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The Effect of Outsourcing Pricing Algorithms on Market

Competition
Joseph E. Harrington, Jr.

To cite this article:

Joseph E. Harrington, Jr. (2022) The Effect of Outsourcing Pricing Algorithms on Market Competition. Management Science
68(9):6889-6906. https://doi.org/10.1287/mnsc.2021.4241

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 MANAGEMENT SCIENCE
Vol. 68, No. 9, September 2022, pp. 6889–6906
http://pubsonline.informs.org/journal/mnsc ISSN 0025-1909 (print), ISSN 1526-5501 (online)

The Effect of Outsourcing Pricing Algorithms on Market

Competition
Downloaded from informs.org by [103.197.36.6] on 20 August 2023, at 09:33 . For personal use only, all rights reserved.

Joseph E. Harrington, Jr.a

a
Department of Business Economics & Public Policy, The Wharton School, University of Pennsylvania, Philadelphia, Pennsylvania 19104
Contact: harrij@wharton.upenn.edu, https://orcid.org/0000-0002-2686-4021 (JEH)

Received: February 26, 2021 Abstract. A third party developer designs and sells a pricing algorithm that enhances a
Revised: July 19, 2021 firm’s ability to tailor prices to a source of demand variation, whether high-frequency de-
Accepted: October 4, 2021 mand shocks or market segmentation. The equilibrium pricing algorithm is characterized
Published Online in Articles in Advance: that maximizes the third party’s profit given firms’ optimal adoption decisions. Outsourc-
January 31, 2022 ing the pricing algorithm does not reduce competition but does make prices more sensitive
https://doi.org/10.1287/mnsc.2021.4241 to the demand variation, and this is shown to decrease consumer welfare and increase in-
dustry profit. This effect is larger when products are more substitutable.
Copyright: © 2022 INFORMS
History: Accepted by Joshua Gans, business strategy.

Keywords: pricing algorithm • competition • competition policy • outsourcing

1. Introduction firms). The use of a third party to provide pricing

As a result of Big Data and algorithmic pricing, firms services is common on platforms such as Amazon
can condition prices on high frequency data, tailor Marketplace and Airbnb and more broadly in retail
prices to narrow submarkets, and engage in more markets (e.g., Assad et al. (2020) offer an analysis of
effective learning to discover the most profitable the use of third party pricing algorithms in retail gaso-
pricing rules. Although there are potential efficiency line markets).
benefits from these advances, concerns have been Although a firm may find it attractive to use a third
raised about possible consumer harm. Enhanced party’s pricing algorithm, the possibility of consumer
price discrimination due to rich customer data may harm has been voiced by various competition authori-
increase total welfare but could result in a transfer of ties. For example, the United Kingdom’s Competition
surplus from consumers to firms. Automated pricing & Markets Authority (2018, pp. 26–27) expressed con-
with high frequency data could make markets more cern about the anticompetitive risk when “competitors
efficient by increasing the speed of response to de- decide … that it is more effective to delegate their pric-
mand changes, but it is unclear how it will affect ing decisions to a common intermediary which pro-
price competition. Learning algorithms fueled by AI vides algorithmic pricing services” and noted that “[i]f
could deliver more profitable pricing rules but that a sufficiently large proportion of an industry uses a
could be because they facilitate collusion. An active single algorithm to set prices, this could result in …
competition policy debate has arisen regarding algo- the ability and incentive to increase prices.”
rithmic pricing and whether legal and enforcement An open question in the area of competition policy
regimes are equipped to deal with the associated is what the adoption of third party pricing algorithms
challenges.1 means for consumers. Of particular relevance is that a
One of the primary implications of Big Data and al- third party’s incentives when it comes to designing
gorithmic pricing is that it has become more attractive the pricing algorithm are likely to differ from those of
for a firm to outsource pricing. With prices deter- a firm in a market, for the latter is interested in selling
mined more by data and less by the judgment of those more units at a higher price, whereas a third party de-
employees in the firm with the best soft information, veloper is interested in selling more pricing algo-
pricing can be delegated to a third party or to a pric- rithms at a higher fee. What difference does it make if a
ing algorithm developed by a third party. A third pricing algorithm is designed by a firm interested in maxi-
party developer is likely to have better pricing algo- mizing profit from the sale of the pricing algorithm rather
rithms than would be created internally because it has than from the sale of the product which the algorithm is to
more expertise and experience, access to more data, price?
and stronger incentives to invest in their development To address this question, a stylized model is devel-
(as the pricing algorithm can be licensed to many oped with a monopoly developer of pricing algorithms.

6889
 Harrington: Outsourcing Pricing Algorithms and Market Competition
6890 Management Science, 2022, vol. 68, no. 9, pp. 6889–6906, © 2022 INFORMS

Although, in practice, there are many developers of et al. 2021), and how firms learn the best pricing algo-
pricing algorithms, it makes sense to first explore the rithms (Salcedo 2015, Calvano et al. 2020b, Asker et al.
monopoly case before examining the additional implica- 2021, Klein 2021). All of those studies assume the pric-
tions of competition among developers. The setting is ing algorithm is designed by the firm itself and thus
one where the pricing algorithm’s comparative advan- do not consider the implications of it being designed
tage is allowing price to condition on a source of de- by a third party with different incentives than that of
mand variation such as high-frequency demand shocks the firm. For a more detailed literature review, the
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(or what industry refers to as “dynamic pricing”) or reader is referred to Appendix A.

finely grained market segments (“personalized pricing”). The second literature to which this paper contrib-
A third party designs the pricing algorithm and offers it utes is one that examines the welfare effects of third-
to firms for a fee. Firms then decide whether to adopt it degree price discrimination in oligopolistic markets.4
and, given those adoption decisions, prices are set. Past research has focused on comparing welfare when
The paper delivers new insight into pricing algo- a firm adopts a uniform price (so all markets are
rithms that are summarized here for when demand charged the same price) and when it engages in third-
variation is significant. A critical distinction between a degree price discrimination. The current paper consid-
pricing algorithm designed by a third party developer ers third-degree price discrimination and compares
who intends to sell it and a firm who intends to use it welfare when outsourcing it to a third party (so it is
is that the third party will take account of the possibil- designed to maximize the third party’s profit) to
ity that the algorithm might compete against itself; when it is internally developed by the firm (so it is de-
that is, competitors might adopt the pricing algorithm. signed to maximize the firm’s profit). As we show, the
This could lead the third party to make the pricing al- deleterious welfare effects of third-degree price dis-
gorithm less competitive in order to enhance the algo- crimination are accentuated when the pricing algo-
rithm’s performance and thus the demand for it. rithm is outsourced.
However, I do not find that to be the case in that out- Section 2 provides the general model and character-
sourcing the pricing algorithm does not result in izes equilibrium conditions both in the market for
higher average prices. If the third party had designed pricing algorithms and the market for firms’ products.
the pricing algorithm to set a higher average price, it Under the assumption of linear demand, Section 3
would make it more attractive not to adopt the pricing derives a closed-form solution for the equilibrium
algorithm when a firm’s rival does, which would pricing algorithm and Section 4 explores some impli-
harm demand for the algorithm. What the third party cations of outsourcing the design of a firm’s pricing
does instead is make price more sensitive to demand rule. Section 5 summarizes and offers some directions
variation, thereby generating more profit when and for future research.
where demand is strong. By making price more sensi-
tive to demand variation, the third party improves the
2. General Model and
profit from joint adoption without making it more at-
tractive not to adopt. Though average price is not Equilibrium Conditions
higher, outsourcing still harms consumers because 2.1. General Model
of increased price variability. Furthermore, this harm Consider a collection of duopoly markets with differ-
is greater when products are more similar because entiated products that differ in cost and demand con-
prices remain highly sensitive to demand variation ditions.5 The set of market types is the finite set H and
rather than becoming more closely tied to cost, which λ(h) is the number of markets of type h ∈ H: For a
is what would occur if there was no outsourcing. In type h market, ch is the common and constant mar-
sum, it does make a difference for the design of a pric- ginal cost, and Di (p1 , p2 , a, h) : R2+ × Λ × H → R+ is the
ing algorithm that it is done by a third party devel- (symmetric) firm demand function, which depends on
oper and, furthermore, consumers are harmed.2 firms’ prices (p1 , p2 ) and a demand variable a ∈ Λ with
This paper contributes to two literatures. Its pri- cdf G : Λ × H → [0, 1]: The demand variable a has two
mary contribution is to the emerging theoretical litera- interpretations. A firm in market h could be facing a
ture exploring the effects of algorithmic pricing.3 A single demand curve and a is a demand shock with
defining feature of these papers is how Big Data and distribution G. In that case, price may condition on
algorithmic pricing are represented in the model. It the current demand shock a. Alternatively, a firm in
can affect how much information firms have on de- market h faces a collection of market segments repre-
mand (Miklós-Thal and Tucker 2019, O’Connor and sented by G. In that case, price may condition on the
Wilson 2022), how rapidly firms can respond to rivals’ market segment a. We will generally use the “demand
prices (Brown and MacKay 2022, Leisten 2021), how shock” interpretation in our exposition. Further de-
firms simultaneously learn about their demand func- mand assumptions will be made in Section 3. It is
tions and optimal prices (Cooper et al. 2015, Hansen generally assumed Λ  [a, a] and G is continuously
 Harrington: Outsourcing Pricing Algorithms and Market Competition
Management Science, 2022, vol. 68, no. 9, pp. 6889–6906, © 2022 INFORMS 6891

differentiable though all analysis goes through if in- is Subgame Perfect Equilibrium. In that final stage, if
stead Λ is a finite set.6 both firms did not adopt the pricing algorithm, then
Let us initially suppose the demand shock a equilibrium prices are pN (h). If both firms adopted,
occurs at a higher frequency than a firm’s pricing then they price at φ(a, h). If one firm adopted and the
decisions (or, when G represents a collection of mar- other firm did not adopt, then the former prices at
ket segments, the firm cannot distinguish among φ(a, h) and the latter, which cannot condition its price
them). In that case, a firm is incapable of condition- on a, chooses a best response to φ(a, h) given its beliefs
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ing price on it and, therefore, its price depends only G on a. The assumption that φ(·) is public information
on G. A symmetric Nash equilibrium price pN (h) is is clearly stylized but allows firms to form accurate be-
defined by: liefs on the profit associated with adoption and for a

  nonadopting firm to form accurate beliefs on an adopt-
pN (h) ≡ arg max (p − ch )D1 p, pN (h), a, h G (a, h)da: ing firm’s price.10
p∈R+
Before moving on, it is worth noting that an adopt-
As a convention, G (a, h) ≡ ∂G(a, h)=∂a: ing firm is unable to modify the pricing algorithm. I
The comparative advantage of the third party de- am presuming the pricing algorithm is a “black box”
veloper is that it can offer a pricing algorithm capable to the firm so it cannot disentangle the demand state
of tracking the high-frequency demand shock (or mar- and start changing the price attached to it. Although I
ket segment) a so a firm’s price can then condition on believe there are situations where such an assumption
it. This algorithm is denoted φ : Λ × H → R+ : When a is appropriate (as the firm lacks the necessary knowl-
is a demand shock then φ assigns a price to each pos- edge), there are also situations where some modifica-
sible demand shock in Λ, and when a is a market seg- tion would be possible. In exploring the latter, one
ment then {φ}a∈Λ is the vector of prices assigned to would want to consider the constraints on a firm’s
the set of market segments Λ. Once adopted by a firm, ability to modify the algorithm lest one trivializes the
the algorithm is assumed to “learn” the firm’s de- role of a third party developer. This extension of the
mand parameters, while a firm can program in its model is left for future research.
cost. As a result, φ conditions on h even though the
third party may not know a particular market’s type. 2.2. Equilibrium Conditions
Let Φ denote the space of mappings from Λ × H into Toward specifying the conditions defining the equi-
the price space R+ . librium pricing algorithm, we will begin by charac-
The third party chooses a fee f, which a firm pays in terizing the market demand for pricing algorithms.
order to adopt φ. As the fee is set ex ante, it is uniform For that purpose, let V(Ii , Ij , φ, h) denote gross profit
across markets. For reasons of competition law, the (before netting out the third party’s fee) for a firm
fee is not tied to an adopting firm’s profit. Given that with adoption decision Ii ∈ {A, NA} given the other
both firms may adopt the algorithm, a third party that firm’s adoption decision Ij ∈ {A, NA}, where A refers
was compensated based on competitors’ profits could to adoption and NA to no adoption. The explicit ex-
effectively act as a cartel manager and coordinate pressions for V(Ii , Ij , φ, h) are provided later. A firm’s
firms’ prices. It is also for this reason that the algo- total cost of adoption is f + ε where f is the fee
rithm is not permitted to condition on the adoption charged by the third party and ε is a market-specific
decision of another firm in the market. If that were adoption cost, which is observable to the firms but
allowed, the algorithms could be programmed to not to the third party. ε is introduced so the proba-
“communicate” and coordinate their prices in the bility of adoption is a smooth function of φ and
event that both adopted, which the third party may be f. ε has a continuously differentiable cdf K : R+ →
inclined to do in order to generate more value for [0, 1] and K puts sufficient mass near zero so that, at
firms, which would then allow it to charge a higher the equilibrium design and fee, there is positive ex-
fee.7, 8 As I’ll later show, in spite of these restrictions, pected demand.11
outsourcing can cause consumer harm. Derivation of the demand for pricing algorithms re-
Next to be described is the sequence of moves and quires characterizing equilibrium adoption decisions.
what agents know. In the first stage, the third party It is an equilibrium for a market to have zero adop-
designs the pricing algorithm and sets the fee; that is, tions when:
it chooses (φ(·), f ) ∈ Φ × R+ : In the second stage,
(φ(·), f ) is publicly revealed and firms make simulta- V(NA, NA, φ, h) ≥ V(A, NA, φ, h) − ( f + ε) ⇐
⇒
neous adoption decisions. After adoption decisions
f + ε ≥ V(A, NA, φ, h) − V(NA, NA, φ, h);
are made and publicly observed, the third stage has
the high-frequency demand shock realized and firms that is, the incremental value of adoption conditional
make simultaneous price decisions.9 The solution concept on the rival firm not adopting is less than the cost of
 Harrington: Outsourcing Pricing Algorithms and Market Competition
6892 Management Science, 2022, vol. 68, no. 9, pp. 6889–6906, © 2022 INFORMS

adoption. It is an equilibrium for a market to have one adoptions, an equilibrium selection is made that both
adoption when: adopt.12
The market demand for the third party’s pricing
V(A, NA, φ, h) − ( f + ε) ≥ V(NA, NA, φ, h) and
algorithm is composed of those markets for which
V(NA, A, φ, h) ≥ V(A, A, φ, h) − ( f + ε), one firm adopts—(3) is satisfied—and those for which
where the first inequality says it is optimal for a firm two firms adopt—so (2) is satisfied. The resulting de-
to adopt given the rival firm does not adopt and the mand is
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second inequality says it is optimal for a firm not to 

[1 × max{K(V(A, NA, φ, h) − V(NA, NA, φ, h) − f )
adopt given the rival firm does adopt. Those condi- h∈H
tions are equivalent to:
− K(V(A, A, φ, h) − V(NA, A, φ, h) − f ), 0}
V(A, NA, φ, h) − V(NA, NA, φ, h) ≥ f + ε + 2 × K(V(A, A, φ, h) − V(NA, A, φ, h) − f )]λ(h) (4)
≥ V(A, A, φ, h) − V(NA, A, φ, h):
where recall λ(h) is the number of type h markets. The
Finally, it is an equilibrium for a market to have two first term in brackets is demand coming from markets
adoptions when: where one firm adopts the pricing algorithm. When
V(A, A, φ, h) − ( f + ε) ≥ V(NA, A, φ, h) (1) holds, it equals
⇐
⇒ V(A, A, φ, h) − V(NA, A, φ, h) K(V(A, NA, φ, h) − V(NA, NA, φ, h) − f ) − K(V(A, A, φ, h)
≥ f + ε, − V(NA, A, φ, h) − f ),

so the incremental value of adoption conditional on

and when (1) does not hold then it is zero. The second
the rival firm adopting exceeds the cost of adoption.
term in (4) is demand coming from markets where both
If
firms adopt, and is the probability that (2) is satisfied.13
V(A, NA, φ, h) − V(NA, NA, φ, h) > V(A, A, φ, h) The third party chooses the design and fee to maxi-
mize its expected revenue.14 The equilibrium design
− V(NA, A, φ, h),
and fee are the solution to:
(1)
(φ∗ , f ∗ )
so adoptions are strategic substitutes, then, generi- 
 arg max f × [1 × max{K(V(A, NA, φ, h)
cally, there is a unique equilibrium number of adop- (φ, f )∈Φ×R+ h∈H
tions. If
− V(NA, NA, φ, h) − f )
V(A, A, φ, h) − V(NA, A, φ, h) ≥ f + ε (2) − K(V(A, A, φ, h) − V(NA, A, φ, h) − f ), 0}
+ 2 × K(V(A, A, φ, h) − V(NA, A, φ, h) − f )]λ(h) (5)
then the market has two adoptions; if
V(A, NA, φ, h) − V(NA, NA, φ, h) ≥ f + ε where
(3)
 
> V(A, A, φ, h) − V(NA, A, φ, h),
V(A, A, φ, h)  φ(a, h) − ch
then the market has one adoption; and if f + ε > V(A, (6)
NA, φ, h) − V(NA, NA, φ, h), then the market has zero D1 (φ(a, h), φ(a, h), a, h)G (a, h)da

 
adoptions.
If instead V(A, NA, φ, h)  φ(a, h) − ch

V(A, A, φ, h) − V(NA, A, φ, h) > V(A, NA, φ, h) D1 (φ(a, h), p∗ (φ, h), a, h)G (a, h)da

 
− V(NA, NA, φ, h),
V(NA, A, φ, h)  p∗ (φ, h) − ch
so adoptions are strategic complements, then the equi-
librium number of adoptions is zero or two. If V(A, D1 (p∗ (φ, h), φ(a, h), a, h)G (a, h)da

 
NA, φ, h) − V(NA, NA, φ, h) > f + ε then the market has V(NA, NA, φ, h)  pN (h) − ch
two adoptions, and if
 
V(A, A, φ, h) − V(NA, A, φ, h) ≥ f + ε ≥ V(A, NA, φ, h) D1 pN (h), pN (h), a, h G (a, h)da
− V(NA, NA, φ, h)

p∗ (φ, h)  arg max (p − ch )D1 (p, φ(a, h), a, h)G (a, h)da
p
then the market has either zero or two adoptions.
When it is an equilibrium to have either zero or two (7)
 Harrington: Outsourcing Pricing Algorithms and Market Competition
Management Science, 2022, vol. 68, no. 9, pp. 6889–6906, © 2022 INFORMS 6893

Using the expression for market demand in (4), (5) is because a firm is able to raise price when demand is
expected revenue. Given its rival adopts φ, the opti- stronger (and more price-inelastic) and lower price
mal price for a firm that does not adopt is p∗ (φ, h) as when demand is weaker (and more price-elastic). By
defined in (7). p∗ (φ, h) along with φ are used to define comparing the third party’s equilibrium pricing algo-
firms’ values when one adopts and the other firm rithm φ∗ with φI , we will identify the difference due
does not—V(A, NA, φ, h) and V(NA, A, φ, h)—as pro- to outsourcing or, alternatively stated, to the pricing
vided in (6). Equation (6) also defines a firm’s value algorithm having been designed to maximize the
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when both adopt, V(A, A, φ, h), and when neither profit from selling the algorithm (i.e., the developer’s
adopts, V(NA, NA, φ, h). We next turn to deriving a profit) rather than from using the algorithm (i.e., a
closed-form solution for (5). firm’s profit).

3. Equilibrium Under Linear Demand 3.2. Characterization Theorems

For purposes of tractability, from here on, linear de- In solving (5), the focus is on affine pricing algorithms:
mand is assumed: φ(a)  α + γa for some (α, γ):15 Prior to presenting our
characterization theorems, it will prove useful to ini-
D1 (p1 , p2 , a, h)  a − bp1 + dp2 , tially consider a constrained problem for the third
party. Suppose the third party chose to optimize while
where b > d ≥ 0: Recall that a ~ G and let a have mean ensuring that adoptions are (weakly) strategic com-
μ and variance σ2 : Assume a − (b − d)c > 0 ∀a ∈ Λ so plements, which means φ satisfies:
demand is always positive. To save on extraneous no-
tation, the market type h is dropped though it should V(A, A, φ) − V(NA, A, φ) ≥ V(A, NA, φ) − V(NA, NA, φ):
be remembered that the pricing algorithm is designed
In that case, the expression in (4) for expected demand
for each market type.
is K(V(A, A, φ) − V(NA, A, φ) − f ). As long as K(V(A,
A, φ) − V(NA, A, φ) − f ) > 0 for some φ, then maximiz-
3.1. Benchmark Equilibria
Suppose there is no third party and the firms are un- ing that expression is equivalent to maximizing the in-
able to condition price on the demand shock. In that cremental value of adoption conditional on the rival
μ+bc firm adopting, V(A, A, φ) − V(NA, A, φ). Consequently,
case, the symmetric Nash equilibrium is pN  2b−d with
the pricing algorithm is designed to solve:
expected profit of
max V(A, A, φ) − V(NA, A, φ)
φ∈Φ
 b(μ − (b − d)c)2
G (a)da  s:t: V(A, A, φ) − V(NA, A, φ) ≥ V(A, NA, φ)
(2b − d)2

    − V(NA, NA, φ): (8)
μ + bc μ + bc
π ≡
N
− c a − (b − d)
2b − d 2b − d It is shown in Appendix B (Lemma B.1) that the solu-
tion to (8) is:
Although this is a relevant benchmark, it is not the ap-  
2(b − d)bc − dμ 1
propriate one for assessing the effect of outsourcing φsc (a) ≡ + a:
because the use of a third party’s pricing algorithm 2(b − d)(2b − d) 2(b − d)
confounds outsourcing with engaging in third-degree Now let us turn to solving (5). We’ll initially consider
price discrimination (with respect to the demand vari- the case when demand is relatively stable, which in-
able a). In order to separate these effects, the proper cludes σ2  0 so there is no demand variation. Theorem 1
benchmark is when firms condition price on the de- shows that the third party designs the pricing algorithm
mand shock but use a pricing algorithm that each firm to make one firm into a price leader (which is why the
internally develops in order to maximize its profit. pricing algorithm is denoted φpl ). By adopting the third
This is the standard Nash equilibrium with third- party’s pricing algorithm, a firm commits to a higher av-
degree price discrimination, φI (a) ≡ 2b−d
a+bc
where I refers erage price, which induces its nonadopting rival to raise
to “internal development.” its price, and that is what makes it profitable for the firm
Although expected price is the same as when to adopt. Proofs are in Appendix B.16
firms do not condition on the high-frequency de-
μ+bc Theorem 1. If σ2 is sufficiently low, then the unique equi-
mand shock, E[φI (a)]  2b−d , expected profit is higher librium pricing algorithm is
with φI ,  
(b + d)((2b − d)bc + dμ) 1
φpl (a) ≡ + a,
b(μ − (b − d)c)2 bσ2 b(4b − 2d )
2 2 2b
πI ≡ + > πN ,
(2b − d)2 (2b − d)2 and either no firm adopts or one firm adopts.
 Harrington: Outsourcing Pricing Algorithms and Market Competition
6894 Management Science, 2022, vol. 68, no. 9, pp. 6889–6906, © 2022 INFORMS

It is straightforward to show that the pricing algo- only one firm to adopt and accordingly the third party
rithm commits the adopter to a higher average price designs the pricing algorithm to take advantage of the
than when pricing algorithms are not adopted: commitment that adoption delivers.
  Although the preceding analysis is interesting, the
(b + d)(2cb2 − cdb + dμ) 1
pl
E[φ (a)] − p N
+ μ more relevant setting is when σ2 is not low so high-
b(4b2 − 2d2 ) 2b
frequency demand shocks are a significant factor in
μ + bc the market. That is the setting that is made more com-
−
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2b − d mon with Big Data and which makes it especially at-

d2 (μ − (b − d)c) tractive for a firm to turn to a third party so that it can
 > 0:
4b(b − d)(2b + d) + 2d3 condition price on demand fluctuations. Our next re-
sult characterizes the equilibrium pricing algorithm
As a result, the nonadopting firm prices higher be-
when the demand variance is high. Now we find the
cause its price is the best response to E[φpl (a)],
pricing algorithm is designed to make adoptions into
μ + bc + dE[φpl (a)] strategic complements and the third party sells its
,
2b pricing algorithm to both firms.
and Theorem 2. If σ2 is sufficiently high, then the unique equi-
    librium pricing algorithm is
(b+d)(2cb −cdb+dμ)
2
μ + bc + d b(4b2 −2d2 )
+ 2b
1
μ μ + bc  
− 2(b − d)bc − dμ 1 μ + bc a − μ
2b 2b − d sc
φ  + a + ,
2(b − d)(2b − d) 2(b − d) 2b − d 2b − 2d
d3 (μ − (b − d)c)
 > 0:
4b(2b(b − d)(2b + d) + d3 ) and either no firm adopts or both firms adopt.
In general, two forces are at play when it comes to When demand significantly fluctuates, it is attrac-
adoption of a third party’s pricing algorithm. First, tive to a firm to be able to condition price on those de-
adoption means committing to a pricing rule, which mand movements and, consequently, equilibrium has
has strategic value but only when the rival firm does both firms adopting the pricing algorithm (assuming
not adopt and thus can respond to that commitment. f + ε is low enough). Anticipating both firms may
Second, adoption means having price condition on adopt, the third party designs the pricing algorithm to
the demand shock. When a market’s demand variance maximize each firm’s willingness-to-pay (WTP) when
is low, the second force is sufficiently weak that the both adopt, as that will maximize the likelihood that it
third party designs the pricing algorithm so as to ex- exceeds the adoption cost f + ε and thereby result in a
ploit commitment.17 sale. Thus, the design is chosen to maximize the incre-
To elaborate on this explanation, consider σ2  0. mental value of adoption, which is V(A, A, φ) −
Without any demand variation, commitment to a pric- V(NA, A, φ): This then creates a design challenge for
ing algorithm is equivalent to commitment to a partic- the third party developer: make it profitable to adopt
ular price. Thus, adoption of the third party’s pricing the pricing algorithm—which encourages designing φ
algorithm creates a sequential-move price game where so that V(A, A, φ) is high—while not making it exploit-
the adopter is the price leader and the nonadopter is able by a nonadopting rival firm—which encourages
the price follower. As we know that the follower’s designing φ so that V(NA, A, φ) is low.
profit is higher than the leader’s in such a game, it is In solving this challenge, note that the third party
then an equilibrium for only one firm to adopt (as- does not design the pricing algorithm to maximize
suming f + ε is sufficiently low and, otherwise, no firm firms’ joint profit, which would mean maximizing
adopts). When σ2 > 0, adoption no longer creates a V(A, A, φ). Consider the joint profit-maximizing (or
sequential-move price game (as the adopting firm’s monopoly) price as a candidate pricing algorithm:
price responds to the demand variable at the same
a + (b − d)c
time as the nonadopting firm is choosing its price), pM (a) ≡ arg max(p − c)(a − (b − d)p)  :
but there is still a benefit from committing to a higher p 2(b − d)
average price since the nonadopting firm’s price is a If the third party were to use this design, it would result
best response to its rival’s average price. As σ2 is in- in a high value of V(A, A, φ) but also cause V(NA, A, φ)
creased from zero, the expected profit from adoption to be relatively high as a nonadopting firm would be
rises because a firm can adjust its price to demand able to exploit the high average price of a rival firm that
shocks, whereas the expected profit from nonadoption has adopted. By instead having the pricing algorithm
is unchanged (as it is based on the adopting firm’s ex- price slightly less than pM (a), there is no first-order ef-
pected price, which is independent of σ2 ). As long as fect on V(A, A, φ), but there is a first-order decrease
σ2 is not too high, it will still be an equilibrium for of V(NA, A, φ) because the nonadopting firm finds its
 Harrington: Outsourcing Pricing Algorithms and Market Competition
Management Science, 2022, vol. 68, no. 9, pp. 6889–6906, © 2022 INFORMS 6895

rival’s price to be lower; hence, it raises the incremental and, at the equilibrium pricing algorithm, V(A, A, φ)−
value of adoption. Consequently, the third party’s de- V(NA, A, φ) ≤ 0, which implies expected demand sim-
sign will have the pricing algorithm price below that plifies from (9) to K(V(A, NA, φ) − V(NA, NA, φ) − f ).
which maximizes joint profit. Thus, when the demand variance is low, the equilib-
Although the price level is less than the monopoly rium price algorithm maximizes the incremental value
price, the sensitivity of the pricing algorithm to the de- of adoption given the rival firm does not adopt,
mand shock is the same as for the monopoly price, V(A, NA, φ) − V(NA, NA, φ). When the demand vari-
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∂φsc (a) 1 ∂pM (a) ance is neither low nor high, it is possible the third
  : party chooses the pricing algorithm so adoptions are
∂a 2(b − d) ∂a
strategic substitutes and V(A, A, φ) − V(NA, A, φ) > 0,
Thus, the third party’s pricing algorithm shifts down which means the solution that maximizes (9) will de-
the pricing rule that maximizes the profit of both firms pend on K. That is why equilibrium has only been
adopting: characterized when demand variance is low or high
d(μ − (b − d)c) and not for the intermediate case.18 However, if the
φsc (a)  pM (a) − : fee (as well as the design) is tailored to the market
2(2b − d)(b − d)
type and the adoption cost shock ε is eliminated, then
Toward understanding the optimality of this rule, the equilibrium pricing algorithm can be character-
consider how changing the pricing algorithm affects ized for all demand variances. That is what we do in
the incremental value of adoption, V(A, A, φ) − V(NA, this section.
A, φ). Given that a nonadopting firm does not condi- Without the adoption cost shock, the third party
tion on the high-frequency demand shock, the ex- will know exactly how many firms will adopt depend-
pected profit of a nonadopter V(NA, A, φ) depends ing on φ and f, where both are now set for a market
only on the expected price of its adopting rival, which type. Thus, we can think of the third party deciding to
is the expectation of φ: In contrast, given adoption sell the pricing algorithm to one firm or two firms. If it
means conditioning price on the realization of the decides to sell it to two firms, then the optimal design
demand shock, V(A, A, φ) depends on the entire distri- is φsc as that maximizes the WTP and thus maximizes
bution of price (based on φ). Making the responsive- the fee that can be charged. In that case, the WTP is
ness of φ to a closer to the responsiveness of the the incremental value of adoption conditional on the
monopoly price, while keeping the expectation of φ rival firm adopting, which (as shown in the proof of
fixed, raises the expected profit from adopting with- Lemma B.1) is 4(b−d)σ2 σ2
so the optimal fee is f  4(b−d) . The
out affecting the expected profit from not adopting; third party’s revenue from selling φsc to two firms
hence, the incremental value from adoption increases. σ2
at that fee is 2(b−d) : If it decides to sell the pricing algo-
This explains why ∂φsc (a)=∂a ∈ (∂pN (a)=∂a, ∂pM (a)=∂a],
and it may be due to linearity that it results in a corner rithm to only one firm, then it will want the design to
solution with ∂φsc (a)=∂a  ∂pM (a)=∂a. This simple maximize the incremental value of adoption condi-
tional on the rival firm not adopting, as again that will
modification of the monopoly price yields high profit
maximize the fee that can be charged. The solution to
when both firms adopt without creating high profit
for a firm foregoing adoption and pricing competi- that problem is φpl , and the WTP (and fee) can be
d4 (μ−(b−d)c)2
tively against a rival firm that did adopt. shown to be +σ ,
2
which is also the third
8b(2b−d)2 (2b2 −d2 ) 4b
party’s revenue.
3.3. Market-Specific Fee
The difference between the revenue from selling
When the demand variance is high, the third party
two units of φsc and one unit of φpl is
optimally designs the pricing algorithm so that adop-

tions are strategic complements (Theorem 2). Thus, σ2 d4 (μ − (b − d)c)2 σ2
expected demand is 2 × K(V(A, A, φ) − V(NA, A, φ) − − +
2(b − d) 8b(2b − d) (2b2 − d2 ) 4b
2
f ) and the equilibrium price algorithm is that which
maximizes the incremental value of adoption given
the rival firm adopts, V(A, A, φ) − V(NA, A, φ). When (b + d)σ2 d4 (μ − (b − d)c)2
 − :
the demand variance is low, the third party opti- 4b(b − d) 8b(2b − d)2 (2b2 − d2 )
mally designs the pricing algorithm so that adop-
It is then optimal for the third party to choose φsc (φpl )
tions are strategic substitutes (Theorem 1). Expected
and sell to two firms (one firm) when
demand is
K(V(A, NA, φ) − V(NA, NA, φ) − f ) (b − d)d4 (μ − (b − d)c)2
σ2 > (<) :
+ K(V(A, A, φ) − V(NA, A, φ) − f ) (9) 2(2b − d)2 (2b2 − d2 )(b + d)
 Harrington: Outsourcing Pricing Algorithms and Market Competition
6896 Management Science, 2022, vol. 68, no. 9, pp. 6889–6906, © 2022 INFORMS

4. Effect of Outsourcing a Firm’s whether the pricing algorithm is developed internally

Pricing Rule or externally. The second point is that the pricing al-
In the remainder of the paper, we’ll focus on the more gorithm is more sensitive to the demand shock when
interesting and relevant case when the demand vari- it is developed by a third party:
ance is high enough that the third party’s equilibrium
∂φsc (a) 1 1 ∂φI (a)
pricing algorithm is φsc and both firms adopt. In order  >  :
to assess the effect of outsourcing, I will compare the ∂a 2(b − d) 2b − d ∂a
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third party’s pricing algorithm φsc with φI , which is the In response to stronger demand, a firm raises its price
pricing algorithm that conditions on the demand shock
but φI limits the amount of that price increase because of
but is internally developed by the firm so as to maxi-
the prospect of losing demand to the other firm. How-
mize its expected profit. In that way, the effect of out-
ever, the third party’s pricing algorithm internalizes that
sourcing is disentangled from the effect of third-degree
effect—as it responds to demand variation as would a
price discrimination. Our primary goal is to shed light
monopolist—and, consequently, price rises more in re-
on the structure of the pricing algorithm, which is at-
sponse to stronger demand. Outsourcing the pricing
tributable to it being developed by a third party.
algorithm then results in greater price sensitivity to de-
mand shocks and, therefore, greater price volatility.
4.1. Sensitivity of Price to Demand Shocks
A numerical example illustrates the effect of out-
It will prove instructive to rearrange the equilibrium
sourcing on price variability. Assume μ  100, b  1,
pricing algorithm into the following expression:
c  10, d  0:6: In the proof of Theorem 2, it is shown
a + bc d(a − μ) that a sufficient (but not necessary) condition for φsc
φsc (a)  + : (10) to be the equilibrium pricing algorithm is
2b − d 2(b − d)(2b − d)

By comparison, the pricing algorithm that conditions d3 (b − d)(μ − (b − d)c)2

σ2 ≥  123:86:
on the demand shock that firms would develop on 2(2b − d)2 (2b2 − d2 )
their own is
If a is uniformly distributed on [80, 120], then σ2 
a + bc 133:33 and the above condition is satisfied. The equi-
φI (a)  : (11)
2b − d librium pricing algorithm is φsc (a)  −46: 43 + 1:25a
and, consequently, price is uniformly distributed on
μ+bc
Recall that firms price at pN  2b−d when they do not [53:57, 103:57]: By comparison, φI (a)  7:14 + 0:71a and
condition price on the high-frequency demand shock. price is uniformly distributed on [63:94, 92:34]: Out-
The three pricing rules are depicted in Figure 1. sourcing increases the range of prices by 76% from
In comparing these pricing rules, the first point to 28.40 to 50.00 and more than triples the variance from
μ+bc 67.21 to 208.33.
note is that average price is 2b−d whether firms condi-
tion price on the high-frequency demand shock and Before netting out the fee for using the pricing algo-
rithm, expected profit is higher to firms when the pric-
ing algorithm is developed by a third party. Expected
Figure 1. Effect of Outsourcing on Prices profit under external development is

b(μ − (b − d)c)2 σ2
+
(2b − d)2 4(b − d)

and with internal development is

b(μ − (b − d)c)2 (b + d)σ2
2
+ :
(2b − d) 4b2
The former is larger:
 
b(μ − (b − d))2 σ2 b(μ − (b − d)c)2 (b + d)σ2
+ − +
(2b − d)2 4(b − d) (2b − d)2 4b2
d2 σ2
 > 0: (12)
4b2 (b − d)
As explained above, the attractiveness of the third
party’s pricing algorithm is that it internalizes the
 Harrington: Outsourcing Pricing Algorithms and Market Competition
Management Science, 2022, vol. 68, no. 9, pp. 6889–6906, © 2022 INFORMS 6897

effect of one’s firm price on the other’s demand and conduct, which would harm consumers. First, the fee
profit. Even if firms are capable of developing their was not allowed to be tied to an adopting firm’s profit.
own pricing algorithms at a comparable cost to the Without that restriction, the third party could claim a
third party, it is collectively advantageous to have share of firms’ profit and thus be incentivized to have
the third party design them.19, 20 the pricing algorithm charge the monopoly price so as to
maximize industry profit. Second, the pricing algorithm
4.2. Consumer Welfare was not allowed to condition on how many firms
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To analyze the consumer welfare effects of a third adopted. Without that restriction, the third party could
party’s provision of a pricing algorithm, I draw on some design the algorithm to price at the monopoly level but
standard methods developed for third-degree price dis- only when both firms adopted, and otherwise price
crimination. The price discrimination literature has competitively. In spite of those efforts to prevent out-
shown that a sufficient condition for third-degree price sourcing from causing consumer harm, we see that
discrimination to lower consumer welfare, relative to a harm still occurs though it is not through higher prices
uniform price, is that total supply does not increase but rather more volatile prices.
(Varian 1989). Intuitively, for a given aggregate total
supply, consumer welfare is maximized by equating 4.3. Product Differentiation
marginal utility across markets (or, in the current model, In order to consider the effect of product differentia-
equating it across demand states), which can only be tion, assume a representative agent’s utility function:
achieved with a uniform price. Thus, holding total sup-  
1
ply constant across price regimes, price discrimination θ1 q1 + θ2 q2 − (β1 q21 + β2 q22 + 2ηq1 q2 )
2
lowers consumer (and total) welfare compared with a
uniform price. If total supply is lower under price dis- where η is the degree of product similarity. Firms’
crimination, then consumer (and total) welfare is even products are independent when η  0 and identical
less. A corollary of that general finding is that, when when η  β. Solving
comparing two price discrimination schemes that pro-  
1
duce the same total supply, the one with more price dis- max θ1 q1 + θ2 q2 − (β1 q21 + β2 q22 + 2ηq1 q2 ) − p1 q1 − p2 q2
(q1 , q2 ) 2
persion across markets has lower consumer welfare.
Toward applying that insight, consider the three yields firm 1’s demand function (with firm 2’s de-
pricing rules: (1) uniform price (which does not condi- mand function analogously defined):
tion on the demand shock); (2) internally developed  
pricing algorithm, which conditions on the demand 1
D1 (p1 , p2 )  2 (θ(β − η) − βp1 + ηp2 )  a − bp1 + dp2
shock; and (3) externally developed pricing algorithm, β − η2
which conditions on the demand shock. Letting p(a)
represent a generic pricing rule, all three pricing rules where
have the same expected quantity (the analogue to θ β η μ
“total supply”), which follows from them having the a ,b  2 ,d  2 ,μ  θ , (13)
β+η β −η 2 β −η 2 β +η
same expected price:

and μθ is the mean of θ. Substituting (13) into
(a − (b − d)p(a)))G (a)da  μ − (b − d)E[p(a)] (10)–(11):
  (2β − η)(θ + c) − η(μθ − c) θ + c η(μθ − c)
μ + bc b(μ − (b − d)c) φsc (a)   −
 μ − (b − d)  : 2(2β − η) 2 2(2β − η)
2b − d 2b − d
(14)
Of course, price dispersion is greater with the inter- (β − η)θ + βc θ + c η(θ − c)
nally developed pricing algorithm than the uniform φI (a)   − (15)
2β − η 2 2(2β − η)
price, so consumer welfare is lower with internal de-
velopment. Furthermore, price dispersion with an ex- Differentiating (14)–(15) with respect to the degree of
ternally developed algorithm exceeds that when it is product similarity parameter η yields:22
internally developed:
∂φsc (a) −β(μθ − c) ∂2 φsc (a)
d(a − μ)  < 0 ; 0
φsc (a) − φI (a) 
0 as a
μ: ∂η (2β − η)2 ∂η∂θ
2(2b − d)(b − d)
∂φI (a) β(θ − c) ∂2 φI (a) β
Therefore, outsourcing reduces consumer welfare as it exac- − < 0, − < 0:
∂η (2β − η)2 ∂η∂θ (2β-η)2
erbates the harm from third-degree price discrimination.21
Recall from Section 2 that the design and fee structure Figure 2 depicts the change in the two pricing algo-
were restricted in order not to facilitate coordinated rithms when products are less differentiated. For all
 Harrington: Outsourcing Pricing Algorithms and Market Competition
6898 Management Science, 2022, vol. 68, no. 9, pp. 6889–6906, © 2022 INFORMS

Figure 2. (Color online) Interaction Between Outsourcing role for third party developers, and little is understood
and Product Differentiation about the implications of a firm using a pricing algo-
rithm whose design was outsourced. With that moti-
vation, this paper investigated how design incentives
differ between a firm interested in selling the pricing
algorithm and a firm interested in using the pricing algo-
rithm, and what this means for consumers. In conclud-
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ing, I summarize some general insight and offer some

directions for future research.
Though the model of the paper has a specific and
stylized structure, it delivers insight that seems intui-
tive and broadly relevant to the outsourcing of pricing
algorithms. In maximizing its profit, a third party de-
veloper designs its pricing algorithm so as to increase
demand for it. In order to encourage multiple firms in
a market to adopt the algorithm, one might think that
a third party developer would make it less competi-
tive. Indeed, a number of competition authorities have
expressed exactly that concern. What is not appreci-
ated is that, if the pricing algorithm sets higher prices,
demand shocks, prices are lower for both pricing algo- it will also make it more attractive for a firm not to
rithms, per the usual explanation. As products are less adopt because it can profitably undercut the prices set
differentiated, the internally developed pricing algo- by rival firms who did adopt. The challenge for the
rithm becomes less sensitive to the demand shock, third party is to design the pricing algorithm so that it pri-
which is the consequence of more intense price com- ces cooperatively in a way that allows only a firm with the
petition. (Recall that, generally, equilibrium prices pricing algorithm to benefit. In the context of our model
converge to cost when products become homoge- for when there is substantive demand variation, this
neous; hence, they become independent of demand.) tactic manifests itself in making price highly sensitive
to demand variation, which a firm can condition on
In contrast, the sensitivity to demand shocks of the ex-
only if it has the pricing algorithm. Hence, a firm with-
ternally developed pricing algorithm is unaffected by
out the pricing algorithm cannot exploit a firm with
a change in the extent of product differentiation. This
the pricing algorithm. In this way, the third party’s de-
singular property is the result of the pricing algo-
sign raises the profit from adoption without raising the
rithm’s response to demand shocks being designed to
profit from not adopting, and that increases demand
maximize joint profit and that response is indepen-
for the pricing algorithm.
dent of product differentiation.
As this is the first investigation of the outsourcing
In sum, outsourcing the pricing algorithm results in
of pricing rules, there are many research directions.
price being more responsive to demand shocks and,
Building on the model of this paper, one could con-
furthermore, this greater sensitivity is not diminished
sider other demand structures. Our results are de-
when firms’ products are less differentiated. Using
rived under linear demand with an additive source of
(12) and (13), the differential in expected profit be-
demand variation, and we know from the price dis-
tween external and internal development is
crimination literature that welfare results can depend
η2 σ2θ on the curvature of demand. Then there are other
4β2 (β + η) types of demand variation including those that affect
the degree of product differentiation as well as firm-
which is increasing in η (where σ2θ is the variance of specific demand shocks.
θ). As products become more similar, price competi- A critical extension is to allow for multiple third
tion intensifies less when pricing algorithms are exter- party developers who compete in designing and sell-
nally supplied and that enhances the profit from ing their pricing algorithms. The introduction of com-
adoption attributable to outsourcing. petition raises many relevant policy questions. Does
competition exacerbate or mitigate the consumer
5. Concluding Remarks harm when there is a single developer? Does equilib-
The economics and management literature on pricing rium involve firms adopting from the same devel-
is based on the assumption that the firm selling the prod- oper? What is the effect of a policy that limits a third
uct is also the one that designed its pricing rule.23 Con- party to supplying at most one firm in a market? With
trary to that assumption, we are witnessing a growing such a policy, is there a trade-off between reducing
 Harrington: Outsourcing Pricing Algorithms and Market Competition
Management Science, 2022, vol. 68, no. 9, pp. 6889–6906, © 2022 INFORMS 6899

competition in the market for pricing algorithms and Appendix A. Literature Review
increasing competition in the product market? The theoretical literature examining the implications of Big
A more fundamental extension is to specify a dif- Data and algorithmic pricing can be categorized along two
ferent space of pricing algorithms. In this paper, a dimensions: 1) the space of pricing algorithms and 2) the cri-
pricing algorithm conditions on market-specific cost terion for selecting a pricing algorithm. The first dimension
pertains to how Big Data and algorithmic pricing enrich the
and demand parameters, and that could be extended
feasible set of pricing algorithms. One branch is behavior-
to condition on rival firms’ prices, either prices from
based pricing, which allows price to condition on a custom-
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past periods (as in Calvano et al. 2020b) or where er’s history of purchases (or some other behavior such as
there is a sequentiality to pricing (as in Brown and clickstream activity). A second branch focuses on how Big
MacKay 2022). Third party development is very Data and algorithmic pricing allows a firm to be more in-
likely to affect the design of pricing algorithms. For formed of demand when it sets price. This can mean using
example, a firm might design its pricing algorithm to data to have a more accurate demand forecast or more finely
search for the lowest price among its competitors segment the market or better tailor price to current market
and charge a price just below that level.24 However, conditions. A third branch examines how pricing algorithms
a third party’s pricing algorithm is unlikely to have affect the way in which a firm’s price responds to competi-
that property for it would result in a downward spi- tors’ prices in terms of either the speed of response or com-
mitting to a particular response. This review focuses on the
ral of prices should competitors adopt it. The ques-
latter two branches, whereas behavior-based pricing is sur-
tion then is exactly how a third party developer’s
veyed in Fudenberg and Villas Boas (2007, 2012).
pricing algorithm will differ from that which a firm The second dimension is how a firm selects a pricing
itself would design. algorithm. The conventional approach characterizes equi-
Within this broader class of pricing algorithms, one librium pricing algorithms for a well-defined game. An
could also explore whether it is possible to prevent col- alternative approach specifies a learning algorithm; that
lusive pricing when competitors adopt pricing algo- is, how past data (prices, sales, profits) are used to iden-
rithms from the same third party. In the model of this tify a better performing pricing algorithm.25 Two classes
paper, it was prevented by constraining the pricing al- of learning algorithms have been considered: estimation-
gorithm so that it does not condition on another firm’s optimization learning and reinforcement learning. The
adoption decision. One could impose an analogous former embodies two distinct modules. The estimation
constraint on this richer space of pricing algorithms by module estimates the firm’s environment and delivers
predictions as to how the firm’s price or quantity deter-
requiring, conditional on the history, the pricing algo-
mines its profit or revenue. In particular, past prices and
rithm to produce the same price regardless of rival sales are used to estimate a firm’s demand function
firms’ adoption decisions. We already know from (where various papers have used OLS, Maximum Likeli-
Calvano et al. (2020b) that collusion can emerge, but hood, and an artificial neural network), and thereby have
there is the question of whether it becomes easier or an estimate of how price affects a firm’s profit (or reve-
more likely under third party development. A chal- nue). With that estimated environment, the optimization
lenge to the prevention of collusion is that the price his- module selects price to maximize profit (or revenue) us-
tory will vary with rival firms’ adoption decisions and ing the estimated demand function, while adding some
that could provide a path to circumventing the con- randomness to generate experimentation. An example
straint. For example, the algorithms could be pro- discussed below is Cooper et al. (2015), whereas den
grammed to have a pattern in the last two digits of a Boer (2015) provides an overview of this work.
An estimation-optimization learning algorithm separately
price and to shift to “collusive” mode when it observes
estimates the environment and then optimizes in the selection
that pattern in a rival’s prices. of an action for the estimated environment. In comparison, re-
It seems clear there are many fascinating questions inforcement learning fuses estimation and optimization by
associated with the third party development of pric- learning directly over actions; it seeks to identify the best action
ing algorithms and a considerable need for more re- for a particular state based on how various actions have per-
search to address them. formed in the past for that state. Its approach is model-free in
that it operates without any prior knowledge of the environ-
Acknowledgments ment. One common method of reinforcement learning is
The comments of four referees, the associate editor, and the Q-learning. With this approach, there is a value assigned to
editor are gratefully acknowledged. I also appreciate the each action-state pair (e.g., an action is a price and a state is a
comments of seminar participants at Queen’s University, history of prices) and these values are updated based on real-
Düsseldorf Institute for Competition Economics (DICE), ized profit. Given the current collection of values and the
and Norwegian School of Economics (NHH) and the re- current state, the action is chosen that yields the highest value.
search assistance of Haoxiang (Harry) Hou. This paper Recent papers using Q-learning are Calvano et al (2020b),
supersedes Harrington (2020). On the latter paper, I appreci- Asker et al. (2021), and Klein (2021).26 Hansen et al. (2021) uses
ate the comments of Ai Deng, Timo Klein, Jeanine Miklós- the Upper Confidence Bound algorithm that, for each price,
Thal, and participants in the Notre Dame Theory Seminar. keeps track of the empirical average of the profit for that price
 Harrington: Outsourcing Pricing Algorithms and Market Competition
6900 Management Science, 2022, vol. 68, no. 9, pp. 6889–6906, © 2022 INFORMS

and the number of times it was chosen. There is an index that demand forecasting though under imperfect monitoring.
is increasing in the empirical average profit and decreasing in Without Big Data, demand is affected by two unobserv-
the number of times a price was chosen. In any period, the able demand shocks. With Big Data, one of those demand
price with the highest index is chosen, so a price is more likely shocks is observed so price can condition on that shock.
to be selected when it has performed better and has been cho- As with Miklós-Thal and Tucker (2019), the deviation pay-
sen less frequently. off is higher because of the improved demand information
Let me now turn to reviewing those papers that most that makes collusion harder, but monitoring is more effec-
directly examine how Big Data and algorithmic pricing af- tive, which makes collusion easier. The net effect on prices
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fects market competition. The first four papers consider is ambiguous.

the impact of Big Data and algorithmic pricing on the pro- The final paper that explores the implications of Big Data
pensity or extent of collusion. Firms interact in an infi- and algorithmic pricing (along with AI) for collusion is
nitely repeated price game where pricing algorithms can Calvano et al (2020b). This paper assumes each firm uses
arbitrarily condition on the history of past prices. Salcedo Q-learning to discover its pricing algorithm. The central
(2015) modifies the canonical perfect monitoring setting to question is whether collusive pricing rules can emerge un-
allow for commitment to and observability of pricing al- der Q-learning and, if so, how robust a phenomenon it is.
gorithms. A pricing algorithm is a finite automaton that For the infinitely repeated price game with differentiated
maps price histories into the set of feasible prices. A firm’s products, they find it is quite common for prices to converge
pricing algorithm is a state variable in that it can be to levels well above static Nash equilibrium levels. Further-
changed only during stochastic revision opportunities. At more, pricing algorithms evolve to having properties of col-
such an opportunity, a firm is assumed to know its rival’s lusive pricing rules.27 For example, one pricing algorithm
pricing algorithm. Thus, in selecting its pricing algorithm that emerged has firms settle on a supracompetitive price
at a revision opportunity, a firm recognizes it will be com- and, in response to a rival undercutting it, firms’ prices sig-
mitted to it until the next revision opportunity and, nificantly drop and then gradually climb back up to supra-
should its rival have a revision opportunity in the mean- competitive levels. The paper considers many variants of
time, that rival will observe the firm’s pricing algorithm the basic model in concluding that collusion is a robust out-
and know it is committed to it. A striking result is de- come of Q-learning. Firms whose pricing algorithms are de-
rived: under certain conditions, all subgame perfect equi- termined by a general form of reinforcement learning can
libria result in prices close to monopoly prices. However, learn to collude. For a critical analysis of Calvano et al
a word of caution, for this result is erected on the untena- (2020b), the reader is referred to Asker et al. (2021).
ble assumption that a firm observes a rival’s pricing The remaining papers show how Big Data and algorith-
algorithm. The presumption is that past price data would mic pricing can result in supracompetitive prices under
allow a firm to “decode” its rival’s pricing algorithm, static optimization. In Brown and MacKay (2022), the profit
though that cannot generally be possible (e.g., when the function is fixed and known, and they focus on the implica-
number of observations are fewer than the number of tions of firms being able to respond more rapidly to rivals’
states in the finite automaton). prices. In the context of a duopoly game with differentiated
Miklós-Thal and Tucker (2019) considers a duopoly with products, firms can be heterogeneous in the frequency with
homogeneous goods where there is one consumer type with which they can change price. For example, one firm may be
fixed demand. A consumer’s maximum willingness-to-pay able to change price every hour, whereas the other firm can
(WTP) can take two possible values and is iid over time. In only change price once a day. This heterogeneity introduces
each period, firms receive a common signal of the WTP prior commitment in that the firm that is locked into its price over
to choosing price. There are two possible signals and ρ ≥ 1=2 a longer period is effectively a price leader with respect to its
is the probability that the signal is accurate. The influence of rival. Allowing firms to choose their pricing technologies,
Big Data is captured by a higher value of ρ; hence, a firm has firms are shown to select different frequencies because creat-
better demand information when it chooses price. The anal- ing a leader-follower relationship yields higher prices and
ysis focuses on grim trigger strategy equilibria under perfect profits for both firms compared with when they simulta-
monitoring. A higher value of ρ has two counteracting ef- neously choose prices (which, by the model’s timing struc-
fects on the maximal collusive equilibrium price. More accu- ture, occurs when they choose the same frequency). So as to
rate demand information allows the cartel to better predict ensure itself of being the follower (which is more profitable
the joint profit-maximizing price and that increases the col- than being a leader), one of the firms chooses the most rapid
lusive value, which makes collusion less difficult. However, pricing technology. By allowing firms to commit to a pricing
more accurate demand information also increases the maxi- frequency, Big Data and algorithmic pricing produce higher
mal deviation profit by better informing a prospective devi- prices. Leisten (2021) offers a novel extension whereby a
ator when deviation profit is high, which makes collusion manager can overrule the algorithm at a cost. There it is
more difficult. When the discount factor is sufficiently high, shown the result of Brown and MacKay (2022) is robust, but
more accurate demand forecasting harms consumers. When there are also some grounds for more collusive outcomes to
the discount factor is sufficiently low, it is possible for con- emerge.
sumers to benefit from firms being better informed of Cooper et al. (2015) and Hansen et al. (2021) consider a du-
demand. opoly setting with differentiated products, where firms do
Closely related in motivation is O’Connor and Wilson not know their demand or profit functions and are endowed
(2022), which also considers the implications of enhanced with a learning algorithm. The only available data to a firm
 Harrington: Outsourcing Pricing Algorithms and Market Competition
Management Science, 2022, vol. 68, no. 9, pp. 6889–6906, © 2022 INFORMS 6901

are its own past prices and profits, which means, in estimat- which is φsc . Referring to the objective in (B.3) as W,
ing the relationship between its price and profit, the firm has second-order conditions are satisfied:
a misspecified model that does not take account of the other
firm’s price. With an omitted variable that is endogenous to ∂2 W (2b − d)2 ∂2 W (4b(b − d)(σ2 + μ2 ) + d2 μ2 )
 − < 0, − <0
what the pricing algorithm does, estimates will be biased. For ∂α2 2b ∂γ2 2b
example, if, when a firm raises its price, the other firm also
happens to raise its price, then the firm’s demand will be esti-    2
mated to be less price-elastic than it actually is. Underestimat- ∂2 W ∂2 W ∂2 W σ2 (b − d)(2b − d)2
−  > 0:
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ing the price elasticity of demand would cause firms to set ∂α 2 ∂γ2 ∂γ∂α b
higher prices than would be achieved for a full-information
equilibrium. Both papers find that this misspecification re- The final step is to show that φsc satisfies (B.2). It is
sults in supracompetitive prices. Cooper et al. (2015) assumes straightforward to show:
prices are set optimally given an OLS-estimated demand
curve. Hansen et al. (2021) view it as a multiarmed bandit b(μ − (b − d)c)2 σ2
V(A, A, φsc )  +
problem where a pricing algorithm is chosen to minimize sta- (2b − d) 2 4(b − d)
tistical regret (i.e., the difference between average profit
achieved with the algorithm and ex-post optimal profit). b(μ − (b − d)c)2
They find that when the signal-to-noise ratio for sales is high V(NA, A, φsc )  ,
(2b − d)2
(i.e., sales are relatively more responsive to price changes
than to demand shocks), firms’ prices are supracompetitive so the incremental value of adoption when the other firm
and positively correlated. It is when learning results in a high adopts is
positive correlation that a firm finds a high price relatively
profitable because the rival also tends to set a high price. σ2
V(A, A, φsc ) − V(NA, A, φsc )  : (B.4)
4(b − d)
Appendix B. Proofs
Analogously, it can be shown the incremental value of
Lemma B.1. The unique affine solution to adoption when the other firm does not adopt is
max V(A, A, φ) − V(NA, A, φ) (B.1) (b − 2d)σ2
φ∈Φ
V(A, NA, φsc ) − V(NA, NA, φsc )  : (B.5)
4(b − d)2
s:t: V(A, A, φ) − V(NA, A, φ)
(B.2)
≥ V(A, NA, φ) − V(NA, NA, φ) Inserting (B.4) and (B.5) into (B.2),
is σ2 (b − 2d)σ2
V(A, A, φsc ) − V(NA, A, φsc )  ≥
2bc(b − d) − dμ a 4(b − d) 4(b − d)2
φsc  + :
4b(b − d) − 2d(b − d) 2(b − d)  V(A, NA, φsc ) − V(NA, NA, φsc )

Proof. Our approach is to solve the unconstrained problem which holds because
(B.1) and then show the solution satisfies the constraint
(B.2). Given linear demand and cost and affine pricing algo- σ2 (b − 2d)σ2 dσ2
rithms, (B.1) takes the form: −  > 0: w
4(b − d) 4(b − d)2 4(b − d)2


max (α + γa − c)(a − (b − d)(α + γa))G (a)da
(α, γ) Lemma B.2 The unique affine solution to

 
μ + bc + d(α + γμ) max V(A, NA, φ) − V(NA, NA, φ)
− −c φ∈Φ
2b
   
μ + bc + d(α + γμ) is
a−b + d(α + γa) G (a)da:
2b
(b + d)(2cb2 − cdb + dμ) a
Taking the integral in the first term and simplifying the φpl  + :
second term yields: b(4b2 − 2d2 ) 2b

max(α − c)μ − (α − c)(b − d)(α + γμ) − γμ(b − d)α Proof. First note that maxφ∈Φ V(A, NA, φ) − V(NA, NA, φ) is
(α, γ)
equivalent to maxφ∈Φ V(A, NA, φ) given that φ does not ap-
(μ − bc + d(α + γμ))2 pear in V(NA, NA, φ). With affine pricing algorithms,
+γ (1 − γ(b − d))(μ2 + σ2 ) − : (B.3)
4b maxφ∈Φ V(A, NA, φ) takes the form:
Solving the first-order conditions to (B.3) yields:
  
μ + bc + d(α + γμ) 
max (α + γa − c) a − b(α + γa) + d G (a)da:
2bc(b − d) − dμ 1 (α, γ) 2b
α ,γ 
4b(b − d) − 2d(b − d) 2(b − d) (B.6)
 Harrington: Outsourcing Pricing Algorithms and Market Competition
6902 Management Science, 2022, vol. 68, no. 9, pp. 6889–6906, © 2022 INFORMS

Taking the integral yields: strictly higher expected demand than φo (σ2 ): This contra-
   diction will prove the result.
μ + bc + d(α + γμ) First suppose limσ2 →0 φo (a, σ2 )  φpl (a) pointwise in a for
(α − c) μ − b(α + γμ) + d
2b all but a set of measure zero. Thus, φo (σ2 ) is close to φpl
  
μ + bc + d(α + γμ) when σ2 is close to zero. Note that
+ γ(μ2 + σ2 )(1 − bγ) + γμ −bα + d :
2b
0 > V(A, A, φpl , 0) − V(NA, A, φpl , 0) (B.9)
Solving the first-order conditions delivers:
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because, given the adopting rival firm chooses φpl , V(NA,

(b + d)(2cb2 − cdb + dμ) 1
α , γ , A, φpl , 0) is the profit from choosing the unique best re-
b(4b2 − 2d2 ) 2b
sponse to φpl and V(A, A, φpl , 0) is the profit from choosing
which is φpl . Referring to the objective in (B.6) as W, φpl , which is not the best response to φpl . By the continu-
second-order conditions are satisfied: ity of V(A, A, φ, σ2 ) − V(NA, A, φ, σ2 ) in σ2 and φ and that,
by supposition, limσ2 →0 φo (σ2 )  φpl , it follows from (B.9)
∂2 W 1 ∂2 W 1 that
 − (2b2 − d2 ) < 0,  − (2b2 σ2 + (2b2 − d2 )μ2 ) < 0
∂α 2 b ∂γ2 b
0 > V(A, A, φo (σ2 ), σ2 ) − V(NA, A, φo (σ2 ), σ2 )
   2
∂2 W ∂2 W ∂2 W for σ2 close to zero, which then implies
−  2σ2 (2b2 − d2 ) > 0: w
∂α 2 ∂γ2 ∂α∂γ
K(V(A, A, φo (σ2 ), σ2 ) − V(NA, A, φo (σ2 ), σ2 ) − f )  0
Proof of Theorem 1. For the purpose of the analysis, the
dependence of values on σ2 is made explicit (where recall for σ2 close to zero. Consequently, if φo (σ2 ) is the solution
that σ2 is an element of a market’s type). Given f, the to (B.7), then it maximizes
equilibrium algorithm maximizes expected demand (for a
market type): K(V(A, NA, φ, σ2 ) − V(NA, NA, φ, σ2 ) − f )

max 1 × max{K(V(A, NA, φ, σ2 ) − V(NA, NA, φ, σ2 ) − f ) (B.7) and, equivalently, maximizes V(A, NA, φ, σ2 ) − V(NA, NA,
φ
φ, σ2 ):29 However, φo (σ2 ) ≠ φpl for a set of positive mea-
− K(V(A, A, φ, σ2 ) − V(NA, A, φ, σ2 ) − f ), 0} sure delivers a contradiction because φpl is the unique (af-
fine) maximum of V(A, NA, φ, σ2 ) − V(NA, NA, φ, σ2 ). This
+ 2 × K(V(A, A, φ, σ2 ) − V(NA, A, φ, σ2 ) − f ):
completes the proof that, for σ2 close to zero, there cannot
Consider σ2  0: It is shown in the proof of Lemma B.1 be a solution to (B.7) in a neighborhood of φpl that is dif-
that (for affine pricing algorithms): ferent from φpl .
σ2 Now suppose limσ2 →0 φo (a, σ2 ) ≠ φpl (a) for a set of posi-
max V(A, A, φ, σ2 ) − V(NA, A, φ, σ2 )  (B.8) tive measure of values for a so the claimed optimum is
φ 4(b − d)
not in a neighborhood of φpl . We know that
which implies maxφ V(A, A, φ, 0) − V(NA, A, φ, 0)  0. Hence,
at the solution to (B.7) for σ2  0, lim V(A, NA, φpl , σ2 ) − V(NA, NA, φpl , σ2 ) > 0,
σ2 →0
V(A, A, φ, 0) − V(NA, A, φ, 0) − f  −f ≤ 0 while
which implies
lim V(A, A, φpl , σ2 ) − V(NA, A, φpl , σ2 ) < 0 (B.10)
σ2 →0
K(V(A, A, φ, 0) − V(NA, A, φ, 0) − f )  0:

Thus, the solution to (B.7) maximizes which follows from (B.9) and continuity of V(A, A, φpl , σ2 ) −
V(NA, A, φpl , σ2 ) in σ2 . As σ2 → 0, we then have30
K(V(A, NA, φ, 0) − V(NA, NA, φ, 0) − f )
K(V(A, NA, φpl , σ2 ) − V(NA, NA, φpl , σ2 ) − f ) > 0
or, equivalently, maximizes V(A, NA, φ, 0) − V(NA, NA, φ, 0):
It is shown in Lemma B.2 that the (affine) solution to maxφ  K(V(A, A, φpl , σ2 ) − V(NA, A, φpl , σ2 ) − f ): (B.11)
V(A, NA, φ, σ ) − V(NA, NA, φ, σ ) is φ . Hence, if σ  0,
2 2 pl 2
Given that φo (σ2 ) is bounded away from φpl as σ2 → 0
then the equilibrium pricing algorithm is φpl .
then, given φpl is the unique maximum of V(A, NA, φ,
Now suppose σ2 is close to zero. Define φo (a, σ2 ) as the
σ2 ) − V(NA, NA, φ, σ2 ),
solution to (B.7) for a sequence of values for σ2 that go to
zero: {φo (a, σ2 )}σ2 →0 . As σ2 → 0, assume φo (a, σ2 ) differs lim V(A, NA, φpl , σ2 ) − V(NA, NA, φpl , σ2 )
σ2 →0
from φpl (a) for a set of positive measure of values for a.28
> lim V(A, NA, φo (σ2 ), σ2 ) − V(NA, NA, φo (σ2 ), σ2 ): (B.12)
I will show ∃η > 0 such that if σ2 ∈ (0, η], then φpl yields a σ2 →0
 Harrington: Outsourcing Pricing Algorithms and Market Competition
Management Science, 2022, vol. 68, no. 9, pp. 6889–6906, © 2022 INFORMS 6903

It then follows: as σ2 → 0, Consider the optimal algorithm subject to making adop-

  tions strategic substitutes:
K V(A, NA, φpl , σ2 ) − V(NA, NA, φpl , σ2 ) − f
φss  arg maxss 1 × (K(V(A, NA, φ, σ2 ) − V(NA, NA, φ, σ2 ) − f )
  φ∈Φ
− K V(A, A, φpl , σ2 ) − V(NA, A, φpl , σ2 ) − f
  − K(V(A, A, φ, σ2 ) − V(NA, A, φ, σ2 ) − f ))
> K V(A, NA, φo (σ2 ), σ2 ) − V(NA, NA, φo (σ2 ), σ2 ) − f + 2 × K(V(A, A, φ, σ2 ) − V(NA, A, φ, σ2 ) − f )
 
− K V(A, A, φo (σ2 ), σ2 ) − V(NA, A, φo (σ2 ), σ2 ) − f  arg maxss K(V(A, NA, φ, σ2 ) − V(NA, NA, φ, σ2 ) − f )
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(B.13) φ∈Φ

+ K(V(A, A, φ, σ2 ) − V(NA, A, φ, σ2 ) − f ): (B.15)

because the first term on the LHS > first term on the RHS

by (B.12) and the second term on the LHS  0 ≤ second Strategic substitutes and K > 0 imply
term on the RHS by (B.10). Given (B.13) and K(V(A, NA, φ, σ2 ) − V(NA, NA, φ, σ2 ) − f )
K(V(A, NA, φpl , σ2 ) − V(NA, NA, φpl , σ2 ) − f ) > K(V(A, A, φ, σ2 ) − V(NA, A, φ, σ2 ) − f ) ∀φ ∈ Φss

− K(V(A, A, φpl , σ2 ) − V(NA, A, φpl , σ2 ) − f ) > 0 and, therefore,

2K(V(A, NA, φss , σ2 ) − V(NA, NA, φss , σ2 ) − f )
then the first term in (B.7) is higher with φpl than φo :
> K(V(A, NA, φss , σ2 ) − V(NA, NA, φss , σ2 ) − f )

K(V(A,NA,φ ,σ ) − V(NA,NA,φ ,σ ) − f )
pl 2 pl 2
lim 1 × max ,0 + K(V(A, A, φss , σ2 ) − V(NA, A, φss , σ2 ) − f ): (B.16)
σ2 →0
−K(V(A,A,φ ,σ ) − V(NA,A,φ ,σ ) − f )
pl 2 pl 2
Note that the RHS of (B.16) is the maximal value of the

K(V(A,NA,φo (σ2 ),σ2 ) − V(NA,NA,φo (σ2 ),σ2 ) − f ) objective in (B.15).
> lim 1 × max ,0 : From Lemma B.2,
σ2 →0 −K(V(A,A,φo (σ2 ),σ2 ) − V(NA,A,φo (σ2 ),σ2 ) − f )
φpl  arg max V(A, NA, φ, σ2 ) − V(NA, NA, φ, σ2 )
φ∈Φ
Hence, a necessary condition for the optimality of φ is o

that the second term in (B.7) is higher with φo than φpl : which means 2K(V(A, NA, φpl , σ2 ) − V(NA, NA, φpl , σ2 ) − f ) is
an upper bound on the LHS of (B.16). Hence,
lim K(V(A, A, φo (σ2 ), σ2 ) − V(NA, A, φo (σ2 ), σ2 ) − f )
σ2 →0 2K(V(A, NA, φpl , σ2 ) − V(NA, NA, φpl , σ2 ) − f )
> lim K(V(A, A, φ , σ ) − V(NA, A, φ , σ ) − f ):
pl 2 pl 2
(B.14) > K(V(A, NA, φss , σ2 ) − V(NA, NA, φss , σ2 ) − f )
σ2 →0
+ K(V(A, A, φss , σ2 ) − V(NA, A, φss , σ2 ) − f ): (B.17)
However, given (B.8) then
In sum, 2K(V(A, NA, φ , σ ) − V(NA, NA, φ , σ ) − f ) is an
pl 2 pl 2
lim K(V(A, A, φo (σ2 ), σ2 ) − V(NA, A, φo (σ2 ), σ2 ))  0 upper bound on the expected demand from any φ ∈ Φss .
σ2 →0
To complete the proof, we want to show: if σ2 is suffi-
which implies (B.14) cannot be true. In sum, if φo (σ2 ) is ciently high, then the expected demand from φsc exceeds
bounded away from φpl then, for σ2 close to zero, φpl has the expected demand from φss ,
higher expected demand. Again, we have a contradiction
to the claim that φo (σ2 ) is optimal. 2K(V(A, A, φsc , σ2 ) − V(NA, A, φsc , σ2 ) − f )
We conclude ∃η > 0 such that if σ2 ∈ [0, η] then the solu- > K(V(A, NA, φss , σ2 ) − V(NA, NA, φss , σ2 ) − f )
tion to (B.7) is φpl . w
+ K(V(A, A, φss , σ2 ) − V(NA, A, φss , σ2 ) − f ), (B.18)
Proof of Theorem 2. For the ensuing analysis, we will
break the third party’s optimization problem into two so φsc is the equilibrium pricing algorithm. Given (B.17), a
subproblems. Partition the set of pricing algorithms into sufficient condition for (B.18) is
those that result in adoptions being strategic comple-
ments—call that subset of pricing algorithms Φsc —and 2K(V(A, A, φsc , σ2 ) − V(NA, A, φsc , σ2 ) − f )
those for which adoptions are strategic substitutes, Φss . ≥ 2K(V(A, NA, φpl , σ2 ) − V(NA, NA, φpl , σ2 ) − f )
Φsc ≡ {φ ∈ Φ : V(A, A, φ, σ2 ) − V(NA, A, φ, σ2 )
or, equivalently,
≥ V(A, NA, φ, σ2 ) − V(NA, NA, φ, σ2 )}:
Φss ≡ {φ ∈ Φ : V(A, A, φ, σ2 ) − V(NA, A, φ, σ2 ) V(A, A, φsc , σ2 ) − V(NA, A, φsc , σ2 ) ≥ V(A, NA, φpl , σ2 )

< V(A, NA, φ, σ2 ) − V(NA, NA, φ, σ2 )}: − V(NA, NA, φpl , σ2 ): (B.19)

We have already shown φsc is the solution when the choice One can show the RHS of (B.19) is
set is Φsc . Note that the associated expected demand is
 2
  d4 μ − (b − d)c  
σ 2
+
1 2
σ :
2K(V(A, A, φsc , σ2 ) − V(NA, A, φsc , σ2 ) − f )  2K −f :
4(b − d) 8b(2b − d)2 (2b2 − d2 ) 4b
 Harrington: Outsourcing Pricing Algorithms and Market Competition
6904 Management Science, 2022, vol. 68, no. 9, pp. 6889–6906, © 2022 INFORMS

Using (B.4), (B.19) takes the form: Given that the quantity is chosen to maximize net sur-
plus U(q, a) − p(a) then ∂U(q(a), a)=∂q  p(a): As a result,
 2
(C.1) becomes:
σ2 d4 μ − (b − d)c σ2
≥ + 
m 
m
4(b − d) 8b(2b − d) (2b2 − d2 ) 4b
2
ρ(aj )U(q (aj ), aj ) ≤ ρ(aj )U(q (aj ), aj )
j1 j1

or 
m
+ ρ(aj )p (aj )(q (aj ) − q (aj )),
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2 j1
d (b − d)(μ − (b − d)c)
3
σ2 ≥ : (B.20)
2(2b − d)2 (2b2 − d2 ) which delivers an upper bound on the change in con-
sumer welfare:
Thus, a sufficient condition for φsc to be the equilibrium 
m 
m
pricing algorithm is that σ2 satisfies (B.20). w ρ(aj )U(q (aj ), aj ) − ρ(aj )U(q (aj ), aj )
j1 j1

Appendix C. Consumer Welfare 

m
≤ ρ(aj )p (aj )(q (aj ) − q (aj )): (C.2)
Suppose there are two products. It will be sufficient to fo- j1
cus on when a consumer consumes equal amounts of
them. Let U(q, a) be a consumer’s utility function when Given that, by construction,
consuming q units of each of the two products and the 
m
state is a. U is assumed to be concave in q. Assume Λ  p (ao ) ρ(aj )(q (aj ) − q (aj ))  0
{a1 , : : : , am } where a1 < ⋯ < am and, for convenience, ∃ao ∈ Λ j1

such that m j1 (1=m)aj  a . The m states are the analogue

o
then
to m markets from the perspective of the price discrimina- 
m
tion literature. Expected utility (or, equivalently, weighted ρ(aj )p (aj )(q (aj ) − q (aj ))
aggregate utility over the m markets) is j1

m 
m

m  ρ(aj )p (aj )(q (aj ) − q (aj )) + p (ao ) ρ(aj )(q (aj ) − q (aj ))
ρ(aj )U(q(a), a) j1 j1
j1

m
 ρ(aj )(p (aj ) − p (ao ))(q (aj ) − q (aj )): (C.3)
where ρ(a) is the probability of utility state a. Concavity of j1
U implies concavity of its expectation.
Using (C.3), (C.2) becomes
Consider two price and quantity vectors: (p (a), q (a))a∈Λ
and (p (a), q (a))a∈Λ : Assume p (a) is nondecreasing in a 
m 
m 
m
and p (a) is increasing in a. Further suppose p (a) is ρ(aj )U(q (aj ), aj ) − ρ(aj )U(q (aj ), aj ) ≤ ρ(aj )(p (aj )
greater (less) than p (a) as a is greater (less) than ao, the j1 j1 j1

quantities satisfy the relationship implied by these price − p (ao ))(q (aj ) − q (aj )) (C.4)
vectors under decreasing demand, and the expected quan-
tities are equal in the two configurations: The RHS of (C.4) is negative because p (aj ) − p (ao ) < 0 and
q (aj ) − q (aj ) > 0 when aj < ao , and p (aj ) − p (ao ) > 0 and
p (a)
p (a) as a
ao q (aj ) − q (aj ) < 0 when aj > ao .
It has then been shown that consumer welfare is highest
with the uniform price, next highest with the internally
q (a)  q (a) as a
ao
developed pricing algorithm, and lowest with the exter-

m 
m
nally developed pricing algorithm. Given constant and
ρ(aj )q (aj )  ρ(aj )q (aj ):
j1 j1
common marginal cost, the same ordering applies to total
welfare (consumer welfare plus industry profit).
Assuming the quantities correspond to the associated de-
mands, note that these three properties hold when: 1) p is
Endnotes
the uniform price pN and p is the internally developed 1
Some of that discussion can be found in Mehra (2016), Ezrachi and
pricing algorithm φI ; and 2) p is the internally developed Stucke (2017), Johnson (2017), Oxera (2017), Deng (2018), Harring-
pricing algorithm and p is the externally developed pric- ton (2018), Gal (2019), Schwalbe (2019), and Calvano et al (2020a).
ing algorithm φsc . 2
Outsourcing does result in higher average prices in Harrington
By concavity, (2020) where it is assumed adoption is exogenous and the third
party designs the pricing algorithm to maximize the expected profit

m 
m
of an adopter. Here we show that higher prices do not emerge
ρ(aj )U(q (aj ), aj ) ≤ ρ(aj )U(q (aj ), aj )
j1 j1 when adoption is endogenous and the third party designs the pric-
ing algorithm to maximize its expected profit.

m ∂U(q (aj ), aj ) 
+ ρ(aj ) (q (aj ) − q (aj )): 3
There is a small empirical literature comprising Chen et al. (2016),
j1 ∂q Assad et al (2020), Brown and MacKay (2022), and Leisten (2021).
4
(C.1) For a survey, see Stole (2007).
 Harrington: Outsourcing Pricing Algorithms and Market Competition
Management Science, 2022, vol. 68, no. 9, pp. 6889–6906, © 2022 INFORMS 6905

5 18
The duopoly case reduces the notational burden and is not essen- This is also the analytical impediment to allowing for firm-
tial to the paper’s main insight. specific adoption costs as then either one or two firms may adopt in
6
For the welfare analysis, Λ is a finite set so that some previous re- equilibrium, which causes the characterization of the pricing algo-
sults in the literature can be used. rithm to depend on K.
19
7
This restriction is motivated by concerns expressed by various This finding offers an interesting contrast from Corts (1998). In the
authorities. The German Monopolies Commission (2018, p. 23) has setting of Corts (1998), price discrimination lowers firms’ profits and
warned that a third party, in its design of a pricing algorithm, “could that creates an incentive for firms to adopt practices—such as every-
contribute to a collusive market outcome [and] it is even conceivable day low prices—so as make price less variable across demand states.
Downloaded from informs.org by [103.197.36.6] on 20 August 2023, at 09:33 . For personal use only, all rights reserved.

that [they] see such a contribution as an advantage, as it makes the al- We have a setting whereby price discrimination raises firms’ profits
gorithm more attractive for users interested in profit maximization.” and that creates an incentive for them to outsource their pricing algo-
While the OECD (2017, p. 27) has warned: “concerns of coordination rithms so as to make price more variable across demand states.
20
would arise if firms outsourced the creation of algorithms to the same If a firm is given the option to develop its own pricing algorithm,
IT companies and programmers. This might create a sort of ‘hub and it remains an open question whether a firm would do so or instead
spoke’ scenario where co-ordination is, willingly or not, caused by purchase the third party’s. That question is left to future research.
competitors using the same ‘hub’ for developing their pricing algo- The maintained assumption of this paper is that the third party is
rithms and end up relying on the same algorithms.” the sole innovator of a pricing algorithm that can condition on this
8
Given the private information in the market, the third party might source of demand variation.
want to offer a menu of algorithms and fees so that firms from dif- 21
A more detailed proof of this result is provided in Appendix C.
ferent market types could self-select. That possibility is left to future As we know from the price discrimination literature, welfare results
research. can be sensitive to properties of the demand function; see, for exam-
9 ple, Cowan (2016). Thus, it is an open question as to whether this
Even if adoption decisions are not observed, a firm could eventu-
ally infer that the other firm adopted from its high-frequency price finding extends to some nonlinear demand functions.
changes. φ (a) and φI (a) are derived assuming an interior solution, and
22 sc

10
The assumption that φ(·) is observed should not be taken too lit- that condition is violated when η is close enough to β. Thus, in re-
erally and instead seen as a proxy for the steady-state information ducing the degree of product differentiation, it is assumed products
that firms would have. For example, a firm that adopts would even- remain sufficiently differentiated.
tually learn the profit from adoption. A firm that does not adopt 23
Useful starting references for this immense literature are Wald-
would eventually learn the distribution of a rival firm’s price that man and Johnson (2007) and Özalp and Phillips (2012).
did adopt, which is what a nonadopting firm needs to know to set 24
Such a rule was used by a poster seller on Amazon Marketplace
its optimal price and to know the profit from nonadoption.
(U.S. v. Topkins, U.S. Department of Justice, 2015). That there was
11
Obviously, the third party will set f so that expected demand is collusion in that case is not relevant to the point being made.
positive. 25
Almost exclusively, the behavior-based pricing literature is based
12
Our later analysis will show that this is part of the Pareto domi- on the equilibrium approach.
nant equilibrium. It is generally in the interests of the third party to 26
Earlier papers using Q-learning in an environment where multi-
persuade firms to coordinate on the equilibrium with two adop-
ple firms choose prices or quantities include Tesauro and Kephart
tions as it prefers positive demand to zero demand. As shown in
(2002), Xie and Chen (2004), Waltman and Kaymak (2008), Dogan
Sections 3 and 4 when there is linear product demand, a firm’s gross
and Güner (2015), and Hilsen (2016).
equilibrium profit is higher when both adopt than when neither
adopts. As a firm’s net profit (gross profit less the equilibrium fee)
27
“Collusion is when firms use history-dependent strategies to sus-
must be at least as great as the profit from not adopting and that tain supracompetitive outcomes through a reward-punishment
profit will be shown to exceed the Nash equilibrium profit when scheme that rewards a firm for abiding by the supracompetitive
neither adopts then firms’ net profit from adoption is higher com- outcome and punishes it for departing from it.” Harrington (2017),
pared with when both do not adopt. In sum, the third party and the p. 1.
28
two firms prefer the equilibrium when both firms adopt. Actually, they must differ by all but a set of measure zero since
13
Without ε, there could be many equilibrium designs only because algorithms are limited to affine functions.
there are many designs sufficient to result in the incremental profit 29
This equivalence does require ∃φ such that K(V(A, A, φo (σ2 ), σ2 ) −
from adoption exceeding f. Introducing ε rids the model of this V(NA, A, φo (σ2 ), σ2 ) − f ) > 0, which is presumed to be true. Other-
indeterminacy. wise, expected demand is zero ∀φ in which case any φ is a solution.
14
The design cost of the pricing algorithm is assumed to be indepen- 30
This does presume f is small enough that K(V(A, NA, φpl , σ2 ) −
dent of the design. The cost is also assumed to be sufficiently small V(NA, NA, φpl , σ2 ) − f ) > 0. Otherwise, there is a degenerate solution
so it is exceeded by the equilibrium revenue. Otherwise, the third as expected demand is zero for all φ.
party would not be in the market of supplying pricing algorithms.
15
Although it has not been shown that a solution must be an affine
function, it would be surprising if that were not the case.
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