Regulating Algorithmic Pricing Part 2

3 Model I present a sequential search model with two firms and a unit mass of consumers uniformly distributed on a Hotelling line from 0 to 1, which captures differences in search cost. The model demonstrates how the three effects of algorithm shape firm pricing strategies. The equilibrium results of the model offer an analytical characterization to the lessons narrated earlier. 3.1 Firms and Consumers Two retailers i ∈ {A, W } are located at opposite end of the Hotelling line. They compete to sell a maximum of one unit of homogeneous product per consumer. But differing search costs make shopping at the two retailers’ products differentiated to consumers. I normalize marginal cost to zero for both firms. Retailer’s objective is to maximize expected total profit across all potential consumers. With an algorithm, firm i can charge two sets of prices: pi1 for the consumers who search i first and pi2 (p−i1 ) for the consumers who search i second. Its rival with an algorithm can also charge two prices: p−i1 and p−i2 (pi1 ), analogous to i’s prices. Without an algorithm, the firm charges a uniform price of pi,−i to all consumers as it cannot distinguish search sequence. Consumers of type k are defined by their location on the Hotelling line: θk ∈ [0, 1]. Each consumer incurs a distinct search cost for A and for W depending on their location on the line if they decide to search. • The consumer’s search cost for A is sAk = σA θk • The consumer’s search cost for W is sW k = σW (1 − θk ) Consumer with 0 search cost for A lives at θk = 0; consumer with 0 search cost for W lives at θk = 1. A consumers’ search cost for A is strictly increasing in θk while her search cost for W is strictly decreasing in θk and increasing in (1 − θk ). 11 Every consumer has the same exogenous value v of obtaining the good. Their objective is to maximize their surplus, which also incorporates any search cost and purchase price incurred. Consumers are naive expected price-takers. In other words, they do not conjecture how their actions impact firm’s pricing strategies.12 Before searching, consumer’s expectation of the price firm i will offer is modeled as a given fraction of her value: p̄i = (1 − γi )v. Her expected payoff of consuming the good (inclusive of purchase price but not search cost) is conveniently v − p̄i = γi v. 3.2 Sequential Search Game The model consists of the following sequential stages. • Stage 0: Firms simultaneously choose whether to invest in algorithm or not. Investment in algorithm requires a fixed capital cost K. Firms with algorithm are able to obtain a σi that is smaller than the default; tell whether the firm is first-searched or secondsearched by the consumer; track rival prices. • Stage 1: Observing the outcome to Stage 0, firms simultaneously choose their pricing strategy. • Stage 2: Consumer chooses whether to search at all, and, if so, which retailer to search first. Searching at retailer i for consumer k costs sik . • Stage 3: For those consumers who had chosen to search, choose whether to search again at the rival firm with an additional search cost; if not, choose whether to accept the first search firm’s offer or exit the market without purchase. • Stage 4: For those consumers who had chosen to search twice, choose among three options: accept the first offer with an additional small cost to return e > 0, accept the 12. This assumption simplifies the calculation of consumers’ reservation values. It is not unrealistic to assume that consumers are naive price takers. See conclusion for extensions. 12 second offer, or exit the market without purchase. Above is a visual representation of the sequential search game from Stage 2 to Stage 4. 3.3 Consumer Payoffs The firms decide pricing strategy based on the consumers’ (expected) payoffs at each stage. • Stage 2: – Payoff from not searching: 0 – Expected payoff from buying from first search at retailer i: v − p̄i − sik – Expected payoff from buying from first search at −i: v − p̄−i − s−ik • Stage 3: Assume the consumer first searches at i, – Payoff from exiting the market after searching once: −sik – Payoff from buying from the first-searched i: v − pi1 − sik – Expected payoff from searching twice and buying from second search: v − p̄−i2 − sik − s−ik 13 • Stage 4: – Payoff from searching twice and not buying: −(sik + s−ik ) – Payoff from searching twice and buying from second retailer: v −p−i2 −(sik +s−ik ) – Payoff from searching twice and buying from first retailer: v − pi1 − e − (sik + s−ik ) 3.4 Algorithmic Effects An algorithm confers three advantages to a firm. • Search Cost Reduction. Through targeted advertising and automatic recommendations, a firm with algorithms can lower the perceived effort it takes for a consumer to shop her desired product. This is reflected in a smaller σi . Since sAk = σA θk , sW k = σW (1 − θk ), as σi decreases, sik decreases, too. A smaller σi makes consumers more likely to be willing to search at i. • Search Sequence Detection. By tracking Cookies, a firm with algorithm learns about a consumer’s browsing history. If the consumer has visited rival’s sites before on a similar product, the firm knows that it is being second-searched. Otherwise, the firm is firstsearched. This distinction allows the firm to price-discriminate effectively between first-searchers and second-searchers and strategize with two sets of contingencies (firstsearched case and second-searched case). Without such algorithm, the firm faces one contingency, charging uniform price for all. • Rival Price Monitoring. Similarly, a firm with algorithm can access a second-searcher’s browsing history, including the historical price its rival offered. This feature allows the firm to offer the second search consumer a price responsive to the earlier price after the consumer received. 14 4 Analysis I adopt a Perfect Bayesian Equilibrium framework and use backward induction to analyze four possible pricing game scenarios: both firms have algorithm (but possibly asymmetric demand), denoted as (Alg, Alg); A has algorithm but W does not, denoted as (Alg, Non); W has algorithm but A does not, denoted as (Non, Alg); and neither have algorithm, denoted as (Non, Non). 4.1 Disjoint Markets or Overlapped Markets For a sequential search model, it is critical to test if a market allows some consumers to benefit from second search. Otherwise, it turns into a model with two regional monopolies. The lack of contested market space eliminates strategic interaction between the two firms. Consider the consumer located at θk∗ who is indifferent between searching at A or W first, but possibly choosing to not search at all. Location θk∗ is defined by: v − p̄A − sAk = v − p̄W − sW k ⇒ (1 − γW )v − (1 − γA )v = σA θk∗ − σW (1 − θk∗ ) θk∗ = (γA − γW )v + σW σA + σW As the expected payoff from searching at A (W) strictly increases (decreases) with respect to θk , markets for A and W overlap only when this indifferent consumer receives a strictly positive payoff from searching at A: v − p̄A − σA θk∗ > 0 ⇒ γA v > σA (γA − γW )v + σW σA + σW 15 ⇒v> 1 σA 1 σA σW (γA − γW )v + γA σA + σW γA σA + σW We call this threshold v. Note: This can be explained as the middle person’s surplus from buying A’s products. If γA = γW , σA = σW , then γA v > 21 σA , the indifferent consumer is at the middle of the Hotelling line. For γA ̸= γW and σA ̸= σW , the term σA σA +σW corrects for asymmetric σ, and (γW − γA ) corrects for asymmetric γ. If the consumer’s value exceeds v, then it is potentially beneficial for some consumers to search twice. We say a market is overlapped when the indifferent consumer faces a positive expected payoff from searching at A or W. Anatomy of an Overlapped Market When the markets are overlapped, the four potential consumer groups are • Search A only (AO): θk ∈ [ 0 , θ̃Ak ] • Search A then W (AW): θk ∈ [ θ̃Ak , θk∗ ] • Search W then A (WA): θk ∈ [ θk∗ , θ̃W k ] • Search W only (WO): θk ∈ [ θ̃W k , 1 ] I will use the following terms to characterize market demand and firms’ pricing strategies. I will calculate each term explicitly in scenario analyses: • θk∗ is the location of the consumer indifferent between searching A or W • θ̃Ak is the location of the consumer indifferent as to searching W after having already searched A given first price pA • θ̃W k is the location of the consumer indifferent as to searching A after having already searched W given first price pW 16 • p̃A1 is the highest A price at which all first search A consumers only search A • p̃W 1 is the highest W price at which all first search W consumers only search W • p∗A1 is the price that maximizes A’s profits from just AO consumers • p∗W 1 is the price that maximizes W’s profits from just WO consumers Consumers of each search group are located in an interval over the Hotelling line. Assuming a uniformly distributed mass of consumers, we can represent quantity demanded by calculating the metrical distance from one end of the interval to another. For example, the demand for AO is (θ̃Ak − 0 = θ̃Ak ), while the demand for AW is (θk∗ − θ̃Ak ). Later in my analysis, I will explicitly refer to the four groups of consumers by their acronym. To visualize the demand, consider the following Hotelling line illustration. Strategy for Disjoint Markets Each firm’s strategy when the markets are disjoint should be consistent across three scenarios as the two firms do not interact strategically. With none of their searchers willing to search at the rival firm, they essentially operate as monopolies. They should charge pi = v−ϵ, ϵ > 0. Note that search costs are sunk. They are too exorbitant for consumers to search at the other 17 firm. As such, consumers would accept any offer as long as the first price offering is below their value: p < v. Moreover, their initial search decision is based on their expected price rather than actual price, which empirical evidence has documented to be driven primarily by idiosyncratic belief. Hereafter, we focus on the situation when markets are overlapped. 4.2 (Alg, Alg) In this scenario, both retailers have algorithms. With overlapped markets, some consumers may choose to search for the rival product after observing the price offered by the firstsearched firm. Their desire to search twice orients from a cost-benefit analysis of the marginal benefit and marginal cost of second search. Therefore, a firm faces two sets of contingencies: being first-searched and being second-searched. The firm chooses a price it offers all firstsearched consumers and a price response to rival’s first price offered to all second searchers. If firm i is second-searched, it could observe the first-searched firm’s price offering p−i1 and decide what price to offer as a best response. Firm i will undercut the price as long as the price lies below v and above the marginal cost, which I normalize to 0. • If p−i1 > v, then its best response is to offer a price slightly below v, just like a monopoly pi2 = v − ϵ, ϵ > 0; • If 0 < p−i1 < v, then its best response is to match the rival price pi2 = p−i1 . I have assumed that in the case of a tie in first-searched and second-searched prices, a consumer would always take the second-searched price, because it is more convenient for them to stay at the current retailer and finish the transaction. • If p−i1 = 0, Bertrand’s Paradox, charge pi2 ≥ 0. If a firm is first-searched, then it has to account for the fact that consumers might opt to search at the rival firm, too. 18 A’s Strategy as first-searched First observe that the set of consumers who search at A first with overlapped markets are located at the interval [0, θk∗ ]. Consider the highest price pA1 such that all consumers in this set will opt not to search at the rival firm. This can be solved by finding the pA1 at which the marginal indifferent consumer θk∗ is indifferent to accepting pA1 and searching twice at W after A. v − p̃A1 − s∗Ak = v − p̄W − s∗Ak − s∗W k ⇒ p̃A1 = p̄W + σW (1 − θk∗ ) = (1 − γW )v + σW (1 − θk∗ ) For any price pA1 > p̃A1 , some consumer would search twice and take the second price whenever the second price matches the first, which is the best response for W, for reasons analogous to those shown for A. A faces the tradeoff between charging a higher per-unit price to lower quantity sold, similar to a monopolist’s problem. Suppose that A charges a price slightly higher than p̃A1 , pA1 = p̃A1 + δ, δ > 0. More consumers would choose to search twice and take W’s price– a loss of quantity demanded for A. In other words, an increase in the first price A offers expands AW and shrinks AO. More specifically, the consumer indifferent between searching A only and continuing to search W is located at p̃A1 + δ − p̄W = σW (1 − θk ) | {z } | {z } MB of 2nd search MC of 2nd search Inserting p̃A1 = p̄W + σW (1 − θk∗ ), we get the range of consumers given the price increase, θ̃Ak : θ̃Ak = θk∗ − δ σW To derive the demand for the first search firm in (Alg, Alg), start with recognizing that the 19 person at point 0 of the type θk = 0 is indifferent between accepting A’s offer and searching at W when pA1 is v − pA1 − σA (0) = v − p̄W − σA (0) − σW (1 − 0) ⇒ pA1 = p̄W + σW which is the highest price A could offer without having everyone also searching at W. Thus, A chooses some price pA1 ∈ [p̃A1 , p̄W + σW ]. Note that p̄W + σW can be rewritten as a function of p̃A1 . Recall that p̃A1 = p̄W + σW (1 − θk∗ ): p̄W + σW = p̄W + σW (1 − θk∗ ) + σW θk∗ = p̃A1 + σW θk∗ The inverse demand A faces as the first-searched retailer for pA1 is: pA1 = (p̃A1 + σW θk∗ ) − σW θ̃Ak ⇒ θ̃Ak = θk∗ − pA1 − p̃A1 σW Thus we have the quantity demanded given A’s price offering. A’s problem is to choose a pA1 that maximizes the expected profit: max ΠA1 = pA1 θ̃Ak such that pA1 ∈ [p̃A1 , p̄W + σW ] pA1 pA1 − p̃A1 ) σW σW θk∗ + p̃A1 pA1 = pA1 ( − ) σW σW = pA1 (θk∗ − with FOC: σW θk∗ + p̃A1 2p′A1 − = 0 as interior solution σW σW 20 ⇒ p′A1 σW θk∗ + p̃A1 = 2 Therefore, A chooses to offer the maximum between p′A1 and p̃A1 . Call this maximum p∗A1 . If p̃A1 > σW θk∗ , choose p̃A1 . Otherwise, choose p′A1 .13 W’s Strategy as first searched Using analogous arguments as with A, consider the best-case scenario for W: there exists a p̃W 1 such that the marginal consumer is indifferent between taking W’s price and continuing to search A. v − p̃W 1 − s∗W k = v − p̄A − s∗Ak − s∗W k For any price pW 1 > p̃W 1 , some consumer would search twice and take the second price whenever the second price matches the first, which is the best response for A. W faces the tradeoff between charging a higher per-unit price to lower quantity sold, similar to a monopolist’s problem. Suppose that W charges a price slightly higher than pW 1 = p̃W 1 +δ, δ > 0. Then more consumers would choose to search twice and take A’s price instead – a loss of quantity demanded for W. WA expands, and WO shrinks. More specifically, the consumer indifferent between searching W only and continuing to search A is located at p̃W 1 + δ − p̄A = {z } | MB of 2nd search σA θk | {z } MC of 2nd search The range of consumers given the price increase, θ̃W k , is: θ̃W k = δ + θk∗ σA 13. Note: If p̃A1 = σW θk∗ , then p′A1 = p̃A1 . 21 The highest price W can charge when the consumer indifferent between accepting W’s offer and searching at A when pW 1 is: pW 1 = p̄A + σA W chooses some price pW 1 ∈ [p̃W 1 , p̄A1 + σA ]. The demand function when W charges a price above p̃W 1 and below p̄A + σA is: θ̃k = θk∗ + pW 1 − p̃W 1 σA W’s problem is to choose a pW 1 that maximizes the expected profit: max ΠW 1 = pW 1 (1 − θ̃W k ) such that pW 1 ∈ [p̃W 1 , p̄A1 + σA ] pW 1 with FOC: p′W 1 = σA (1 − θk∗ ) + p̃W 1 2 W chooses to offer the maximum between p′W 1 and p̃W 1 . Call this maximum p∗W 1 . If p̃W 1 < σA (1 − θk∗ ), choose p̃W 1 . Otherwise, choose p′W 1 . 4.3 (Alg, Non) In this scenario, A is equipped with an algorithm, but W is not. A faces two sets of contingencies but W only faces one. A’s Strategy Since A has the algorithm as in the previous case, it faces the same sets of contingencies. Recall that p̃A1 is the highest A price at which all first search A consumers are AO and none are AW. 22 When pW ≤ p̃A1 , A’s profit maximization problem is analogous to the one from (Alg, Alg). A would never offer a price lower than p̃A1 , in which case no AW consumers exist, making pW irrelevant. But A may choose to charge a price higher than p̃A1 , in which case it loses all AW consumers but extracts more surplus from AO consumers. Thus, we need to compare the expected profit A receives when A charges p∗A1 (monopolist’s price) versus pW − η (undercut price). When A charges p∗A1 , it only retains the AO. When A charges pW − η, it retains all AO+AW consumers.14 If pW > p̃A1 , p∗A1 > pW > p̃A1 , then we have (pW − η)θk∗ | {z } Profit for undercut price > p∗A1 θ̃Ak | {z } Profit for monopolist price ∗ ⇒ pW p∗ −p̃A1 θk − A1σW θ̃Ak > p∗A1 ∗ = p∗A1 [ θk θk∗ ] Therefore, A’s profit-maximizing price is still p∗A1 = max{p̃A1 , p′A1 }.15 When pW > p̃A1 , then A can charge a price higher than p̃A1 and still retain all of AO+AW consumers. A knows that with the algorithm, it can charge a price slightly lower than W’s first price offering and retain AW consumers. This makes A’s response price pW − η, small η > 0. Alternatively, A could willingly forego some sales in AW and extract a higher surplus from the remaining consumers. This makes A’s response price p∗A1 . A wants to charge p∗A1 if pW < p∗A1 θ̃θK∗ , pW − η otherwise. By construction, we know that k p̃A1 ≤ p∗A1 , as the latter is the maximum between the former and the interior solution. So, 14. Note: AO depends on pA . As p∗A1 ≥ pW , the AO for p∗A1 will be smaller than the AO for pW − η. 15. Note that the actual values of (p̃A1 , p∗A1 ) differs between (Alg, Alg) and (Alg, Non) as σW differs in value across those two algorithm scenarios. The way to calculate these values remains the same. 23 if p∗A1 > pW > p̃A1 , A still wants to charge p∗A1 . But if pW > p∗A1 ≥ p̃A1 , then A wants to charge pW − η to undercut. Overall, A’s best response function is    p∗ if pW ≤ p∗A1 θ̃θAk ∗ A1 k BRA (pW ) =   pW − η if pW > p∗A1 θ̃Ak θ∗ k W’s Strategy Without an algorithm, W can only offer a uniform price pW to all consumers. Regardless of what price W offers for WA, it will always be undercut by A for any pW ≥ 0. Therefore, to maximize expected profit, W has to focus on WO and AW consumers. If W considers pW > p∗A1 , W knows that it will gain no sale from AW consumers, as its price is too high. Any profit W can expect to gain will come from WO consumers. W’s maximization problem is thus max ΠW = max pW (1 − θ̃W k ) such that pW > p∗A1 pW pW θ̃W k θk∗ We have studied a similar problem in (Alg, Alg). W’s best response is to charge p∗W . If W considers pW ≤ p∗A1 θ̃θW∗k instead, W may get some AW consumer sales in addition to k WO sales. More concretely, its expected profit maximization problem is θ̃W k max ΠW = pW (1 − θ̃W k ) + pW (θk∗ − θ̃Ak ) such that pW ≤ p∗A1 ∗ pW | {z } | {z } θk WO sales AW sales Note that p∗W is the optimal price W can charge for the WO sales only. Therefore, W chooses between maximizing sales for WO at the expense of losing sales in AW or lowering the price to retain both WO and AW sales. But then we have the two following cases: 24 If p∗W ≥ p∗A1 θ̃θW∗k , then W needs to compare profits from WO consumers at p∗W versus profits k from WO consumers plus all AW consumers at a price p∗A1 . Note that W would never charge a price below p∗A1 as it can get all AW consumers with pw = p∗A1 . Because A’s best reponses depends on whether pW ≥ p∗A1 and not by how much lower, the size of AW consumers will not increase even if pw were lower. Similarly, if W charges a price above p∗W , it loses all AW consumers anyways, so its best strategy is still to maximize the profit from WO consumers. If p∗W < p∗A1 θ̃θW∗k , W faces two options: charging p∗W or charging slightly higher than p∗W but k just below p∗A1 . If it charges p∗W , it maximizes profits from WO consumers. If it charges p∗W + δ, where 0 < δ < p∗A1 − p∗W , it does not maximize profit from WO consumers but possibly retains more sales from AW consumers. To characterize this tradeoff, I set up the maximization function as δ max ΠW (p∗W + δ) = max(p∗W + δ)[(1 − θk∗ − ) + (θk∗ − θ̃Ak )] | {z } δ δ σA | {z } retained AW shrunk WO such that 0 < δ < p∗A1 − p∗W = p̃W + (1 − θk∗ )σA − p∗W with FOC: δ′ = σA (1 − θ̃Ak ) − p∗W 2 Suppose such value exists in the defined range. When p∗W < p∗A1 , W compares the expected profit when pW = p∗W versus when pW = p∗W + σA (1−θ̃Ak )−p∗W 2 = p∗W 2 + σA (1−θ̃Ak ) . 2 We know W will choose the latter because, had p∗W become the optimized value, δ ′ would be 0 or negative. 25 Overall, W’s best response function is    p∗A1    p∗ BRW (pA ) = max{p∗W , 2W +      p∗ w 4.4 if pW < p∗A1 ≤ p∗W σA (1−θ̃Ak ) } 2 if p∗W ≤ pW < p∗A1 if pW > p∗A1 (Non, Alg) This scenario flips the asymmetry in algorithm investment from the previous scenario: A does not have the algorithm but W does. As such, the best responses for A and W are derived based on analogous reasoning as in (Alg, Non). A’s Strategy Without an algorithm, A can only offer pA to all consumers, as it does not distinguish firstsearchers from second-searchers. Regardless of what price A offers for AW, it will always be undercut by W for any pA ≥ 0. Therefore, to maximize expected profit, A has to focus on AO and WA consumers. If A offers pA > p∗W 1 , A’s best response is to charge p∗A . If A offers pA ≤ p∗W 1 instead, its expected profit maximization problem is max ΠA = pA θ̃Ak + pA (θ̃W k − θk∗ ) such that pA ≤ p∗W 1 | {z } | pA {z } AO sales WA sales Note that p∗A is the optimal price A can charge for the AO sales only. Therefore, A chooses between maximizing sales for AO at the expense of losing sales in WA or lowering the price to retain both AO and WA sales. But then we have the two following cases: If p∗A ≥ p∗W 1 , then A needs to compare profits from AO consumers at p∗A versus profits from 26 AO consumers plus all WA consumers at a price p∗W 1 . If p∗A < p∗W 1 , A faces two options: charging p∗A or charging slightly higher than p∗A but just below p∗W 1 . It maximizes expected profit as δ ) + (θ̃W k − θk∗ )] max ΠA (p∗A + δ) = max(p∗A + δ)[(θk∗ − | {z } δ δ σW | {z } retained WA shrunk AO such that 0 < δ < p∗W 1 − p∗A = p̄W + σW − p∗W with FOC: δ′ = −p∗A + σW θ̃W k 2 Overall, A’s best response function is    if pA < p∗W 1 ≤ p∗A p∗W 1    p∗ +σ θ̃ BRA (pW ) = max{p∗A , A 2W W k } if p∗A ≤ pA < p∗W 1      p∗ if pA > p∗W 1 A W’s Strategy Since W has the algorithm it faces two sets of contingencies. Recall that p̃W 1 is the highest W price at which all first search W consumers are WO and none are WA. When pA ≤ p̃W 1 , W’s profit maximization problem is analogous to the one from (Alg, Alg). If pA > p̃W 1 , p∗W 1 > pA > p̃W 1 , then we have (pA − η)(1 − θk∗ ) > {z } | Profit for monopolist price p∗W 1 (1 − θ̃W k ) | {z } Profit for undercut price ∗ p′ −p̃W 1 θk + W 1σA (1 − θ̃W k ) ∗ ⇒ pA > p∗W 1 = p [ W1 (1 − θk∗ ) 1 − θk∗ 27 ] Overall, W’s best response function is   θ̃W k  p∗ if pA ≤ p∗W 1 1− W1 (1−θk∗ ) BRW (pA ) =   pA − η if pA > p∗W 1 1−θ̃W∗k 1−θ k 4.5 (Non, Non) Both A and W will charge the same price to all consumers as neither can distinguish whether some consumer is searching them first or second. So, the pricing strategy for each retailer reduces to a single uniform price choice: pA and pW respectively. A’s Strategy A faces three potential sets of consumers: AO, AW, and WA. The profit it receives from each set of consumers depends on pA relative to pW .16 ΠA         pA               pA  =              pA         θ̃Ak + (θk∗ − θ̃Ak ) + (θ̃W k − θk∗ )  if pA < pW |{z} | {z } | {z } AO AW WA   θ̃Ak + (0) + (θ̃W k − θk∗ )  |{z} |{z} | {z } AO AW WA  if pA = pW  θ̃Ak + (0) + (0)  |{z} |{z} |{z} AO AW WA if pA > pW Suppose the conjectured pW = p∗A1 . Charging pA > pw = p∗A1 is weakly dominated by charging pA = pW = p∗A1 as profits from AO consumers are maximized at pA = p∗A1 by construction. If A were to consider charging pA < pW = p∗A1 , then • Profits from AO consumers are maximized at pA = p∗A1 • Profits from AW consumers are maximized at pA = pW − η = p∗A1 − η and drops to zero if any higher 16. Note that θ̃Ak differs across the three scenarios as the underlying pA differs. 28 • Profits from WA consumers are maximized at pA = pW = p∗A1 and drops to zero if any higher Therefore, as long as there are some AW consumers (i.e. p∗A1 > p̃A1 ), the only pA < pW = p∗A1 that A would consider is pA = pW − η = p∗A1 − η. But if p∗A1 = p̃A1 and hence no WA consumers, A will prefer to charge pA = p∗A1 . Whether A wants to charge pA = p∗A1 − η or pA = p∗A1 depends on whether there exist AW consumers. If none, A’s best response is to charge p∗A1 . If some, A should charge pA = p∗A1 −η as there exists some small η > 0 such that profits from AW more than offset profits lost from AO and WA from charging η lower than the otherwise optimal p∗A1 . Now suppose the conjectured pW > p∗A1 . Any price pA < pW gives A all the sales from AO, AW, and WA consumers. If A were to charge pA = pW , A gets the sales from AO and WA only but loses sales to AW consumers. But if there are no AW consumers at pA = pW , A would be better off charging pA = pW . This is not possible, as pw > p∗A1 ≥ p̃A1 . Therefore, at pA = pW > p∗A1 ≥ p̃A1 , there are some AW consumers. Therefore, A should charge pA = pw − η as there exists some small positive η such that profits from AW gained from reducing price by η swamps the slightly higher profits from AO and WA consumers stemming from charging η higher price. Now suppose the conjectured pW < p∗A1 . There is a tension between profit gained from AO and profit from AW and WA as profit from AO is maximized at pA = p∗A1 and profits from AW and WA require pA ≤ pW for WA and pA < pw for AW in order to be positive. As profit from AO declines monotonically as pA falls from p∗A1 and profit from AW declines monotonically as pA falls from pW − η and profit from WA declines monotonically as pA falls from pW , A should be choosing pA from among {p∗A1 , pw , pw − η}. If pW ≤ p̃A1 then charging pA = pW dominates charging pA = pW − η as at pA = pW ≤ p̃A1 there are no AW consumers; 29 neither are there AW consumers for the even lower pA = pW − η. If pW = p̃A1 = p∗A1 then pA = pW = p∗A1 as A gets sales from AO and WA at the best price possible for those two consumers and there are no AW consumers. If pW < p̃A1 ≤ p∗A1 then compare p∗A1 · θ̃Ak |{z} evaluated at pA = p∗A1 to pW · θ̃W k |{z} evaluated at pW A will choose whichever of the two pA has higher profits above pA    p∗ if p∗ · θ̃Ak |p∗ ≥ pW · θ̃W k |p A1 A1 W A1 =  p  if p∗A1 · θ̃Ak |p∗A1 < pW · θ̃W k |pW w If pW > p̃A1 then there are some AW consumers even when pA = pW > p̃A1 (or the even lower pA = pW − η). Therefore, charging pA = pW − η for a sufficiently small positive η dominates charging pA = pW . • If pW ≥ p∗A1 > p̃A1 , then A will charge pA = pW −η as A gets all three sets of consumers at that high price (and any higher would lead to losing AW consumers of which there will be some given pW − η > p̃A1 . Note that A is fine with pA = pW − η > p∗A1 as it captures all AW consumers even at that higher price • if pW > p∗A1 ≥ p̃A1 , then A will charge pA = pW − η for similar reasons as above 30 • If p∗A1 > pW > p̃A1 , then compare p∗A1 · θ̃Ak |{z} evaluated at pA = p∗A1 to (pW − η) · θ̃W k |{z} evaluated at pW A will choose whichever of the two pA has higher profits above pA    p∗ if p∗A1 · θ̃Ak |p∗A1 ≥ pW · θ̃W k |pW A1 =   pw − η if p∗A1 · θ̃Ak |p∗ < pW · θ̃W k |pW A1 Altogether    p∗A1        p∗A1 − η       pW − η    BRA (pw ) = p∗A1      pW        p∗A1       p −η W if pW = p∗A1 = p̃A1 if pW = p∗A1 > p̃A1 if pW > p∗A1 ≥ p̃A1 if pW < p̃A1 ≤ p∗A1 and p∗A1 · θ̃Ak |p∗A1 ≥ pW · θ̃W k |pW if pW < p̃A1 ≤ p∗A1 and p∗A1 · θ̃Ak |p∗A1 < pW · θ̃W k |pW if p∗A1 > pW > p̃A1 and p∗A1 · θ̃Ak |p∗A1 ≥ pW · θ̃W k |pW if p∗A1 > pW > p̃A1 and p∗A1 · θ̃Ak |p∗A1 < pW · θ̃W k |pW 31 W’s Strategy Using analogous arguments to those developed for A’s strategy, I derive W’s best response to pA . The profit W receives from each set of consumers depends on pA relative to pW .17 ΠW         pW               pW  =               pW        (θk∗ − θ̃Ak ) + (θ̃W k − θk∗ ) + (1 − θ̃W k )  if pW < pA | {z } | {z } | {z } AW WA WO   (θk∗ − θ̃Ak ) + (0) + (1 − θ̃W k )  | {z } |{z} | {z } AW WA WO  if pW = pA  (0) + (0) + (1 − θ̃ )  |{z} |{z} | {z W k} AW WA WO if pW > pA Altogether    p∗W 1        p∗W 1 − η       pW − η    BRW (pA ) = p∗W 1      pA        p∗W 1       p −η A if pA = p∗W 1 = p̃W 1 if pA = p∗W 1 > p̃W 1 if pA > p∗W 1 ≥ p̃W 1 if pA < p̃W 1 ≤ p∗W 1 and p∗W 1 · (1 − θ̃Ak |p∗A1 ) ≥ pW · (1 − θ̃W k |pW ) if pA < p̃W 1 ≤ p∗W 1 and p∗W 1 · (1 − θ̃Ak |p∗A1 ) < pA · (1 − θ̃W k |pW ) if p∗W 1 > pA > p̃W 1 and p∗W 1 · (1 − θ̃Ak |p∗A1 ) ≥ pA · (1 − θ̃W k |pW ) if p∗W 1 > pA > p̃W 1 and p∗W 1 · (1 − θ̃Ak |p∗A1 ) < pA · (1 − θ̃W k |pW ) 17. Note that θ̃W k differs across the three scenarios as the underlying pA differs. 32