Reduced demand uncertainty and the sustainability of collusion: How AI could affect competition

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Information Economics and Policy journal homepage: www.elsevier.com/locate/iep

Reduced demand uncertainty and the sustainability of collusion: How AI could affect competition ✩ Jason O’Connor a, Nathan E. Wilson b,∗ a b

University of Pittsburgh, United States Federal Trade Commission, United States

a r t i c l e

i n f o

Article history: Received 29 August 2019 Revised 7 August 2020 Accepted 22 August 2020 Available online xxx JEL classification: K12 L13 L40 Keywords: Artificial Intelligence Uncertainty Collusion Price Discrimination Antitrust

a b s t r a c t We model how a technology that perfectly predicts one of two stochastic demand shocks alters the character and sustainability of collusion. Our results show that mechanisms that reduce firms’ uncertainty about the true level of demand have ambiguous welfare implications for consumers and firms alike. An exogenous improvement in firms’ ability to predict demand may make collusion possible where it was previously unsustainable or more profitable where it previously existed. However, an increase in transparency also may make collusion impracticable where it had been possible. The intuition for this ambiguity is that greater clarity about the true state of demand raises the payoffs both to colluding and to cheating. Our findings on the ambiguous welfare implications of reduced uncertainty contribute to the emerging literature on how algorithms, artificial intelligence (AI), and “big data” in market intelligence applications may affect competition. Published by Elsevier B.V.

1. Introduction ✩

The views expressed in this article are those of the authors. They do not necessarily represent those of the Federal Trade Commission (FTC) or any of its Commissioners. Neither author has ever worked for, or conducted research sponsored by, a major technology company. We note, however, that both authors have affiliations with the FTC (current in the case of Wilson and previously for O’Connor), which has investigated various technology companies as part of both its antitrust and consumer protection missions. We thank Yossi Spiegel and two referees for their constructive feedback. In addition, we are grateful for comments from Rob Porter, Scott Stern, Yu-Wei Hsieh, Debbie Minehart, Alex MacKay, Daniel Sokol, Justin Johnson, Catherine Tucker, Jeanine Miklos-Thal, Joe Harrington, and attendees of the 2019 International Industrial Organization, 2019 DC IO Day, 2019 Georgetown Center for Economic Research Biennial, and 2020 Allied Social Sciences Associations conferences. Finally, we gratefully acknowledge comments from Jeremy Sandford, Dave Balan, Patrick Degraba, Dan Hosken, David Schmidt, Charles Taragin, Tom Koch, and others at the FTC. The usual caveat applies. ∗ Corresponding author. E-mail addresses: [email protected] (J. O’Connor), [email protected] (N.E. Wilson).

In the real world, competition has always occurred under conditions of at least somewhat imperfect information. Firms have not been able to observe the true level of demand nor all competitively relevant actions by their rivals. However, the recent proliferation of large data sets and the availability of algorithmic tools to analyze them have caused some commentators to conclude that the prevalence of uncertainty may be changing, and that these developments have potentially baleful implications for competitive intensity. For example, in October 2018, the US Assistant Attorney General for antitrust intimated that a price-fixing case involving algorithmic price-setting tools might be brought soon.1 At roughly the same time, the UK’s Competition and Markets Authority released a white 1 See https://www.broadcastingcable.com/news/delrahim- criminal- case - against- anti- competitive- search- algorithms- coming#disqus_thread.

https://doi.org/10.1016/j.infoecopol.2020.100882 0167-6245/Published by Elsevier B.V.

Please cite this article as: J. O’Connor and N.E. Wilson, Reduced demand uncertainty and the sustainability of collusion: How AI could affect competition, Information Economics and Policy, https://doi.org/10.1016/j.infoecopol.2020.100882

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paper describing risk factors associated with the use of data-driven algorithms.2 In this paper, we bring the tools of formal theory to bear on the question of how real-time collection of big data, plus techniques to make timely, more accurate predictions from these, could alter the incidence and character of collusion. To do this, we develop a framework derived from the seminal Green and Porter (1984) model of dynamic competition to consider how changes to firms’ ability to observe the true level of demand may affect market outcomes.3 In our model, a market with two firms is subject to two independent negative demand shocks. We consider the sustainability of collusive equilibrium strategies depending on whether firms have access to a technology enabling them to perfectly forecast or observe one of the shocks. We generally refer to the uncertainty-reducing technology as “AI” in keeping with how the recent literature (e.g., Agrawal et al., 2018) describes the use of artificial intelligence algorithms and data to improve “nowcasting” capabilities. Like all oligopoly models that fix the number of competitors and abstract from entry and exit, our framework is a stylized abstraction from the “real world.” However, our model of competition with separate sources of demand uncertainty could characterize many industries where final prices and quantities are imperfectly observable by rivals. Indeed, the possibility that a market may experience multiple independent shocks can arise any time demand for a given product is a multivariate function of (at least partially) stochastic factors. One common demand factor is consumer income, which is subject to shocks that are likely independent from shocks to a different typical factor, the availability of complementary or substitute products. This rough pattern characterizes disparate industry settings. For example, the spot market for coal is affected by uncertain demand factors like the weather, which shifts demand for electricity. An unexpectedly cool summer will depress coal demand relative to the norm by reducing the use of air conditioners. Simultaneously, demand for coal will also be impacted by fluctuating supply conditions in alternative sources of power. An unexpectedly wet winter may mean that hydro power is likely to take a larger than typical share of the overall electricity market. Similar dynamics can be seen in as different a market from coal as online retail. There, demand will be affected by shocks to the overall economy that affect consumers income and consumption priorities. Simultaneously, changes affecting the desirability of buying through brick and mortar outlets will affect demand for products available via online distribution, as seen in the 2020 COVID-19 pandemic. In both the coal and online retail examples, market participants likely can infer something about the incidence of demand shocks, but there will remain some degree of uncertainty about their realized magnitudes that leaves open other possible explanations for lower than expected sales. In particular, it will be difficult to rule out that low sales

reflect undercutting on price by a rival firm. For example, an individual coal seller may have priors about the prices offered by rival mining companies but will not be able to observe the outcome of individual sales to utilities. Similarly, even if an online retailer’s list prices are public, they may engage in large-scale targeted private discounts that enable them to capture more of the market. Thus, in either setting, a firm’s failure to achieve the sales it expected may reflect an unobserved decline in demand or undercutting. Sellers will not be able to perfectly distinguish these possibilities insofar as they only directly observe their own sales and profits. Algorithms that can reliably collect and analyze data correlated with the true state of demand could reduce some of this uncertainty. The model that we develop in this paper shows that the exogenous adoption of nowcasting AI by sellers has ambiguous implications for both firms and consumers. On the one hand, reduced demand uncertainty may benefit firms in many cases. By clarifying what the true demand conditions are, AI enables firms to more precisely differentiate rivals’ cheating from unobserved negative demand shocks. Furthermore, increased clarity about the true state of demand allows colluding firms to better tailor their collusive prices to demand conditions, increasing the average per period profit earned during periods of collusion. These factors may make collusion sustainable when it previously was not, and also may cause the duration of punishment to shrink relative to the duration required in the pre-AI period (assuming collusion was possible then). On the other hand, once firms have better knowledge of the true state of demand, they may better time their decision to deviate from a collusive agreement. All else equal, this knowledge can make collusion unsustainable even if it had been sustainable before AI. The net effect of the coordinationfacilitating and coordination-inhibiting effects of greater transparency will depend on market characteristics as described by the market’s location in the model’s parameter space. To provide greater insight into how different market characteristics interact in sustaining collusion in the two information environments, we implement our model in a linear demand setting. In this context, we show that consumers gain most from the introduction of AI when the probability of the now forecastable shock is large relative to the probability of the one that is still unobservable. Returning to the coal example, this might occur if weather prediction algorithms end up being better able to predict local temperatures than rainfall in areas with hydro plants. If the likelihood of cool temperatures is large relative to the likelihood of higher rainfall, then our model suggests the introduction of AI is likely to make collusion unsustainable. Specifically, the firms will want to take advantage of rare hot (i.e., high demand) seasons and undercut the collusive price. As in other treatments of collusion, we also find that consumers are more likely to gain, all else equal, when firms’ (common) discount rate is lower. Finally, we show that whenever collusion is sustainable both before and after AI, producer surplus increases while consumer (and total) surplus declines.

2 See https://www.gov.uk/government/publications/pricing-algorithmsresearch- collusion- and- personalised- pricing. 3 In future work, we hope to explore how changes in firms’ ability to observe rivals’ actions impact competition.

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In extensions, we show that our baseline results are qualitatively robust to modifying our model to allow colluding firms to optimize simultaneously over the collusive prices in addition to the length of punishment periods. However, there are some differences when prices and punishment are optimized in tandem relative to our baseline model. In particular, if firms are allowed to collude at prices below the monopoly level in the post-AI environment, the possibility of a total breakdown in coordination is dramatically reduced. That said, in some cases where collusion is sustainable pre-AI the prices needed to sustain collusion after AI adoption must be set so low that consumer surplus increases. Overall, we contribute to the large literature on conditions conducive to collusion and coordinated conduct (Tirole, 1988; Ivaldi et al., 2003; Kovacic et al., 2011), particularly with respect to the role played by uncertainty (Robson, 1981; Green and Porter, 1984; Kandori, 1992; Raith, 1996; Athey and Bagwell, 2001). Our work shows that reducing uncertainty has ambiguous effects, making collusion more attractive in some cases, but also sometimes more difficult to sustain. These results dovetail with the equally ambiguous results in an emerging empirical literature considering the connection between transparency and collusion. In this literature, some papers, like Lemus and Luco (2019), find that increased transparency may hinder coordination. However, other contributions, such as Byrne and De Roos (2019), have found evidence that increased transparency may facilitate coordination. In addition, our paper contributes to the ongoing debate about how antitrust policy should address the competitive effects of recent developments in data analytics. This literature already contains contributions reflecting a wide variety of opinions (see, e.g., the partial bibliography of Ritter (2017)), but many of the most vocal commentators have suggested that these phenomena threaten consumer welfare. In our view, these critics have focused on two related, but distinct, theories of harm: (1) an increased ability to personalize prices will allow firms to better extract consumer surplus (Ezrachi and Stucke, 2016; Ezrachi and Stucke, 2017), and (2) the use of algorithms will facilitate collusion (Ezrachi and Stucke, 2016; Mehra, 2015).4 While our paper only addresses the latter theory, we worry that the critics have proceeded to judgment without necessarily establishing a rigorous basis for either theory of harm in the economic or computer science literatures.5

By rooting consideration of AI in the game theoretic treatment of uncertainty and coordination, our paper shows that the effects of AI and related technologies are more nuanced than most commentators have heretofore considered. Our results show that even without endogenizing entry, exit, and repositioning on the supply-side, let alone demand-side technological adaptation (Gal and Elkin-Koren, 2016), the implications of improved analytical capabilities are varied, and depend heavily on market primitives. This nuanced conclusion stands in contrast to much of the discussion in the policy-oriented literature. Within the literature on algorithms and competition, our paper fits with other emerging contributions taking a more technical approach (Ittoo and Petit, 2017; Kuhn and Tadelis, 2017; Calvano et al., 2019; Miklós-Thal and Tucker, 2019; Harrington, 2020). In particular, our work is similar in motivation to that of Miklós-Thal and Tucker (2019), who also assume algorithms allow firms to better predict demand rather than directly set prices. However, their model extends the Rotemberg and Saloner (1986) model of collusion where firms rely on a signal of future demand in setting prices, while we modify Green and Porter (1984). An important difference between our modeling frameworks is that we make different assumptions about competitors’ ability to monitor adherence to the collusive strategy. Whereas Miklós-Thal and Tucker (2019) focus on markets where firms can perfectly monitor each other, we consider situations where firms can only imperfectly infer each others’ behavior by considering market outcomes. This means that the arrival of uncertainty-reducing technologies not only enables more tailored pricing but also improves monitoring. Despite our distinct modeling frameworks, both we and Miklós-Thal and Tucker (2019) find that AI has ambiguous effects on consumer welfare and profits. That our respective papers reach broadly similar conclusions supports a cautious policy approach on the potential threat of AI to consumer welfare. The paper is organized as follows. Section 2 outlines the modeling framework. Section 3 compares the equilibria that will result in the pre- and post-AI states. Section 4 presents the results of a less restrictive version of our model, which allows firms to set collusive prices and punishment periods simultaneously. Section 5 briefly discusses the validity of thinking of AI adoption as exogenous. Finally, Section 6 concludes. 2. The theoretical model 2.1. Setting

4

An additional element may be that adopting algorithmic pricing policies can serve as a credible commitment to different pricing strategies, which may serve to soften competition. See, e.g., Brown and MacKay (2019). 5 Salcedo (2015) is a notable exception in the literature on algorithmic collusion. However, the necessary and sufficient conditions for collusion in the model are restrictive and not obviously consistent with the approach taken by firms actually engaged in algorithmic pricing. Most of the assessments of price discrimination ignore the fact that economic theory indicates that the practice has ambiguous effects, particularly in the presence of competition (Cooper et al., 2004; Carlton and Perloff, 2015). Consistent with this, recent empirical work on the use of algorithmic pricing has found that most consumers benefit (Dubé and Misra, 2017) or that overall welfare substantially increases (Reimers and Shiller, 2018).

We consider an infinite horizon, discrete-time model of repeated Bertrand duopoly competition with homogeneous products. That is, firms simultaneously choose a price for their good and sell as much quantity of the good as is demanded at their price.6 The marginal costs of production for both firms are zero, as are the fixed costs of produc6 We believe the assumption of homogeneous products is without loss of generality so long as the extent of differentiation across firms is public and the shocks reduce demand equally for both products. Allowing for differentiation simply changes the profit level of the firms in the different

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tion for both firms.7 This process of market competition repeats each period, though firms’ choice of prices may be affected by profits received in previous periods. The downward sloping market demand curve is identical in each period, so that the base level of demand does not change from period to period. However, demand is subject to two random and independent shocks, M and  , each period. If either of these shocks occurs, it reduces demand in that period by an equivalent, constant amount. Both shocks occur with known, strictly positive probabilities μ and σ , respectively. We only consider cases where there is some uncertainty about the shocks: 0 < μ < 1 and 0 < σ < 1. If both shocks occur, then firms are unable to sell their product at any price, and, therefore, make zero profits for that period.8 As in the canonical Bertrand model, in the event that one firm has a lower price, it supplies the entire market that is willing to purchase the product at that price. The other firm makes no sales and earns no profits. In short, we assume there is no way for a firm to undercut its rival and limit its sales to less than the entirety of the market.9 When the two firms choose the same price, they divide the market equally. Firms observe only their own prices, sales, and profits in each period. They do not directly observe the conduct of their rivals. Thus, if firm i makes zero profits in one period, it could indicate that both negative demand shocks occurred or that its price had been higher than that of firm j. We model the impact of AI as a means of reducing demand uncertainty by assuming that one of the shocks becomes perfectly predictable to firms in the post-AI state of the world. Once firms have adopted AI, they are able to perfectly predict whether the M demand shock will occur. Adoption of AI has no other impact on the model framework.

a firm realizes zero profits in some period t, it will price at the competitive level, pc , for the next T periods before returning to the collusive price in period t + T + 1. Because both M and  occur with positive probability, price wars (i.e., the punishment phase where both firms price at the competitive level) will occur even if neither firm ever deviates from the collusive strategy. The optimal collusive price reflects a balancing of the profits to be earned in the different states where demand is greater than zero for some positive price:

 p ≡ argmax p (1 − μ )(1 − σ ) m

 +(μ + σ − 2μσ )

π

m l

2

πhm



2

 ,

(1)

where πkm indicates the joint profits earned if both firms charge the collusive price, pm , given demand condition k. k can take two levels: high (h) and low (l), reflecting whether no shocks occurred or only one shock occurred, respectively.11 We assume that πhm > πlm as would be the case if demand is linear and the shocks simply shift the demand curve’s intercept down. When calculating profits, the identity of what shock may have occurred is irrelevant, as they are equivalent in magnitude. Equation 1 shows that the determination of the optimal collusive price will be analogous to that made by monopolists using a single price to sell to a pool of different consumer types who cannot be separated. In other words, it resembles the price-setting problem when there are two types of consumers but third degree price discrimination is not possible. If the likelihood of being in the “low” demand state is high, then the optimal price should be substantially discounted relative to the monopoly price during the high period, even if that means failing to extract significant surplus from consumers willing to purchase during the high demand state. Similarly, if the likelihood of a low demand state is small, then the optimal collusive price may be close to the full-demand monopoly level even though that means sacrificing some potentially profitable sales during low demand periods. For example, if demand is linear, the derivative of pm with respect to μ is always negative, so the price increases as μ decreases, which obviously reduces sales in the low demand state.12 Assuming a common discount rate of δ , the expected value of the discounted stream of profits for each firm from participating in the collusive arrangement, V, is onehalf the expected value of the single-period monopolist profits plus the discounted continuation value when either zero or one of the shocks occurs, plus the collusion continuation value discounted by the length of the punishment period, T,weighted by the probablity of both shocks occurring:13

2.2. Collusive equilibrium before the adoption of AI Given the nature of competition and the form of demand uncertainty, we consider coordination in a manner akin to that developed by Green and Porter (1984) in the context of quantity-setting firms and re-formulated for Bertrand competition in Tirole (1988, Section 6.7.1).10 Firms’ collusive equilibrium strategy takes the following form: the competitors agree ex ante to set their price every period at a defined collusive level, pm (i.e., the monopolist’s price given the existence of uncertain demand), until observing a period in which they earn zero profits. When

states but does not qualitatively change the welfare effects of introducing AI. 7 In the case of linear demand (as analyzed in Section 3), the zero marginal cost assumption is without loss of generality as it is simply a normalization of demand. However, this assumption is a limitation with other forms of demand. 8 In the context of linear demand, this could either be because each shock is at least half of the intercept for demand or that there is some compounding of demand reduction when both occur. 9 We discuss the justifications for this common assumption below in Appendix A. 10 As noted above, Miklós-Thal and Tucker (2019) consider an alternative approach to coordination between firms competing in prices.

11

The identity of the specific shock does not matter in the pre-AI world. Eq. (E.3) of Appendix E shows how pm is derived in the linear demand case. 13 Since this is a homogenous Bertrand model, the firms revert to pricing at marginal costs and make zero profits while in the price war phase. 12

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 V = (1 − μ )(1 − σ )

 +σ ( 1 − μ )

πlm 2

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πhm 2

 + δV

 + μ (1 − σ )



+ δV

+ μσ δ

T +1

πlm 2



pression holds with equality. Doing this yields:

+ δV

T = V.

(2)

(1 − μ )(1 − σ )πhm + (μ + σ − 2μσ )πlm . 2(1 − (1 − μσ )δ − μσ δ T +1 )

(3)

Thus, the payoffs to colluding are declining in the length of the punishment period since the denominator of V is strictly increasing in T. Collusion will be sustainable only if neither firm would prefer to undercut the collusive price to earn monopolylevel profits today and trigger a punishment phase with certainty over continuing to split the collusive profits until a punishment phase is triggered by the arrival of both shocks. If a firm deviates from collusion by undercutting its rival’s price slightly, the expected value of this deviation from the price-cutting firm is:

(1 − μ ) (1 − σ )πhm + μ(1 − σ )πlm + (1 − μ )σ πlm +δ

T +1

V.

2μσ δ −δ

log(δ )

.

(5)

We now suppose that both firms adopt an AI technology that allows them to perfectly predict the outcome of the M shock and set their prices accordingly.16 However, the incidence of the  shock remains unobserved. The introduction of the new technology can be seen as having reduced – but not completely removed – uncertainty in the market, allowing all firms to have greater insight into the demand conditions facing them in any given period. Returning to our example markets, it would be as though coal sellers gained a better ability to anticipate temperature variation (but not rainfall swings) or online retailers improved their ability to gauge consumer appetite (but not the competitiveness of offline sales). In the post-AI setting, in each period, pricing will be optimized based on the common revelation of one of the demand shocks. Given that firms’ information sets and optimal competitive choices change as a result of the introduction of AI, so, too, will their coordinated strategy. Whereas they had previously coordinated on one collusive price, pm , now it is optimal to agree on two separate prices that depend on what is commonly observed about the underlying demand conditions. Specifically, we consider coordination when the firms charge pl if M occurs and ph otherwise, where ph ≥ pm ≥ pl with at least one of the inequalities being strict. The collusive prices are chosen so as to maxi-

V ≥ (1 − μ )(1 − σ )πhm + μ(1 − σ )πlm +(1 − μ )σ πlm + δ T +1V. Rearranging the terms of this inequality yields:

(1 − μ )(1 − σ )πhm + (μ + σ − 2μσ )πlm . (1 − δ T +1 )

 1−2δ+2δμσ 

2.3. Collusive equilibrium after adoption of AI

The first three terms represent the expected profits from being a full monopolist in the period when deviation occurs. The fourth term is the continuation value of collusion, which will resume after the T periods of punishment via Nash-Bertrand pricing. The punishment period is guaranteed in the deviation case, even if both shocks occur and the deviating firm makes no profits when it cheats on the collusive agreement. Incentive compatibility for the collusive agreement requires that the continuation value of collusion, V, be higher than the deviation value; this condition can be expressed as:

V ≥

log

As competition takes place in discrete time, T will actually be the smallest positive integer greater than (5). Examining (5) shows that it produces negative values of the punishment period T for seemingly plausible values of the market parameters μ, σ , and δ .14 However, in the context of sustaining collusion in repeated Bertrand duopoly competition, T must always be a positive number.15 If T = 0, a firm that deviates from collusion faces no punishment. In this case, the deviating firms continuation value is δ V, just as if the firm did not deviate and chose the collusive price. Thus, a firm would always deviate; the expected value of the deviating firm would have the same continuation value as collusion but would yield full expected monopoly profits instead of half the expected monopoly profits that the firm receives in collusion. Setting T = 0, therefore, cannot sustain a collusive equilibrium. If a set of parameter values has an optimal T less than 0 according to (5), collusion is unsustainable for this market.

Inequality Rearranging terms in Eq. (2) to show the value of coordinating as a function of the single-period monopolist profits and the length of the punishment period yields:

V =

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(4)

Inequality 4 shows that the sustainability of collusion will be a function of the discount rate as in the classic modeling of collusion under conditions of no uncertainty. However, it will also be affected by the probability of shocks and the length of the punishment period. All else equal, it is easier to satisfy inequality 4 and sustain collusion as the duration of the price war goes to infinity. However, since V is decreasing in T (from (3)), the optimal punishment period for the firms is the smallest value of T that still satisfies the incentive compatibility constraint in inequality 4. The smallest compatible T value can be calculated by substituting for V from (3) into inequality 4 and solving for the value of T for which the ex-

14 For example, with μ = 0.75, σ = 0.75, and δ = 0.7, according to (5), T is approximately −4.172. 15 T cannot be less than zero for the obvious reason that firms cannot implement a negative punishment period. 16 We use “observe” and “perfectly predict” interchangeably. In the context of the model, where firms perfectly predict the M shock before setting prices, the use of AI is equivalent to observing whether the shock has occurred before setting prices. We discuss the possibility of endogenous adoption of AI in Section 5.

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mize expected static single-period profits given what is observed about M.17 j Let πk indicate industry profits if both firms charge

an agreed upon number of periods, S, in the event they earn no profits in a period after the M shock has occurred. Importantly, for a given set of market parameters, the duration of the punishment phase S may be different than the pre-AI punishment period T. This is because we assume that the punishment periods are chosen to maximize firm profits while keeping collusion incentive compatible. We now derive the discounted ex-ante value of coordination after adopting AI, denoted as U. The new discounted value of coordination is, as in the pre-AI case, one-half the expected single-period monopolist profits plus the discounted continuation value when either zero or one of the shocks occurs, plus the discounted punishment continuation value weighted by the probability that both shocks occur:

price p j , j = l, h in demand state k. The demand state in the post-AI environment differentiates between the two shocks (the observable M state is listed first): k = hh with probability (1 − μ )(1 − σ ), k = hl with probability (1 − μ )σ , and k = lh with probability μ(1 − σ ). The optimal collusive price will depend on whether the M shock will not occur ( j = h) or will occur ( j = l). If M is observed to occur, then the optimal pl will simply be the monopoly price assuming  does not occur because if it does, then no sales will be made at any price. Conversely, if M does not occur, then the optimal price is defined πh

πh

as ph ≡ argmax p (1 − σ ) 2hh + (σ ) 2hl . Appendix E presents the derivation of the collusive prices when the M shock does not occur for the case of linear demand. Thus, there are now four possible profit states. If the M shock does not occur, then either  occurs and demand is reduced or  does not occur and there is no reduction in demand. If the M shock does occur, then either  occurs and there is no demand or  does not occur and demand is not reduced to 0 at every positive price. Note that, unlike in the pre-AI world, the two reduced demand states will not have the same profits for firms. Since firms can perfectly predict (or observe) the M shock before setting their price, they will tailor their price to account for this information. If the M shock is the cause of the reduced (but not zero) demand state, prices will be lower than if  causes the reduced demand (and the M shock does not occur). This is strictly due to firms updating their expectations of demand and adjusting prices accordingly. As a result, there are four possible profit states in a colluding equilibrium, one of which is still zero profits due both shocks occurring and reducing demand to zero. The non-zero profit states h ≥ πl ≥ πh . can be ordered as follows: πhh lh hl The elimination of uncertainty around one of the shocks alters more about the structure of coordination than just the number of prices to be selected ex ante. The firms can now perfectly infer if cheating has taken place when M has not occurred.18 As a result, we assume that there will be two different punishment rules. We assume that the punishment rule when a firm knows with certainty that its rival has cheated (a scenario unique to when firms observe that the M shock does not occur) is an infinite reversion to competitive pricing. In contrast, we assume coordination will be maintained in the “low” demand states when M is known to occur in much the same manner as before the arrival of AI: if there is zero demand for a firm, the evidence suggests that cheating may have occurred, but the firm does not know this with certainty. Therefore, as in the pre-AI case, we assume colluding firms revert to Nash-Bertrand pricing for

U = (1 − μ )(1 − σ )



h πhh

+μ ( 1 − σ )

2





+ δU + (1 − μ )σ

πlhl 2





πhlh

+ δU + μσ δ S+1U.

2

+ δU



(6)

Equation 6 can be rearranged to express the value of coordinating in the underlying parameters:

U=

h l (1 − μ )(1 − σ )πhh + μ(1 − σ )πlh + (1 − μ )σ πhlh . 2(1 − δ (1 − μσ ) − μσ δ S+1 )

(7) Like the discounted value of collusion before AI ((3)), (7) shows that the value of collusion U is decreasing in the length of the punishment period, S. Therefore, as in the pre-AI case, the optimal duration of punishment periods from the perspective of the firms is the shortest length that still satisfies the firms’ incentive compatibility constraints. Comparing the value of collusion after AI, (7), to the value of collusion before AI, (3), shows that AI changes the value of collusion in several ways. The first difference is the probability-weighted monopoly profits: AI allows firms to better optimize their collusive prices to one of the two shocks, which increases their profits compared to the preAI case, whether one shock occurs or no shocks occur. The second difference is a change in the length of the punishment period from T to S that appears in the denominator of each term. We show below in Section 3.1.3 that for the case of linear demand, if collusion is sustainable in both information regimes, the length of the punishment period will be weakly smaller after AI adoption. All else equal, this also increases the value of collusion, suggesting that colluding firms benefit from AI in this case. As in the pre-AI world, a collusive equilibrium requires that neither firm would prefer to seize all of the profit in a given period by deviating via choosing a lower price even if it triggers a certain price war. However, while there was previously only one incentive compatibility constraint per firm, with AIthere are two. This is because the relative payoffs to deviating from the collusive strategy differ depending on what is observed about the incidence of M. For the incentive compatibility constraints, first, it must be the case that firms would not cheat on the collusive agreement if they learn that the M shock will not happen. Cheating in this case would, by assumption, lead to

17 As discussed below in Section 4, there may be circumstances where these statically optimal prices may be inferior to those chosen to maximize dynamic profits. These dynamically optimal prices will not generally have a closed form solution. 18 The fact that improved prediction capabilities improves a firms ability to monitor the rival firm is one of the ways in which our model differs from that of Miklós-Thal and Tucker (2019).

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to make deviating from the collusive strategy unappealing (i.e. S ≤ T).19 Comparing the two incentive compatibility constraints in inequality 8 and inequality 9 shows that they are in tension. On the one hand, higher values of S make it more likely that the constraint holds when the M shock will occur, because higher values of S decrease the right hand side of inequality 9 faster than they decrease U (ceteris paribus). However, higher values of S make it less likely that the incentive compatibility constraint holds when M will not occur (inequality 8). This is because (7) shows that U is declining in S. These relationships are broadly intuitive. If S is small, then when a firm knows that M has occurred, meaning that there is a significant chance that it will be punished anyway if  also occurs, it may prefer to try to obtain some profit by cheating, because the cost of guaranteeing punishment through deviating is relatively light. By contrast, if S is high, then when a firm knows that it faces a good demand state because M will not occur, it will be inclined to deviate, recognizing that the value of coordinating will be undercut by the likelihood of future prolonged price wars that will arise in equilibrium. The intuition about the contrasting effects of the two constraints on S is formalized by setting the value of coordination, (7), equal to the two constraints to establish a lower and an upper bound on the values S may take in order to sustain collusion. The constraint in the high demand state determines the upper bound of S, and the constraint in the low demand state will determine the lower bound of S. Specifically, substituting for U from (7) into the two compatibility constraints (inequality 8 and inequality 9) and taking the value of S where these hold with equality gives these bounds in terms of model parameters and per-period profits in each of the three possible states with positive demand:

a permanent breakdown in coordination - the continuation value is zero. All of the value from deviation comes from the immediate expected per-period profits. Since in this case only the  shock is possible, these payoffs are the probability of no  shock times the monopoly profits with full demand plus the probability of the  shock times the monopoly profits with the level of demand reh + σ π h . Therefore, the relduced by the shock: (1 − σ )πhh hl evant inequality for sustaining collusion when it is known that M will not occur is that the expected collusive profits plus the value of continuing in collusion be greater than the deviation profits:

(1 − σ )(

h πhh

2

+ δU ) + σ (

πhlh 2

h + δU ) ≥ (1 − σ )πhh + σ πhlh

By rearranging terms, this can be simplified to show that the crucial constraint is that the long run value of coordinating must exceed the scaled static payoffs to coordinating:

U≥

1 h ((1 − σ )πhh + σ πhlh ). 2δ

(8)

The second constraint stems from the requirement that firms will not deviate from colluding when they know that the M shock will occur; this condition is closely analogous to the pre-AI incentive compatibility constraint, inequality 4. If a firm chooses to deviate, unlike in the case where M does not occur, it can still resume coordination after the punishment period. Thus, the payoff to deviating from collusion in this case is the probability of no  shock times the sum of monopoly profits with reduced demand (since M will occur with certainty) demand and the value of collusion discounted by the punishment period plus the probability of the  shock times the sum of zero (zero demand when both shocks occur) and the value of collusion discounted by the punishment period: (1 − σ )(πlhl + δ S+1U ) + σ δ S+1U. The value of collusion is given by the sum of half the monopoly profits with reduced demand and the discounted continuation value of collusion times the probability that  does not occur plus the probability that  does occur times the continuation value of collusion discounted by the punishment period S. Therefore, the relevant inequality to sustain coordination when M will occur is given when the value of collusion outweighs deviating:

(1 − σ ) (

πlhl

 log S=

log(δ )

≤S

and: 

l + δU ) + σ δ S+1U ≥ (1 − σ )(πlh + δ S+1U )

log S≤

(−π l δμ(−1+σ )+π h (−1+σ )(1+δ (−2+μ+μσ ))−π h σ (1+δ (−2+μ+μσ ))) lh

hh

(δμσ (π h (−1+σ )−π h σ )) hh log(δ )

hl



hl

= S.

(11)

Once more, we can rearrange terms to express the incentive compatibility constraint in relation to the long run value of coordinating:

πl 1 ( lh ). S+1 2 δ−δ



(10)

2 +σ δ S+1U.

U≥

h (δ (−1+μ )(πhh (−1+σ )−πhlh σ )+πlhl (−1+δ (1+μ−2μσ ))) h (−1+μ )(−1+σ )−π h (−1+μ )σ +π l (μ−2μσ ))) (δ (πhh hl lh

As in the pre-AI case, the punishment duration must be a positive integer value. In the post-AI case, however, in order for collusion to be sustainable, there must be a positive integer that satisfies both bounds and the positivity requirement. Section 3.1.1 provides a detailed analysis of these requirements in the context of linear demand.

(9)

Inequality 9 shows that when M occurs, the long run value of coordinating must exceed a scaled function of the static payoffs to coordinating when M has occurred. These static profits will be lower than in the pre-AI world since they exclude all high demand periods, which, all else equal, should imply that fewer punishment periods are required

19 Consistent with this intuition, we show this formally for the case of linear demand below.

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3. Comparing equilibria before and after AI adoption with linear demand

Solving for the lower bound of S (S) in the linear demand environment leads to the equation:



log

Above, we pointed out that casual inspection of the equilibrium conditions in the post-AI world suggests that AI could support or impair collusion relative to the pre-AI world.20 In this section, we clarify which of the two possibilities will dominate depending on where in the parameter space markets may be. Our analysis is designed to address a central concern of policy-makers: that AI may both widen the scope of collusion and reduce consumer surplus even further where collusion is already possible. For both tractability and ease of interpretation, we restrict attention to a linear demand setting. This specification simplifies the interactions of the parameter values but retains the essential strategic elements of our general model. Our specific implementation assumes that the inverse market demand function takes the familiar form of P = a − bQ. In the event that either the  or M shocks occur, then demand shifts down by d, i.e., P = (a − d ) − bQ.21 We further simplify the dimensionality of the parameter space by assuming that the magnitude of the demand shock is d = 2a . As indicated above, both duopolists can produce a potentially infinite quantity at zero marginal cost.

S=

δ (μ(σ 2 −2σ +3 )−σ 2 +4σ −5 )+1 δ (μ(σ 2 −2σ +3 )−(σ −2 )2 )

log(δ )



.

(12)

Meanwhile, the formula for the upper bound of S (S) is: 1 −2

S=

log(

δ

σ 3 −3σ 2 +σ +3 μ + (σ −2 )2

σ

log(δ )

)

.

(13)

Unfortunately, there are not closed form expressions delineating the region in the δ , μ, and σ parameter space where the requirements for collusion, ∞ > S > S > 0 hold. However, one can use a numerical model to map collusion regions by calculating the relevant functions of S and T in the parameter space to identify where the collusion criteria are met. Focusing first on the impact of the different uncertainty parameters by holding δ constant, we show how the likelihoods of the two demand shocks affect the sustainability of coordination pre- and post-AI in Fig. 1. For this figure, we fix δ = 0.7, a = 10, and b = 5. The results are qualitatively robust to alternative choices of the linear demand parameters. Specifically, we found (in figures not included here but available upon request) that pre-AI and post-AI collusive equilibria do not change over a range of values for a and b, keeping d = 2a .24 This result is consistent with Ivaldi et al. (2003), who note that changes in the shape of demand have no effect on the sustainability of collusion. The effect of different values of δ is considered below. For both information regimes, Fig. 1 shows that the sustainability of collusion declines with higher values of the uncertainty parameters μ and σ . However, post-AI, the relationship between the two probabilities and the sustainability of coordination is not symmetric as it is in the preAI world. As indicated by the loss of collusion in much of the bottom right quadrant of the Figure, sustainability after the adoption of AI is substantially more sensitive to the probability of the demand shock predicted by AI, μ. Specifically, when σ < 12 , and μ is significantly larger than σ , collusion will only be sustainable in the pre-AI world.25 Thus, for a non-trivial share of the parameter space, the adoption of AI and the attendant increase in transparency actually limits the scope for coordination. In contrast, when μ > 14 but σ is larger than μ, there is a chance that the introduction of AI enables collusion when it was previously impossible. Overall, the Figure shows that the probabilities of the negative demand shocks, especially the AI-predicted shock, can lead to very different competitive outcomes. We next consider how the firms’ identical discount rate affects the sustainability of collusion pre- and postAI. Fig. 2 illustrates this effect by duplicating the analysis shown in Fig. 1 but for four levels of δ .26 The Figure

3.1. Changes in the sustainability of collusion 3.1.1. Overview Before turning to how variation in the market characteristic state space changes the welfare implications of AI adoption, we begin by analyzing when collusion will be sustainable under the different information regimes. Pre-AI, collusion will be sustainable if and only if there is a finite, positive price war length, T. The expression for T when demand is linear is no different from (5) as the formula does not contain profit terms. Proposition 1 follows from combining T > 0 with (3) and inequality 5.22 Proposition 1. Pre-AI, collusion is sustainable only if (1) δ > 1 2δ −1 2 and (2) μσ < 2δ . In short, the sustainability of collusion in the pre-AI information regime requires a non-trivial discount factor and a limited likelihood that both shocks occur in the same period, which triggers a price war even when no firm deviates. Post-AI collusion requires three separate conditions on the price war length bounds, S and S, to be true. First, S must be positive and finite. Second, S must also be positive and finite. Finally, it must be the case that S > S.23 20 As noted in the model setup, we do not analyze the special case when behavior in the pre-AI and post-AI settings will be identical. This will occur when μ = 1 and σ = 0. For those parameter values, the post-AI and pre-AI expressions are identical. For example, the same prices will be chosen, and the lower bound of S is equivalent to the expression for T. As in the case for the pre-AI world, the upper bound on S for these parameter values is infinity. 21 We present expressions for elements of interest (e.g., prices, quantities, and profits) of interest in Appendix E. 22 The proof is provided in Appendix B. 23 Given the discrete time nature of the environment, the third constraint is really that floor(S ) ≥ ceiling(S ).

24 We calculated equilibria for a over [2,20] and b over [0.4,10] each in increments of 0.2, for each of δ ∈ {0.6, 0.7, 0.8, 0.9} and found no changes in equilibria for each δ , μ, σ tuple. 25 It is worth noting that as δ approaches 1, the area where collusion is sustainable pre-AI but not post decreases. 26 As in Fig. 1, we fix a = 10 and b = 5 for these plots.

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Fig. 1. Sustainability of Collusion as a Function of Demand Shock Probabilities and AI (δ = 0.7)

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Fig. 2. Sustainability of Collusion as a Function of δ , Demand Shock Probabilities, and AI

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shows that the same qualitative patterns exist across discount rates. However, though collusion is sustainable in more of the μ and σ parameter space with higher levels of δ in both information regimes, the number of points in the parameter space where sustainability changes is more substantial in the post-AI world. For δ of 0.9, there is virtually no part of the parameter space where collusion is sustainable in the pre-AI world but not the post-AI one. In contrast, for close to half of the area where collusion was sustainable pre-AI with δ = 0.6, it would not be sustainable post-AI. The Figures presented above show that collusion’s sustainability may vary – substantially – across the different information regimes. However, the Figures do not provide clear insight into why competition’s character may change as AI improves firms’ ability to diagnose their demand environment. Therefore, we now consider how changes to the discount rate and demand uncertainty parameters affect the value of colluding as well as the firms’ different incentive compatibility constraints, which ultimately determine if collusion is sustainable. 3.1.2. Greater likelihoods of negative demand shocks hinder coordination – But asymmetrically post-AI We turn first to understanding the impact of the likelihood of negative demand shocks on collusion in the preAI and post-AI worlds.27 Because there are no closed form solutions for where collusion is sustainable post-AI, our analysis must be somewhat indirect. However, the underlying intuition for the patterns observed in Figs. 1 and 2 is straightforward. All else equal, higher likelihoods of the two demand shocks negatively affect the value of coordinating both preand post-AI. Setting aside – for the moment – their impact on the payoffs to deviating, the reduced incidence of high payoff states due to higher σ and μ levels makes sustaining coordination more difficult because it reduces the continuation value. However, the impact of these factors will differ across the information environments. In particular, while the regions that support pre-AI collusion are symmetric around the line μ = σ in Fig. 1 and Fig. 2, higher likelihoods of the different shocks have quite different implications once they can be distinguished and one is perfectly observable.28 After the adoption of AI by the firms, there is no symmetry between the μ, σ values and the region that supports collusion. The differences across the two information regimes stem from the fact that greater transparency both pushes firms towards a coordinated equilibrium and pulls them away. To understand the shape of post-AI collusion regions in Figs. 1 and 2, we note that for any given length of price war S, the ability to set a price with less uncertainty about demand increases expected static profits, and thus the payoffs to coordinating. This will be true in every part of the parameter space; inspection of the numerators of (3) and

(7) reveals this point. Ceteris paribus (in particular, ignoring the differences in likelihood of the observable and unobservable shocks), the higher continuation values of coordination that follow from higher static profits will tend to lead to an expanded scope for coordination (and allow shorter price wars in a collusive equilibrium). Thus, it is not surprising that we observe that parts of the parameter space that could not sustain coordination pre-AI become able to so post-AI. However, to understand the overall effect of μ and σ on the sustainability of collusion after the adoption of AI, we must also address the fact that the increased availability of information on demand conditions also affects firms’ incentives to deviate from a collusive strategy. We discuss the intuition of this consideration and then the formal basis of it, all of which follows from inequality 4, inequality 8, and inequality 9. To start, the change in the value of deviation after the adoption of AI will be greater in some parts of the parameter space than others. We focus on parameter values where collusion most often changes with the introduction of AI, those where the likelihood of the observed shock, μ, is high. Keeping the other model parameters constant, high μ values lower the value of colluding. Inequality 9 shows that long penalty periods may be necessary to keep firms from deviating in the often occurring “low” demand states. Otherwise, it would not be worth forgoing a chance at receiving the monopolist’s profit level during the periods when the unobservable shock does not occur. However, a prolonged punishment period is not necessarily available as a means of deterring deviation after the firms adopt AI. A large S may cause the incentive compatibility constraint for the “high” state when the M negative shock is known to have not occurred (i.e., inequality 8) to be violated. This is because if the punishment period is long enough, then the continuation value will be smaller than the expected gains from deviating and seizing the entire market when demand is observed to be high, even if it means a permanent end to coordination. Therefore, it is intuitive that we observe that coordination post-AI may not be sustainable for high values of μ even when it was sustainable pre-AI (holding δ constant).29 The δ = 0.6 plot in Fig. 2 illustrates this dynamic given the large “Pre-AI only" collusion region in the bottom right quadrant. Formally, the tension created by the increase in transparency is apparent in the inequalities that determine the bounds on price war duration in the post-AI worlds, inequality 8 and inequality 9. As we show in Appendix C, the value of collusion, U, is decreasing as μ increases, but the deviation values are not decreasing in μ (as the M is observed before a firm would make the choice to deviate). The dynamic in the pre-AI world is different: both the value of collusion V and the value of deviating are decreasing in μ. The change in the underlying effect of μ on the value functions and deviation values means that collusion

27 In Appendix C, we derive expressions to analyze how marginal changes in σ and μ affect the value of coordinating as well as the incentive compatibility constraints. 28 For any two (μ, σ ) pairs, if σ1 = μ2 and μ1 = σ2 the sustainability of collusion will be the same pre-AI.

29 The same logic does not apply for large values of σ , because in addition to negatively affecting the continuation value, thereby putting upward pressure on S, larger values of σ also negatively affect the payoff to deviating when M does not occur, mitigating downward pressure on S.

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 1−2δ+2δμσ 

in high μ, low σ parts of the parameter space may be lost after the adoption of AI.

T −S =

Log

2μσ δ −δ

Log(δ )

(−5+4σ −σ +μ(3−2σ +σ )) Log[ 1+δδ(− (−2+σ )2 +μ(3−2σ +σ 2 )) ] 2

−

3.1.3. Higher discount rates facilitate coordination – especially post-AI In this section, we analyze why changes in the discount rate, δ , alter collusion sustainability as observed in Fig. 2. As usual in models of collusion, the Figure shows that holding the information regime constant, an increase in δ makes coordination sustainable in more (μ, σ ) pairs. However, Fig. 2 shows that higher discount rates disproportionately increase the scope for collusion post-AI. Because of the lack of closed form solutions for where collusion is sustainable in the post-AI period, we again emphasize intuition for this result with references to derivations that show how sensitive the value of coordinating is to how much firms value the future. Given our linear demand setting, the value of collusion in the pre-AI world, (2), can be re-expressed as:

V =

≥ 0.

(16)

Given the bounds on δ identified above in Section 3.1.1 and the non-negativity of S and T, this can be simplified to:

1 − 2δ + 2δμσ ≤ 2μσ δ − δ 1 + δ (−5 + 4σ − σ 2 + μ(3 − 2σ + σ 2 )) . δ (−(−2 + σ )2 + μ(3 − 2σ + σ 2 ))

(17)

Imposing the constraints defined above in Section 3.1.1 for T reveals that this relationship will always be true. Thus, when collusion is sustainable in both information regimes, price wars are weakly shorter in the post-AI world, benefiting firms at the expense of consumers, particularly as the discount rate grows large. The above analysis shows that placing more value on the future through a higher δ means that, all else equal, a world with more information on demand (through AIenabled anticipation of M) will be more valuable for the firms than one without. This dynamic thus explains why the panels of Fig. 2 show that as δ increases both the area where collusion can only be sustained post-AI grows and the area where collusion is only sustainable pre-AI shrinks. However, it does not necessarily clarify why the shape of the region where coordination is sustainable postAI changes with higher values of δ , as Fig. 2 indicates that the asymmetric effects of the different uncertainty parameters very depending on the level of the discount rate. The intuition for why post-AI coordination becomes substantially more sustainable for parameter combinations involving higher levels of μ as δ increases is as follows. As discussed above, when the observable negative demand shock, M, is quite likely (high values of μ) relative to the probability of the unobservable shock (σ ), it may be difficult to sustain coordination post-AI. This reflects the need for a long price war to deter deviation in “low” demand states when M has occurred, which may cause to S > S, precipitating a breakdown in coordination. However, as δ increases, the net present value of the expected gains from coordinating in the future increase, meaning that a less stringent penalty length is required. This will tend to increase the likelihood that S > S, enabling coordination to be sustained.

a2 (−2 + μ + σ )2 , (14) 64b(−1 + δ (1 + (−1 + δ T +1 )μσ ))(−1 + μσ )

while the value of collusion after both firms adopt AI, (6), will be:

U=

Log[δ ]

2

a2 (−(−2 + σ )2 + μ(3 + (−3 + σ )σ ))(−1 + μσ ) . 64b(−1 + δ (1 + (−1 + δ S+1 )μσ ))(−1 + μσ ) (15)

Comparing (14) and (15), several things are clear. First, the denominators are identical but for the duration of the punishment period. Second, the numerators of (15) and (14) only matter multiplicatively in the analysis of this subsection since they do not contain δ . Moreover, both are greater than 0, which is intuitive since they represent combinations of probabilities and profits, both of which are weakly greater than 0. Since the denominators of these value functions are decreasing in δ (but remain positive), both U and V are increasing in δ . These factors imply that the value of coordinating rises faster with increases in δ post-AI. This can be shown as follows. We focus first on the numerators, ignoring any possible difference between T and S. The numerator of the post-AI collusion continuation value U (15) is greater than the numerator of pre-AI collusion continuation value (14): their difference is a2 (μ − 1 )μ(σ − 1 )3 , which will always be greater than 0. For cases where S = T , the increase in U must, therefore, be greater than the increase in V for a given increase in δ . Next, we turn to the denominators, which only differ in the lengths of the price wars. Per the envelope theorem, S and T should not change in response to marginal changes in the parameters. Thus, the crucial question is if we can affirmatively state whether S is always less than or equal to T. When coordination is sustainable in both the pre-AI and post-AI worlds, then the price war durations will be T as defined by (5) and S = S as defined by (12). Therefore, T ≥ S requires that:

3.2. Implications for welfare Overall, our analyses have shown that, in a linear demand setting, both the coordination fostering and coordination impairing impulses of transparency matter. We turn now to assessing how increases in the information available to firms, and their impact on coordination, affect welfare. Once more, due to a lack of closed form solutions for the change in welfare with AI adoption as a function of market parameters, we use figures to consider how various 12

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Fig. 3. Change in Consumer Surplus as a Function of Demand Shock Probabilities and AI (δ = 0.7)

measures change in relation to combinations of the parameter values of μ and σ .30 Fig. 3 shows how consumer surplus changes due to the implementation of AI by the firms as a function of μ and σ . The Figure is divided into two subplots to show the areas where consumer surplus decreases (left panel) separately from the areas where consumer surplus increases (right panel) after AI adoption by the firms. In terms of the regions, the Figure closely mirrors Fig. 1, with consumer surplus declining in parameter regions where coordination is always sustainable or becomes newly sustainable, and consumers gaining only when coordination becomes nonviable with the introduction of AI. This result implies that consumer surplus does not improve with the implementation of AI unless AI makes collusion unsustainable.31 In line with expectations, we find that the highest percentage of lost consumer surplus occurs where collusion is possible only after the adoption of AI. Fig. 4 shows the percentage change in total welfare due to the implementation of AI. As with Fig. 3, the Fig-

ure is divided into two subplots to show the areas where welfare increases (right panel) separately from the areas where welfare decreases (left panel) after AI adoption by the firms. We find that total welfare changes closely track changes in consumer surplus. That is, the magnitude of changes in consumer surplus swamp the magnitude of the changes in producer surplus. Where consumers are worse off, total welfare falls and vice-versa. Fig. 5 shows the change in optimal price war length that makes collusion sustainable for parameter values where collusion is sustainable regardless of the presence of AI. This figure represents the value (S − T ) at each point in the parameter space when δ = 0.7. Interestingly, changes in consumer surplus do not necessarily track changes in war length. In particular, in the parts of the parameter space where war length drops significantly (e.g., by more than 10 periods), the percentage change in consumer welfare is not noticeably outsized. Relative to other factors, a significantly longer price war length does not significantly increase consumer surplus. Finally, we provide further details on the welfare effects of changes in Appendix D. In that section, we analyze the switch from a pre-AI equilibrium to a post-AI one for approximately 4 million markets, each a specific combination of the model parameter values. We then estimate the average effect of “all else equal " changes in each of these parameters on the difference in pre-AI and post-AI producer surplus and consumer surplus across the parameter value combinations we analyzed.

30 For these Figures, we fix the values of the model parameters at the same values as used in Fig. 1. Intuitively, the results are qualitatively similar for other values of the discount rate δ ; however, higher levels of δ are associated with better outcomes for firms and worse outcomes for consumers. 31 It is possible that consumers could be better off even under post-AI collusion if the ability to better “price discriminate” led to customers being served in low demand states that would have otherwise gone completely unserved.

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Fig. 4. Change in Total Surplus as a Function of Demand Shock Probabilities and AI (δ = 0.7)

We find that, on average, larger δ values, in combination with the introduction of AI, lead to larger reductions in consumer surplus and larger increases producer surplus. This is not surprising, as larger δ values generally facilitate collusion all else equal, but the introduction of AI magnifies this effect on average. Importantly, this analysis also finds that if a higher value of a parameter is associated with larger consumer surplus post-AI on average, then it is associated with lower producer surplus on average. The reverse is also true. No parameter ever moves the difference in consumer surplus and producer surplus after the adoption of AI in the same direction on average. Also, a higher likelihood of the observable shock, μ, is associated with better consumer outcomes post-AI, which is consistent with our analysis showing that collusion is difficult to sustain when a high likelihood negative shock becomes observable. Ultimately, we find these results to be informative of average effects but it should not to be confused with a comparative statics analysis. Indeed, any policy analysis of the effect of AI depends on specific market parameter values, not the average effect across the full range of parameter values we consider here. Overall, consistent with some policy-makers’ fears, our analysis shows that there are parts of the parameter space where AI enables collusion and reduces welfare. However, we also find that there are parts of the parameter space that can no longer sustain collusion after AI’s adoption,

leading to substantial gains in consumer and total surplus. Thus, the implications of reduced demand uncertainty are not unambiguously better for firms than they are for consumers. 4. Extension: What if firms simultaneously set S and the collusive prices? Pre-AI, the sustainability of collusion does not depend on firms’ static payoffs because no profit terms appear in (5), which provides the criteria for the existence of a finite punishment period. Therefore, firms would never have an incentive to choose prices other than the optimal price given the possibility of demand shocks given in (1). PostAI, however, the same may not be true. Because different profit terms appear in (10) and (11), choosing different collusive prices may have the effect of changing the value of the profit terms that determine the optimal price war length in a way that admits a finite price war period and sustains collusion. That is, if firms set a different collusive price than the monopoly price when they are unable to sustain collusion at the monopoly prices, the firms may be able to collude after all. By selecting different collusive prices and “restoring collusion,” firms could reduce the amount of the parameter space where AI actually works to consumers’ benefit, providing further support for some of the more forceful policy concerns associated with AI. 14

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Fig. 5. Change in Equilibrium Punishment Duration as a Function of Demand Shock Probabilities and AI (top coded at a reduction 10 periods, δ = 0.7)

increments. We let a = 10 and vary b between 1 and 10 in increments of 1, which is that same as in the analysis of Appendix D. Fig. 6 shows how the scope for post-AI collusion may be expanded when sub-monopoly prices are chosen.32 It indicates that when sub-monopoly pricing is possible (indicated by "Below" in the Figure), collusion becomes sustainable in a large portion of the parameter space where it was not in both the pre-AI and post-AI with monopoly prices settings. Indeed, we find that allowing firms to simultaneously choose prices and price war durations means that there is no longer any part of the parameter space where collusion was sustainable pre-AI, but not post-AI. All else equal, this appears to support the more pronounced

To address this question, we numerically examine alternative equilibrium outcomes in our linear demand setting. Specifically, we consider how price war lengths and firms’ profits change when different combinations of collusive prices are chosen. We select alternative collusive equilibria on the basis of maximizing firm values when colluding. We assume in this case that if there exist prices below the monopoly price that sustain collusion, firms will choose to collude at the price from this set that produces the largest U, or the largest continuation value. We assess this using a brute force, grid search approach. Because of the computational burden of this process, we restrict the considered parameter space to a subset of that used in our full linear numerical work discussed above and at length in Appendix D. Specifically, we set the grid search tick size between possible sub-monopoly collusive prices to 0.01 and evaluate different values of σ and μ at 0.002

32 For this and the subsequent Figure, we fix the values of the model parameters at the same values as used in Fig. 1.

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Fig. 6. Sustainability of Collusion as a Function of Demand Shock Probabilities and AI with sub-Monopoly Pricing (δ = 0.7)

concerns observed in the legal commentary about the impact of AI on competition. However, all else may not be equal. While Fig. 6 shows that allowing collusive prices to be simultaneously determined with punishment periods leads to the unambiguous prediction that AI expands the scope for some degree of collusion, it does not necessarily undermine the conclusion that AI’s welfare implications are ambiguous. To consider this, we plot the sign of changes in consumer welfare from pre-AI to post-AI when prices are allowed to vary in Fig. 7. The Figure indicates that relative to the pre-AI world, consumer welfare still increases in parts of the parameter space even when some form of collusion is sustainable via non-monopoly pricing in the post-AI world. In these cases, although the post-AI price is not reduced to marginal cost as it is when collusion is not sustainable, the sub-monopoly price is so much less than the pre-AI monopoly price that consumers are better off.

Therefore, our underlying conclusion that exogenous reductions in demand uncertainty have ambiguous implications for firms and consumers alike remains robust. This conclusion is not an artifact of restricting firms to monopoly prices, which might be easily identified ex ante but sub-optimal from the firms’ perspective. 5. Discussion: Why adopt AI if it renders collusion unsustainable? Given that AI adoption renders collusion unsustainable (or unprofitable relative to non-adoption) for some parameter values, it is reasonable to ask why firms would ever adopt AI in these scenarios. We now discuss several reasons why AI might be adopted even if it led to reductions in producer surplus. First, competitors may find themselves with a coordination problem akin to the standard “entry” game 16

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Fig. 7. Change in Consumer Welfare as a Function of Uncertainty Levels and AI with sub-Monopoly Pricing (δ = 0.7)

(Farrell, 1987). Consider a duopoly market where the two firms have successfully colluded without AI. Now, it becomes possible to adopt AI. It is observable that if both firms adopt the technology, they will no longer be able to successfully collude. However, each firm may privately have an incentive to adopt AI. This is because for some regions of the parameter space, a firm with AI may find it profit-maximizing to depart from the non-AI collusive strategy if the other firm does not have AI. The other firm will not necessarily alter its behavior, even though it now earns lower profits, because it still earns positive profits during “low” demand phases. With no means of coordinating over which firm will adopt and increase its profits at the rival’s expense, both might choose to do so. Second, firms may choose to invest in developing or adopting AI if it is uncertain ex ante which of the different

shocks will become publicly observable. Thus, firms may invest in AI expecting to be able to coordinate significantly better ex post, but subsequently learn that the technology actually makes it no longer sustainable. We believe such misplaced expectations are not unreasonable in connection to AI and other digital means of reducing demand uncertainty. The technology space is evolving extremely rapidly, plausibly making predictions about the capabilities of new innovations difficult. Third, we believe there may be circumstances where AI adoption may be an ancillary effect of decisions driven by other factors influencing firms’ objective functions. For example, multi-division firms may have an incentive to adopt more sophisticated analytical tools to improve operations that have the additional effect of reducing the scope for coordination in some markets.

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In future work, we hope to probe the robustness of our results to more fully endogenizing the development and adoption of the demand reducing technologies.

Because of these factors, we do not see that meaningful additional insight would be gained by allowing metering in the model.

6. Conclusion

Appendix B. Proof of parameter restrictions on sustainability of Pre-AI collusion Proposition 1 : Collusion is sustainable only if (1) δ > 12 and (2) μσ < 2δ2−1 δ . The incentive compatibility constraint (inequality 5) and the value of collusion in the pre-AI world requires:

A number of works from antitrust scholars have claimed that AI and algorithms will lead to a dramatic transfer of surplus from consumers to producers. Often, this is driven by the belief that the algorithms will “figure out” a way to collude. Other parts of the literature have pushed back, outlining the many practical impediments that the original criticism missed. While we are sympathetic to the counterargument, we think it is important not to lose sight of the fact that the use of AI and/or data processing algorithms could nevertheless affect the incidence and costs of collusion. In this paper, we show how improved predictions about the state of demand, a plausible outcome of greater analytical sophistication, can influence firm conduct. Our results imply that the effect is ambiguous, but that there are parts of the parameter space where the adoption of improved analytical tools could harm consumers when the market structure is held constant and demand-side changes are not allowed. We await the development of more evidence that may show where – in practice – the different impulses we identify may dominate.

(1 − δ T +1 )((1 − μ )(1 − σ )πhm + (μ + σ − 2μσ )πlm ) 2(1 − (1 − μσ )δ − μσ δ T +1 ) ≥ (1 − μ )(1 − σ )πhm + (μ + σ − 2μσ )πlm . (B.1) (1 − μ )(1 − σ )πhm + (μ + σ − 2μσ )πlm > 0 because μ and σ are probabilities. Moreover, 2(1 − (1 − μσ )δ − μσ δ T +1 ) > 0, for the same reason and that 1 > δ > 0. Therefore, we can simplify inequality B.1 to: 1 − δ T +1 ≥ 2(1 − (1 − μσ )δ − μσ δ T +1 ) by dividing by the right hand side of inequality B.1 and multiplying by the left hand side. In turn, this expression can be further simplified by adding and subtracting terms to both sides to yield:

2δ (1 − μσ ) − 1 ≥

δ T +1 (1 − 2μσ ).

(B.2)

To gain further insight into the relevant parameter space, we consider two cases, μσ < 12 and μσ > 12 . If μσ < 12 , then the right hand side of the inequality above is positive but can be made arbitrarily close to zero by increasing T. Collusion will thus be sustainable for some value of T if the left-hand side of the inequality is not negative. The left-hand side will be positive when 2δ (1 − μσ ) − 1 > 0, which can be rewritten as μσ < 2δ2−1 δ . This, in turn, implies that δ > 1/2, because otherwise the left hand side would have to be less than zero, which is not possible for probability parameters. Thus, if collusion is sustainable when μσ < 12 then δ > 12 .33 Now, let us consider whether collusion can be sustained if μσ > 12 . If this were true, then both the left-hand and right-hand sides of inequality B.2 are less than zero. All else equal, smaller values of T make this constraint easier to meet because they make the right hand side larger in negative magnitude. Therefore, setting aside the fact that as mentioned above in Section 2.2 it would not be a punishment length that can sustain collusion, if one sets T = 0, one would obtain the widest range of parameter values capable of meeting the constraint. By setting T = 0, inequality B.2 is transformed to 2δ − 2δμσ − 1 ≥ δ − 2δμσ . However, inspection shows that

Appendix A. Discussion of metering We believe the common assumption in models of collusion that metering cannot take place is appropriate in our context. If relaxed, the nature of the game would either become quite complicated without necessarily adding much additional intuition or would result in an uninteresting solution. We note that metering while deviating can be profitable only if there is a possibility that neither demand shock will occur. (If demand is low at best, then the deviating firm cannot gain any sales as half the sales have to go to the rival lest a price war be initiated.) The metering firm wants to leave half of the low demand quantity for the rival. We do not think there is a sustainable equilibrium here if firms are basing their decisions solely on what they observe in a given period: if one firm can improve profits by lowering price while metering quantity sold, then both will want to do so. However, if both do this, then their mutual cheating would be revealed in good states of the world. This would seem to lead to a breakdown in coordination. At this point, then, we have the standard Bertrand dynamic that each firm will want to undercut its rival in equilibrium (all the time if pre-AI and all the time that the observable shock does not occur post-AI). Alternatively, to sustain collusion, one might base continuing to coordinate on what was observed over time about the incidence of different demand states, possibly through an expanded set of punishment rules. Such a game seems feasible in theory but excessively complicated and probably dependent on very particular sets of parameter values.

33 For this range of δ , the two shock probability parameters must meet one of the following conditions:

0<σ ≤

2δ − 1 2δ

(B.3)

or:

2δ − 1 2δ − 1 < σ < 1 and 0 < μ < . 2δ 2δσ

18

(B.4)

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this equation is satisfied only if δ ≥ 1, which is outside the range of a well-defined discounting parameter. Therefore, collusion is not sustainable for μσ > 12 . Thus, collusion in the pre-AI period will be feasible only if μσ < 12 and μσ < 2δ2−1 δ .

Table D1 Descriptive Statistics of Linear Demand Parameterizations

Appendix C. Analyzing the value of coordinating and defecting Focusing on the value of collusion in the pre-AI (14) and post-AI (15) worlds in our linear demand example, we observe that the denominators of the different terms are equal but for the different price war duration terms, T and S. Per the envelope theorem, these should not change in response to marginal changes in the parameters. Therefore, for the sake of tractability, we focus on differences in the changes to the numerators of U and V, which we label as Uˆ and Vˆ , respectively. Differentiating with respect to μ, the probability of the negative demand shock that is perfectly predicted by AI, we find that:

dVˆ a2 (−2 + μ + σ )(2 + σ 2 − σ (2 + μ )) = dμ 32b(−1 + μσ )2

Statistic

N

Mean

St. Dev.

Min

Max

CSpre CSpost V U CS PS

3,936,600 3,936,600 3,936,600 3,936,600 3,936,600 3,936,600

16.043 15.356 2.048 2.162 - 0.688 0.229

19.140 17.842 4.152 4.433 4.218 1.374

0.172 0.172 0.000 0.000 - 49.140 - 19.183

179.044 171.550 55.430 56.132 57.549 15.447

In contrast, differentiating the right hand side of inequality 8 with respect to μ shows: M=h dICC post

dμ

= 0,

(C.7)

where M = h denotes the fact that M did not occur. Likewise, the effect of μ on the right hand side of inequality 9 is 0. The lack of impact of μ is intuitive. The postAI world is one where decision-making takes place after M has been observed. Therefore, the profit terms affecting participation do not contain μ. Variation in σ will, however, affect the profitability of deviation in the post-AI world. Differentiating the right hand side of inequality 8 with respect to σ shows:

(C.1)

and:

dUˆ a2 (−3 + 3σ − σ 2 ) = . dμ 32b

[m3Gdc;September 24, 2020;9:10]

M=h dICC post

dσ

(C.2)

=

a2 (−2 + σ ) , 16b

(C.8)

The signs of both equations are negative. Focusing on σ , we find that the first derivatives of the value functions are:

which is negative. Thus, observing that M has not taken place, a higher likelihood that the second shock will take place tends to make it easier to sustain coordination. By contrast, the effect on the right hand side inequality 9 is:

dVˆ a2 (−2 + μ + σ )(2 + μ2 − μ(2 + σ )) = dσ 32b(−1 + μσ )2

M=l dICC post

(C.3)

dσ

and:

dUˆ a2 (−4 + 3μ + 2σ − 2μσ ) = . dσ 32b

(C.4)

Appendix D. Numerical exercises with generalized linear demand model To further disentangle the relative importance of different parameters to the sustainability and impact of coordination, we perform a series of numerical exercises. For these analyses, we fix a = 10 and explore what happens as the other parameter values in the linear demand framework change. Specifically, we allow b to vary between 1 and 10, d to vary between 2.5 and 8, μ and σ (the independent probabilities of the M and  shocks respectively) to vary between 0.1 and 0.9, and δ to vary between 0.5 and 0.9. Specifically, we vary μ and σ uniformly by 0.002; δ uniformly by 0.1; and b uniformly by 1. We define d as 10 x with x varying from 1.25 to 4 by 0.25. Each unique combination of these values appears exactly one time in our “data” set. Table D.1 shows descriptive statistics for key outcome variables of the numerical models. CSpre and CSpost represent the net present values of consumer surplus in the preand post-AI worlds. U and V are, as above, the per-firm amount of discounted profit streams,  indicates change, and PS represents total producer surplus. The Table indicates that on average firm values are higher after the adop-

(C.5)

Similarly, the effect of a change in σ on the pre-AI incentive compatibility constraint is:

dICC pre a2 (−2 + μ + σ )(2 + μ2 − μ(2 + σ )) = . dσ 32b(−1 + μσ )2

(C.9)

where M = l indicates that M did occur.

Once more, the sign in both cases is negative. In short, higher likelihoods of the two shocks reduce the value of coordinating since they raise the likelihood of being in low profit states. We now consider the impact of the uncertainty parameters on the participation constraints. Taking the derivative of the right hand side of the incentive compatibility constraint (i.e., the present value of deviating) in the pre-AI world (4) with respect to μ we find:

dICC pre a2 (−2 + μ + σ )(2 + σ 2 − (2 + μ )σ ) = . dμ 32b(−1 + μσ )2

= 0,

(C.6)

Both of these terms will always be negative in sign. Thus, ignoring the impact on long-run collusive profits, a higher probability of a negative shock makes coordination easier to sustain because the expected profits from deviating are lower in the pre-AI world. 19

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with superior coordination. However, greater levels of μ are associated with better outcomes for consumers, and worse ones for producers. We interpret this result as indicating that, on average, sustaining collusion is harder when a high likelihood negative shock becomes observable, consistent with our analysis in Section III.

Table D2 Decomposition of Economic Outcomes on Parameter Values Dependent variable:

PS

CS

(1)

(2)

1(CS > 0) (3)

b

- 0.058 0.174 0.00000 (0.0002) (0.001) (0.00004) d 0.044 - 0.123 0.012 (0.0004) (0.001) (0.0001) δ 3.072 - 9.275 - 0.106 (0.005) (0.014) (0.001) μ - 0.644 1.900 0.154 (0.003) (0.009) (0.0004) σ 0.248 - 0.796 - 0.225 (0.003) (0.009) (0.0004) Constant - 1.598 4.831 0.108 (0.004) (0.013) (0.001) Observations 3,936,600 3,936,600 3,936,600 2 0.131 0.126 0.100 R Note: Standard errors in parentheses beneath coefficients.

Appendix E. Linear demand results when collusion is sustainable Below, we provide the analytical formulas for key elements of interest for a general linear model, where (inverse) market demand has the following form P = a − bQ and the occurrence of a shock leads the intercept to decline by d. E1. Pre-AI First, we derive the monopoly price given the possibility of negative demand shocks. The optimization problem will be:

tion of AI leads to reduced uncertainty (i.e. U > V). In contrast, the average amount of consumer surplus declines. However, relative to the pre-AI mean levels of surplus, neither average change is particularly large. The relationships between the various elements of the parameter space and economic outcomes are non-linear. However, intuition about modest changes in the different elements that influence consumer and producer surplus can be gleaned by regressing them on a linearly separable function of the different model parameters. The results of these regressions on changes in producer surplus, consumer surplus, and an indicator for an improvement in consumer surplus (1(CS > 0)) are shown in Table D.2. Table D.2 helps to clarify the relative salience of different elements. The first two columns suggest that, on average, the interests of consumers and firms are never aligned about the desirability of AI when demand is linear. Any parameter that is associated with an average increase in producer surplus following the arrival of AI is also associated with an average decline in consumer surplus, and vice versa. In general, the regressions indicate that when demand is more steeply sloped (as measured by b), consumers benefit from AI. Conversely, when the negative shocks are larger, on average, the introduction of AI is associated with greater consumer harm. These results connect to the size of profits in the different states of the world. As b increases, the gaps between the different profit states decrease. Interestingly, the final two columns indicate that d’s impact is non-linear. In expected value terms, a higher probability of the negative shock revealed by AI leads to lower consumer surplus. However, it increases the odds that consumer surplus rises relative to the pre-AI world. On average, δ has a positive impact on producer surplus and a negative one on consumers. This suggests that, all else equal, collusion is more easily sustained, more lucrative, or both when firms highly value future profits and there is decreased demand uncertainty. The two demand shock probabilities have opposite effects. The estimates for σ suggest that larger absolute changes in the level of demand uncertainty are associated



p2 a max (1 − μ )(1 − σ ) p− p b b



  p2 a−d +(μ + σ − 2μσ ) p− . b

(E.1)

b

The first order condition (FOC) for this is:

a

F OC : 0 = (1 − σ − μ + μσ )

b

−

2p b



  2p a−d +(μ + σ − 2μσ ) − . b

(E.2)

b

Solving for p yields:

pm =

a(1 − μσ ) − d (μ + σ − 2μσ ) 2(1 − μσ )

(E.3)

If both firms charge the monopoly price, the market quantity if both shocks do not occur can take two different values. These are found by replacing p in the relevant demand function with the monopoly price shown above. First, if no shocks occur, then the market quantity will be:

Qhm

1 a = − b b =



a(1 − μσ ) − d (μ + σ − 2μσ ) 2(1 − μσ )



a + dμ + dσ − aμσ − 2dμσ . 2b − 2bμσ

(E.4)

If one shock occurs, then the market quantity will be:

Qlm = =

1 a−d − b b



a(1 − μσ ) − d (μ + σ − 2μσ ) 2(1 − μσ )

a − 2d + dμ + dσ − aμσ . 2b − 2bμσ



(E.5)

As with quantities, there will be two possible profit outcomes if both shocks do not occur. For a given firm playing the collusive strategy, these are determined by dividing the market quantity equally. 20

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Meanwhile, if  does occur, then quantity demanded will equal:

Thus, if neither shock occurs, a colluding firm’s static profits will be:

πhm

=

2

Qhm 2

pm =

[m3Gdc;September 24, 2020;9:10]

(−a + (a + 2d )μσ − d (μ + σ ))(d (μ + σ − 2μσ ) + a(−1 + μσ )) . 8b(−1 + μσ )2

(E.6)

Conversely, if one shock occurs, a colluding firm’s static profits will be:

πlm 2

=

Qlm 2

pm =

(−d (−2 + μ + σ ) + a(−1 + μσ ))(d (μ + σ − 2μσ ) + a(−1 + μσ )) . 8b(−1 + μσ )2

With a linear demand structure assumed, we are able to analytically solve for expected consumer welfare in each period. This will be equal to: Qm a − pm 2h + Qm − d − pm 2l .

CSno = (1 − μ )(1 − σ )(

(μ + σ − 2μσ )(a

Qlh =

)

)

E2. Post-AI

πll

Once the AI technology has been widely adopted, optimal prices will be chosen based on firms’ observation of M. First, we consider the optimal price if M is observed to have occurred. In that case, firms will choose the monopoly price for P = a − d − bQ. Thus, the collusive price will be:

2

πhh πlh

If firms know that M has not occurred, then under our modeling assumptions they will choose the price that maximizes expected profits across the scenarios when  occurs and does not occur. Thus, their objective function is:

p2 a max (1 − σ ) p− p b b





+σ

The FOC for this problem is:

a



2p 0 = (1 − σ ) − +σ b b



p2 a−d p− b b

2

(E.10)

a − σd 2

a−d . 2b

a + dσ . 2b

.

(E.16)

= phh

Qhh 2

=

(a − dσ )(a + dσ ) 8b

.

(E.17)

= phl

Qlh 2

=

(a + d (−2 + σ ))(a − dσ ) 8b

.

(E.18)

Qll 2

.

(E.19)

Qhh 2

+ σ (a − d − ph )

Qlh 2

.

(E.20)

The expected consumer surplus will just be the weighted average of these two: h l CSAI = (1 − μ )CSAI + μCSAI .

(E.12)

(E.21)

Supplementary material Supplementary material associated with this article can be found, in the online version, at 10.1016/j.infoecopol. 2020.100882

(E.13) CRediT authorship contribution statement

Conversely, when M does not occur, there will be two different cases. If  also does not occur, then quantity demanded is equal to:

Qhh =

8b

h CSAI = (1 − σ )(a − ph )

(E.11)

Once we have determined the optimal prices, the implied market quantities demanded can be calculated as above. When M has occurred, quantity demanded will be:

Qll =

( a − d )2

If M will not occur, then consumer surplus will be:

Therefore, rearranging terms yields the optimal price:

ph =

2

=

l CSAI = (1 − σ )(a − d − pl )



2p a−d − . b b

Qll

As in the Pre-AI context, consumer surplus has an analytic expression, which can be calculated depending on whether or not M is observed. If M is observed, then consumer surplus will be:



.

= pl

If  does occur, then the firm’s profits would be:

(E.9)



(E.15)

Meanwhile if neither shock happens, then a colluding firm’s profits will be:

2

a−d p = . 2

a + d (−2 + σ ) . 2b

As in the Pre-AI case, once optimal prices and quantities demanded at those prices are determined, colluding firms’ per-period profits in the different possible states may be calculated. We begin with the case when M is observed. In this case, one of the duopolists’ static profits will be:

(E.8)

l

(E.7)

Jason O’Connor: Conceptualization, Methodology, Software, Formal analysis, Writing - original draft. Nathan E. Wilson: Conceptualization, Methodology, Software, Formal analysis, Writing - original draft.

(E.14) 21

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