Equilibrium Characterization for Data Acquisition Games
Jinshuo Dong, Hadi Elzayn, Shahin Jabbari, Michael Kearns, Zachary Schutzman
EEquilibrium Characterization for Data Acquisition Games
Jinshuo Dong , Hadi Elzayn , Shahin Jabbari , Michael Kearns and
Zachary Schutzman
University of Pennsylvania { jinshuo, hads } @sas.upenn.edu, { jabbari, mkearns, ianzach } @cis.upenn.edu Abstract
We study a game between two firms in which eachprovide a service based on machine learning. Thefirms are presented with the opportunity to pur-chase a new corpus of data, which will allow themto potentially improve the quality of their products.The firms can decide whether or not they want tobuy the data, as well as which learning model tobuild with that data. We demonstrate a reductionfrom this potentially complicated action space to aone-shot, two-action game in which each firm onlydecides whether or not to buy the data. The gameadmits several regimes which depend on the rela-tive strength of the two firms at the outset and theprice at which the data is being offered. We analyzethe game’s Nash equilibria in all parameter regimesand demonstrate that, in expectation, the outcomeof the game is that the initially stronger firm’s mar-ket position weakens whereas the initially weakerfirm’s market position becomes stronger. Finally,we consider the perspective of the users of the ser-vice and demonstrate that the expected outcome atequilibrium is not the one which maximizes thewelfare of the consumers.
Recent years have seen explosive growth in the domain ofdigital data-driven services. Search engines, restaurant rec-ommendations, and social media are among the many prod-ucts we use day-to-day which sit atop modern data analysisand machine learning (ML). In such markets, firms live anddie by the quality of their models; thus success in the ‘race fordata’, whether acquired directly from customers or indirectlyvia acquisition of rival firms or purchasing data corpuses, iscrucial. In this work, we study two questions: whether suchmarkets tend towards monopoly, and how competition affectsconsumer welfare. Importantly, we consider these questionsin light of the modeling choices that firms must make.In our model, two firms compete for market share (util-ity) by providing identical services that each rely on an MLmodel. The firms’ error rates depend on their choices of al-gorithms, models and the volume of available training data.Each firm’s market share is proportional to the error of its model relative to the model built by its competitor. This ismotivated by the observation that the services built using MLare highly accurate, so users are more conscientious of themistakes the service makes, rather than the successes. A com-petition exponent measures relative ferocity of competitionand maps to a plausible Markov model of consumer choice.See Section 2.2 for more details.The firms initially possess (possibly differing) quantities ofdata, and are given the opportunity to buy additional data ata fixed price to improve their models. Since data is costlyand relative (rather than absolute) model quality determinesmarket share, each firm’s best course of action may dependon the actions of its rival. Hence, each firm acts strategicallyand faces two decisions: whether to buy the additional data,and what type of model to build in order to produce the bestproduct given the data it ends up with.The decision of what model to build seems to complicatethe firms’ action space greatly; there is a very large set ofmodel classes to select from, and different classes have dif-ferent efficiencies. For example, when restricting attention toneural networks, the choices of depth and number of nodesper layer produce different hypothesis classes with differentoptimal models. Thus, in principle, the decisions of whatmodel class to select and whether to purchase additional datamust be made jointly. However, learning theory allows usto greatly reduce this large action space. In Section 2.1, weshow that the game in which firms jointly choose a modeland whether to attempt to buy the additional data reduces toa strategically equivalent game in which firms first choosewhether to buy the data and then choose optimal models.In Section 3, we characterize the Nash equilibria of ourgame for different parameter regimes. For no combinationof parameters does exactly one firm wish to buy the data;unsurprisingly, for very high prices, neither firm buys data,and for very low prices, both firms do. In the middlingregime, the competitive aspect of the game imposes a ‘pris-oners' dilemma’-like flavor: both firms would prefer neitherfirm buy the data, but each do so in order to prevent the otherfrom strengthening its position. Moreover, the unique mixedstrategy Nash equilibrium in this regime involves firms in-creasing their probability of buying data as price increases .This counterintuitive result follows from the logic of equilib-rium: firms playing mixed strategies must be indifferent tobuying and not buying the data, and as the price rises, the a r X i v : . [ c s . G T ] M a y robability that a competitor acquires the data must rise inorder to make investing in data acquisition a palatable option.Finally, we study whether any of the dynamics of the gamepush the market towards a monopoly. Perhaps counter to a‘rich-get-richer’ feedback loop that might be expected in dataraces, we observe that in all equilibria, the data gap (and thus,market share gap) always narrows (in expectation). As mea-sured by consumer welfare, this is actually undesirable . Boththe direction of the data gap as well as the welfare impli-cation may be counterintuitive, particularly with respect tothe well-known stylized fact that market concentration is badfor consumers. However, consumer data that improves a ser-vice can be viewed as exhibiting a form of network effects, inwhich case perfect competition can result in inefficiency andunder-provisioning of a good [14]. In other words, a greaterdata gap would result in more consumers using a less error-prone service. As for the data race, anecdotal evidence, suchas GM’s acquisition of automated driving startup Cruise, de-spite Waymo’s earlier market entry and research head-start,are suggestive (though not conclusive) that these predictionsmay be indicative of real-world dynamics [18].We view our work as a first step towards modeling andanalyzing competition for data in markets driven by ML. Un-der our simplifying assumptions, we derive concrete resultswith relevance both for policymakers analyzing algorithmicactors as well as engineering or business decision-makersconsidering the tasks of data acquisition and model selection.Our results are qualitatively robust to other natural modelingchoices, such as allowing both firms to purchase the data, aswell as treating the data seller as a market participant; how-ever, more significant departures may lead to different con-clusions. See Sections 4 and 5 for more details. The theory of ML from a single learner’s perspective is welldeveloped, but until recently, little work had studied compe-tition between learning algorithms. Notable exceptions in-clude [2, 16]. We differ from both works by exploring thecomparative statics and welfare consequences of a single de-cision (data acquisition). Concurrent work [3] studies a gamein which learners strategically choose their model to competefor users, but users only care about the accuracy of predic-tions on their particular data. In contrast, users in our modelchoose based on the overall model error.Our work also intersects with several strains of economicliterature, including industrial organization and network ef-fects [7, 9, 14]. We differ from such models in two key ways.First, in contrast to assuming a static equilibrium [14] or fix-ing a dynamic but unchanging process at the outset [10], ourwork can be viewed as an analysis of a shock to a given po-tentially asymmetric equilibrium in the form of the availabil-ity of new data. Second, the consumers in our model do notbehave strategically (see e.g. [5, 16] for more discussion).Finally, our work is related to spectrum auctions, competi-tion with congestion externalities [5], and the sale of informa-tion or patents [12, 13]. Our results primarily share qualita-tive similarities: the choice of one firm to buy data (spectrum)forces the other to do so to avoid losing market share, thoughit would not have been profitable absent the rival, and actual outcomes run counter to consumer preferences (see e.g. [5]).
We formally motivate and model the ML problem of the firmsand demonstrate how this reduces to a game in which thefirms can either buy or not buy the new data.
Consider a firm using ML to build a service e.g. a recom-mendation system. The amount of data available to the firmis a crucial determinant to the effectiveness of the predictiveservice of the firm. Fixing the amount of data, the firm facesa fundamental tradeoff; it can use a more complex model thatcan fit the data better, but learning using a complicated modelrequires more training data to avoid over- or underfitting.We can formally represent this tradeoff as follows. Let H denote the hypothesis class from which the firm is selectingits model and assume the data is generated from a distribu-tion D . Then given m i.i.d. draws from D the error of thefirm when learning a hypothesis from H can be written as err D ( H ) = err( m, H ) + min h ∈H err D ( h ) [19].The first term, known as estimation error , determines howwell in expectation a model learned with m draws from D canpredict compared to the best model in class H . The secondterm, known as approximation error , determines how well thebest model in class H can fit the data generated from D .The approximation error is independent of the amount oftraining data, while the estimation error decreases as the vol-ume of training data increases. The choice of H affects botherrors. In particular, fixing the amount of training data, in-creasing the complexity of H will increase the estimation er-ror. On the other hand, the additional complexity will de-crease the approximation error as more complicated data gen-erating processes can be fit with more complicated models.Once the amount of data is fixed, the firm can optimize overits choice of model complexity to achieve the best error. Weexamine a few widely used ML models and their error forms.As a first example, consider the case where the firm isbuilding a neural network and has to decide how many nodes d to use. d is the measure of the complexity of the model classand given m data points, the error of the model can be writtenusing the following simplification of a result from Barron [1]. Lemma 1 (Barron [1]) . Let H be the class of neural networkswith d nodes. Then for any distribution D , with high proba-bility, the error when using m data points to learn a modelfrom H is at most c d/m + c /d, for constants c and c . Fixing m , the choice of d that minimizes the error can becomputed by minimizing the bound in Lemma 1 with respectto d . This corresponds to d = c √ m/c and we get that theerror of the model built by the firm is (cid:112) c c /m .As another example, consider the very simple setting of realizable PAC learning where the data points are generatedby some hypothesis in a fixed hypothesis class. Lemma 2 (Kearns and Vazirani [15]) . Any algorithm for PAClearning a concept class of VC dimension d must use Ω( d/(cid:15) ) examples in the worst case. hus in this setting, in the worst case, firms need Θ(1 /(cid:15) ) training data points to achieve error (cid:15) . A similar bound givesthat with high probability, the firms can guarantee error of Θ(1 /m ) (see [15]).In the examples above the error of a firm with m data pointstakes the form of either Θ( m − / ) or Θ( m − ) after the firmoptimizes over the choice of model complexity. Importantly,the error in both cases (and more generally) degrades as thenumber of data points increases. The rate at which the errordegrades is commonly known as the learning rate .There are other learning tasks with learning rates differ-ent than the examples above. Consider a stylized model of asearch engine where the set of queries is drawn from a fixedand discrete distribution over a very large or even infinite set,and the search engine can only correctly answer queries thatit has seen before. If, as is often assumed, the query distri-bution is heavy-tailed, then the search engine will require alarge training set to return accurate answers.In this framework, the probability that a search engine in-correctly answers a query drawn from the distribution is ex-actly the expectation of the unobserved mass of the distribu-tion given the queries observed so far. This quantity is knownas the missing mass of a distribution (see e.g. [4, 8, 11, 17]).Lemma 3 shows how to bound the expected missing mass forthe class of polynomially decaying query distributions. Lemma 3 (Decrouez et al. [8]) . Let P k for k > be adiscrete distribution with polynomial decay defined over i ∈ N ≥ such that Pr x ∼ P k [ x = i ] = i − k / Σ ∞ j =0 j − k . Then the ex-pected missing mass given m draws from P k is Θ( m /k − ) . By varying k in the query distribution of Lemma 3, thelearning rate in the search problem can take the form of Θ( m − i ) for any i ∈ (0 , . Thus, the learning rate for searchmay be much faster or slower compared to the previous ex-amples, and the exact rate depends on the value of k .We saw that given a fixed amount of data, a firm using MLcan optimize over its learning decisions to get the best possi-ble error guarantee. Furthermore, while error decays as moredata becomes available, the rate of decay can vary widely de-pending on the task. We next see how various learning ratescan be incorporated into the parameters of our game. Consider two competing firms (denoted by Firm 1 and 2) thatprovide identical services e.g. search engines. We assume themarket shares of the firms depend on their ability to make ac-curate predictions e.g. responding to search queries. As dis-cussed above, the quality of their models is determined ulti-mately by the size of their training data with a task-dependentlearning rate. Each firm trains a model on its data and usesits model to provide the service. Let err and err denote the excess error of the firms for the corresponding models. Intu-itively, these errors measure the quality of the firms’ services,so a firm with smaller error should have higher market share.We assume each firm captures a market share proportionalto the relative errors of the two models. Formally, we defineFirm 1 and 2’s error-based market share as µ = 1 − err a err a + err a = err a err a + err a and µ = 1 − µ . (1) The constant a ∈ N , which we call the competition expo-nent (inspired by Tullock contest [20]), indicates the ferocity of the competition, or how strongly a relative difference inthe errors of the firms’ models translates to a market advan-tage. As a gets closer to 0, the tendency is towards each firmcapturing half of the market, and thus a large difference inthe models’ errors is needed for one firm to gain a signifi-cant advantage in the market share. Conversely, as a growslarger, even tiny differences in the models’ errors translate tomassive differences in the market share. (See Figure 1.)Figure 1: Plot of f = (1 − r ) a / ( r a + (1 − r ) a ) for various a values. When r = err / (err + err ) , f is the market shareof Firm 1.An error-based model reflects markets for services whichdemand extremely low errors, such as vision systems for self-driving cars. Under the error-based model, if Firm 1 has . accuracy and Firm 2 has accuracy, Firm 1 willcapture of the market share. By contrast, an accuracy-based model (i.e. when the market share of Firm 1 is de-fined as acc a / (acc a + acc a ) ) would suggest much less real-istic near-even split.Firm 1 Firm 2 err err acc acc Figure 2: Vertices denote thefirms and the directed arrowsdenote the probability of transi-tion. acc is shorthand for accu-racy and err is shorthand for er-ror.We provide another justification suggesting that an error-based market can arise even when the learned model is usedto provide an everyday service in which high accuracy is nota strict requirement. Consider a customer who, each day, usesthe service. She begins by choosing the service of one of thefirms uniformly at random. As long as the answers she re-ceives are correct, she has no reason to switch to the otherfirm’s service, and uses the same firm’s service tomorrow.However, once the firm makes an error, the customer switchesto the other firm’s service. The transition probabilities areherefore given by the accuracy and error of each firm. SeeFigure 2 for the Markov process representing this example.We can think of the market share captured by each firm asthe proportion of the days on which each firm saw the cus-tomer. This is exactly the stationary distribution of the asso-ciated Markov process as stated in Lemma 4.
Lemma 4.
Let µ and µ denote the probability mass thatthe stationary distribution of the Markov process in Figure 2assigns to Firms 1 and 2. Then µ = err / (err + err ) , and µ = err / (err + err ) . Sketch of the Proof.
By the definition of a stationary distribu-tion, µ and µ should satisfy the following conditions: (1 − err ) µ + err µ = µ err µ + (1 − err ) µ = µ . Given that µ + µ = 1 by definition, we can solve the systemof linear equations to compute the market shares.Lemma 4 states that the market share of each firm in theMarkov process is exactly the error-based market share as de-fined in Equation 1 when setting a = 1 . A similar argumentmotivates an error-based market share for values of a ∈ N ,where the customer switches firms after experiencing a mis-takes in a row. The probability of making a mistakes in a rowis just err ai for Firm i , so the stationary distribution of theMarkov process is exactly the two error-based market sharesas defined in Equation 1.Using our observations from Section 2.1, we can write theerror-based market share in the large-data regime as follows. Theorem 1.
Let m and m denote the number of data pointsof Firm 1 and 2, respectively. Then for some b ∈ R + , themarket share of Firm 1 can be written (asymptotically) as µ = m b / ( m b + m b ) .Sketch of the Proof. Depending on the task at hand, we canwrite the err of a firm as Θ( m − r ) for some r ∈ (0 , , where err refers to the excess error of the model with the smallestworst-case error. Substituting this into Equation 1 and ignor-ing lower order terms, which vanish asymptotically, we get µ = m − ra m − ra + m − ra = m ra m ra + m ra . Now let b = ra . Since a is a natural number and r is a realnumber in (0 , , the combined competition exponent is a realnumber strictly larger than 0.Because a can be any integer and there exists a correspond-ing learning problem for any learning rate in (0 , , Theo-rem 1 implies that the combined competition exponent in ourgame can be any positive real number, motivated by the initialchoice of a and the learning rate of the firms’ ML algorithms.The reductions and derivations in Sections 2.1 and 2.2 al-low us to simplify the acquisition games as follows. We firstsimplify the actions of each firm to only decide whether tobuy the data or not, since model choice can be optimizedonce the number of available data points is known. Moreover,Theorem 1 not only allows us to simplify the form of marketshare, but also provides us with a meaningful interpretationfor any positive (combined) competition exponent. Given the reductions so far, we model our game as a two-player, one-shot, simultaneous move game. Firms 1 and 2 be-gin the game endowed with an existing number of data points,denoted by m and m , respectively. Without loss of gener-ality, we assume m ≥ m . Each firm must decide whetheror not to purchase an additional corpus of n data points at afixed price of p . The firm can either Buy (denoted by B ) orNot Buy (denoted by N B ) the new data. If both firms attemptto buy the data, the tie is broken uniformly at random (Sec-tion 4 discusses relaxing the assumption that only one firmmay buy the data). After the purchase, each firm uses its datato train an ML model for its service.We assume the particular form of the market share of Firm1 using the reduction in Theorem 1. The market share of Firm2 is defined to be one minus the market share of Firm 1.A strategy profile s is a pair of strategies, one for each ofthe firms. Fixing s , the utility of Firm i (denoted by u i ( s ) )is its market share less any expenditure. The utility of Firm1 in all of the strategy profiles of the game is summarized inTable 1 (rows and columns correspond to the actions of Firm1 and 2). The utility of Firm 2 is defined symmetrically. Firm 1/Firm 2 Buy (B) Not Buy (NB)Buy (B) ( µ ( m + n, m , b )+ µ ( m , m + n, b ) − p ) µ ( m + n, m , b ) − p Not Buy (NB) µ ( m , m + n, b ) µ ( m , m , b ) Table 1: u ( s ) in all of the strategy profiles of the game.A strategy profile is a pure strategy Nash equilibrium (pureequilibrium) if no firm can improve its utility by taking a dif-ferent action, fixing the other firm’s action. A mixed strategyNash equilibrium (mixed equilibrium) is a pair of distribu-tions over the actions (one for each firm) where neither firmcan improve its expected utility by using a different distribu-tion over the actions, fixing the other firm’s distribution. Weare interested in analyzing the Nash equilibria (equilibria). We now turn to finding and analyzing the equilibria. First, weintroduce some additional notation. Let A = ( m + n ) b ( m + n ) b + m b − m b m b + ( m + n ) b ,C = ( m + n ) b ( m + n ) b + m b − m b m b + m b ,D = ( m + n ) b m b + ( m + n ) b − m b m b + m b . These parameters have intuitive interpretations. A/ is theexpected change in Firm 1’s (or Firm 2’s) market share whenmoving the outcome from ( N B, B ) (or similarly ( B, N B ) )to ( B, B ) . C is the change in market share that Firm 1 re-ceives if it moves from ( N B, N B ) to ( B, N B ) , and D is thesymmetric relation from the perspective of Firm 2. For simplicity we assume this data is independent of and iden-tically distributed to the data in possession of the firms. e observe that A = C + D . Moreover, since C and D arenonnegative, it is immediately clear that A > max { C, D } .Finally, when m > m (i.e. Firm 1 starts with strictlymore data), Firm 2 experiences a larger absolute change in itsmarket share when the outcome changes from ( N B, N B ) to ( N B, B ) than to ( B, N B ) . In other words, Firm 2 experi-ences a larger increase in market share when it buys the datacompared to the decrease it experiences when Firm 1 receivesthe data. We defer all the omitted proofs to Appendix A. Lemma 5. If m > m then for all n and b we have that C < D . The equilibria of the game clearly depend on the values ofthe parameters m , m , n , p and b . For example, if p > ( p ≤ ), then neither firm should ever want to (not) buythe data. We observe that, fixing the values of m , m , n and b , there is a range of values for p where the data is tooexpensive ( too cheap ) and N B ( B ) is a dominant strategyfor both firms. There is also an intermediate range of valuesfor p where more interesting behaviors emerge, as formallycharacterized in Theorem 2. Theorem 2.
1. When p ≤ max { C, D } , ( B, B ) is the unique equilib-rium.2. When p ≥ A , ( N B, N B ) is the unique equilibrium.3. When max { C, D } < p < A , ( B, B ) and ( N B, N B ) are both equilibria. Furthermore, there exists a (unique)mixed equilibrium (( α, − α ) , ( β, − β )) such that α − α ) = p − DA − p and β − β ) = p − CA − p , where α and β denote the probabilities that Firms 1 and2 select the action B , respectively.Proof. We use flow diagrams to analyze the equilibria of thegame (see e.g. [6] for more details on this technique). As a tu-torial of this flow diagram argument, we carefully analyze thediagram for the regime of our game in which p < min { C, D } as depicted in the top left panel of Figure 3.In a flow diagram, each vertex corresponds to a strategyprofile. An arrow indicates that one player changes its strat-egy while the other’s action is fixed. In particular, in Figure 3vertical (horizontal) arrows demonstrate the change of strat-egy for Firm 1 (Firm 2). The numerical value above the arrowindicates how much a player gains by a deviation, and arrowsare oriented so that they always point in the direction of non-negative gain. The leftmost vertical arrow indicates that Firm1 increases its utility by ( A − p ) / by changing its decisionfrom N B to B , fixing that Firm 2 is committed to playing B .Similarly, the rightmost vertical arrow indicates that Firm 1increases its utility by C − p when it makes this change, fix-ing that Firm 2 is committed to playing N B . The horizontalarrows are the symmetric results for Firm 2, fixing the actionof Firm 1. The topmost arrow indicates the increase in utilitywhen moving from
N B to B when Firm 1 plays B , and the bottommost corresponds to the increase in utility for the samechange of action when Firm 1 plays N B .This particular flow diagram models the regime of thegame where the price is sufficiently low such that ( B, B ) is the unique pure equilibrium. Consider the profile ( B, B ) .Since arrows only point at, rather than originate from, ( B, B ) ,unilateral deviations from ( B, B ) are unprofitable for bothplayers. Hence, ( B, B ) is a pure strategy equilibrium in thisregime. Furthermore, there is no other pure equilibrium be-cause there are no other ‘sinks’ in the top left panel of Fig-ure 3. Moreover, no mixed equilibrium exists. To see this,note that in a mixed equilibrium, a player mixing can onlymix over best responses. But since the arrows representingFirm 2’s deviations both point towards ( B, N B ) is dominatedby B ; hence N B cannot be a best response, so Firm 2 can-not be mixing. But since Firm 1 is not indifferent between B and N B if Firm 2 chooses B , Firm 1 will not mix either.More generally, this logic means that mixed equilibria requirearrows pointing in opposite directions.Similar logic allows us to easily analyze the continuum ofgames induced as allow p varies monotonically. Every valueof p induces exactly one of the flow diagrams in Figure 3.Thus, characterizing the equilibria in each flow diagram char-acterizes the equilibria of the different parameter regimes. (1) p ∈ ( −∞ , min { C, D } ) : The top left panel of Figure 3represents the flow diagram in this regime and we can see thatthe only equilibrium is the pure strategy of ( B, B ) . (2) p ∈ (min { C, D } , max { C, D } ) : The top middle panelof the Figure 3 represents the flow diagram in this regime. ByLemma 5, (min { C, D } , max { C, D } ) ≡ ( C, D ) . Again wecan see that the only equilibrium is the pure strategy ( B, B ) . (3) p ∈ (max { C, D } , A ) : The top right panel of Figure 3represents the flow diagram in this regime. There are twopure equilibria: ( B, B ) and ( N B, N B ) . There also exists amixed equilibrium. In a mixed equilibrium, both players arerandomizing, and thus must be indifferent between the purestrategies they are randomizing over; this condition allows usto solve for the mixed strategies.Let α denote the probability that Firm 1 is playing B. Thenin a mixed equilibrium, Firm 2 is indifferent between the twoactions. Therefore, α (cid:18) ( m + n ) b m b + ( m + n ) b + m b ( m + n ) b + m b − p (cid:19) + (1 − α ) (cid:18) ( m + n ) b m b + ( m + n ) b − p (cid:19) = α (cid:18) m b ( m + n ) b + m b (cid:19) + (1 − α ) (cid:18) m b m b + m b (cid:19) . By rearranging we get that α − α ) = p − DA − p , as claimed.Similarly let β denote the probability that Firm 2 is playingB. Then in a mixed equilibrium, Firm 1 is indifferent betweenthe two actions. With a similar calculation we can show that β − β ) = p − CA − p , B, B ) (
B, N B )( N B, B ) (
N B, N B ) ( A − p ) ( A − p ) D − p C − p ( B, B ) (
B, N B )( N B, B ) (
N B, N B ) ( A − p ) p − C ( A − p ) D − p ( B, B ) (
B, N B )( N B, B ) (
N B, N B ) ( A − p ) p − C ( A − p ) p − D ( B, B ) (
B, N B )( N B, B ) (
N B, N B ) ( p − A ) ( p − A ) p − Cp − D Figure 3: The flow diagrams for different parameter regimes. Top left panel when p ∈ ( −∞ , min { C, D } ) . Top middle when p ∈ (min { C, D } , max { C, D } ) . Top right panel when p ∈ (max { C, D } , A ) . Bottom panel when p ∈ ( A, + ∞ ) .as claimed. (4) p ∈ ( A, + ∞ ) : The bottom panel of Figure 3 representsthe flow diagram in this regime and we can see that the onlyequilibrium is the pure strategy of ( N B, N B ) .Theorem 2 allows us to make several key observationsabout the market structure of this game. First, since C and D represent the maximum increase in the market sharefirms could achieve by buying the data, the fact that the onlyequilibrium when p ∈ [min { C, D } , max { C, D } ] is ( B, B ) means that both firms buy the data despite the fact that the best-case improvement in market share is less than what theypay. This ‘race for data’ thus has the character of a prisoner’sdilemma – if both firms could agree not to buy the data, theywould be better off, but either would be tempted to buy thedata and improve market share.Second, Theorem 2 illustrates how several features of equi-librium depend on the ferocity of competition, as determinedby the exponent a ; as a varies, the frontiers of the regimes de-scribed in Theorem 2 shift too. For example, in the case that a = 0 , market share is split evenly between the two firms,regardless of error or accuracy; unsurprisingly, as a → (which implies b → ), A, C , and D also approach , sothe payoff difference between strategy profiles becomes neg-ligible. As a consequence, the regimes (1), (2), and (3) col-lapse, and all but very small p induce regime (4), where ( N B, N B ) is the only equilibrium. Thus, for small a , unless p is very close to zero, ( N B, N B ) is the only equilibrium.We observe similar behavior when a is large. Assuming that m > m + n , then a → ∞ implies that A → (and hence b → ∞ ), again implying that regimes (1), (2), and (3) col-lapse. Thus again, unless p is very close to , ( N B, N B ) isthe unique equilibrium. This is for a different reason than the a small case, however: Firm 1 now has no incentive to buy,since it is guaranteed almost the whole market share usingits current model. Moreover, in this scenario, Firm 2’s initialdisadvantage is too great to be overcome by buying the data.If a is in between these two extremes, many choices of m and m lead to a non-empty interval (max { C, D } , A ) , withendpoints far from and . When p falls in this interval,regime (2) holds, so a mixed equilibrium exists; we discusssolving for this mixed equilibrium in Section 3.2. The com-plete characterization of the equilibria for all regimes of p inTheorem 2 allows us to pin down the optimal fixed price from the perspective of maximizing seller’s revenue. However, infull generality, the seller’s problem encompasses further pos-sibilities like auction pricing; hence, we defer this calculationto future work. See Section 5 for a discussion. Next, we carefully examine the mixed equilibrium and studythe relationship between the weights each firm places on eachaction and the parameters of the game.Recall that α and β in Theorem 2 denote the probabilitythat Firms 1 and 2 purchase the data in the mixed equilib-rium. When m > m , then α < β which implies that thesmaller firm will succeed more often in purchasing the datain the mixed equilibrium. The relationship of α and β withthe number of data points n and the price p is as follows. Lemma 6.
Let (( α, − α ) , ( β, − β )) denote the mixed equi-librium in the the regime where max { C, D } < p < A . Then α and β , both increase when p increases or n decreases. Lemma 6 may seem counterintuitive as it implies that asthe price p rises through the range in which a mixed equilib-rium exists, the probability that any of the firms want to buythe data also increases . However, once the price p crossesthe threshold A , the unique equilibrium is the pure strategy ( N B, N B ) . This gives rise to a discontinuity. See Figure 4.Of course, this all says nothing about the equilibrium util-ities for the firms; as long as the equilibrium utilities arenot identical, players will naturally have ordinal preferencesover the set of equilibria. We analyze these preferences inLemma 7, which elucidates the discontinuity at p = A . Lemma 7.
When p ∈ (max { C, D } , A ) , u ( N B, N B ) ≥ u ( s ) and u ( N B, N B ) ≥ u ( s ) for all strategy profiles s . However, u ( B, N B ) ≥ u ( B, B ) ≥ u ( N B, B ) , while u ( N B, B ) ≥ u ( B, B ) ≥ u ( B, N B ) . While both firms agree that ( N B, N B ) is the most pre-ferred outcome, their preferences over the remaining threeoutcomes are discordant. In particular, given that at least onefirm will try to buy the data, each firm would prefer itself tobe the buyer rather than the opponent. If either firm believesthe other may try to buy the data, it will put positive weight onthe action B in the mixed equilibrium. Once the price crosses A , both firms know that it would be irrational for the other tobuy, so we see a unique pure equilibrium of ( N B, N B ) .igure 4: The probability of selecting B in equilibrium forFirm 1 as a function of p . The blue, red and yellow linescorrespond to the ( B, B ) , mixed and ( N B, N B ) equilibria. We now analyze the change in the market shares.
Lemma 8.
When m ≥ m , the only strategy profile thatstrictly increases the market share of Firm 1 is ( B, N B ) . So while only ( B, N B ) leads to an increase in the marketshare of Firm 1, it is not a pure equilibrium. We show thateven when firms play according to the mixed equilibrium, theexpected market share of Firm 1 does not strictly increase. Theorem 3.
When m ≥ m and p ∈ (max { C, D } , A ) , theexpected market share of Firm does not strictly increase ifboth firms play according to the mixed equilibrium. Together, Lemma 8 and Theorem 3 demonstrate that thenatural forces of the interaction on the market are, perhapssurprisingly, antimonopolistic. Since we assume that Firm 1enters the game with a greater market share than Firm 2, butthat no equilibrium allows Firm 1 to increase its market share,the game disfavors the concentration of market power. Thisraises the question of whether this antimonopolistic tendencyis good for the users. We analyze this next.
We now consider the perspective of the users of the firm’s ser-vice. We show that consumers prefer the outcome ( B, N B ) ,in which the initially stronger firm concentrates its marketpower. This is not supported by a pure equilibrium in anyregime, nor is it the most likely outcome generated by mixedequilibrium; hence, we will see that the interests of the firmsdo not align with the interests of the consumers. We definethe consumer welfare as follows. Definition 1.
Let m ( s ) and m ( s ) denote the (expected)number of data points that Firm 1 and 2 posses when playingaccording to strategy profile s . Then the consumer welfare is CW ( s ) = µ ( s )(1 − err ( m ( s )))+ µ ( s )(1 − err ( m ( s ))) . The welfare definition arises from assuming consumers re-ceive unit of utility for accurate predictions and for er-roneous ones. Notice that maximizing this definition of con- sumer welfare is exactly equivalent to minimizing the market-share weighted error probability. This leads to Theorem 4. Theorem 4.
Suppose m > m . Then the consumers havethe following preferences over the strategy profiles. CW ( B, NB ) > CW ( B, B ) > CW ( NB, B ) > CW ( NB, NB ) . Note that consumers’ preference for the outcome in whichFirm 1 concentrates its market power is not the same as say-ing that the consumers prefer a monopoly. Rather, the con-sumers have a preference for higher quality services. When m > m , Firm 1’s model before acquiring the data has alower error rate than that of Firm 2, and so, of all the possibleoutcomes, the one which leads to a product with the lowesterror rate is the one in which Firm 1 is able to improve onits already superior product. But if Firm 2 were not a playerat all, then a monopolistic Firm 1 would have no incentive to buy the new data. Therefore, a monopoly without the threatof competition will not lead to the best outcome from the con-sumer’s perspectives. Next, we consider robustness to three simple extensions.
Firm Acquisition
We treat the data seller as a market par-ticipant with its own customers and market share. This allowsus to model firm acquisition: buying the data translates to ac-quiring the firm and its customers, and neither firm buying thedata corresponds to the third firm remaining in the market.
Simultaneous Sale
Rather than the data being exclusivelysold to one firm in the case that both firms buy, we allow theseller to sell the data to both firms at the same fixed price.
Correlated Equilibria
We consider the richer concept of correlated equilibrium and search for additional equilibria.In each of these extensions, we can again derive the quan-tities A , C , and D ; while the precise quantities change, theirrankings and relationships do not. Thus, in the first two ex-tensions, the general phenomenon of three regimes, with mix-ing over the middle regime, remains unchanged. In the thirdextension, some new correlated equilibria exist, but none in-clude the qualitatively different result of coordinating pur-chase of the data by a single firm. Moreover, in expectation,the market share becomes less asymmetric in all extensions. We view our work as a first step towards modeling and ana-lyzing competition for data in markets driven by ML. Thereare several directions for further investigation. First, we mod-eled the data to be acquired as having a fixed size and a fixedprice, but real datasets can be divisible. One further directionto consider is a game in which we expand the strategy spaceof the players to include buying any number of data points ata fixed price per data point or nonlinear function of the num-ber of data points purchased. More generally, treating theseller of the data as an additional player in the game allowsfor further questions, such as: what is the optimal revenue-generating mechanism to sell the data? And does the optimalmechanism maximize social welfare?dditionally, many firms that provide learning-based ser-vices acquire their data through their customers that use theservice. In this way, capturing a larger market share inducesa feedback loop which allows a firm to iteratively improve itsproduct. What can be said about our game in a repeated set-ting with dynamic feedback effects? Furthermore, firms thatprovide digital services often operate in a secondary marketin which other firms pay for advertising spots in their prod-uct. Improving one’s market share should in principle allow afirm to charge advertisers a higher price, but we do not knowto what extent this affects the analysis of the equilibria of thegame. Incorporating advertiser behavior would greatly com-plicate the model but provide potentially interesting results.
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A Omitted Proofs
Proof of Lemma 5.
Define u = m /n and v = m /n . Then u > v and we also have that C = (1 + u ) b (1 + u ) b + v b − u b u b + v b ,D = (1 + v ) b u b + (1 + v ) b − v b u b + v b . Next let U = u b , V = v b , W = (1 + u ) b − u b and Z =(1 + v ) b − v b . Notice that W > and Z > for all b > .Algebraic manipulations show that C < D ⇔ (1 + u ) b (1 + u ) b + v b − u b u b + v b < (1 + v ) b u b + (1 + v ) b − v b u b + v b ⇔ U + WU + W + V − V + ZV + Z + U < U − VU + V ⇔ V W ( U + V + Z ) < UZ ( U + V + W ) . Fix a pair of ( u, v ) with u > v , there are two cases toconsider : (1) W ≥ Z and (2) W < Z .In case (1) U + V + Z ≤ U + V + W . Notice that V W
V W < UZ = ⇒ V W ( U + V + W ) < UZ ( U + V + W )= ⇒ V W ( U + V + Z ) < UZ ( U + V + W ) . These two cases correspond to b (cid:62) and < b < , but thiscorrespondence is irrelevant. hich is the last condition in the chain of double implications.In fact, V W < U Z does hold, because u > v = ⇒ v (1 + u ) < u (1 + v )= ⇒ v b (1 + u ) b < u b (1 + v ) b = ⇒ v b ((1 + u ) b − u b ) < u b ((1 + v ) b − v b )= ⇒ V W < UZ.
Now we turn to the second case. Suppose
W < Z . Weagain must show that
V W ( U + V + Z ) < U Z ( U + V + W ) .Notice that the following implication holds. V W ( U + V + Z ) < UW ( U + V + W ) = ⇒ V W ( U + V + Z ) < UZ ( U + V + W ) , since UW ( U + V + W ) < UZ ( U + V + W ) . Hence we show that the first inequality is true. Note that
V W ( U + V + Z ) < UW ( U + V + W ) ⇐⇒ V ( U + V + Z ) < U ( U + V + W ) ⇐⇒ V ( V + Z ) < U ( U + W ) ⇐⇒ v b (1 + v ) b < u b (1 + u ) b , which is trivially true by u > v and b > .Hence in both cases, we have that V W ( U + V + Z ) < UZ ( U + V + W ) = ⇒ C < D, concluding the proof. Finally, we note that symmetry yieldsthe corresponding claim m > m = ⇒ D < C . Proof of Lemma 6.
The left hand sides of the equations char-acterizing the mixed equilibrium in the statement of Theo-rem 2 have the form c/ (2(1 − c )) which is increasing in c when c ∈ (0 , . If we call the left hand side of either ofthese equations (cid:96) , we can solve for c = 2 (cid:96)/ (1 + 2 (cid:96) ) which isalso monotonically increasing in (cid:96) over (cid:96) ∈ [0 , . Hence, toanalyze the monotonicity of the c , it is suffices to analyze themonotonicity of (cid:96) . But since (cid:96) must also equal the right handside, it suffices to analyze the monotonicity of the right handsides of these equations.Since p does not appear in b, C or D , it is easy to see thatboth fractions ( p − D ) / ( A − p ) and ( p − C ) / ( A − p ) increaseas p increases. Hence, both α and β increase as p increasesin the regime p ∈ (max { C, D } , A ) . The parameter n on theother hand appears in A , C and D . All of these parametersincrease as n increases. It is then similarly easy to see thatboth fractions ( p − D ) / ( A − p ) and ( p − C ) / ( A − p ) decreaseas n increases. Proof of Lemma 7.
We claim that the ordinal preferences ofFirm 1 over the outcomes are as follows. u ( NB, NB ) ≥ u ( B, NB ) ≥ u ( B, B ) ≥ u ( NB, B ) . Since u ( B, B ) = ( u ( B, NB ) + u ( NB, B )) / , it suffices toshow that u ( NB, NB ) ≥ u ( B, NB ) ≥ u ( NB, B ) . We first show that u ( N B, N B ) ≥ u ( B, N B ) . p − C ≥ ⇒ p − ( m + n ) b ( m + n ) b + m b + m b m b + m b ≥ ⇒ m b m b + m b ≥ ( m + n ) b ( m + n ) b + m b − p = ⇒ u ( NB, NB ) ≥ u ( B, NB ) . We then show that u ( B, NB ) ≥ u ( NB, B ) . A − p ≥ ⇒ ( m + n ) b ( m + n ) b + m b − m b m b + ( m + n ) b − p ≥ ⇒ ( m + n ) b ( m + n ) b + m b − p ≥ m b m b + ( m + n ) b = ⇒ u ( B, NB ) ≥ u ( NB, B ) . Moreover, the ordinal preferences of Firm 2 are as follows. u ( NB, NB ) ≥ u ( NB, B ) ≥ u ( B, B ) ≥ u ( B, NB ) . Again note that u ( B, B ) = ( u ( N B, B ) + u ( B, N B )) / .So it suffices to show that u ( N B, N B ) ≥ u ( N B, B ) ≥ u ( N B, B ) .We first show that u ( N B, N B ) ≥ u ( N B, B ) . p − D ≥ ⇒ p − ( m + n ) b m b + ( m + n ) b + m b m b + m b ≥ ⇒ m b m b + m b ≥ ( m + n ) b m b + ( m + n ) b − p = ⇒ u ( NB, NB ) ≥ u ( NB, B ) . We wrap up by showing that u ( N B, B ) ≥ u ( B, N B ) . A − p ≥ ⇒ ( m + n ) b ( m + n ) b + m b − m b m b + ( m + n ) b − p ≥ ⇒ (cid:18) − m b ( m + n ) b + m b (cid:19) − (cid:18) − ( m + n ) b m b + ( m + n ) b (cid:19) − p ≥ ⇒ ( m + n ) b m b + ( m + n ) b − p ≥ m b ( m + n ) b + m b = ⇒ u ( NB, B ) ≥ u ( B, NB ) . Proof of Theorem 3.
Let α and β be as in Theorem 2. Firstobserve that by rewriting the conditions for the mixed equi-librium, we get that α = 2 ( p − D ) A + p − D and β = 2 ( p − C ) A + p − C .
Then in the mixed equilibrium, the four outcomes occur withthe following probabilities:1. ( B, N B ) with probability α (1 − β ) ,2. ( N B, B ) with probability (1 − α ) β ,3. ( N B, N B ) with probability (1 − α )(1 − β ) , and4. ( B, B ) with probability αβ .The expected change in Firm 1’s market share can be calcu-lated by summing over the change in its market share in eachoutcome (see the proof of Lemma 8), weighted by how of-ten the outcome occurs in the mixed equilibrium. Thus theexpected change in Firm 1’s market share is α (1 − β ) C + (1 − α ) βD + (1 − α ) (1 − β ) 0 + αβ C − D
2= 2 (cid:18) ( C − D ) ( p ( A − C − D ) + CD )( A + p − C ) ( A + p − D ) (cid:19) . ince we are only interested in whether the market share in-creases or decreases, we only care about the sign of the aboveterm. The denominator is always positive, as both terms inthe denominator are positive when p ∈ (max { C, D } , A ) . Soit suffices to show that the numerator is non-negative.When m = m then the first term in the numerator iszero, so the expected market share is the same as the initialmarket share. On the other hand when m > m , Lemma 5implies that the first term in the numerator is negative. Weclaim that the second term in the numerator is always posi-tive. To see this, first observe that p ( A − C − D ) + CD isa linear function of p and it is strictly positive at both endpoints p = max { C, D } = C (by Lemma 5) and p = A . Bythe properties of linear functions, the term is positive for allvalues of p between max { C, D } and A , which is exactly theregime we are interested in. Proof of Lemma 8.
The change in the market share of Firm1 in strategy profiles ( B, N B ) compared to the beginning ofthe game is the parameter C , which is always positive. Sim-ilarly, the change in the market share of Firm 2 in strategyprofile ( N B, B ) compared to the beginning of the game isthe parameter D . The change in the market share of Firm 1in this strategy profile is − D since the sum of market sharesis always one, and since D is always positive, − D is nega-tive. Moreover, the expected change in Firm 1’s market sharefor ( B, B ) is ( C − D ) / because we decide which firm pur-chases the data by a fair coin toss. By Lemma 5, D ≥ C , so ( C − D ) / ≤ . Finally, there is no change in the marketshares in the outcome ( N B, N B ) . Thus, only for ( B, N B ) does Firm 1’s market share strictly increase. Proof of Theorem 4.
We first simplify the consumer welfarefor a strategy profile s by shorthands err ≡ err ( m ( s )) , err ≡ err ( m ( s )) , µ ≡ µ ( s ) and µ ≡ µ ( s ) . CW ( s ) = µ (1 − err ) + µ (1 − err )= err b err b + err b (1 − err ) + err b err b + err b (1 − err )= 1 − err b err + err b err err b + err b . So the strategy profile that maximizes the social welfare of theconsumers equivalently maximizing the following equation max s CW ( s ) ≡ max s err b + err b err b err + err b err . We take the following three steps to prove the statementof the theorem: (1) CW ( B, N B ) > CW ( N B, B ) , (2) CW ( B, B ) = ( CW ( N B, B ) + CW ( B, N B )) / and (3) CW ( N B, N B ) < CW ( s ) for all s (cid:54) = ( N B, N B ) . For sim-plicity in the rest of the proof we assume that the error scaleswith the square root of the number of data points.To prove part (1), first, observe that ( m + n ) ( b − / m b/ (cid:0) √ m + n − √ m (cid:1) − ( m + n ) ( b − / m b/ (cid:0) √ m + n − √ m (cid:1) > , since ( m + n ) ( b − / m b/ > ( m + n ) ( b − / m b/ , when m > m and also (cid:0) √ m + n − √ m (cid:1) > m b/ (cid:0) √ m + n − √ m (cid:1) , since the function f ( w ) = √ w + b − √ w is increasing in w when b > . Adding a positive term to the expression above,we get that ( m + n ) ( b − / m b/ (cid:0) √ m + n − √ m (cid:1) − ( m + n ) ( b − / m b/ (cid:0) √ m + n − √ m (cid:1) + (cid:16) ( m + n ) ( b − / ( m + n ) ( b − / − m b − / m b − / (cid:17) × ( √ m − √ m ) > ⇒ ( m + n ) b/ m b − / + ( m + n ) b/ ( m + n ) ( b − / + m b − / m b/ + ( m + n ) ( b − / y b/ > ( m + n ) ( b − / m b/ + ( m + n ) ( b − / ( m + n ) b/ + m b/ m b − / + ( m + n ) b/ m b − / = ⇒ ( m + n ) b/ + m b/ ( m + n ) ( b − / + m b − / > m b/ + ( m + n ) b/ m b − / + ( m + n ) ( b − / = ⇒ ( m + n ) − b/ + m − b/ ( m + n ) − b/ √ m + m − b/ (cid:112) ( m + n ) > m − b/ + ( m + n ) − b/ m − b/ √ m + n + ( m + n ) − b/ √ m = ⇒ CW ( B, NB ) > ( NB, B ) . In the penultimate step we multiplied the numerator and de-nominator of the left and right hand fractions by ( m + n ) − b/ m − b/ and ( m + n ) − b/ m − b/ , respectively.Part (2) is trivial given that in our model when bothplayers choose B one of the strategy profiles ( B, N B ) and ( N B, B ) is chosen with equal probability. Since CW ( B, N B ) > CW ( N B, B ) by part (1) then part (2) im-plies that CW ( B, N B ) > CW ( B, B ) > CW ( N B, B ) . Toprove part (3), we show that CW is increasing in m . Thisimplies that CW ( N B, B ) > CW ( N B, N B ) . To show thatthe function CW is increasing in m we show that the partialderivative with respect to m is positive. ∂CW ( NB, NB ) ∂m = ∂∂m (cid:18) m − b/ + m b − / m − b/ √ m + m − b/ √ m (cid:19) = (cid:16) m − b/ √ m + m − b/ √ m (cid:17) − × (cid:16) − b m − ( b +1) / (cid:16) ( m − b/ √ m + m − b/ √ m (cid:17) + b (cid:16) ( m − b/ + m − b/ (cid:17) + x − / (cid:16) m − b/ + m − b/ (cid:17) (cid:17) = 12 m ( m − b/ √ m + m − b/ √ m ) × (cid:16) bm b (cid:18) − √ m (cid:19) + bm − b/ y − b/ (cid:18) − √ m (cid:19) + 1 √ m (cid:16) m − b/ + m − b/ (cid:17) (cid:17) > , which is positive since both m ≥ and m ≥1