORITHMIC TACIT COLLUSION

In their 2016 landmark work “Virtual Competition”, Ezrachi and Stucke introduced certain scenarios in which the use of pricing algorithms may lead to a tacit collusion scenario. Their work has in the following years been cited in this context by numerous scholars and institutions.74 While Ezrachi and Stucke also present scenarios where algorithms facilitate and foster explicit collusion in the legal sense,75 this thesis focuses on the tacit collusion scenarios.

3.2.1. PREDICTABLE AGENT

Tacit collusion can be achieved if competitors each use distinct, individual pricing algorithms with a similar design that observe the market and adapt prices.76 Algorithms can be programmed to follow price increases when it is sustainable, that is, when others follow promptly.77 On the other hand, the algorithms can react to deviations by retaliating.78 This could then lead to an alignment of prices without the need for prior contact between the undertakings, let alone a clear indication for the algorithms to collude.79 This has been referred to as the “Predictable Agent scenario” or “Tacit collusion on Steroids”.80

3.2.2. SELF-LEARNING ALGORITHMS

Also, dynamic, self-learning algorithms can be programmed to achieve a general target such as maximizing profits and then find enhancing transparency and thereby sustaining tacit collusion as the optimal strategy themselves.81 Such machine learning algorithms can independently adjust the means for achieving the formulated goal by processing input data.82 Advanced

74 Eg. OECD (2017).

75 So-called messenger and hub-and-spoke scenarios, see Ezrachi/Stucke (2016), 36.

76 AdlC/BKartA (2019), III.

77 Ezrachi/Stucke (2016), 61.

78 OECD (2017), 27.

79 AdlC/BKartA (2019), III.

80 Ezrachi/Stucke (2016), 37.

81 Ibid.

82 Beneke/Mackenrodt (2018), 111; OECD (2017), 31.

(19)

algorithms could be able to decode each other and predict the competitors market conduct.83 While this could in theory be prevented by coding the algorithm accordingly, undertakings naturally do not have an interest in preventing tacit collusion as long as it is perceived perfectly legal.

3.2.3. CONCLUSION

In both scenarios tacit collusion may arise through a “feedback loop” of algorithms observing market conditions and other algorithms strategies and adjusting their own strategy in turn as a rational response to the market conditions.84

The difference between the scenarios lies in the level of sophistication of the algorithms. In the Predictable Agent scenario, the reaction of an algorithm is based on specific previously coded rules and thus predictable, while the algorithms output in the second scenario is determined by machine learning.85

The market outcome is the same (tacit collusion), but the second scenario may include further liability issues.86 While in Predictable Agent, the programmer knows that the likely outcome is tacit collusion and thus higher prices, in the second scenario there is neither positive knowledge nor intent of the human programmers.87

3.3. EMPIRICAL EVIDENCE

Although some economists see tacit algorithmic collusion as not or just theoretically possible,88 empirical research shows the contrary.

83 Bernhardt/Dewenter (2020), 322.

84 Dorner (2021), 10.

85 Gautier et al. (2020), 418.

86 Dorner (2021), 2.

87 Ezrachi/Stucke (2020), 220; Of course, it is possible that the programmers speculate on a corresponding output and therefore deliberately rely on machine learning to avoid liability.

88 Schwalbe (2019), 599, deems algorithmic collusion possible in theory but unlikely in practice;

Bernhardt/Dewenter (2020), 341, “rather unlikely”.

(20)

For instance, a study by Calvano et al. showed that reinforcement-learning algorithms learned to avoid lower profits as the result of a price war and instead did stick to the cartel price, while not being designed or instructed to collude.89

A study by Assad et al. on retail gasoline markets in Germany also showed increased margins resulting from industry-wide adoption of Artificial Intelligence.90 The algorithms in both studies did not fail to learn how to compete, but rather learned how not to compete, meaning that they identified collusive supra-competitive pricing as the most efficient policy.91

Salcedo stated that collusion between algorithms is “inevitable” if the pricing policies embedded in the algorithms are fixed in the short-term and can only be revised over time.92 The algorithms then learn to decode each other and interpret price increases as a proposal to collude, ending up at a supra-competitive equilibrium.93

In any case, being sufficiently equipped for even highly speculative situations is necessary, especially recognizing the quickly emerging digital economy. One should not be blinded by the high requirements or rare occurrence of traditional tacit collusion.

3.4. DIFFERENCES FROM TRADITIONAL TACIT COLLUSION

In this part, I will illustrate the effects of pricing algorithms on the general economic conditions that sustain tacit collusion. Even though algorithms will not transform every market towards tacit collusion, the following section shows how especially digital markets with their advanced levels of transparency and data availability are prone to algorithmic tacit collusion.

89 Calvano et al. (2020).

90 Assad et al. (2020).

91 Ibid, 38; Calvano et al. (2020), 3295.

92 Salcedo (2015), 3.

93 Ibid, 3, notes that this equilibrium is not necessarily close to the monopoly level; However, Ittoo/Petit (2017), 4 f., criticize that Salcedo does not consider competitive restraints through market entry and that the assumption of the algorithms’ ability to decode may be unrealistic.

(21)

First, the real-time pricing through algorithms enables firms to “test” prices in their level without the risk of losing customers and suffering losses.94 The price increase and the responses by competitors take place within so little time that this risk is eliminated. The risk inherent in competition resembled by the degree of uncertainty as regards the future conduct of the competitors is then limited to an extend where one cannot talk of “competition” in a narrow sense anymore. Thus, the enhanced speed of monitoring and adapting prices could enable competitors to choose a common equilibrium price level.

Furthermore, algorithms with enough input data could even predict the price that every oligopolist would accept as a focal point.95

The collusive equilibrium in traditional tacit collusion is even more difficult to identify if the products of the competitors are heterogenous.96 Through the use of algorithms, it could be possible to analyze the levels of prices in relation to their differentiation more easily, allowing for a stable collusive regime even in markets with more differentiated products.97 The level of prices is then aligned but still reflects the product differentiation.

Second, human biases such as loss aversion, sunk cost fallacy, or framing effects are eliminated.98 A computer does not “respond in anger”.99 This reduces the risk of price wars triggered by emotional irrational behavior.

Third, the increased frequency of interactions allows for instant retaliation and removes the incentive for deviations in the first place, so that no competitor can build a reputation as a price discounter.100 The faster a deviation from the collusive equilibrium is detected and responded, the more stable collusion gets.101

94 This may take place through unilateral price signals and the subsequent choice of the corresponding Nash response by the competitors. If, however, the preferences for the equilibrium level differ, the agreement on a common equilibrium may take several “rounds” of bargaining. See Wagner-von Papp (2012), 9; See also Stewart- Moreno (2020), 61 f.; OECD (2017), 30.

95 Beneke/Mackenrodt (2021), 163.

96 Wagner-von Papp (2012), 20, calls heterogeneity a „disruptive factor”.

97 AdlC/BKartA (2019), 19.

98 Ezrachi/Stucke (2016), 77.

99 Ibid.

100 Ibid, 63; AdlC/BKartA (2019), 18.

101 Kaplow/Shapiro (2007), 1105.

(22)

However, the ability to quickly react to competitors’ price changes depends on sufficient price transparency in the market. By using pricing algorithms, this transparency is increased since more market data gets digitalized and accessible.102 This makes it easier to detect deviations from the collusive equilibrium.103

Fourth, to implement pricing algorithms, input data is needed.104 While Ezrachi and Stucke state that algorithms will not be able to overcome instability resulting from low entry barriers105, in my opinion the data power may eventually drive less equipped players out of the market, thereby raising entry barriers for non-technologically equipped new players.106 It cannot be assumed that every undertaking will be able to follow suit and use pricing algorithms due to their costly implementation and maintenance.

Concluded, while the necessary economic conditions for tacit collusions traditionally merely occur in oligopolies, the use of algorithms may facilitate every single condition in a way that could lead to the expansion of the oligopoly problem. Firms may get enabled to replace explicit collusion with tacit coordination even in non-oligopolistic markets.107 Hence, tacit collusion could be made “less dependent on its traditional preconditions and, overall, more likely”.108

4. An Ordoliberal Doctrine of Algorithmic Exploitation

The previous remarks have shown that the economic outcomes of traditional and algorithmic tacit collusion are the same and may be even more severe for the latter one. This leads us to the

102 Ibid.

103 BKartA (2020), 4.

104 Ibid.

105 Ezrachi/Stucke (2020), 241.

106 Increased data power has recently played a role in the Commissions assessment of the Facebook/Whatsapp merger, EC, COMP/M.7217, 03.10.2014, para. 164, and in the determination of Facebooks dominant position by the German Federal Court of Justice (Bundesgerichtshof) 23.06.2020, KVR 69/19, para. 95.

107 Ezrachi/Stucke (2016), 77; OECD (2017), 33, 35.

108 Picht/Loderer (2018), 19.

(23)

conclusion that, from an economic welfarist standpoint, algorithmic tacit collusion is, in principle, undesirable.109

However, this is not an innovative thought as it also applies to traditional tacit collusion.

Furthermore, it does not help in closing the apparent legal gap.

In the following sections, I will take an ordoliberal point of view, disregarding the focus on the obvious economic results for consumers for the moment and rather turning the lens to the competitive process, which has immense significance for ordoliberals. I argue that this process is severely harmed by a feature of algorithms considered as central for algorithmic tacit collusion: High-frequency price setting.

My proposed theory of harm is based on Art.102(2)(a) TFEU and shows that the legal gap of tacit collusion turns out to be, when it comes to algorithmic tacit collusion, in fact only an enforcement gap, which can be closed using current competition law instruments.

4.1. INTRODUCTION –ORDOLIBERALISM

The ordoliberal school of thought emerged in the 1930s in Freiburg, Germany. Legal and economic academics started to reconsider economic policy and became known as the

“Freiburger Schule” (Freiburg School). While there are conceptual differences in the theory of different generations of ordoliberal thinkers,110 this thesis is restricted to key ordoliberal concepts as regards the competitive process and the role of competition law.

Ordoliberalism centers around the establishment of a socio-economic “Wettbewerbsordnung”

(competitive order) to balance economic efficiency and social goals like equality.111

Unique and of paramount importance is the relationship between competition and democracy in ordoliberal writings. Having experienced two world wars, many of the ordoliberal thinkers

109 See, eg., Thomas (2019), 162, who perceives consumer welfare as the only relevant standard for the assessment of algorithmic collusion.

110 See Behrens (2014), 17 ff.; Anchustegui (2015), 144 ff.

111 Wörsdörfer (2014), 239, 240; Deutscher/Makris (2016), 189.

(24)

of the first generation have seen how excessive concentrations of private power can be translated into political influence and interest capture.112 They have also seen how laissez-faire liberalism has failed to prevent such concentrations in the Weimar Republic and has led to the rise of a centrally planned economy under the Nazi-Regime, undermining checks-and-balances in the political sphere.113 Hence, ordoliberalism proposes a “third way” between classic laissez- faire liberalism and central planning.114

The competitive process is considered an instrument of disempowerment in the sense that it prevents the excessive concentration of economic power.115 Just as a State based on the rule of law, the competitive order provides for a rule-based framework in which individual freedoms are ensured and limited by each other.116 Without such a framework, competition would

“degenerate into a vulgar brawl”117 since absolute economic freedom tends to destroy itself.118 The framework is to be legally manifested in an economic constitution.119

4.2. THE ORDOLIBERAL UNDERSTANDING OF COMPETITION &CONSUMER SOVEREIGNTY

Late Ordoliberalism influenced by FA v. Hayek understands the market as an information producing, dynamic process.120

This process consists of constant interaction of individuals making their choices and thereby revealing their economic preferences, which in turn produces information that other individuals

112 See Podszun (2022), who labels the ordoliberal competitive order an “order of peace”.

113 Deutscher/Makris (2016), 186, 188; see also Behrens (2014), 5, 11, and Andriychuk (2012b), 111.

114 Kamecke (2001), 23; Anchustegui (2015), 145; Wörsdörfer (2014), 245, places ordoliberalism between “social Darwinism of laissez-faire” and Hobbes’ Leviathan; Bonefeld (2012), 641, calls laissez-faire liberalism a “doctrine of faith” and ibid., 639, quoting Rüstow (1942): classic laissez-faire liberalism as a “superstitious belief” in the automatism of market economy.

115 Böhm (1961), 22, „Wettbewerb als Entmachtungsinstrument“; Wörsdörfer (2014), 240, 244; Deutscher/Makris (2016), 190.

116 Ahlborn/Grave (2006), 201, quoting Eucken.

117 Röpke (1982), 188.

118 Behrens (2014), 15; Andriychuk (2012b), 111; Ahlborn/Grave (2006), 201; Bonefeld (2012), 634, 639.

119 Anchustegui (2015), 147.

120 Behrens (2014), 20; Grewe (2020), 88, states that v Hayek shares the process-oriented perception of the competitive process; see Andriychuk (2012b), 110, on differences between ordoliberal and Austrian school.

(25)

need to make their economic decisions.121 Hence, competition can be described as an interplay between individual market players influencing each other through their economic choices.122

The consumer exerts a special role, because his decisions steer the market and ultimately lead to a demand-driven supply of goods and services. Consumer’s choice is the “driving force of competition”.123 The suppliers subjugate themselves to “the rule of demand and the consumer as the conductor of the economic process”.124 This paramount importance of the consumers economic freedom of choice is reflected in the ordoliberal concept of so-called “consumer sovereignty”.125 The independent decisions of consumers fill out the abstract notion of

“welfare” with specific substance.

Hence, ordoliberals view the market outcome as the genuine expression of consumers preferences which, if undistorted, does not require correction under any ideal. The outcomes of this process are not predictable as the preferences of the consumers are only revealed through their choices in the competitive “discovery process”. 126

The consumer thus assumes the key role of an arbiter in the competition between companies, indirectly guiding their business decisions.127 Consumer sovereignty is an expression of economic liberty.128 A good illustration is the comparison of the impact of consumers choices on markets with the impact of a vote in elections, where the voter reveals his political preference.129 Consequently, just as in an election, the undisturbed environment for making a choice is of crucial importance. Individual, independent, and autonomous decisions are

121 Behrens (2018), 6; Behrens (2014), 20; Grewe (2020), 90.

122 Deutscher/Makris (2016), 198, call it a „checks-and-balances system” leading to an interdependence between the market players. This loops back to the beginning of this thesis where I outlined the extraordinarily high level of interdependence in oligopolies, where the checks and balances can turn into a system disadvantaging consumers.

123 Behrens (2014), 8.

124 Wörsdörfer (2014), 19.

125 Ibid, 249, with more references.

126 Behrens (2014), 20, quoting v Hayek; Andriychuk (2012a), 378, notes that the concept of competition as a spontaneous order is associated with Hayek, but dates back to ordoliberal thinkers.

127 Behrens (2014), 17, with reference to Böhm, Eucken; Grewe (2020), 174; Anchustegui (2015), 171.

128 And contrasts authoritarian central coordination of markets, see Wörsdörfer (2004), 249.

129 Deutscher/Makris (2016), 191.

(26)

essential and need to be safeguarded. 130 If the consumer can act in his role as a sovereign, ordoliberals refer to a “market democracy”.131 The results of competition then constitute an expression of the “volonté Générale”.132

4.3. RESTRICTION OF CONSUMER SOVEREIGNTY THROUGH HIGH-FREQUENCY PRICING

After having outlined the paramount importance of consumer sovereignty for the competitive process, I will now demonstrate how this sovereignty is restricted in algorithmic tacit collusion scenarios, more specifically through algorithmic high-frequency pricing.

As shown, the concept of consumer sovereignty builds upon rational and independent decisions by consumers. I will first assess how consumers make economic decisions, drawing insights from both neo-classical and behavioral economics.

4.3.1. DECISION MAKING FRAMEWORK –PRICE AS AN INDICATOR

The neoclassic model of the homo oeconomicus outlines the concept of a fully rational decision maker who strives for utility-maximizing decisions.133 It is assumed that there is complete information which enables market participants to make the best possible decision.134

However, behavioral economics have shown that the assumption of perfectly informed market actors cannot hold true. Rather, consumers suffer from bounded rationality.135 This is

130 Ibid, 198.

131 Wörsdörfer (2014), 219; See also Deutscher/Makris (2016), 191; Bonefeld (2012), 638, with reference to Röpke, calling the market a “continuous consumer plebiscite”.

132 Deutscher/Makris (2016), 191, quoting Böhm.

133 Zafirovski (2018), 194; Wurmnest (2012), 118.

134 Wurmnest (2012) 119.

135 Behrens (2014), 33; Klement (2015), 329; Wurmnest (2012), 196, who, however, notes that “it is too early to send the homo oeconomicus into retirement”; And Schmidt/Haucap (2013), 37, who argue that the ideal of fully rational decisions should not be called into question, „an ideal does not lose its validity if it can only be approximately realized“; Bar-Gil (2011), explores how sellers may adapt their pricing policy and products to the imperfect rationality and thus exploit consumers.

(27)

necessarily the case and cannot be fully eliminated since human capacity of processing information is limited.136 Consumers thus act rationally only within the framework of the information available to and processable by them.

This reveals a need to narrow the ordoliberal understanding of “rational decisions” down for the sake of keeping the theory put forward here realistic. It can never be assumed that a decision is fully rational in the neoclassical sense, but always the second-best, being rational to a point where all available information is factored in to the individually desired extend.

Furthermore, the rationality of a decision depends highly on individual preferences. Holding a decision irrational therefore runs the risk of being paternalistic. This is in line with the ordoliberal argument of competition as a discovery process and the unpredictability of consumers decisions because this unpredictability means that the self-determination of consumers must be respected by not dictating them a decision, but rather safeguarding an environment in which they are able to make rational decisions themselves.

Welfare-centered approaches contrarily measure market outcomes with a self-defined ideal, which can mean that outcomes are corrected even though they stem from independent and rational decisions if this serves the achievement of the ideal, for instance if it increases overall welfare. Here, the difference between ordoliberalism and utilitarian approaches gets clear.

Ordoliberalism respects the consumers decision and guarantees him the framework to being able to decide rationally.137

To safeguard the environment for decision making in a market, one must focus on the information that the consumer can in fact base his decision on. The information must enable the consumer to compare the benefits and costs and (subjectively) conclude which product has the best value.138 Several information parameters, such as price, quality, and service, are available.

However, quality and service are not quantifiable, and the comparison of goods based on these parameters may require special expertise.139

136 Helleringer/Sibony (2017), 612.

137 See in this regard also section 4.5.1 on welfare costs of this ordoliberal approach.

138 Bar-Gil (2011), 14.

139 Schmidt/Haucap (2013), 90; see also Bar-Gil (2011), 14, “complexity stands in the way of effective competition”.

(28)

Price, on the other hand, can be an important information parameter for the following reasons:

First, a working price mechanism reveals information about the scarcity of a good.

Second, consumers may draw conclusions the other information parameters from the price. It has been empirically proven that consumers perceive the price as an important indicator for quality.140

This especially applies to online markets, where the consumer has no physical pre-purchase possession of the good. He is thus unable to physically examine the product in any way and thus is reliant on the indicator function of the price. Online markets are also the most prone to the use of pricing algorithms, as changing physically displayed prices is more complicated than changing a digital display.141 Since the consumer will explore information to the point where the marginal costs of collecting more information are higher than the expected marginal benefit of doing so,142 the price will often constitute the only objective decision parameter in online markets.143

Hence, it can be concluded that price as a decision parameter is the main point of reference for rational purchase decisions by consumers, especially in digital marketplaces.

Furthermore, a rational purchase decision must necessarily be based on the unconsciously assumed fact that the price is the product of a competitive price-setting process.144 If this is not given, the price logically cannot be a valid indicator, so that the drawing of conclusions based on the price would be irrational, rendering the subsequent purchase decision irrational as well.

Alternatively, the consumer can rationally include the knowledge of the price setting process being uncompetitive in his decision making, but that requires that he is aware of it.

140 Price thus not only indicates monetary sacrifice; Erickson/Johansson (1985), 196, note that the extend of use as a quality indicator relates to the availability of alternative information. This availability is restricted in online markets, see above; See Völckner/Hofmann (2007), for more recent research (“effect of price on perceived quality has decreased but is still an important indicator”).

141 Brick-and-Mortar stores may use algorithmic pricing to determine the optimal prices but would in most cases still have to adjust them manually.

142 Bar-Gill (2011), 14.

143 “Objective” meaning that relying on other information such as quality descriptions made available by the seller requires a subjective assessment of the trustworthiness of the supply side or the source of information respectively.

144 Importantly, I do not equal competitive price setting with rational price setting since a rational pricing policy may consist of charging supra-competitive prices, as demonstrated by game theory. Rationality on the demand side thus does not require the assumption of rationality on the supply side.

(29)

Concluded, consumer sovereignty depends on rational decisions by consumers. Consumers are boundedly rational and mainly rely on the market price as an information parameter. A rational decision relying on the price as an information parameter requires the knowledge of the price setting process. This knowledge is in most cases limited to the unconscious assumption of competitiveness. If the price-setting process was not competitive and the consumer is not aware of that, the consumer’s decision based on price as a wrongful information parameter is objectively irrational.

In the next section, I will apply these findings to algorithmic high-frequency pricing and explore how this conduct imposes an information asymmetry on the consumer by eliminating price competition.

4.3.2. ASYMMETRIC INFORMATION AND RATIONALITY

As seen above, real-time pricing enables competitors to attune to a non-cooperative, collusive equilibrium without the risk of incurring profit losses during this process.

In contrast to traditional tacit collusion, there is at no point a possibility for consumers to

“intervene” in this process with a purchase decision and profit from temporarily lower prices.

The same applies to subsequent game periods under the now settled collusive equilibrium.

While the loss of incentives for deviations from the equilibrium due to the potential subsequent retaliations is the driving force of traditional tacit collusion, it is even more severe for algorithmic tacit collusion. This can be shown with an example:

As Chamberlin stated, the significance of the time lag between a price cut and the price adjustments by competitors must be considered in relation to the expected market presence.145 For instance, sellers that are only temporarily active on the market may value short term gains by price cuts higher than permanently present sellers since the former do not have to fear retaliation. Hence, they may still cut the price if there is at least some time gap between the cut and the retaliation.

If, however, time lags are completely diminished, the incentive for price cuts as regards short- term profits is eliminated for every seller, no matter how long his market presence may be.

145 Chamberlin (1929), 90.

Referenties

GERELATEERDE DOCUMENTEN

CCL21 stimulation of the high affinity state of LFA-1 on mDCs led to a significant reduction of LFA-1 lateral mobility and an increase on the fraction of stationary receptors,

25: Originele tekening 1965-1966 uit het Provinciaal Archeologisch Depot van de provincie Noord-Holland te Castricum.. 26: Originele tekening 1965-1966 uit het Provinciaal

Having analysed the research question and the many variables using neoclassical realist lenses, we notice there are both systemic and domestic factors that influence the

They complement each other as they have chosen different globalization themes to investigate the impact on the nature of sovereignty; international cooperation

Maar nu blijkt dat de vakken die helemaal geen uitwisseling meer hebben met het buitenwater van het Ilperveld, - het water komt rechtstreeks uit het kanaal -, meteen al een

If we compute for each location the accident rate (the ratio between the number of accidents and the measure of exposure, the number of encounters in this

Figure 19: Time weighted recent insulin pump (actual GRIP and advised total OLVG) per glycemic range....

There is a probabilistic algorithm that, given a positive integer k, a squarefree positive integer b coprime to a, the q-expansion of a Hecke eigenform f of weight k for Γ 1 (ab) up