Homogeneous Algorithms Can Reduce Competition in Personalized Pricing
Authors: Nathanael Jo
Ashia C. Wilson
Kathleen Creel
Manish Raghavan
Venue: NeurIPS 2025 poster ()
Keywords: Algorithmic Pricing
Personalized Pricing
Collusion
Competition
Outcome Homogenization
Algorithmic Monoculture
Antitrust
Tacit Collusion
Primary area: social_and_economic_aspects_of_machine_learning
TL;DR: Correlated algorithms can reduce competition and result in higher prices, posing antitrust implications

Abstract: Firms' algorithm development practices are often homogeneous Whether firms train algorithms on similar data or rely on similar pre-trained models, the result is correlated predictions. In the context of personalized pricing, correlated algorithms can be viewed as a means to collude among competing firms, but whether or not this conduct is legal depends on the mechanisms of achieving collusion. We investigate the precise mechanisms through a formal game-theoretic model. Indeed, we find that (1) higher correlation diminishes consumer welfare and (2) as consumers become more price sensitive, firms are increasingly incentivized to compromise on the accuracy of their predictions in exchange for coordination. We demonstrate our theoretical results in a stylized empirical study where two firms compete using personalized pricing algorithms. Our results demonstrate a new mechanism for achieving collusion through correlation, which allows us to analyze its legal implications. Correlation through algorithms is a new frontier of anti-competitive behavior that is largely unconsidered by US antitrust law.

Homogeneous Algorithms Can Reduce Competition in Personalized Pricing
Nathanael
Kathleen A. Creel Northeastern University k. creel@hortheasten. edu
Massachusetts Institute of Technology nathan ciiiaanjomitteee
Ashia Wilson
Manish Raghavan Massachusetts Institute of Technology mragh@mit.edt
Massachusetts Institute of Technology ashia@mit.. edu
Abstract
Firms' algorithm development practices often homogeneous. Whether firms train algorithms on similar data or rely on similar pre-trained models, the result is correlated predictions. In the context of personalized pricing, correlated algorithms can be viewed as means to collude among competing firms, but whether or not this conduct is legal depends on the mechanisms of achieving collusion. We investigate the precise mechanisms through formal game- theoretic model. Indeed, we find that (I) higher correlation diminishes consumer welfare and (2) as consumers become more price sensitive, firms are increasingly incentivized to compromise on the accuracy of their predictions in exchange for coordination. We demonstrate our theoretical results in a stylized empirical study where two firms compete using personalized pricing algorithms. Our results demonstrate new mechanism for achieving collusion through correlation, which allows us to analyze its legal implications. Correlation through algorithms new frontier of anti-competitive behavior that is largely unconsidered by US antitrust law.
1 Introduction
Firms increasingly use algorithms for personalized pricing, charging some consumers more than others the same good or service based on their perceived willingness to pay [59] 34 For example, recent lawsuit against DoorDash alleged that the company charges Apple users more in delivery fees [51] and travel websites use browser information to charge US- US-based customers more [40] Rather than targeting prices to individuals, strategy called "first-degree price discrimination." firms segment consumers based on their characteristics and charge different prices to each segment, ie., "third-degree price discrimination"
Today, firms machine learning to segment consumers into more targeted categories 18 For example, ridesharing competitors Uber and Lyft
[56]
48
offer discounts subset of consumers based algorithmic segmentation (see Figure 7
Competing firms seek accurate personalized prices in order to better predict customers' willingness to pay. Offering a low price to consumer willing to pay more means missing surplus value; offering high price to consumer unwilling to pay means missing a sale altogether. However, if personalized prices correlated between firms, firms insulated from the competitive cost of their mistakes: if a firm failed to offer a discount, a correlated competitor likely to also not offer a discount to that consumer [14]. In other words, firms can benefit from correlated prices.
39th Conference on Neural Information Processing Systems (NcurIPS 2025).

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The rise of algorithmic pricing makes correlated predictions easier to achieve than ever Correlated predictions can occur when firms deploy algorithms made with "shared components such as al- gorithms trained on similar datasets, aimed at similar benchmarks, or based on similar pre-trained models [8] In other words, model development practices can be and indeed are fairly homogeneous. At the extreme, firms can adopt the same algorithm from third party, creating an "algorithmic monoculture" with perfect predictive correlation across firms 31.
When firms benefit from correlated predictions, we might expect consumer welfare to be harmed. This naturally raises questions about how antitrust laws, which are designed to protect consumers from collusion, apply in a regime of algorithmic competition. Specifically, (when) does algorithmic correlation constitute illegal collusion on prices?
Correlation and competition law. An agreement between competitors to fix prices is per se illegal in the US, meaning that further inquiry is needed as to the action's 's effect on the market the parties' However, multiple firms choose similar pricing models because they are the best models for business purposes without intention to correlate they are unlikely to be in violation of competition law. Actions such as changing prices in response to the market conditions are typically considered "parallel business behavior" and are not per se illegal without proof of intent to form an agreement or intent to reach supra-competitive prices [50] 33
intent in reaching such an agreement [49 54
In the absence of explicit agreement to collude, courts often consider "plus factors" that can tip the scales from acceptable business behavior to illegal conduct. For example, evidence that firms against their own economic self-interest is common plus factor 32). Critically, which pricing model would serve a firm's is economic self-interest may depend on the model choices of other firms. A firm might profit from deploying a worse-performing model, apparently against its own self-interest, because it aligns with a competitor. Does this raise concerns about illegal conduct [29]? At this level of analysis, it remains unclear if or when adopting correlated algorithms constitutes illegal the need for a formal model to capture the mechanisms at play.
Our contributions. We build game- theorethheeicccc model of competition between two firms who use algorithms to price-discrmminae. We find that, indeed, firms can benefit from correlating with competitors when consumers are sufficiently price sensitive (Theorem 4.2
We quantify this value as 2 Firms are sometimes willing to give
a trade-off between correlation and predictive performance up performance in exchange for correlation (Theorem (.3),, which is of concern because consumers are always worse off (more likely to pay higher prices) when competing firms' algorithms are more correlated (Theorem 4.1) We also conduct empirical analyses demonstrating that firms choosing different model classes or choosing to share training data may lead to correlated models in equilibrium.
Our findings allow to develop insights about how current antitrust laws must evolve in the era of digital markets. In particular, the specter of frictionless algorithmic price correlation without overt communication should raise concerns about a new frontier of anti-competitive behavior. While the Federal Trade Commission (FTC) has brought multiple recent cases against parties using the same pricing algorithm to allegedly maintain inflated prices for hotel stays [52]. rent [53, and pork (55), these cases involve traditional plus factors such as parties expressing their interest to collude in writing. Our work suggests the possibility of new plus factors such broadcasting choice of model as an invitation to collude. We discuss these legal implications in Section 7
In sum, we analyze algorithmic homogenization as a novel mechanism for collusion, which sets us apart from existing works on algorithmic collusion that often focus on reinforcement learning algorithms that naturally find collusive equilibria (30 42
16 23 2). Specifically, our results do not require assumptions about the particular learning dynamics of competing algorithms: even in the absence of threats of pricing retaliation or adaptive learning, the mere reduction of strategic uncertainty via correlated predictions is enough to drive higher prices in equilibrium.
'When we refer to "correlated algorithms", we mean algorithms that make correlated errors. Two fully independent algorithms with high accuracy will naturally be correlated with each other. but are concerned with predictions that are even more correlated than the independent state affairs.
'Though in practice firms may not directly face this trade-off, we compare correlation with performance because both ways in which firms extract more surplus from consumers. See Section 5.2 for discussion.

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2 Related Work
Homogeneity and monoculture. Our work builds on recent work in machine learning on "al- gorithmic monoculture", namely the state of affairs in which "many decision- maakrss rely on the same algorithm" and in doing so correlate their behavior [I. Existing literature focuses on how monoculture harms the welfare of those who are subject to correlated algorithmic errors and face "homogeneous outcomes" 8 28 41 Our work spotlights the harm to consumers that comes from higher prices in the context of personalized pricing.
Economic models of oligopoly pricing. We consider competition under a duopoly, which has been extensively studied in the economics literature. The works most related to ours are came-theoretic models of duopolies under Bayesian uncertainty 37 12 57 26 45| [. Much of this literature considers whether firms have incentives to collude by sharing information with one another. Whether a model will suggest that firms are rewarded for sharing information depends on a variety of modeling choices including whether firms compete over production quantity [[5]] or price [6]. In our model, as in these infommation-shariii models, firms' information is parameterized by its performance and degree of correlation. This allows us to reason about strategic decisions firms make regarding shared data, model components, or predictive algorithms.
Personalized pricing, A growing body of empirical, theoretical, and legal literature considers how personalized pricing interacts with concepts like competition and privacy [5 21
60
13 43).
Most related to our work are theoretical models of personalized pricing in the context of competition. Both Rhodes and Zhou M4 and Baik and Larson [3] consider models of competition in personalized pricing under first-degree price discrimination. In contrast, our model is designed to provide insights when firms have imperfect but potentially correlated information.
Algorithmic collusion. The spirit of modern antitrust law is to promote competition. There generally broad consensus that an open and free market economy - which at its core fosters competition benefits consumers by lowering prices, spurring innovation, and increasing the quality of goods and services [46] 38 25 9. In the United States, antitrust enforcement relies on three sets of federal laws: the Sherman Act, the Clayton Act, and the FTC Act, each prohibiting different actions that harm competition. In this work, we will focus our attention to the parts of the Sherman Act and the FTC Act that are intended to delineate which forms of collusion among competitors are illegal.
In general, legal scholars consider three mechanisms for algorithmic collusion. First, an algorithm can be a tool that aids humans in explicitly sustaining cartel-like behavior. Second, an algorithm can be a hub that coordinates actions or be the sole algorithm used among competitors. Third, highly sophisticated algorithms can learn each other's behaviors and achieve supra-competitive prices without explicit communication. From the first to third category, the likelihood that the behavior is illegal decreases at best, the action becomes more likely to fall into a contested grey area [47]. This is because humans become less involved in achieving collusion, making it harder to prove an intent or conspiracy to agree to fix prices. Absent proof of intent to form an agreement, firms are considered to engage tacit collusion, which is generally not illegal without plus factors.
Recent legal scholarship has raised concerns about the potential for algorithms to facilitate tacit collusion. Various works have proposed legal and legislative pathways expand the powers of regulatory agencies [33] or methods to more effectively screen and audit for tacit collusion (36] 27]. However, the mechanisms for algorithmic tacit collusion have not been extensively studied. Several theoretical papers have found collusive outcomes under the third mechanism for algorithmic collusion,
2]. In the repeated-game -ame setting, it is well-known that collusive equilibria exist [24] since players can credibly threaten future retaliation. Past works have shown that standard learning paradigms can find such collusive equilibria. Our work, in contrast, studies the existence of collusive equilibria in the long-run given algorithms are ex ante correlated, meaning threats are not necessary to sustain them.
where reinforcement learning models compete in a repeated pricing setting (30 42 16 23
3 Model
We consider duopoly model where two firms sell identical goods. For each consumer, firm decides whether to offer a default price H" or discounted price L Both firms incur the same unit costs C,
'This is consistent with pricing via "couponing" [e.g.. 19 a strategy according to which firms target offers of fixed discounts (e.g.. 20% off) consumers.
3

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Firm 2
Firm 2
H
L
H
T
H
H 2: 5
(o o)L)
H
(0,0)
(0, L)
[TH] Firm 1
[
Firm
L
(1-o)L,oM)]
(L,0)
(5,5)
Table I: Payoff matrices for both firms when the consumer is willing to pay the high price (TH, top) and low price (TZ, bottom). Within each cell, we denote (Firm 1 payoff, Firm 2 payoff).
leading to per-unit profit of H H and L = L" - C' when pricing high and low, respectively. Without loss of generality, we assume that C' O. We ignore consumers whose valuation V for the good is less than L, and we define 0 to be the fraction of consumers with valuation at least H H We will use TH and Tr. refer consumers with valuations at least or strietly less than H respectively
Consumer behavior. Consumers can purchase from either firm. Under perfect Bertrand competition, each consumer would simply choose purchase the lower-priced good. The economics literature often relaxes the perfect competition assumption such that firms that price higher experience non-zero demand [see, e.g., [88] This may be because firms have finite supply, meaning consumers forced to purchase at higher price when the low price goods sell out, or because some consumers are lazy and take the first price they encounter that is below their valuation. We parameterize this model as follows: When consumer of type TH is offered price L' by one firm and H" by the other, they purchase at price H" with probability E [o, 0.5) and I with probability a. Thus, for larger values of a, consumers are less price- sencessensitive.
We assume that consumers never pay price above their valuation and always make purchase as long as at least one firm offers a price below their valuation. Further, when a consumer is offered two identical prices, they choose firm purchase from uniformly at random. An intuitive example of this consumer behavior is riders choosing between ridesharing apps. When prices are the same, potential riders are indifferent between two rideshare services (i.e., they do not have brand loyalty). However, when prices differ and consumers are willing to pay the higher price. models the friction consumers face in comparing the two options. Perhaps some consumers check both apps to shop for the lowest price, but others choose one app at random and take the first price below their valuation.
Firms' utility and information structure. Our consumer choice model yields the payoff matrices for the two firms for each consumer type shown in Table D. Note that from firms' perspective, their utilities are with respect to unit profit as opposed to sale price. We will denote U+(:T) as the utility/payoff for firm i for a given action profile and for consumer's type E (TI,, }. For example, U, ((H, L); TH) aHl and U2((H, L); TH) (1 L.L.
Firms do not have perfect information and their algorithmic predictions make mistakes. We as- sume that when consumer arrives with features each firm produces algorithmic predic- tion P1 (x), (2) E {0, 1}. segmenting users into types {TL, ] For simplicity, we assume the algorithm has equal true positive and true negative rates, which
will denote d1 for firm 1: P(p (x) 1 | TH] = Plp (x) = 0 Tz] = a1- We define the same quantity for firm and will refer to the model's performance. We will drop and simply refer the algorithmic prediction as P1,22
An important feature of our model is that p1 and p2 need not be independent conditioned user type. If, for example, both firms purchase data from a third party, their predictions may be correlated. Throughout the paper, the terms "correlated predictions" and "correlated algorithms" strictly mean that algorithms make correlated errors. Two fully independent algorithms with high accuracy will naturally be correlated with each other, but are concerned with algorithms that become even more correlated than the independent state of affairs. In the extreme case of algorithmic monoculture, firms use the same model, i.e.,, their predictions implies independence (p1 IL P2 7) and = 1 implies
identical regardless of accuracy. We parameterize
their correlation by p E j0, 1]. where maximal correlation 5) When a a2 p 1 if and only if p1 p2 deterministically. For now,
'We treat H and L as exogenous to this model. We interpret H as a posted price (e.g.. the nominal fare offered by a ride-sharing app) and L as fixed discount (e.g., coupon) on that posted price.
'While a more sophisticated model might directly estimate consumer willingness to pay. firms in practice simplify continuous prediction problems into discrete ones (39) and collect data at discrete price points (18. When A a2, p1 and p2 cannot be perfectly correlated. See Appendix A for formal definition of p.

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treat p as exogenous, we will consider strategic choices impacting in Section We assume all parameters are known to both firms total, our model has five free parameters (see Table 2
Equilibrium concept. A firm's strategy space is simple: for each binary prediction (p E (0, 13) given by the algorithm, parameters are known, firms know the joint distribution on P1. P. Our analysis will focus on Bayes Nash Equilibria (BNE). We do not require that firms price based on the algorithm's 's predictions; for some parts of the parameter space, firms may ignore the algorithm and either always or never offer the discount. We will focus on the region where both firms follow their algorithms at equilibrium (i.e., ditrr-discriminate). formally: s (p) = H if p = 1, L if 0.
price (T,, H7). This results in 4 possible (pure) strategies. Because all
The strategy profile (s", (both firms price- pnice-dsscriminate is an equilibrium if and only if the conditions below hold:
T. [U ((H, 8" (p=))==) P 1] 2 E [U (p2)) =
P2.
E. [U [U, (P2)):T) pi 0]
E
U (H, 8 0].
P2.7
pa
Analogous conditions must hold for player 2. Intuitively, expected util- ity when both firms follow the algo- rithm's 's recommendation (both when Pi and when Pl 0) must be higher than when one firm deviates. We are only interested in the condi- tions where (s", is a BNE because firms should follow the algorithm they adopt it in the first place. In Ap- pendices D and we expand on our model by investigating settings where consumers have brand loyalty to firm and where n firms compete in a market.
Parameter
Interpretation
E [0, 1]
Frac. of consumers w/ demand 2 H" Model performance for firms & 2 Consumers' indifference to price Degree of model correlation ho- mogenization
a1 02 [0.5,
σ E [0, 0.5] P [0,1]
Table 2: List of free parameters in the model.
Importantly, we note that BNE captures the long-run behavior of firms' algorithmic choices rather than the outcome of a single static pricing round. We never specify how firms might converge to that equilibrium, unlike previous works 33
2 that model reinforcement learning or repeated-gants dynamics. Collusive equilibria always exist in repeated games (as folk theorems setting. Further, while these works specify conditions for convergence, our equilibrium concept uniquely allows us to isolate the precise mechanism that yields supra- cuppetcoeeeiitive prices: the degree of correlation between competing algorithms.
42
16
23
suggest 24) ) but do not necessarily exist
4
Main Results
We find that (1) consumers are worse off as algorithms become more correlated; and (2) firms exhibit stronger preferences for correlation as consumers are price sensitive.
(1) Consumers are always worse off when pricing algorithms are correlated. When firms' pricing strategies are correlated (e.g., they price identically), consumers have less choice and must accept the given price or forgo the good. Conversely, when firms price independently, consumers are more likely to have the option to choose lower price. We formalize this in Theorem 4.1
Theorem 4.1. Fix a, 01, 02 0, and H/ For all p such that (s*,*") is BNE, consumer welfare is decreasing in p.
All proofs can be found in Appendix Our next few results describe when firms benefit by choosing correlated algorithms (and thereby harming consumers).
(2) Higher consumer price sensitivity leads to stronger firm preference for correlation. As consumers become more price sensitive (a decreases), firms increasingly prefer to use more correlated algorithms over independent ones.
Theorem 4.2. Suppose, for fixed f, 01, 022 and H/L, (s" s') is a BNE when p = PA and = PB. with PB > PA. Assuming both a1, 02 < 1, firms have higher utility under PB when a < o (O,R), where a² (0, R)
RB -1-1. Otherwise, firms have weakly higher utility under p --
This assumption especially common in oligopolies with few players that interact with each other frequently.
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Price Sensitivity
H/L Ratio of High to Low Price
8 % Population Willing to Pay
0.75
Independent Correlated
H/L 0.8
0.10
0.05
H/L 4
0.8
0.00
-0.05
0.6
0.0
0.1
0.2
0.3
0.4
0.5
0.0
0.1
0.2
0.3
0.4
0.5
-0.00
Figure 1: Regions where firms following the algorithm's recommendation is a Bayes Nash Equilibrium (BNE) for independent models only (p = 0, light gray). identical models only (p = 1. dark gray). and both (gradient). The gradient represents the difference in firm utility when p relative p O: blue (red) signifies positive (negative) difference. Columns represent two values of 0 E (0.5, (0.5. 0.75, while rows represent values of H/LE {2, 4). The x-axis in each subfigure and y-axis
01 02
Intuitively, when consumers they may get undercut by their competitor and only attain a minority the market. In these situations, firms prefer correlation because there is no risk of undercutting; both firms receive similar prediction and therefore price similarly. On the other hand, conditioned on pricing low, firms prefer independence: firm would rather be undercutting its competitor than pricing identically. The balance between these two competing forces preference for correlation when pricing high and a preference for independence when pricing low-deeermine whether firm prefers correlation overall.
price sensitive, firms have higher risk in pricing H because
The tension between these forces is mediated by a, which determines the relative risk from being undercut. Indeed, in Figure [ we observe that within the gradient region (where both independent and correlated models are equilibria), preference for correlation monotonically decreases (from blue to red) increases. In the extreme case when firms always prefer independence. When a firm predicts p. = and prices H accordingly, there is risk in being undereut: the firm receives 0.5HT if ти and otherwise, regardless of their opponent's price. However, when a firm predicts p; = and prices L, they would in fact prefer that their opponent prices H so that they guarantee sale when the consumer's valuation is low (7 TL).
0.5 (consumers completely price insensitive),
Strategic Choices in Algorithm Development
have shown that correlation can raise firm utility. Next, we examine impact on strategic decisions in algorithm development before price competition, illustrating one possible strategic choice among many.
5.1 Model
Two firms choose between two model development processes: (I) collecting their own training data, yielding a model y; with performance a, or (2) purchasing training data from the same vendor, producing a model P. with performance de- When both firms buy data, their models are correlated at Pe 0. If firm collects its own data, we assume their model's errors are independent of their competitor's (p = Po = 0), though in practice, independent data may not guarantee uncorrelated errors. More broadly, any shared component in model novelopment--ot just data procurement- induce correlation. For example, our experiments in Section alllow firms to choose between model classes with varying levels of correlation. To summarize, firms have the following payoff matrix:
Firm
Pe
Pi
where, for instance, Ep. (pe,Pe) Epopper:pppp (pe)) (p))) Epeperr:pma UU((( (pa), sp.))
Firm
Pc
F., (Pe, Pc)
Epp (Per Pi)
Pr
Epo (pr,Pe)
Epn (ps.
We are interested in analyzing conditions under which equilibria exist. Two possible equilibria are (I) both firms choose algorithm Pe with correlation Pe. and (2) both firms choose algorithm p; with

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to=
Price Sensitivity
Pr Corr. of correlated models
Perf. of independen
0= p.1.).]
d= p. 0.5
o= 0.3, p.
0.1
(a))
0.8
0.0
0.6
0.8 -0.1
0.6
0.8
0.6 0.8
0.6
Performance correlated model (a,)
Figure 2: Regions where firms using both correlated models and independent models are Pure Nash Equilibra (first-stage game). additional condition that firms following the algorithm's recommendation must be Bayes Nash Equilibrium (second-star -tanod game). The x-axis is the performance of the correlated algorithm de, and the y-axis is the performance of the independent algorithm a,, The gradient represents the difference firm utility when p Pe (correlated) at performance relative to the utility at = 0 (independent) at performance as; blue (red) signifies positive (negative) difference. All subfigures show parameters for which firms have preference for correlation at a. a, as per Theorem with H
0.75.
independent outcomes p - Pu = O. From hereon, we will refer to scenario (1) and (2) respectively as "correlated" and "independent", ignoring that other actions can also lead to independence.
Formally, the following conditions must hold for correlation or independence be Pure Nash Equilibria (PNE):
correlated in equilibrium:
F. (Per Po) 2 F1, (pi. Pe) and F.? (Per Pc) 2 F2 (pe:
independent cquilibrium: E% (pt, pi) 2 Ep (Pei p.) and E (p., p.) 2 E (pr.Pe). Pe).
As in the previous section, we focus on the strategy s" of price-disrimminating As such, additional condition for equilibrium is that (s", is a BNE in the downstream second-stage game.
5.2
Results
Our main result shows that under certain conditions, firms may prefer correlation over independence, even when the correlated algorithm performs worse. Crucially, the following theorems apply only in parameter spaces where firms adopt the price-discriminating strategy s".
Lemma 5.1. When 0; >> Oc. both firms choosing independence is always a PNE
We next establish the conditions under which both firms correlating are in equilibrium, which comes from Theorem 4.2
Corollary 5.2 (Corollary Thm 4.2 For a; a. and < a² (0, correlating is strictly PNE.
With Lemma 5.1 and Corollary 5.2 in hand, we can now state our final result.
Theorem 5.3. For < (6, R) and any oc, there exists a, such that both correlation and in- dependence are PNE and firms have higher utility under correlation than under independence.
Theorem 5.3 says that given a preference for correlation at a; = az, there are settings where firms derive strictly higher utility from equilibrium with correlated but less accurate models. We will demonstrate this effect in Section
Figure shows the various regions where both correlation and independence are PNE. All subfigures depict model parameters where firms prefer correlation at a; = de- As expected, all subfigures have region at 0g + e, where correlation is still prefemed to independence despite having a lower performance (blue gradient region). seems that higher price sensitivity and a higher correlation option tend to increase the valid region of For example, when 0 0.75, H L 3, 0.1, and P. = 1, firms would rather correlate at a performance of a = 0.6 than have much more informative independent model of a; 0.72.
Trade-off between correlation and performance. While firms may not have direct information about model's correlation with competitors, development practices can still favor correlation. Beyond simply measuring predictive performance, firms typically A/B test new models when deploying them. Having developed seemingly more accurate model, a firm might run an A/B test that reveals that leads to lower profits, since happens to correlate less with competitor. Thus, market signals suffice to enable firms to trade off accuracy for correlation.
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(a)
(b)
Model
Models
Ta- 1 Price Senstivily
Logistic Regression (LA)
LR-LR
UNIT (LR-LR) Uttl InF
0.1
Random Forest
RF-RF
0.02
0.8
0.00
0.4
0.8
Acc. Precision Recall
TNR AUC
Correlation
2.5
5.0
7.5
10.0
12.5
Matric
Ratio High Low (HIL)
Figure 3: (a) [Left] Accuracy, precision, recall, true negative rate (TNR), and area under ROC curve (AUC) for given firm deploying a logistic regression (LR) random forest (RF) model. Since both firms and face the same model options, their results are identically distributed. [Right] Correlation hetween both firms' models when they both logistic regression (I.R-I.)) both random forests (RF-RF). Error bars indicate 95% confidence intervals over seeds. (b) Utility when firms use logistic regression models (IRR-LR) subtracted by utility when both firms use random forests (RF-RF). Greater than indicates a preference for correlation at the expense of predictive performance. z-axis varies the proportion of H (high price) to (low price), and line colors indicate different values of a, where lighter color means higher consumer sensitivity price. Shaded region indicates 95% confidence intervals over sceds.
Ta- Price Sonsibiity
HR Ratio of High to LOw Price
10,
RF
RF
Firm Model
Firm Model
Figure 4: Best response matrices or random forest (RF) model, over five model parameters. Best response for Firms and 2 are highlighted in blue and red. Nash equilibria exist when both blue and red are highlighted in the same box (e.g.. (LR, LR) in the middle subfigure). When both (LR, LR) and (RF, RF) equilibria, yellow square indicates higher firm utility between the two. Grey boxes are "invalid" regions because (s" ss") would not have been BNE in the downstream game where firms compete on prices. These results use the average firm utility over 15 seeds.
two firms where the action space is to deploy a logistic regression (LR)
Empirical Study
We demonstrate our theoretical results stylized game between two firms that predict income based demographic attributes. We use ACSIncome data [17], which contains US Census data from 2018. The task is to predict whether an individual earns over $50,000, which aligns with pricing via "couponing" [9]
6.1 Setup: Different Model Classes
To further illustrate strategic choices in our model, we consider how firms select different model classes. In Appendix C.2, conduct experiments varying the degree which firms train their models on shared data. Our experiment involves two firms. Each firm chooses between a better performing (random forests) and worse performing (logistic regression) model. However, logistic regression - despite having worse performance has lower variance, meaning that it is likely to be more correlated when the opposing firm also chooses the same model class. We will show empirically that firms may prefer to sacrifice predictive performance in exchange for correlation.
Both firms train and test on Census data in California. The test set is 30% of the data (n 58, 700) and is fixed across both firms. We randomly split half of the remaining 70% as the training set for Firm 1, and the other half for Firm 2, each having 35% of the entire data to train (n 68, 482). We repeat the training data splits over 15 random seeds.
Figure 3 a) shows the performance and correlation for both firms when using a logistic regression (LR) and random forest (RF) model. RF outperforms LR across many metries: accuracy, precision, recall, true negative rate, and area under ROC curve. This is by design our goal is to simulate

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scenario where firms have a choice between more correlated model with worse performance (LR) and better performing but less correlated model (RF). See Appendix C.T for details.
6.2 Results
Preference for correlation. Figure f shows best response matrices for both firms choosing between LR or RF, over various values of and H/L. T. Cells with blue and red cross indicate Pure Nash Equilibrium (PNE) for that action profile. When H/ I. is too low or too high, firms will never choose to follow their algorithms to begin with (grey cells) because always pricing low or high will give higher expected utility. When H/ L is moderate, less correlation (RF, RF) is always a PNE as per Theorem 5.1 More correlation (LR, LR) is PNE under the condition outlined in Corollary 5.2 Finally, when both (LR, LR) and (RF, RF) are PNE, the difference in performance between LR and RF are small enough such that (LR, LR) is higher in utilityy (yellow cell) than (RF, RF) as per Theorem 5.3 This preference for correlation (LR. LR) occurs when H/L is large and ar low (consumers more price sensitive), most heneficial to firms when there is a high risk of being undercut by the opponent; therefore, firms would rather have certainty about the other firm's actions than better performing model.
Figure b)) illustrates. As per Theorem 4.2 correlation is
Firms prefer lower variance models under competition. Lower variance models have less predic- tive multiplicity 77I and thus predictive errors more correlated. Our empirical study suggests that competing firms are pushed to adopt simpler (higher bias, lower variance) models on the margins.
7 Discussion
Taken together, our results suggest that firms will sometimes prefer a less accurate personalized pricing algorithm when doing so allows them to better correlate their behavior with their competitors (Theorem 5.3) and this behavior reduces consumer welfare (Theorem 4.1). Furthermore, firms are more likely to prefer correlated algorithms when consumers are price sensitive (Theorem 4.2p and the consumers most likely to suffer are those to whom the price matters most.
Correlation is a mechanism reduce competition and sustain higher prices. When firms make up duopoly, using more correlated algorithms allows firms to reduce competition, which increases prices. When algorithms are not correlated, firms naturally attempt to undercut their opponent in order to extract more surplus, so the high price equilibrium cannot be sustained. This undercutting will continue to lower prices until firms reach new equilibrium.
Models can become correlated when any part of the development pipeline is homogeneous, such as using similar pre-trained models, lower variance models (Section (6) or training on similar data (Appendix C.2
We empirically demonstrate that these settings lead to higher prices.
We note several limitations. Our findings and discussions on antitrust law apply specifically to algorithmic pricing. Moreover, our stylized model intended to illustrate the possibility of correlated outcomes occurring under reasonable conditions. Future work may attempt to investigate more mechanisms or even empirically investigate this possibility in real-life markets.
7.1 Legal Implications
Our results add to the growing body of work suggesting that the ease of collusion that algorithmic price-setting facilitates may support revision of traditional anti-trust standards 35
22 33
Publicly signaling models may invite collusion. Mazumdar [33 suggests that adopting a pricing algorithm that "broadcasts" its intentions can signal invitation to collude. While there is little precedent for firms publicly committing to a model, our findings suggest this may pose a risk.
In particular, our model suggests that publicly adopting a less accurate model could be considered an invitation to collude. Assume that both correlated models (LR, LR) and independent models (RF, RF) are equilibria (e.g.. middle subfigure of Figure 2). and firms initially use (RF, RF). In order for firms to reach the collusive outcome of (LR, LR) without explicit communication or agreement, one firm must unilaterally switch to LR, sacrificing its own utility (by leaving an equilibrium) in the hope that its competitor will follow. This costly action functions as a signal- demonstraing willingness to reduce competition despite short-term losses. Antitrust law should determine whether

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public announcement of model choice can be an anti-competitive "plus factor" in the same way that public announcement of intent to price high can be competiieeee.
Choosing a less accurate model is not the only way to collude by correlating. Among equally accurate models, firm selecting the model most correlated with a competitor would necessarily constitute a "plus factor" since choosing a best-in-class model aligns with a firm's economic self-interest, As discussed, however, intent achieve supracompetitive prices is sufficient establish illegal collusion. Thus, choosing correlated model may be concerning if firms intended to reduce competition.
Intentional choice of correlated models prices has long been a comnertoone of anti-true trust law. We propose that intentionally adopting correlated algorithms can constitute illegal collusion, just intent to coordinate higher prices does. Previous legal cases alleging algorithmic collusion like RealPage [53] and AgriStats (55) have relied on a hub-and-spoke structure, where firms share information centrally, receive price recommendations, and face enforcement for deviations. Recently, the state of California updated its antitrust laws to prohibit the use of common pricing algorithm that intends restrain trade (10). However, our model demonstrates that horizontal collusion can occur even without a central coordinator - firms merely need knowingly homogenize.
frontier of competition law. Intention to collude on
Acknowledgments and Disclosure of Funding
This research was supported in part by an MIT Social and Ethical Responsibilities of Computing (SERC) seed grant.
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2 A Model (Continued)
A.1 Correlation Parameter
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0
(1-8)(1 [)[[1-a) -a-
(1, p(min(01, -
010g
TI
0
(1 [-0)ar (2) p(mim((),
d102)
0
0
(1 1--0) 00)a p(min((1,, 02) - 0102)
Table 3: Joint distribution P1,
B
Proofs
B.1
Consumer Welfare and Proof of Theorem 4.1
Before proving the theorem, we will introduce some additional notation. Let ((()): denote consumer welfare under the action profile (.) and demand state т. We define consumer welfare as the consumer valuation of the good subtracted by the cost of the good. As such, we define two additional variables Vz, VH to consumers' expected valuation under Tы and TH, respectively. Let òg = Vz - L" and similarly for бн Vи H". By definition, 8,, 8H 2 0 otherwise, consumers will not purchase the good. Consumer welfare under the various actions and demand states can be summarized in Table
Tи Firm
Tr. Firm
H
H
L
8 + (1 ( - 13)
H
0
85
Firm
L
8H (1
(H' L")
6н + (H" L')
L
8g
85
Table
Consumer welfarr under all action possibilities and both demand states (TH,7t).
Proof. We will denote expected consumer welfare for given value of (pt). p1 P1.pPTTp
E
Our goal is to show that
d
W((s*((i): (p2))):: <.
dp P1.Pa.Tip
Our approach will be to show that increasing p increases the likelihood that p1 P2. which in turn reduces consumer welfare. First, observe that
I3 P1:P2-TiP
[W((s (pi), 8" (pz)):t)
[W((s* (p)). 8 (p2)):T) P2] Pr [pt p2]
P1-P2:T:P
+ E (W((s (p2)))=7 p * p2]
Pr (p) + p2]
P1PPa.T:p
We will show that
E PL.P2.TiP
( P2]
21

---

and
E PL
(piss))), s" (p2)): pr + p2]
do not depend on p.
[W((s (pr), s" (p2))it) 1p = =2]
E
(P2)):T) p1 P2, TH] Pr
[TH P1 P2]
P1.P2,7:p
PL.P2.TiP
PI-P2ITiP
E
P1 = =P2.7 Tz.
Pr
[TL | P1 = =P2]
PL.P2.73P
PiiPa.Tip
= W[((s (pt), (p2) TH)
Pr PLIP2.TP
[T P1 = =22
W[((* (p1), s (p2) TL)
Pr [TL P1 P2]
P1-P2.TiP
Note that by definition, Prpp.paiolP Pr Piprapat:oPp =p2]. Therefore,
P2 T = Tm] = Prppraiilp = 12 = TI.] =
Pr.:elT r TL. Pr P2pp (11 P2 T= TL] 0
Pr PL.PTT
TT. Pi =pal
Prir P1 = P2]
Prr:pT IT -H] Prp: p2oplp1 P2 T TH]
Pr TT TH 1 P1 p2l PL-P2:T:P
Pro = =p
This implies
E. [W((s (p1), (P22)):T) 1m =p2 = (W (p1),, (p2)); TH) + (1- s (p2)); Pr.Pa.7:p
A similar argument shows that Pp. ((s s (p2)):T) p1 * p2] does not depend on p.
Next, we will show that that
[Ws (pt). (0 (p2)): pm = S Pr P2.T
p2l
E P1.P2.T
[ (p1), s" (p2)): p + Pal. (1)
meaning that consumers have higher expected utility when offered different prices. Because T is independent of the event p1 = P2. we can analyze each 7 E {TL, TH separately, For TH
[W((s" (P1), s" (P²)) P1 P2, T P1.P2
TH]
Pr [Pi P2 p1 P2, (H)
-88)
P1-P2
+ Pr [p1
P2 = 0 P1 = P2. T TH] ([L(WL, LH: - 8n)
Pr [pi = p2
P1 = p2, T -
Pi:P
+ Pr [p
p2 = 0 P1 p2, = -1'
1- 01 02 (1 p)1100 + pmin (H" L')
01 02 + 2( p)1;02 2pmin(a),,))
(H" - L')
2
because d1 and dg are both at least 0.5. Similarly,
E [W((s (p1), s" (H) p1 * p2 TH] P1.P2
6H (1 o)(" - L')
2 (1- -H---")) L')
1
since S 0.5, meaning
E (Wi(s (p2)) = P2, - = [m] S E
[W((* (pt), ) 8 (p2)); p1 # Tw].
P1-P2
P1-P2
(2)
Next, observe that
E
[W((s* (pi), 8" (p2))5TT.) 1p1 = P2, T2 S E s P1 + p2, T = T..
P1.P2
22

---

simply because the left hand side is at most εд and the right hand side is deterministically dτ. Combining 2 and and using the fact that 7 independent of the event p1 = P2 proves ① As
result,
d ( E [ (p2)):: pi = P2l
E (po)): dp pI.
Pr
[pi = pzl
dp P1.P2.
Pr.PasT:p
E (W((s"())) (12))) pl # p2l (1
pI = pm)
P1.P2.T:0
E (p((s (p2))): pr = p2]
PI-P2.
(p2)); pı + (pal) dp p pu.pppppp
Pr
(p = =pa]
P1-P2.T
E (p( (p2))): = p2]
P1.-2..
E
[W((s (p2)); Pı + p2l
P1-P2.T
d
ai
02 (1 a2 pmin(ai, 02)
dp
$ 0,
where the last line follows by I and using the fact that
1 - + (1 p)e102 + pmin((1, 2 0.
dp
This inequality is strict as long as a1, 02 (otherwise p has impact on the joint distribution of
P1,P2, 7).
B.2
Proof of Theorem
Proof. The following condition for Firm must hold for them to prefer prefer p PB over p PAi
E (Us (p2))s+)) (pa)))))
E PA.Pa.F.0-PA
((p)))]
PL PL-Pa-TiP-PS
For ease of notation, probabilities cancel out when subtracting p pn to p = PA. leaving only the A and A, terms:
min(a), 02) ai a2 and A, = pn PA > We will see that the
Σ Σ Σ
(.p)).s U.[(s" (pi). s0 (p²): | p:p pnl - PPp,.
T:p P:pp pall >
(IE P2 (0.1) rE (TH.LL)
HOAA, HoBAA, +
L(1 A)
LOAA
L(1 ( A)1 L0(1-o)44,, +
p
2
2
1 AA (HO(1 -27) I(200 1)] > 0
2
We can derive the same exact inequality for firm 2. When min(a1, 02) 0100 # 0 (or, when both
a1, 02 < 1), we get
H8
20(H L)"
We further show that lower monotonically increases preference for correlation:
i AA H0(1 -2o) + L(200 1) = 2A H),
do
which is always negative because L H. by definition.
23

---

B.3 Proof of Theorem 5.1
Proof. The main intuition behind this proof is that an algorithm with performance a, can simulate an algorithm with lower performance d.. Recall that we define s" to be the optimal strategy of following the algorithm. Let s^* be the strategy of doing the opposite of the algorithm's recommendations. We define s' to be the following strategy:
s' s'(a:)
[at)) (a))
w.p.
w.p. -
where scto,= The strategy s' (a,) is equivalent in expectation to s" (ac) in terms of firm utility. To see this, we will prove that the conditional distribution Pl[7|s (a;)] is equivalent to Pa.]ss
P = Tm| = 1 P(T = Tmlss =
Pis'(a,) = 1/7 TF PP TH
PPs (a,) 1 TH [F
P[s((a,) = 1 TH [H] (a;) = =
= 1/T TH [HH] + IP (ao) = 1 TI |P [T.)
and
P sslss(a;) = = P =xlss((a)) 0]
([s'(a,) = 0 TH [H
(a.) = 0 TH PP
P(s'(a,) 0 = TH [P( (Pss ((;))())) 0| P(TL) PP (a.) 0 TH TP [T (Psss(a))
0/7
T [P( [z.
Based the Bayes' Rule expansion above, suffices to prove the following equivalences:
Pls((a.) = 1| TH. (a.) = 1/T = TH]
(4)
Pis'(a;) = 1/7 m2 P|s = 1/7 mz
(5)
Proof of
Pis' (a.) = 1/7 = TH] = 9P16" (ax) = 1/T = mM + (1 P| (ax) = 1/7 = TH]
= (ae) 1/T TH] (1 4) (a.) 0|T = TH]
qu; + (1 () a.)
ae -
2a; 1 a, 20;
ae (1 0;)
(2a; 1)
0g
20,
= (a.)((a)) = 1/7 = TH
Proof of (5)
Pis (a.) 1/7 Tz] qP(s" (a.) 1/7 Tz] + (1 9)PP( 1/7 TL]
qP|s>((a 1|7 = Tz] + (1- - P] 0/T = TL]
0; ag
0. + 0; (1 a;)
(2a; 1)(1 ac)
(1 1--4) q)aa + q(1 - a:) 20
aj
a
20:
2a, 1
(as) = 1/7 Tr.
Note that the space of possible accuracies is 2 0.5 for algorithm to be useful. When a. 0.5. a, >> 0.5 by assumption of the Theorem and therefore q is never undefined. Then,
F (s" (a:), s" (ax))) 2 E1 (ss((a)) s (a:))) F (aolsss())) (ac), 8 sa(a)))
and similarly for Firm 2.
24

---

B.4 Proof of Corollary 5.2
Proof. We will show that the condition for strict preference for correlation (in the second-star stage game) is equivalent to correlation being strictly in equilibrium (in the first-stage game). We first start with the preference correlation in the proof for Theorem 4.2:
E Pp
(U'(( (p1), s" (p2)):+)]
E PL.Pa.TiOPPA
UUi (p))
Since this condition is for any PA < PB. we will let PB Pe and PA = 0. Further, we will change the p notation to a, and a; where relevant, since = by assumption.
E Ro. Re..esTip=pp
Vil(s"(ac)) (
E (c.
(a)))
which is equivalent to the condition that both firms using correlated models is in equilibrium; this strict inequality implies a strict equilibrium. Symmetric argument applies for Firm
B.5 Proof of Theorem
Proof. First, both firms using independent algorithms is always a PNE when o; dg as per Theo- rem 5.1
We will next state what is needed to prove the theorem. When firms have preference for correlation a; ae, both firms using correlated algorithms should be PNE when a; 0. c, for small
enough
3e 0 s.t. E Ep (aes ae) p (a: e,
top of that, firms also prefer correlated algorithms over independence at a; = ag + e, for small enough
3 s.t. Ep (Mes ac) El p (a+ + e, +)).
The proofs for (6) and 7 come from Corollary 5.2 which states that when firms strictly prefer
correlation a; des correlating is 8-strictly PNE:
36 > s.t. El P (ao, 0e) 2 El EJn. (u;; ac)
We define the following shorthand:
A
Peis- ac) E
El
B
Pois (ai, ao)
C(c)
El
El 2o,s (a; + e, ac)
D(e)
po.s (as + e, 04 +
Put another way, Corollary 5.2 states that
38 Ost. A B
(8a)
A2 C(c) +
(8b)
A D(c) +
(8c)
at c = 0 and a, = because B = C((== D 0). Since C(e) and D(e) are continuous in 6,
66 and 7 are true by (8b) and (8c).
C Experiments (continued)
C.1 Additional Details for Model Multiplicity Setup
We chose the following model hyperparameters to simulate higher performance for random forests compared to logistic regression:
25

---

Model
Hy Hyperparameters
Logistic Regression Random Forest
L le-nnllyy saga solver # trees = 9
min # samples in each leaf =
weight: 1.2x for negative class
All unspecified hyperparameters use the default values set by scikit-learn All experiments (including the ones outlined in the following Section) were run using Apple Silicon M2 chip with 16GB. They only require CPUs and are not computationally expensive any modern computer can easily run these experiments.
C.2
Additional Experiments: Data Procurement
C.2.1 Setup
Our experiment involves two firms who may independently choose to correlate with each others' models by using overlapping datasets. Firm trains Census data in Texas while Firm 2 trains on Census data in Florida. They both have the option to purchase supplementary data of worse quality from third-party, which in this case is Census data from California whose labels have been perturbed 25% of the time. In doing so, we are giving firms the choice of correlating their models at the expense of predictive accuracy.
In order to smoothly interpolate between independence and correlation, we define a parameter ",; for instance, Firm 1 can the training data (1 7) TX + CA, and similarly for Firm 2. If both firms use O, there is no overlap in training data and their resulting models will be the least correlated. Conversely, when both firms use 1, their training data is identical and their models will be the most correlated.
We randomly sample effect of y. We then further sample 7 proportion of each dataset to ensure that all training data used have exactly observations. We run this experiment over 15 random seeds, and over y [0, in 0.1 increments. Both firms train the same model class (random forests) and have the same test data: Census data from Illinois.
200, 000 datapoints from TX, FL, and CA in order to standardize the
C.2.2 Results: Second-Stage Game
Figure 5 aa shows the predictive accuracy for both firms and the correlation between both firms as varies. As expected, accuracy monotonically decreases and correlation monotonically increases as 7 increases since firms use more and more of the same lower-quality training data. We observe significant decrease in accuracy for both firms when longer receive the more predictive signal from their original training data.
1, presumably because both models no
Figure5|bb shows the difference in utility between at the x-axis and = 0 (independent datasets). When this difference is above (blue dashed line), firms have a preference for correlation that value. We observe such a preference for correlation when consumers are more price sensitive (lower σ)) and when the ratio between the and I. prices larger, as per Theorem [.22 Firms prefer correlation even when accuracy marginally decreases (subfigure (a)); this happens particularly when there high risk of being undereut, making correlating especially beneficial even at the expense of predictive accuracy. However, firms no longer prefer correlation when the trade-off in accuracy is too high (e.g., Firm 7 = 1). We note that firms are asymmetric: because their models' accuracies differ at various 7, they do not always prefer correlation in the same way, but the general trends remain.
C.2.3 Results: First-Stage Game
We also model firms' decision to correlate at a particular y. In particular, Figure shows the best response matrices for both firms choosing various values of 7, over various model parameters (H/L.o) Cells with red and blue cross indicate Pure Nash Equilibrium. In general. higher correlation (7) only in equilibrium for higher H/T. and lower a, which reflect the same trends the second-stage game. For example, When H/L 6, 0.1. the sole equilibrium exists at
26

---

Firm
Firm 2
Correlation
0.80
Accuracy
Accuracy
0.775
0.775
0.75
0.750
0.750
0.0
0.5
1.0
0.5
1.0
0.0
0.5
1.0
Y
(b)
H/L
H/L
H/L
U(y) Uty 0J (Normalized)
0.00
0.04
Firm
0.00
0.02
0.00
0.00
Fim 2
0.01
-0.01
0.02
0.3
0.00
0.00
0.25 0.50
1.00
0.50 0.75
0.25
1.00
Y
Figure 5: (a) [Left] Correlation between both firms' models the empirical study across various values ? ? (1) corresponds to no overlap (full overlap) training data. [Middle and Right] Accuracy Firm and 2's models over various values of ?. Error bars are 95% confidence intervals over 15 seeds. (b) Difference in utility between at the x-axis and 0 (no overlap in training data) for the empirical study, over various values of H/ L and Top and bottom rows correspond to Firm I and 2's utilities, respectively. Shaded regions indicate 95% confidence intervals over seeds.
HIL = 3, 0=
HIL 0z 0.1
HIL 10 0.11 2 5. 6.7 8.9
0 1 3 44 .5. 5...6.6. 6.7 9
.0 1 4 www.566...... 7.8.9
(Firm
(Firm
Figure 6: Best response matrices for the two firms in the empirical study, over three select model parameters. y 0 mcans overlap in training data (least correlated) while y indicates identical training data (most correlated). Best response for Firms and are highlighted in blue and red, respectively. Nash equilibria exist when both blue and red are highlighted in the same box (c.g., (0, 0) in the left subfigure), Grey boxes are "invalid" regions because following the algorithm would not have been a BNE in the downstream game where firms compete in prices. These results use the average firm utility over 10 seeds.
0.7). When H/L increases to 10, equilibrium is at (1, We note, however, that in extreme H/L values, certain regions are "invalid" in the sense that firms would not follow the algorithm in the downstream second-stage game (grey cells).
C.3 Additional Results
Firms choose to correlate, even when algorithms are uninformative, Figure [ displays regions where firms following the algorithm's recommendation BNE for independent models only (light gray) and correlated models only (dark gray). When 0.5, independent models are never in equlibrium because algorithms are as good as random. However, when a = 0.5 and models are correlated, firms may still choose to follow the algorithm. This region is more likely to be in low regimes where there the highest risk being undercut by one's opponent - and therefore there value in coordinating actions despite the model having no predictive power.
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D Extension: Asymmetric Market Power
In this section, relax the assumption that consumers indifferent firms when firms offer the same price. In particular, we will introduce a parameter E [0, 1] that controls consumers' preference [0, 1J, still capturing price sensitivity. For example,
toward Firm 1. We will also reparameterize in the context of Uber and Lyft, y specifies the proportion of people who check Uber before Lyft. Of the people who check Uber first, a proportion are price-insensitive, meaning that they would still choose Uber even when Lyft offers lower price. When 0.5, we model the same consumer behavior in the original model. The payoff matrices summarized in Table
Firm 2
H
T.
H
H, (1 H)
(Ya L) (vL, (1-7)1
[Ty] Firm 1
L
((1 + 7o L, (1 7)
Firm 2
L
H
(0
(). L)
[TL] Firm
L
(L,0) (7L, (1 (1-7)
Table 5: Payoff matrices for both players when the consumer is willing to pay the high price (TH, top) and low
price (TL. bottom). Within each cell, we denote (Firm 1 payoff, Firm 2 payoff).
28

---

Senartivty
HL
flate
high Low rrcce
Popular Weing Pay
Indeperdent Same xg
HIL
0.50
0.25
Independence
Piefer corslatioo
0.55
56
1.0
02
5.6
1.0
0.75
0.25
0.0
0.2
0.4
5.5
3.5
3.0 0.0
02
0.4
0.5
2.5 3.0
Figure 8: Regions where firms following the algorithm's recommendation is Bayes Nash Equilibrium (BNE) for independent models only (p = 0, light gray), identical models only (p dark gray), and both (gradient). The blue region denotes where both firms have higher utility under = 1, while the red region is where at least one firm has higher utility under O. Columns represent two values of E {0.5, 0.75). while rows represent two values of H/L (2, 4). The x-axis in each subfigure is "y and the y-axis is a. We fix
0.9.
Results. Our main result is that asymmetry lessens the impact of homogenization, Figure 8 shows regions where using fully independent algorithms (light gray) and the same algorithm (dark gray) are BNE. When both are BNE, blue indicates that both firms have higher utility when using the same algorithm, and red otherwise. As 7 increases, firm I increasingly prefers correlation because they benefit from pricing similarly to their opponent (since consumers are increasingly loyal to firm 1). Firm increasingly prefers independence as y increases for the same reason; firm 2 would like opportunities to undercut firm I since consumers prefer firm I when they price similarly. As result, the region where both firms prefer correlation exist within ball around y E [0.5 - e, 0.5 e) for some E.
E
Extension: n Firms
We now extend our model to capture a market with n competing firms. Let Pi,..., Prr be the predic- assume that firms have two choices:
tions of players. For simplicity, algorithm (p 0) or to use the same (fully correlated) algorithm (p = 1). In Appendix [E.4] we extend this to the setting in which firms be partially correlated. We will also assume that all firms have the same model performance a.
fully independent
Fix firms. Let k < be "coalition" of firms who choose employ the same algorithm, while the remaining n - firms are fully independent. Since all firms have the same accuracy, it does not matter which & players distribution over n-bit vectors in which the first coordinates are identical, the last n k are mutually independent and independent from the first k. We explicitly define the joint distribution below:
assign as part of the coalition. Hence, will let Ω2 denote the joint
P(p Pk, Ph........... P.. Pa|TH) a² a)
II #=k+1
a (1 a)²-PA
P(p P(Pi.....PPPPPt11.. Pe.. ,n/FL) = a)*
a 1-pi (1 a)
#=++
where = PPp; 1/TH) P(p: = 0/TT.) as in the main text, and PI P2 ph since the
first ls firms predict same outputs almost surely. In other words, whenever pl, Pk. are different from each other, the joint probability is O.
E.1 Firm Utility
Let ny and na be the number of players that price low and high, respectively, such that ne Rh.
29

---

All other firms but
All other firms but
All same as
Some different
All same as i Some different
H
H/n
o oH/nn. nnp L/n (1 L)/o
H
0
0
[TH] Firm i
[Tr]
Firm
L
L/n
L/ne
Table 6: Payoff matrices for firm i when the consumer is willing to pay the high price (TH, left) and low price (Tz. right).
E.2 First-stage and Second-stage game
We now define our equilibria of interest. For every we establish the following conditions:
Second stage game (Deciding to use the algorithm). The Bayes Nash Equilibrium condition is:
E s" (x
pt 1]
(U.((L,
1], Vi
E-BRITT
E P-rr
(U,(LL, (p ())=)) pi 0] Z E P-r
[Ui((
px =0], vi, ( (p-i)):T)
where p-; denotes the predictions of all n firms save for firm i.
First stage game. We analyze whether or not the coalition is stable, given that in the downstream second-stage game firms follow the algorithm. Let C = (1, .I,..., k} be the set of firms that the same algorithm, and I == (k + ,,. n be the set of firms that are fully independent. We define (k) to be the utility of firm that uses independent model, and Vo(k) the utility of firm in the coalition using the same algorithm, i.e..
Vi(A) p
(U((s* iez
Vc(k) prollet
(p)(s()));+), iEC
The Nash equilibrium condition is:
Vi(k) Vo(k +1) Vc(k) 2 Vi 1).
(9a)
(9b)
In other words, players not in the coalition should have higher utility being independent and not join the coalition. Players in the coalition should be satisfied with staying in the coalition compared to their option of leaving.
Special case when h and k: always an equilibrium, since if no one is using the fully correlated algorithm, no player can strictly increase their utility by switching from being fully independent fully correlated. k also be an equilibrium, but only if none of the n independent players strictly benefit by choosing the correlated algorithm. When a n, only condition (9b) applies.
Note that when k: only condition (9a) applies. It is
E.3 Results
Figure shows regions where firms using their algorithm are in equilibrium for both the first and second stage games, for varying number of competing firms and model parameters. We note first that the regions monotonic in & and thus create disjoint regions.
are disjoint. This is because the first-stage conditions (Equations (b and (aa)
Larger coalitions (k) are only stable when consumers are price sensitive and are willing pay the high price. For example, in Figure 9 the red (k = 4) and purple (f: 5) regions exist only when a is low and is high. Intuitively, this makes sense because the larger the coalition, the more risk there is; deviating means potentially being able to undercut the coalition when they make mistake. Large coalitions are therefore most valuable/preferable when consumers are incredibly price sensitive and most consumers willing to pay the high price to begin with.
30

---

Price Sensitivity
H/L Ratio of High Low Price
0 % Population Wi ling to Pay
HIL = 3
H/L = a
H/L 7
H/L = 5
H/L = a 0.7
0.9
0.8
0.6
0.8
0.6
0.8
4
0.6
0.8
0.6
0.0
0.2
0.4
0.2
0.2
0.4
0.0
0.2
0.4
0.0
0.2
0.4
0
k=2
k=3
k=
Regions where firms are in equilibrium for both the first and second-stage games, for various coalition sizes k. From top to bottom, each row represents an increasing number of competing firms n. Columns represent different model parameters H/L and
Figure 9:
With many competing firms, correlation is only stable with high-performing models and high price differentiation. For example, in Figure the = 4, settings only have significant colored regions when = 0.9 and H/ T. 5, 7 (the highest options). The intuition is that at high H/ T. (e.g.. 7), there is lower risk of being undercut since firm will still get times the surplus when they make the sale, Also, when firms have highly accurate model, they make fewer mistakes and hence have fewer chances of being undercut. Hence, they can correlate without fear of being undercut.
E.4 Modeling Correlations in Full Generality: Gaussian Copula
In our analysis above, we restricted firm interactions to either be fully correlated or fully independent with each other. We also required that there exists only one coalition, and that firms employ algorithms with the same accuracy. Here, we provide a model that relaxes all three restrictions.
Assume that n firms are divided into m disjoint clusters C₁, j E [m]. The main idea is that (I) correlations are constant within each cluster; and (2) we are only modeling pairwise correlations for all players and no higher order terms. In particular, we introduce correlation parameters Pg and PC;, for all j E [m]. which denote inter-cluster correlations and within-cluster correlations in Cj. respectively. Mapping back to our analysis above, we had of n players form one cluster with k players form singleton clusters where the within-cluster correlation
PC. and the remaining
is trivial and Pg = O.
We will model each binary outcome p; using latent Gaussian variable Z;:
1, Z < 0,
1,
Z, t
Pi
TH
Pe
Tr
0,
Z, S t;
where 4 (a;). This way. we still preserve the property that P((: = 1/7)) P(p: 0(Tz) a, To model the joint distribution, we first define multivariate normal distribution 7~ NO
where Σ is 1 in the diagonals, PC2 for pairs within cluster C₁, and Pg for pairs in different clusters.
31

---

For example, if n = 3 and firms and 2 are in coalition, then the covariance matrix E will b
1
PC₁
PC
1
Pg
Pg
Pg
Let the probability distribution function (PDF) of Z be Φr. Then, let
PPll) Plp......PPl S .... ./o $5 2n d21 ... d2n.
P(p ............
where
Bi = (-00,tyl if p; = TH 00)
B; = (-00, til
if p; =
if =
if P:
[1i 00)
if p; =
This ensures that all combinations of pl: gives probabilities that sum up to one.
E.4.1
Converting p from binary space to Gaussian space
Ideally, we would like specify Pg and pc, in binary space because it is outcome of interest. However, as deseribed above, the p need to in the covariance matrix of the multivariate normal. This means need way to translate from Pbinary Peaussian- Since correlations, this mapping can be done for every pair of players. In particular, we will use the polychoric correlation approach (20), which establishes relationship between the correlation of two ordinal variables, each assumed to represent latent bivariate Gaussian variables.
only modeling pairwise
1 1n Pppien (24, 2g )d2,dz 0;ag
Poinary
Va,(1 a, a,)a 0,)
where ppensin the joint bivariate normal PDF with correlation Pgaussian We leave analyses involving these more complex interactions to future work.
"Note that by definition of the multivariate normal, the associated covariance matrix > must be positive semi-definite, which is generally not satisfied if pe = and P, 0. We can set - for some small 0 as approximation for pc in order satisfy the positive remi-definite condition.
32


Reviews:
[{"review_id": "7t6uriqxWM", "reviewer": "Reviewer_84Rt", "summary": "* This paper develops a game-theoretic model of price discrimination and collusion through the choice of predictive models.\n* In the setting, there are two firms selling identical goods. The goods may either sell at a high or low price leading to per-unit profits $H$ or $L$, respectively. A fraction $\\\\theta$ of consumers have valuations higher than the high price, and their sensitivity to price is characterized by a parameter $\\\\sigma$. Firms predict the type of each consumer (high/low valuation) using a classifier $p_i(x)$, with group accuracy characterized by a parameter $a_i$. Predications have correlation parameterized by $\\\\rho$. The strategy space of each firm is the price they offer given a prediction, and the goal is to characterize the equilibria of the resulting game.\n* The main results show that correlation hurts user welfare (Theorem 4.1), and characterize the firm’s preference towards correlation as a function of price sensitivity (Theorem 4.2). The second set of results analyzes the resulting equilibria when firms can decide whether to train prediction models independently or using common data - Showing the firms may benefit from correlated but less accurate prediction models (Theorem 5.3). Empirical analysis is presented based on an income-level prediction task, showing that similar phenomena are also evident in real-world datasets.\n* Finally, broader implications are discussed with an emphasis on legal implications.", "questions": "* In cases where user welfare is negatively affected by the strategic decisions of the firms, is it possible to incentivize the firms to act in a way which benefits the users?\n* How does the work relate to the existing literature on AI and competition? (e.g., Ben-Porat & Tennenholtz, EC 2019; Jagadeesan et al., NeurIPS 2023; Einav & Rosenfeld, ICML 2025; Tsoy & Konstantinov, NeurIPS 2023; Laufer et al., WWW 2024, etc.). Does the interaction model generalize existing models in this literature? Is it possible to interpret some of these results from the perspective of collusion?\n* What happens if other types of agents are introduced to the dynamics? For example, what are the implications of LLM providers being allowed to adjust their price dynamically based on demand?", "limitations": "Limitations and broader impact are discussed.", "rating": 5, "confidence": 4, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 4, "clarity": 4, "significance": 4, "originality": 4, "strengths_and_weaknesses": "Strengths:\n* The paper is well-written and very easy to follow. Formal statements of results are clear, and intuition for the results is provided.\n* The proposed model is intuitive and seems to capture key aspects of collusion dynamics in a concise way.\n* Results offer actionable insights that have potential to influence policy decisions.\n\nWeaknesses:\n* Results seem to rely on the assumption that all the relevant information about the setting is known to all parties, while in practice such information may be hard to obtain. In particular it seems that estimation of the correlation factor $\\\\rho$ in practical settings would require sharing fine-grained model performance metrics between competitors. It’s unclear whether results hold in settings in which the core assumptions are not met.", "was_revised": true, "final_justification": "This is a very interesting paper, and I especially appreciate its clear aim of informing public policy. The reviews have raised concerns about the model possibly being too simplistic, and helpful clarifications were provided in the rebuttal. Given this, I maintain my original score, and I believe that the paper can benefit from a more prominent discussion of the model’s scope, core assumptions, practical applicability, and limitations.", "rebuttal": "Thank you to the reviewer for the comments! We do indeed assume that firms have information about competing models’ accuracies and correlations. This assumption is realistic in oligopolies that frequently interact with each other (i.e., small but frequent purchases from many consumers, as is the case with our motivating scenarios of Uber and Lyft). Firms that frequently observe the market outcomes from their predictions allow them to better estimate the relevant parameters in equilibrium. Another perspective is that when competing firms play a no-regret strategy in this simple game, they will eventually converge to an equilibrium without explicit knowledge of parameters, see Hartline et al. (2015).\n\nQuestion (1). Interesting question. We suspect that the regulator would typically need enough information about firm behavior and the market in order to offer good incentives. However, regulators typically do not have this much information unless they conduct a formal investigation on possible antitrust violations.\n\nQuestion (2). In general, the cited references investigate different mechanisms/scenarios related to competition. For example, Einav & Rosenfeld (2025) show that instead of maximizing accuracy, firms might try to differentiate and segment the market – this outcome is of course possible, but many goods/services (e.g., Uber/Lyft, Amazon goods) are largely similar save for brand loyalty (which we can model via market asymmetry). We see a similar market segmentation effect in Jagadeesan et al. (2023). There are many scenarios that competitive firms face in real life, and we are merely analyzing a novel mechanism – one of many – that has important antitrust implications.\n\nQuestion (3). We believe our model can capture agents with different dynamics as well. The reviewer’s question about LLM providers adjusting prices would fit in our model; instead of making predictions about a user, they could be making decisions based on time, demand, etc. But of course, our model is limited to two possible prices (which can be easily extended to multiple and the same results will hold)."}, {"review_id": "oBoYTvoIr2", "reviewer": "Reviewer_jsLv", "summary": "This paper studies the problem of algorithmic homogeneity in personalized pricing. Specifically, it proposes a game-theoretic model to demonstrate how correlated algorithms can facilitate tacit collusion among competing firms, ultimately harming consumer welfare. Empirical studies on a stylized game between two firms are presented to demonstrate the theoretical results. Finally, the paper connects its technical findings to current antitrust law, suggesting new plus factors for detecting collusion.", "questions": "The model simplifies pricing to a binary choice. How would the results change if firms could set prices on a continuous spectrum, or if there were more than two discrete price points? Would the incentives for correlation still hold?", "limitations": "yes", "rating": 4, "confidence": 3, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 3, "clarity": 3, "significance": 3, "originality": 3, "strengths_and_weaknesses": "On the positive side, this paper presents a timely and important investigation into the anti-competitive implications of algorithmic homogeneity in personalized pricing. The game-theoretic model is well-defined, and the theorems clearly present the core findings: consumer welfare decreases with higher correlation, and firms are incentivized to choose correlated algorithms, especially when consumers are price-sensitive. I found the problem presented by the paper interesting and relevant. In general, this paper proposed an innovative research topic.\n\nMy main concern with this paper is its focus on the oversimplified model with two firms selling identical goods with a binary pricing scheme. While the extensions of asymmetric market power and N-firms are discussed in the appendix, it suggests that they are secondary. Given the importance of real-world market complexities, a more integrated discussion of how firm asymmetry and multiple firms impact the core findings (e.g., Theorems 4.1, 4.2, 5.3) could improve the paper's scope.", "was_revised": true, "final_justification": "The authors satisfactorily addressed all of my questions. My rating remains unchanged.", "rebuttal": ""}, {"review_id": "sMX4BVgS6m", "reviewer": "Reviewer_zrwy", "summary": "This paper studies the possibility of algorithmic collusion through using highly correlated predicting algorithms for price discrimination. The authors consider a duopoly model and model their competition with a few parameters including consumer’s price sensitivity ($\\sigma$), extent of algorithm correlation $\\rho$, price discrimination ratio $H/L$, the competitors’ prediction accuracy ($a_1$) and his own prediction accuracy $a_2$. Under this stylized model with two-firm, binary action, symmetric prediction error, the paper shows the following main results: \n(1) Given these known parameters, when Bayesian Nash equilibrium exists, the authors prove that consumer welfare decreases when $\\rho$ (correlation between two firm’s action) increases. \n(2) When consumers are more price sensitive, firms prefer highly correlated algorithms than independent ones.\n(3) An extended study about the situation where the firm can strategically choose an independent algorithm (by training on their own data) or a correlated algorithm (by buying data from vendors), and identifies conditions under which correlated algorithm is always preferred. \n(4) They verify their sights through simulation on ACSIncome data.", "questions": "See weakness for concrete questions.\n\nSome additional questions\n-\tWhat does the white region in Figure 1 mean?\n-\tIn Table 1 caption, should “top” “bottom” be left, right?", "limitations": "Yes.", "rating": 3, "confidence": 4, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 2, "clarity": 3, "significance": 2, "originality": 2, "strengths_and_weaknesses": "Strength:\n-\tThe problem studied in this paper is very interesting. Algorithmic collusion is indeed a critical, intriguing and difficult problem that touches AI, laws, policy and regulation\n-\t The paper illustrate interesting insights into the issue of price discrimination from a tractable economic model. It highlights how algorithmic correlations, a growing concern in the AI domain, can negatively impact consumer welfare.\n-\tThis paper leverages publicly available data and machine learning algorithms in a very smart way to efficiently demonstrate the insights behind its proposed method.\n\nWeakness:\nThe reviewer has a very mixed feeling. While the reviewer understands that it is difficult to strike a balance between insights and model complexity, the reviewer feels the studied model is a bit too simplistic and several assumptions are too strong – a learning algorithm is summarized into a single number $a$, pricing actions is simplified into two numbers, and correlation is captured by a single number $rho$ (PS: I seem to be missing how explicitly $rho$ affects the correlation of $p_1, p_2$), and all the knowledge is publicly shared. While these simplifications make the models very tractable (I think most quantities can be characterized in closed form under this simple model), I am not sure how generalizable the obtained insights are. \n\nOn the other hand, the reviewer also found the conclusions derived from the model is mostly expected, which reduces the novelty and insightfulness of the paper. \n\nTo concretize my concerns above, the reviewer has the following questions: \n\n-\tTrue positive rate most likely is very different from true negative rate, especially when the class label (customer type) is unbalanced. Will the insights still hold if this assumption has been relaxed?\n-\tThe definition of correlation is not rigorous. What is the mathematical formula of the correlation studied in this paper? Does this author consider the Pearson correlation, which is a ratio between condition covariance over conditional variance? The unrigorous definition leads to unrigorous proof of Theorem 4.1, which treats correlation as an exogenous variable, and independent of a1, a2, the performance of algorithms employed by the companies. Usually, a1, a2, and $rho$ moves at the same time, which the reviewer suggest to add simulations to study this problem", "was_revised": false, "final_justification": "", "rebuttal": ""}, {"review_id": "wxhlWHh8vV", "reviewer": "Reviewer_Z8sD", "summary": "This paper investigates a mechanism for collusion between competing firms using personalized pricing algorithms. The authors propose that firms can reduce price competition by using \"homogeneous\" algorithms—that is, algorithms that produce correlated predictions. The paper develops a formal game-theoretic model of a duopoly where two firms use algorithms to segment consumers and decide whether to offer a high or a low price.\n\nThe main theoretical findings are: (1) higher correlation between firms' algorithms harms consumer welfare, and (2) firms are more incentivized to use correlated algorithms when consumers are highly price-sensitive. The authors show that firms may strategically prefer to use a less accurate but more correlated algorithm over a more performant but independent one. These theoretical results are demonstrated in an experiment using public census data.", "questions": "* How robust are the findings to a more realistic pricing model? If firms could choose prices from a continuous range, would the incentive to sacrifice accuracy for correlation still exist, or would competitive pressures dominate?\n* The sigma parameter for consumer choice is essential to the results. How would the equilibria change under alternative models of consumer behavior, such as those involving search costs, network effects, or reference effects?\n* The paper argues that its static model is a feature that isolates the correlation mechanism. However, couldn't this mechanism be amplified in a dynamic setting? For instance, could RL agents learn to choose simpler, more correlated models as a way to facilitate convergence to a collusive pricing policy?\n* From a practical, regulatory standpoint, how can one distinguish between firms intentionally correlating via a less accurate model (the proposed \"plus factor\") versus firms that are simply less sophisticated and independently choose a simpler (e.g., linear) model that happens to be more correlated? The paper touches on this, but the problem of inferring intent seems very difficult.", "limitations": "The authors correctly note some limitations. My main concerns are stated in the Weaknesses and Questions sections.", "rating": 4, "confidence": 3, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 2, "clarity": 3, "significance": 2, "originality": 3, "strengths_and_weaknesses": "### Strengths\n* The paper studies an important problem in algorithmic fairness and economics—how shared components in ML pipelines can lead to anti-competitive outcomes. The focus on homogeneity as a specific mechanism for collusion is a good contribution.\n* The work is well-motivated by current events and its connection to antitrust law is a strength. The discussion of legal implications is good.\n* The paper effectively combines a formal game-theoretic model with an empirical study. The experiments, while simple, provide a demonstration that the theoretically-derived incentives can manifest with standard machine learning models on real-world data.\n\n### Weaknesses\n* The model's primary weakness is its simplicity. The assumption of a two-price world (High/Low) is a major abstraction from real-world pricing. The consumer choice model, governed by a single price-sensitivity parameter sigma, is also highly simplified. These abstractions make it difficult to be confident that the core results would hold in settings with continuous pricing or more complex consumer search behaviors.\n* The paper relies on a static Bayes Nash Equilibrium analysis. This approach deliberately abstracts away from the learning dynamics through which firms might arrive at a collusive equilibrium. While this isolates the correlation mechanism, it also makes the model less descriptive of the \"algorithmic\" part of algorithmic collusion, which often implies adaptation and learning over time.\n* The underlying principle (reduce strategic uncertainty through correlation can soften competition and lead to higher prices) is a known concept in the economics of oligopolies. The paper's novelty lies in applying this principle to the specific context of ML model development practices, which is a valuable but incremental step.", "was_revised": true, "final_justification": "As explained in the response to the reviewer, my overall assessment lies at the borderline. I think the paper has some merit, and also admit clear limitations. I am willing to follow the AC's final decision.", "rebuttal": ""}]

Author rebuttal:

Meta-review: This paper introduces a novel mechanism of tacit algorithmic collusion in personalized pricing. It shows that correlated algorithms reduce consumer welfare and that firms may strategically prefer less accurate but more correlated models. The work is further supported by a stylized empirical study and connected to antitrust law. 

All reviewers appreciated the clarity of the model, the formal results, and the strong policy relevance. However, some noted weaknesses in the simplicity of the assumptions, the limited generalizability, and the fact that some conclusions echo known results in economics. The rebuttal convincingly clarified definitions (e.g., correlation), addressed robustness (extensions to asymmetric and multi-firm settings), and provided concrete real-world analogies (Uber/Lyft, Expedia/Booking).

Overall, I find the conclusions worth publishing and recommend acceptance.
Decision: Accept (poster)