Coopetition Against an Amazon

A key challenge to firms competing in today’s electronic marketplace is competition against Big Tech companies that have considerably more data and so better predictive models. One way in which smaller firms may overcome this hurdle and survive or even thrive in such a market is to engage in coopetitive strategies—namely, to cooperate with other small firms that are its competitors. Such coopetitive strategies have increasingly become a field of study by both academics and practitioners (see, for example, Brandenburger and Nalebuff (2011) and Bengtsson and Kock (2000)), but they largely focus on industrial applications such as healthcare, IT, and service industries. In this paper, we study coopetition between e-commerce companies, and focus on the possibility of data sharing as a way to deal with their data imbalance vis-à-vis Big Tech.

Such coopetitive data-sharing can be undertaken by the firms themselves or by external service-providers. To facilitate the former, the newly burgeoning area of federated machine learning has as its goal the design of mechanisms that generate predictive models for firms based on such shared data (Yang et al., 2019). The assumption underlying federated learning is that these firms actually want predictive models based on all shared data, but in competitive scenarios this need not be the case. Although each firm certainly desires its own model to be as predictive as possible, it most likely does not wish the same for its competitors. So when is full data-sharing, leading to maximally predictive models for all firms, optimal? When is partial or no sharing better? And do these answers differ when firms compete against an Amazon?

In addition to using federated machine learning, coopetitive data-sharing can also be facilitated by an external service-provider. While tools for data sharing are prevalent—and include platforms such as Google Merchant and Azure Data Share—they are currently undergoing even further development. For example, in its European Strategy for Data, the European Commission plans the following: “In the period 2021-2027, the Commission will invest in a High Impact Project on European data spaces and federated cloud infrastructures. The project will fund infrastructures, data-sharing tools, architectures and governance mechanisms for thriving data-sharing and Artificial Intelligence ecosystems” (European Commission, 2020). But just as with federated learning, the potential benefits from data sharing may arise under partial, rather than full, sharing. For instance, one industry white paper urges service providers to

“…analyze the combined, crowdsourced data and generate benchmark analyses and comparative performance reports. Each participating client gains insights that it could not otherwise access, and each benefits from the service-host provider’s ability to slice and dice the aggregated data and share the results that are relevant to each client” (Loshen, 2014).

What is the optimal way for such service providers to combine, slice and dice, and share e-commerce firms’ data in order to facilitate successful coopetition against an Amazon?

Regardless of how coopetitive data-sharing is achieved—via federated learning or through a service provider—the answer to whether or not firms desire maximally predictive models for all depends on how these predictions are eventually used. If the firms involved act in completely unrelated markets, then of course full data-sharing is optimal—it yields a maximally predictive model for a firm, and that firm is not harmed by others’ better predictions. This is no longer true if the firms are competitors, since then a firm may be harmed by an improvement in its competitors’ predictions. In this case, whether or not the tradeoff is worthwhile depends on the details of the market in which firms compete.

In this paper we focus on one such market, which is motivated by recommender systems for online advertising and Long Tail retail. In our model, there are many types of consumers, and firms use their data to infer and take a tailored action for each consumer type. In the online advertising market, for example, firms use their data to personalize advertisements in order to maximize the probability that a consumer clicks on their ad. Data sharing is common in this market, and platforms such as Google Merchant and Azure Data Share specifically facilitate such sharing across clients in order to improve personalization and increase clickthrough rates.

For another concrete application, consider retailers in a Long Tail market—a market with an enormous number of low-demand goods that collectively make up substantial market share—interested in finding consumers for their products. Success in such markets crucially relies on retailers’ abilities to predict consumers’ tastes in order to match them to relevant products (Anderson, 2006; Pathak et al., 2010). On their own, smaller firms can be destroyed by giants with vastly greater amounts of data and thus a clear predictive advantage. In cooperation with other small firms, however, this data imbalance may be mitigated, possibly granting smaller firms a fighting chance. (We do not discuss here antitrust issues surrounding data sharing; the reader is referred to Martens et al. (2020) for a discussion thereof in the context of the EU’s European Strategy for Data.)

In this paper we study such cooperation between competing firms, and design data-sharing schemes that are optimal—ones that maximize total firm profits subject to each firm having the proper incentives to participate. More specifically, we study two settings, one without and the other with an Amazon. In the former, in which smaller firms compete only with one another, we show that coopetition is beneficial to the extent that firms can share data to simultaneously improve their respective predictions. In the optimal data-sharing scheme firms benefit from sharing because it allows them to better predict the tastes of consumers for whom no firm predicts well on its own.

In the latter setting, in which smaller firms compete with one another but also with an Amazon, sharing data about consumers for whom no firm predicts well can be beneficial, but is not optimal. We show that, in the optimal scheme, firms share data with others and weaken their own market positions on some market segments, in exchange for receiving data from others and strengthening their market positions on other segments.

The following simple example illustrates our model and previews some of our results. The example is framed as retailers offering goods to consumers who may make a purchase, but can also be interpreted as advertisers displaying advertisements to consumers who may click on the ads. In either case, suppose the market consists of two goods, $g_{0}$ and $g_{1}$, that consumers may desire. Each consumer is described by a feature vector in $\{0,1\}\times\{0,1\}$ that determines that consumer’s taste: Consumers of types 00 and 11 are only interested in $g_{0}$, whereas consumers of types 01 and 10 are only interested in $g_{1}$. A priori, suppose the distribution of consumers in the market is such that an $\alpha$ fraction are of type 00, a $\beta$ fraction of type 01, a $\gamma$ fraction of type 10, and a $\delta$ fraction of type 11.

When a consumer shows up to the market, a retailer may offer him one of the goods (as a convention, we use masculine pronouns for consumers and feminine pronouns for retailers/players). If the retailer has data about the consumer’s type, she will offer the good desired by that consumer. Assume that when a consumer is offered the correct good he makes the purchase, and that this leads to a profit of 1 to the retailer. If there are two retailers offering this good to the consumer he chooses one at random from whom to make the purchase. A retailer whose good is not chosen gets profit 0.

Suppose now that there are two retailers pursuing the same consumer, but that they do not fully know the consumer’s type. Instead, the first retailer only knows the first bit of the consumer’s type, and the second retailer only knows the second bit. This situation is summarized in Figure 1, where the first retailer knows the row bit and the second knows the column bit. Which goods should the competing retailers offer the consumer? For this example, suppose for simplicity that $\alpha>2\beta\geq 2\gamma>4\delta$. Then each of the retailers has a dominant strategy: If the bit they know is 0, offer $g_{0}$, and if the bit they know is 1, offer $g_{1}$.

To see that this is a dominant strategy, consider for example the row retailer, and suppose she learns that a consumer’s row bit is 0. She thus knows that the correct good is $g_{0}$ with probability $\alpha/(\alpha+\beta)$ and $g_{1}$ with probability $\beta/(\alpha+\beta)$. Her utility, however, depends also on the good offered by the column retailer. The row retailer’s worst-case utility from offering $g_{0}$ is $\alpha/(2\alpha+2\beta)$, which occurs when the column player also offers $g_{0}$. Her best case utility from offering $g_{1}$ is $\beta/(\alpha+\beta)$. Since we assumed $\alpha>2\beta$, offering $g_{0}$ is best for the row player regardless of the other retailer’s offer, and is thus dominant. The analyses for the case in which the row retailer’s bit is 1, as well as for the column retailer’s strategy, are similar.

The result from deployment of these dominant strategies is that consumers of type 00 will be offered the correct good by both retailers, and will thus choose one at random. Consumers of types 01 and 10 will be offered the correct good by only one of the retailers, whereas consumers of type 11 will not make a purchase as they will not be offered the correct good. The expected profit of the first retailer will thus be $\alpha/2+\gamma$, and of the second retailer $\alpha/2+\beta$.

Does data sharing improve retailers’ profits? Suppose retailers could share their respective information with one another, so that both always knew the consumer’s type. This would lead to both always offering the correct good, and hence to expected profits of $1/2$ for each. However, this would be detrimental to the second retailer whenever $\alpha/2+\beta>1/2$, since her profits with data sharing would be lower than without. Such cooperation is thus not individually rational – the second retailer will not want to cooperate with the first.

Instead, suppose there is a mediator—a trusted third-party that may represent either a service provider or a cryptographic protocol (such as a federated learning algorithm) run by the two retailers. The European Commission calls mediators common data spaces European Commission (2020), and describes them as “third-parties [who] may act as intermediaries and apply new technologies and ways of organising markets in order to…enable transactions that were previously not feasible. They can be private, public or community organisations that are neutral with respect to data uses…” (Martens et al., 2020). Given such a mediator, consider the following data-sharing scheme: both retailers share their data with the mediator, who then passes along the data to all retailers only if the consumer’s type is 11. If the consumer is not of type 11 then each retailer has her original data, and additionally infers that the consumer’s type is not 11 (for otherwise she would have learned that the type is 11).

How does this scheme affect retailers’ strategies? Clearly, when retailers learn that the consumer’s type is 11, both offer $g_{0}$. What happens if a retailer learns that the consumer’s type is not 11? Consider the row retailer. If her bit is 0, then learning that the consumer’s type is not 11 does not provide any new information, and so she still has the same dominant strategy of offering $g_{0}$. If her bit is 1, however, then she learns that the consumer’s type must be 10. But note that in this case, her best strategy is to offer $g_{1}$, which is the same as her dominant strategy absent a mediator. Thus, the mediator changes the row retailer’s behavior only when the consumer is of type 11. A similar analysis and conclusion hold for the column retailer.

Since retailers offer the same goods with and without data sharing in all cases except when the consumer is of type 11, this scheme changes retailer’s profits only in this latter case. In particular, it leads to an additional (ex ante) profit of $\delta/2$ for each retailer beyond her original profit—with probability $\delta$ the consumer is of type 11, in which case the retailers split the additional surplus of 1—and is thus beneficial to both. Observation 1 shows that such a scheme is, in fact, not only individually rational but also optimal.

Suppose now that the two retailers are also competing against an Amazon for the consumer. Since the Amazon is a giant it has more data, and, in particular, we assume that it has complete information about each consumer’s type. In this three-way competition, the original dominant strategies of our two smaller retailers do not perform as well: They lead to profits of $\alpha/3+\gamma/2$ and $\alpha/3+\beta/2$ to the first and second retailer, respectively, since when they offer the correct good the consumer now chooses amongst up to 3 retailers. The mediator described above, which reveals information when the type is 11, leads to higher profits, since now there is an additional $\delta/3$ to each of the smaller retailers. However, that mediator is no longer optimal, and the retailers can actually do better.

To see this, observe that, conditional on consumer type 01, the total profit to the small retailers is $\beta/2$, since only the second makes the correct offer $g_{1}$ and then competes with the Amazon. In contrast, if both retailers were to know the type and offer the correct good, then their total profit would be $2\beta/3$, namely $\beta/3$ each. This is harmful to the second retailer as it involves a loss of profit, but a gain for the first and the sum. But the second retailer can be compensated by getting data from the first elsewhere, for instance on consumer type 10.

For simplicity of this example, suppose that $\beta=\gamma$, and consider a mediator that facilitates full data-sharing, in which both retailers learn each other’s data and thus have complete information about consumers’ types. Here, the profit of each is $1/3>\alpha/3+\beta/2=\alpha/3+\gamma/2$, so both gain from this data sharing. Furthermore, the total profit of the retailers is $2/3$, which is the maximal utility they can obtain when competing against an Amazon. Thus, this data-sharing scheme is individually rational and optimal. Observe that it leads to higher retailer welfare than the optimal scheme absent a mediator.

Full data-sharing is not always individually rational, however. If $\beta$ is much greater than $\gamma$, then the second retailer is not sufficiently compensated by the first for sharing data about consumer 01. The second retailer will consequently be harmed by such data sharing, and so the scheme will not be individually rational. However, in Theorem 4 we show that there is a different data-sharing scheme that is individually rational and optimal, a scheme in which the first retailer shares all her data and the second shares some of her data. Overall, we show that such data-sharing coopetition against an Amazon is beneficial to the small retailers.

There is a set $G$ of goods and a population of consumers interested in obtaining one of them. Each consumer has one of a finite set of types, $\omega\in\Omega$, that describe the good $g_{\omega}\in G$ in which he is interested. There are three players who compete for consumers: two regular players indexed 1 and 2, and an Amazon, a player indexed 0. We will separate the analysis to two settings: first, when the Amazon player 0 is not present and players 1 and 2 compete only with one another; and second, when they additionally compete with Amazon. The model and definitions here apply to both settings.

Player 0 (if present) has complete information of the consumer’s type. In Section 3 we assume that players 1 and 2 have jointly complete information—that, when combined, their respective data uniquely identifies each consumer’s type—but that each player on her own may only be partially informed. In Section 4 we remove this assumption.

To model this informational setting, we represent players’ data using the information partition model of Aumann (1976): Each player $i$ is endowed with a partition $\Pi_{i}$ of $\Omega$, where $\Pi_{i}$ is a set of disjoint, nonempty sets whose union is $\Omega$. For each $\omega\in\Omega$ we denote by $P_{i}(\omega)$ the unique element of $\Pi_{i}$ that contains $\omega$, with the interpretation that if the realized type of consumer is $\omega$, each player $i$ only learns that the type belongs to the set $P_{i}(\omega)$.

Framing the example from the introduction within this model would associate $\Omega$ with $\{0,1\}^{2}$ and the partitions $P_{1}(00)=P_{1}(01)=\{00,01\}$, $P_{1}(10)=P_{1}(11)=\{10,11\}$, $P_{2}(00)=P_{2}(10)=\{00,10\}$, and $P_{2}(01)=P_{2}(11)=\{01,11\}$.

In this model, player 0’s complete information means that $P_{0}(\omega)=\{\omega\}$ for all $\omega\in\Omega$, and players 1 and 2’s jointly complete information means that $P_{1}(\omega)\cap P_{2}(\omega)=\{\omega\}$ for all $\omega\in\Omega$. We further assume that, before obtaining any information, all players have a common prior $\pi$ over $\Omega$.

To model data sharing between players 1 and 2 we suppose there is a mediator that gathers each player’s information and shares it with the other in some way. Formally, a mediator is a function $\mathcal{M}:2^{\Omega}\times 2^{\Omega}\mapsto\Delta\left(M^{2}\right)$, where $M$ is an arbitrary message space. The range is a distribution over pairs of messages, where the first (respectively, second) is the message sent to player 1 (respectively, player 2).

We begin with an informal description of the game: Players offer consumers a good, and consumers choose a player from whom to acquire the good. Consumers are single-minded: For each $\omega$ there is a unique $g_{\omega}\in G$ such that the consumer will only choose a player who offers good $g_{\omega}$. If there is more than one such player, the consumer chooses uniformly at random between them. One interpretation of this consumer behavior is that he chooses by “satsificing”—making a random choice among options that are “good enough” Simon (1956), where $g_{\omega}$ represents these good-enough options. See Ben-Porat and Tennenholtz (2019) for a similar approach.

We assume that prices and costs are fixed, and normalize a player’s utility to 1 if she is chosen and to 0 otherwise. So, for example, in the online advertising application of our model, consumers are assumed to click on a random ad amongst those ads that are most relevant; once they click, the expected profit to the advertiser (conversion rate times profit from a sale) is 1. The normalization to 1 is actually without loss of generality. If we had a different profit $p_{\omega}$ for each consumer type $\omega$, then we could change every $p_{\omega}$ to 1, modify the prior probability $\pi(\omega)$ of type $\omega$ to $p_{\omega}\cdot\pi(\omega)$, and then renormalize the prior to be a proper distribution. This would leave all of our analysis intact.

The following is the order of events, given a fixed mediator $\mathcal{M}$. It applies both to the case in which there are only two players $\{1,2\}$ and to the case in which there is an additional Amazon player, indexed 0.

1. 1.

   Each of players 1 and 2 chooses whether or not to opt into using the mediator.

2. 2.

   If one or both players $\{1,2\}$ did not opt into using the mediator, then each player $i$ obtains her base value, the utility $v_{i}$ (described below).

3. 3.

   If both players $\{1,2\}$ opted into using the mediator, then:

   1. (a)

      A consumer of type $\omega$ is chosen from $\Omega$ with prior distribution $\pi$.

   2. (b)

      If present, player 0 learns $P_{0}(\omega)$.

   3. (c)

      Messages $(M_{1},M_{2})$ are chosen from the distribution $\mathcal{M}(P_{1}(\omega),P_{2}(\omega))$, and each $i\in\{1,2\}$ learns $P_{i}(\omega)$ and $M_{i}$.

   4. (d)

      Each player simultaneously chooses a good to offer the consumer, and then the consumer chooses a player from whom to obtain the good.

There are several ways to interpret the base values $v_{1}$ and $v_{2}$. Our main interpretation is that these are the expected utilities of the players in the game without data sharing. To formalize this, consider the unmediated game $\Gamma=\left(\Omega,\mathcal{P},G^{\left|{\mathcal{P}}\right|},(P_{i}(\cdot))_{i\in\mathcal{P}},(u_{i})_{i\in\mathcal{P}},\pi\right)$, where $\mathcal{P}$ is the set of participating players, and is either $\{1,2\}$ or $\{0,1,2\}$, $G$ is the set of actions, $P_{i}(\cdot)$ is the information of player $i$ (which consists of the partition element of the realized type), $u_{i}$ is $i$’s utility function (described below), and $\pi$ is the common prior over $\Omega$. Given this Bayesian game, let $v=(v_{1},v_{2})$ be the expected utilities in some Bayesian Nash equilibrium (BNE) of $\Gamma$—a profile $(s_{1},s_{2})$ in which $s_{i}$ is the best strategy for each player $i$ conditional on her information and the assumption that the other player plays the strategy $s_{j}$.

Although this is the main interpretation of the base values, our model and results permit additional and more general interpretations as well. One additional interpretation is to suppose each $v_{i}$ is the minimax value of player $i$ in the unmediated game. In this second interpretation, we could imagine player $j$ “punishing” player $i$ if the latter does not opt into using the mediator, by playing a strategy that minimizes the latter’s payoff (off the equilibrium path). A third, more general interpretation is that the base values also depend on factors outside of the specific game, such as firm size, customer base, and so on, in addition to the primitives of the game $\Gamma$.

In our constructions of mediators in Sections 3 and 4, the interpretation of $v$ will not matter—our mediators will be optimal given any such values, regardless of whether they are derived endogenously as the equilibrium utilities of the game absent a mediator, or whether they arise exogenously from factors outside of the specific game. However, in Section 5, in which we compare data sharing across two different environments—with and without a mediator—we will stick with the main interpretation of base values as equilibrium payoffs absent a mediator. This is because, when we compare two different environments, the payoffs players could attain in equilibrium without a mediator will differ across these environments.

Now, as noted above, data sharing is modeled by a mediator $\mathcal{M}:2^{\Omega}\times 2^{\Omega}\mapsto\Delta\left(M^{2}\right)$. Without loss of generality we invoke the revelation principle and assume that $M=G$, the set of possible goods. We interpret the messages of the mediator as recommended actions to the players, one recommendation for each player.

Formally, if both players 1 and 2 opt into using the mediator $\mathcal{M}$, then all play the mediated Bayesian game $\Gamma^{\mathcal{M}}=\left(\Omega,\mathcal{P},G^{\left|{\mathcal{P}}\right|},\mathcal{I}^{\mathcal{M}},(u_{i})_{i\in\mathcal{P}},\pi\right)$. $\mathcal{P}$ is the set of participating players, and is either $\{1,2\}$ or $\{0,1,2\}$. The function $\mathcal{I}^{\mathcal{M}}:\Omega\mapsto\Delta(T^{\left|{\mathcal{P}}\right|})$ denotes the information of players in the game for each state, where $T=2^{\Omega}\times M$ is the set of possible pieces of information a player may have—a partition element and a message from the mediator. For player 0, this information consists of the realized type of consumer only, and so $\mathcal{I}^{\mathcal{M}}(\cdot)_{0}=\left(P_{0}(\cdot),\emptyset\right)$ (where the $\emptyset$ means player 0 gets no message from the mediator). For players 1 and 2, the information consists of both the partition element of the realized type of consumer and the action recommended to her by the mediator, and so $\mathcal{I}^{\mathcal{M}}(\cdot)_{i}=\left(P_{i}(\cdot),\mathcal{M}(P_{1}(\cdot),P_{2}(\cdot))_{i}\right)$.

Next, each player’s set of actions in the mediated game is $G$, and her utility function $u_{i}=u_{i}:\Omega\times G^{\left|{\mathcal{P}}\right|}\mapsto{\mathbb{R}}$. The latter is equal to 0 if player $i$’s action $g\neq g_{\omega}$, and otherwise it is equal to $1/k$, where $k$ is the total number of players who play action $g_{\omega}$. The nonzero utility corresponds to utility 1 if a consumer chooses the player’s good, which occurs if a player offers the consumer the correct good and the consumer chooses uniformly amongst all players that do so. We often write $E[u_{i}(\cdot)]$ when the expectation is over the choice of $\omega$, in which case we omit the dependence of $u_{i}$ on $\omega$ for brevity. Finally, (mixed) strategies of players in $\Gamma$ are functions $s_{i}:T\mapsto\Delta(G)$. We also denote by $s_{i}(\omega)=s_{i}(\mathcal{I}_{i}(\omega))$.

An important note about the mediator is in order. We assume that when players opt into participating, they truthfully reveal their data $P_{i}(\omega)$ to him. Players’ strategic behavior is relevant in their choice of opting in or not, and then in whether or not they follow the mediator’s recommendation in their interaction with the consumer. While truthful reporting is clearly restrictive, it is a natural assumption in our context—for instance, when service providers act as mediators, they are typically also the ones who host retailers’ data in the cloud, and so already have the (true) data on their servers. Similarly, federated machine learning algorithms work based on the assumption that participants share their true data. Furthermore, truthful reporting can be justified by the repeated nature of the interaction. Observe that if a firm is not truthful in its reporting, the other firm will discover this at some point, as the mediator’s recommendations will not pan out as well as they should. Thus, if firms have the ability to opt out of using the mediator at some point in the future, then each firm can use the threat of opting out to incentivize the other to report truthfully.