aa r X i v : . [ ec on . T H ] O c t Naive Analytics Equilibrium (Preliminary Draft)Ron Berman ∗ Yuval Heller † October 30, 2020
Abstract
We study interactions with uncertainty about demand sensitivity. In our solutionconcept (1) firms choose seemingly-optimal strategies given the level of sophistica-tion of their data analytics, and (2) the levels of sophistication form best responsesto one another. Under the ensuing equilibrium firms underestimate price elasticitiesand overestimate advertising effectiveness, as observed empirically. The misestimatescause firms to set prices too high and to over-advertise. In games with strategic com-plements (substitutes), profits Pareto dominate (are dominated by) those of the Nashequilibrium. Applying the model to team production games explains the prevalence ofoverconfidence among entrepreneurs and salespeople.
Keywords : Advertising, pricing, data analytics, strategic distortion, strategic com-plements, indirect evolutionary approach.
JEL Classification : C73 , D43, M37.
Researchers often assume that better measurement and accurate estimations of the sensi-tivity of demand allow firms to improve their advertising and pricing decisions. Arriving at ∗ The Wharton School, University of Pennsylvania. email: [email protected] † Department of Economics, Bar-Ilan University. email: [email protected]. The author is grateful tothe European Research Council for its financial support (ERC starting grant
Highlights of the Model
We present a model where the payoffs of competing players(firms) each depend on her actions and on her demand, where the demand depends on theactions of all players. The players do not know the demand function, but can select actionsand observe the realized demand. The game has two stages. In stage 1 each player hires a(possibly biased) analyst to estimate the sensitivity of demand. An analyst may under- orover-estimate the sensitivity of demand. In stage 2 each player chooses an action taking the2stimate as the true value.Our solution concept, called a Naive Analytics Equilibrium (NAE), is a profile of an-alysts’ biases and a profile of actions, such that (1) each action is a perceived best-replyto the opponents’ actions, given the biased estimation, and (2) each bias is a best-reply tothe opponents’ biases in the sense that if a player deviates to another bias this leads to anew second-stage equilibrium, in which the deviator’s (real) profit is weakly lower than theoriginal equilibrium payoff. The first-stage best-replying is interpreted as the result of agradual process in which firms hire and fire analysts from a heterogeneous pool, and eachfirm is more likely to fire its analyst if its profit is low.
Summary of Results
Our model is general enough to capture price competition withdifferentiated goods (where the goods can be either substitutes or complements), as well asadvertising competition (where the advertising budget of one firm has either a positive or anegative externality on the competitors’ demand). Our results show that firms hire biasedanalysts in any naive analytics equilibrium, and that the direction of the bias is consistentwith the empirically observed behavior of firms: in price competition firms hire analyststhat under-estimate price elasticities, and in advertising competition firms hire analysts whoover-estimate the effectiveness of advertising.We also show that there is a Pareto-domination relation between the naive analyticsequilibrium and the Nash equilibrium (of the game without biases), where its directiondepends on the type of strategic complementarity. In a game with strategic complements(i.e., price competition with differentiated goods) the naive analytics equilibrium Paretodominates the Nash equilibrium, while the opposite holds in a game with strategic substitutes(i.e., advertising competition with negative externalities). The intuition is that in a gamewith strategic complements (resp., substitutes), each player hires a naive analyst that inducesa biased best reply in the direction that benefits (resp., harms) the opponents. This is sobecause these biases have a strategic advantage of inducing the opponents to change theirstrategies in the same (resp., opposite) direction, which is beneficial to the player.Next, we analyze a standard functional form in each type of competition, and show thateach price/advertising competition admits a unique naive analytics equilibrium, and that3he competing firms choose an equal level of biasedness that depends on the parameters ofthe competition.Finally, we demonstrate that our model can be applied in more general settings. Specif-ically, in Section 5.3 we apply the model to a game of team production with strategic com-plementarity (see, e.g., Holmstrom, 1982; Cooper and John, 1988). We present two testablepredictions in this setup: (1) players are overconfident in the sense of overestimating theirability to contribute to the team’s output, and (2) players contribute more than in the (un-biased) Nash equilibrium. These predictions are consistent with the observable behavior ofentrepreneurs and salespeople, who often exhibit overconfidence.
Related Literature and Contribution
From a theoretical aspect, our methodology ofstudying a two-stage auxiliary game where each firm is first endowed with a biased analystand then chooses her pricing/advertising level given results of the analysis is closely re-lated to the literature on delegation (e.g., Fershtman and Judd, 1987; Fershtman and Kalai,1997; Dufwenberg and Güth, 1999; Fershtman and Gneezy, 2001) . The delegation litera-ture shows that in price competition, firm owners would design incentives for managers thatencourage the managers to maximize profits as if the marginal costs are higher than theirtrue value (see, in particular, Fershtman and Judd, 1987, p. 938).Our model contributes to this literature but also differs from it in a few importantaspects. First, in our setup the incentives of all agents are aligned and are based solely onthe firm’s profit. A deviation of the firm from profit-maximizing behavior is due to (non-intentional) naive analytics, rather than due to explicitly distorting the compensation ofthe firm’s manager. Our novel mechanism is qualitatively different (as it relies on biasedestimations rather than different incentives), and it induces testable predictions and policyimplications which are different than those induced by delegation (as further discussed inRemark 1). Second an important merit of our model is its generalizability to a wide varietyof phenomena and its applicability to wide class of games. The concept of biased estimationof sensitivity of demand can be applied in many seemingly-unrelated setups (e.g., price See also the related literature on the “indirect evolutionary approach” (e.g., Güth and Yaari,1992; Heifetz and Segev, 2004; Dekel, Ely, and Yilankaya, 2007; Heifetz, Shannon, and Spiegel, 2007;Herold and Kuzmics, 2009; Heller and Winter, 2020).
Structure
Section 2 presents a motivating example. In Section 3 we describe our modeland solution concept. Our main results are presented in Section 4. Section 5 analyzes threeapplications: price competition, advertising competition, and team-production game. Themain text contains proof sketches and formal proofs are relegated to the appendix.
Consider two firms i ∈ { , } each selling a product with price x i ∈ R + . The products aresubstitute goods. The demand of firm i ∈ { , } at day t is given by: q it ( x i , x − i ) = max (20 − x i + 0 . x − i + z it , , with z it ∼ ǫ . − ǫ . − i denotes the other firm. That is, the expected demand follows Bertrand competitionwith differentiated goods, and the daily demand of each firm has a random i.i.d demandshock, with value either ǫ or − ǫ with equal probability. We assume that the marginal costsare zero, which implies that the profit of each firm is given by its revenue π it ( x i , q it ) = x i · q it .The firm managers do not know their demand functions, and they hire analysts to es-timate the sensitivity of demand to price, in order to find the optimal price. The analystat each firm asks the firm’s employees to experiment for a couple of weeks with offering adiscount of ∆ x in some of the days, and uses the average change in demand ∆ q betweendays with and without the discount to estimate the elasticity of demand.Importantly, the firm’s employees do not choose the days with discounts uniformly. Theemployees observe in each morning a signal that reveals the demand shock (say, the dailyweather), and they implement discounts on days of low demand, possibly due to the employ-ees having more free time in these days to deal with posting the discounted price.There are two types of analysts: naive and sophisticated. A naive analyst does notmonitor in which days the employees choose to give a discount, and he implicitly assumesin his econometric analysis that the environment is the same in the days with discounts asin those without discounts. In contrast, sophisticated analysts either monitor the discountdecisions to enforce uniform distribution of discounts, or correct the correlation between theweather and discounts in their econometric analysis (e.g., by controlling for the weather).A sophisticated analyst correctly estimates the mean change in demand∆ q = (20 − x i + 0 . x − i ) − (20 − ( x i − ∆ x ) + 0 . x − i ) = − ∆ x, and thus he accurately estimates the elasticity of demand η i = − x i q i ∆ q ∆ x = − x i q i ( − ∆ x )∆ x = x i q i . In contrast, a naive analyst under-estimates the mean change in demand to be:∆ q sloppy = (20 − x i + 0 . x − i + ǫ ) − (20 − ( x i − ∆ x ) + 0 . x − i − ǫ ) = − ∆ x + 2 ǫ, η i, naive = x i q i (∆ x − ǫ )∆ x ≡ x i q i α i . Assume, for example, that the parameters ∆ x and ǫ are such that α i = (∆ x − ǫ )∆ x = 0 . α \ α α \ α α \ α In this section we introduce an analytics game in which competing firms hire analysts toestimate the sensitivity of demand, which is then used to determine the strategic choices ofthe firm; importantly, the demand of each firm is also affected by the strategic choices of itscompetitors. Next we present our solution concept of a naive analytics equilibrium. An analytics game is a two stage game in which each of N = { , , ..., n } players (firms)hire an analyst who estimates the sensitivity of demand in the first stage and then make astrategic choice that affects demand in the second stage. We first describe the structure ofthe second stage, which we call the underlying game and denote by G = ( N, X, q, π ). In theunderlying game each firm i ∈ N makes a strategic choice x i ∈ X i that affects the demandsand the profits of all firms, where X i ⊆ R is a (possibly unbounded) interval of feasiblechoices of firm i . Let X = Q i ∈ N X i be the Cartesian product of these intervals.Let − i ≡ N \ { i } denote all firms except firm i and − ij ≡ N \ { i, j } denote all firmsexcept i and j . Let ( x ′ i , x − i ) denote the strategy profile, in which player i plays strategy x ′ i ,while all other players play according to the profile x − i (and we apply a similar notationfor x − ij ). Let q i ( x ) denote the demand of firm i . The (true) payoff, or profit, of each firm i ∈ N , denoted by π i ( x i , q i ( x )), depends on the firm’s demand q i ( x ) and its strategic choice x i . We assume that the demand functions q i ( x ) and payoff functions π i ( x i , q i ) of all firmsare continuously twice differentiable in all parameters. We further assume that the profit isincreasing in demand, i.e., ∂π i ( x i ,q i ) ∂q i > x i ∈ X i and any q i .8e assume that the payoff function of each player is unimodal. Assumption 1 (Unimodality ) . For each player i and each profile x − i ∈ X − i , there exists x ∗ i ∈ X i such that dπ i ( x i ,x − i ) ∂x i > for any x i < x ∗ i and dπ i ( x i ,x − i ) dx i < for any x i > x ∗ i . Unimodality implies that any opponents’ profile x − i ∈ X − i has a unique best-reply, whichwe denote by BR i ( x − i ) = argmax x i ∈ X i ( π i ( x i , x − i )).Next we assume that the player’s strategy influences the demand and the payoff (given afixed demand) in opposing directions. In the motivating example the strategic choice is theprice, which increases the profit per product sold, while decreasing the firm’s demand. Assumption 2 (Opposing effects) . ∂π i ( x i ,q i ) ∂x i · ∂q i ( x ) ∂x i < for any x i ∈ X i and any q i . Let
Int ( X i ) denote the interior of X i . A necessary (and due to unimodality, also suffi-cient) condition for a strategy x i ∈ Int ( X i ) to be a best reply to the opponents’ strategyprofile is that it satisfies the following first-order condition: dπ i ( x i , q i ( x )) dx i = ∂π i ∂x i |{z} ( i ) + ∂π i ∂q i |{z} ( ii ) · ∂q i ∂x i |{z} ( iii ) = 0 (3.1)Sections 5.1–5.2 present two applications of this model. The first application generalizesthe motivating example of price competition. The second application is for advertisingcompetition where the strategic choice of each firm is its advertising spending. In this subsection we describe the first stage of the analytics game, in which each firm choosesan analyst to estimate its demand.In order to maximize their profits when choosing x i , firms need to know or estimatethe impact of their actions on their profits. We assume that each firm knows (or correctlyestimates): (i) the direct effect of its strategy on its profit ∂π i ∂x i ; and (ii) the effect of thefirm’s demand on its profit ∂π i ∂q i . In contrast, we assume it is difficult for the firm to estimate(iii) the response of its demand to marginal changes in its strategy, i.e., to estimate ∂q i ∂x i .For example, during price competition firms know how their product’s prices affect their9rofit margins and how demand affects profit, but might not know how sensitive consumersmight be to price changes. Similarly, in advertising competition firms know how increasingadvertising spending affects their bottom line costs, but might not know the impact of theiradvertising on demand. Each firm therefore hires an analyst in the first stage who is taskedwith estimating ∂q i ∂x i .Let A ⊆ R ++ denote the interval of feasible biases of analysts, and we assume that A includes an open interval around 1. Analysts are characterized by a bias α i ∈ A ⊆ R ++ thatcauses them to estimate the marginal effect of the strategy x i on demand q i as α i ∂q i ∂x i insteadof ∂q i ∂x i . We denote the bias profile of all analysts by α = ( α , . . . , α n ). A sophisticated analystis unbiased, while naive analysts have α i = 1. Consequently, an α i -analyst induces the firmto choose a strategy x i that solves the biased first-order condition ∂π i ∂x i + ∂π i ∂q i · α i · ∂q i ∂x i = 0instead of the unbiased condition in (3.1).There are many reasons why analysts might be biased. One example is inadvertentlycreating endogenous correlation between the firm’s strategy and demand without taking thiscorrelation into account in the analysis. If, as in the motivating example, a firm sets lower saleprices on days of low demand and higher regular prices on days of high demand, estimatingprice elasticities using the resulting data will show that consumers are less price sensitive thanthey actually are. Another example is when firms set their advertising budgets differently inspecific times such as before holidays, or weekends. This would create correlation in the levelsof advertising with those of competitors. Ignoring this correlation during analysis may leadto a biased estimate of advertising effectiveness. We present micro-foundations for biasedanalytics towards the end of Section 5.1 (price competition) and Section 5.2 (advertising). α -Equilibrium In what follows we define the ratio of indirect effect to direct effect (RIDE), and use thisnotion to define an equilibrium of the second-stage, given the analysts’ biased profile.The ratio of the indirect marginal effect to the direct marginal effect (henceforth,
RIDE )10f a firm’s strategy on profit is defined as follows:RIDE i ( x ) ≡ − ∂π i ∂q i ( x ) · ∂q i ∂x i ∂π i ∂x i (3.2)Assumption 2 (opposing effects) implies that RIDE i ( x ) is positive and well-defined. Whena firm changes its strategy, it influences its profit through two channels: the direct effect onprofit ∂π i ∂x i and the indirect effect through the influence on the demand ∂π i ∂q i ( x ) · ∂q i ∂x i . RIDE i measures the ratio between the indirect effect and the direct effect. We note that RIDE i is aunitless measure, and that it coincides with the elasticity of demand | η q i ,x i | when the firm’sprofit is given by the multiplication π i ( x ) = x i · q i ( x ), as in the the motivating example (seeExample 1 in Section 4.1). If the profit function can be written as the difference betweenrevenues R and (demand-independent) costs C , as in π i ( x ) = R ( q i ( x )) − C ( x i ), then RIDE i isthe ratio of marginal revenues to marginal costs. This is the case, for example, in advertisingcompetition where advertising affects revenue only through demand (Section 5.2).Using the definition of RIDE i we observe that the standard (interior) Nash equilibriumsolution to (3.1) is equivalent to solving:0 = dπ i ( x i , q i ( x )) dx i = ∂π i ∂x i + ∂π i ∂q i · ∂q i ∂x i ⇔ RIDE i ( x ) = 1 . When analysts may be biased, we define an α - equilibrium as a strategy profile such thateach firm’s biased first order condition holds: ∂π i ∂x i + ∂π i ∂q i · α i · ∂q i ∂x i ! = 0 ⇔ RIDE i ( x ) = 1 α i . (3.3) Definition 1 ( α − Equilibrium) . Fix a biasedness profile α ∈ A n . A strategy profile x is an α - equilibrium if RIDE i ( x ) = α i for each player i .An implication of biasedness is that an analyst with α i < x i that creates a relatively larger indirect effect, while when α i >
1, the analystwill cause the firm to set x i to have a relatively larger direct effect when comparing to the(unbiased) profit-maximizing value of x i . 11 .4 Naive Analytics Equilibrium (NAE) In what follows we define our main solution concept of a naive analytics equilibrium. Tosimplify the definition and exposition, we assume that the underlying game G has a unique α -equilibrium for every biasedness profile α . Assumption 3 (Uniqueness) . For each α ∈ A n there exists a unique α -equilibrium, whichwe denote by x ( α ) ∈ X . Assumption 3 is satisfied in various economic applications, including price and advertisingcompetitions presented in Sections 5.1 and 5.2. Due to the unimodality assumption theunique ~ x NE ≡ x (cid:16) ~ (cid:17) . In Appendix B we present conditions on the RIDEs of the players thatimply the existence of a unique α -equilibrium.Assumption (3) allows us to define the underlying-game’s payoff of a biasedness profile α ∈ R + n as ˜ π i ( α ) ≡ π i ( x ( α )), which is the payoff of firm i when all firms follow the unique α -equilibrium. In particular, ˜ π i (cid:16) ~ (cid:17) is the payoff of player i in the unique Nash equilibriumof the underlying game.A naive analytics equilibrium is a bias profile and a strategy profile, where (1) the strategyprofile is an α -equilibrium, and (2) each bias is a best-reply to the opponents’ bias profile(i.e., a unilateral deviation to another bias would induce a new α -equilibrium with a lowerpayoff to the deviator). Formally, Definition 2. A naive analytics equilibrium is a pair ( α ∗ , x ∗ ), where:1. x ∗ ∈ X n is the α ∗ -equilibrium of the underlying game G (i.e., x ∗ = x ( α ∗ )).2. ˜ π i (cid:16) α ∗ i , α ∗− i (cid:17) ≥ ˜ π ( α ′ i , α ∗− i ) for each player i and each α ′ i ∈ A . We do not interpret the equilibrium behavior in the first-stage as the result of an explicitmaximization of sophisticated firms who know the demand function and calculate the optimal α -s for their analysts. Conversely, we think of the firms as having substantial uncertainty12bout the demand function and its dependence on the behavior of various competitors. Dueto this uncertainty, the firms hire analysts to estimate the sensitivity of demand. Occasionally(say, at the end of each year) firms consider replacing the current analyst with a new one(say, with a new random value of α i ), and a firm is more likely to do so the lower its profitis. Gradually, such a process would induce the firms to converge to hiring most of the timeanalysts with values of α that are best replies to each other, and thus constitute a naiveanalytics equilibrium ( α, x ).It is important to note that the observed data does not contradict the optimality ofthe strategic choices of the firms or the correctness of the estimations of the sensitivityof demand of the analysts. Consider, for example, a naive analytics equilibrium ( α, x )in the price competition described in Section 2. A firm that wishes to confirm that itsprice is indeed optimal (i.e., that it maximizes its profit given the demand) is likely toexperiment with temporary changes in prices to see its influence on demand. Under thearguably plausible assumption that the analysis of such an experiment will be done withthe same level of sophistication as the one leading to ( α, x ), the firm’s conclusion from theexperiment would be that the sensitivity of demand is exactly as estimated by the firm’snaive analyst, and that the firm’s equilibrium strategy is optimal (e.g., it induces elasticityof − Remark . An alternative interpretation of our model (which wedo not use in the paper) is of delegation. Let π α i i : X → R be a subjective payoff functionsuch that maximizing π α i i with an unbiased estimation is equivalent to maximizing π i witha biased estimation of α i , i.e., for any strategy profile x ∈ X RID E i ( x ) = 1 α i iff π α i i ( x ) = argmax x ′ i ∈ X i π α i i ( x ′ i , x − i ) . Let Π Ai = { π α i i | α i ∈ A } be the set of all such subjective payoff function. One can reinterpreta naive analytics equilibrium as an equilibrium of a delegation game (Fershtman and Judd,1987) in which in the first stage each firm’s owner simultaneously chooses a payment scheme13o its manager, which induces the manager with a subjective payoff function in Π Ai . In thesecond stage the managers play a Nash equilibrium of the game induced by the subjectivepayoffs. Although, both interpretations (namely, naive analytics and delegation) yield thesame prediction about the equilibrium strategy profile, they differ in other testable predic-tions. For concreteness, we focus the comparison for price competition (as in the motivatingexample). The delegation interpretation predicts firms to correctly estimate the elasticityof demand and to pay managers a payoff that increases in the firm profit and decreases inthe firm’s sales (see, Fershtman and Judd, 1987, p. 938). It is seldom observed that firmsencourage managers to decrease the firm’s sales. The naive analytics interpretation predictsthat firms will hire naive analysts that over-estimate elasticity of demand, with a manager’spayoff scheme that depends on the firm’s profit (and is not a decreasing function of thefirm’s sales). As mentioned elsewhere in the paper, we believe this latter prediction is moreconsistent with the empirically observed behavior of firms. We answer 3 questions about firms in a naive analytics equilibrium: (1) what is their directionof deviation from an unbiased best reply to the opponents’ strategies (Section 4.2), (2) whendo they under or over estimate the sensitivity of demand through biased analytics (Section4.3), and (3) when do they achieve payoffs that Pareto dominate the Nash equilibrium(Section 4.3). Our results rely on assumptions of monotone derivatives, which are presentedin Section 4.1. In Section 4.4 we show a Stackelberg-leader representation of our results,which will be helpful in the applications in Section 5. A summary of the results is presentedin Table 2 in the end of the section.
Our next assumption requires the externality of a player’s strategy on her opponent’s payoff dπ i dx j to be monotone (i.e., either always positive or always negative). Assumption 4.
Monotone payoff externalities : dπ i ( x ) dx j are either all positive or all egative for every i = j ∈ N and every x ∈ X . Note that Assumption 4 is equivalent to requiring that demand externalities are monotone(i.e., dq i ( x ) dx j are either all positive or all negative) due to our Section 3.1’s assumptions: (1)an opponent’s strategy influences a player’s payoff only through the player’s demand, and(2) player’s payoff is increasing in her demand.Next we require that the externality of a player on her opponent’s RIDE is monotone. Assumption 5.
Monotone RIDE externalities : d ( RIDE i ( x )) dx j are either all positive or allnegative for every i = j ∈ N and every x ∈ X . Finally, we require that the RIDE of a player is monotone in her own strategy.
Assumption 6’.
Monotone RIDE : d (RIDE i ( x )) dx i are either all positive or all negative forevery i ∈ N and every x ∈ X .The following simple observation shows that the payoffs are unimodal iff the RIDE ex-ternality d (RIDE i ( x )) dx i has the same sign as the player’s direct effect on her payoff. Claim . Let G = ( N, X, q, π ) be an underlying game that satisfies Assumption 2 (opposingeffects) and Assumption 6’. Assume that for each player i and each opponents’ profile x − i ,there is x ∗ i ∈ X i such that RIDE i ( x ∗ i , x − i ) = 1. Then G satisfies Assumption 1 (unimodality)iff d (RIDE i ) dx i · ∂π i ∂x i > i ∈ N and every x ∈ X . Sketch of proof (proof is in Appx. A.2).
In this sketch we show that if d (RIDE i ) dx i , ∂π i ∂x i > dπ i dx i = 0 (resp., dπ i dx i < dπ i dx i >
0) iff ∂π i ∂x i = (cid:12)(cid:12)(cid:12) ∂π i ∂q i · ∂q i ∂x i (cid:12)(cid:12)(cid:12) (resp., ∂π i ∂x i < (cid:12)(cid:12)(cid:12) ∂π i ∂q i · ∂q i ∂x i (cid:12)(cid:12)(cid:12) , ∂π i ∂x i > (cid:12)(cid:12)(cid:12) ∂π i ∂q i · ∂q i ∂x i (cid:12)(cid:12)(cid:12) ), which holdsiff RIDE i ( x ) = 1 (resp., RIDE i ( x ) > RIDE i ( x ) < x i = x ∗ i (resp., x i > x ∗ i , x i < x ∗ i ), which implies unimodality.Claim 1 allows us to replace both Assumption 6’ (monotone RIDE) and Assumption 1(unimodality) with the following assumption that the RIDE is monotone and in the directionequivalent to unimodality. Formally The assumption that there is x ∗ i s.t. RIDE i ( x ∗ i , x − i ) = 1 can be omitted if one assumes compact X i -s. Appx. B presents conditions on the RIDE derivatives that imply unique α -equilibrium (Asm. 3). ssumption 6. Unimodal monotone RIDE : d ( RIDE i ) dx i has the same sign as ∂π i ( x i ,q i ) ∂x i (i.e., d ( RIDE i ) dx i · ∂π i ( x i ,q i ) ∂x i > ) for every i ∈ N and every x ∈ X . Assumptions 4–6 are satisfied in many economic applications (including price competitionand advertising competition, as detailed in the Sections 5.1–5.2).
Example 1 (Motivating example revisited) . Recall that in the motivating example the profitof firm i is π i = x i · q i and its expected demand is q i = 20 − x i + 0 . x j . Observe that theRIDE coincides with the elasticity of demand:RIDE i = − ∂π i ∂q i ( x ) · ∂q i ∂x i ∂π i ∂x i = − x i · ( − q i = x i q i = | η q i ,x i | . Further observe that the payoff externalities are positive: dπ i dx j = x i d (20 − x i + 0 . x j ) dx j = 0 . x i > , and the RIDE externalities and RIDE derivative are negative and positive, respectively: d RIDE i dx j = ddx j x i q i ! = − . x i q i < , d RIDE i dx i = ddx i x i q i ! = q i + x i q i > . Assumptions 4–6 map to eight combinations on the directions of the derivatives. Effec-tively, these eight combinations define four unique classes of games since relabeling strategiesas their negative values (i.e., replacing x i with − x i ) results in essentially the same game withopposite signs to each of three monotone derivatives (see, Fact 1 in Appendix A.3).Next we show that any game with monotone derivatives satisfies either strategic comple-mentarity or strategic substitutability. A game has strategic complements (resp., substitutes)if the players’ decisions reinforce (resp., offset) one another in the sense that increasing aplayer’s strategy increases (resp., decreases) the opponents’ best replies. Formally, Definition 3.
A game with a best-reply function has strategic complements ( resp., strate-gic substitutes ) if BR j ( x ′ i , x − ij ) > BR j ( x i , x − ij ) (resp., BR j ( x ′ i , x − ij ) < BR j ( x i , x − ij )) for Our definition of strategic complementarity/substitutability in term of the best-reply functionfollows Monaco and Sabarwal (2016). It is essentially equivalent in our setup to the commonly-used alternative definitions of increasing differences and the sign of the cross derivative (see, e.g.,Bulow, Geanakoplos, and Klemperer, 1985). i, j ∈ N , strategy profile x and strategy x ′ i > x i .The following observation shows that any game with monotone derivatives has eitherstrategic complements (if the two RIDE derivatives have the opposite signs) or strategicsubstitutes (if the two RIDE derivatives have the same sign). Formally, Claim . Let G be an underlying game that satisfies Assumptions 2 and 4–6. Then:1. The game has strategic complements iff d (RIDE j ) dx j · d (RIDE j ) dx i < d (RIDE j ) dx j · d (RIDE j ) dx i > Proof sketch for the motivating example in which d ( RIDE j ) dx i < < d ( RIDE j ) dx j (proof in Appx. A.4). Let x ′ i > x i . Negative RIDE externalities imply that RIDE j ( x ′ i , x − ij ) < RIDE j ( x i , x − ij ),which, in turn, implies (due to positive RIDE derivative) that BR j ( x ′ i , x − ij ) > BR j ( x i , x − ij ),which shows that G has strategic complements . G In this subsection we show that the direction in which agents deviate from an unbiased bestreply to the opponents’ strategies is the one that induces a beneficial reply by its opponents.We say that a player benefits from an upward commitment if increasing a firm’s ownstrategy induces best-replying opponents to change their strategies in the direction that isbeneficial to the firm. That is, in a game with positive payoff externalities a firm’s decisionto increase its strategy would elicit competitors to pick a higher strategy, while in a gamewith negative externalities, increasing a firm’s own strategy would induce competitors todecrease their strategy. Formally:
Definition 4.
A game with monotone payoff externalities has a beneficial upward (resp.,downward) commitment iff either:1. the game has positive (resp., negative) payoff externalities and BR j ( x ′ i , x − ij ) > BR j ( x − j )for each i, j ∈ N , each x ∈ X and each x ′ i > x i , or17. the game has negative (resp., positive) payoff externalities and BR j ( x ′ i , x − ij ) < BR j ( x − j )for each i, j ∈ N , each x ∈ X and each x ′ i > x i .In a game with beneficial upward commitment, if firm i were a Stackelberg leader itwould deviate from the simultaneous-game Nash equilibrium towards a higher strategy, sincethe competing firms will reciprocate with a deviation in the direction that benefits firm i .Similarly a game with beneficial downward commitment would entice a Stackelberg-leadingfirm to deviate from the simultaneous-game Nash equilibrium towards a lower strategy.Fix a strategy profile x . We say that player i under-replies if her strategy is lower thanthe unbiased ( α i = 1) best reply to the opponents’ strategies x − i , i.e., if x i < BR i ( x − i ) . We say that player i over-replies if x i > BR i ( x − i ). Our first result shows that in anynaive analytics equilibrium all players differ from (unbiased) best replying in the directionof beneficial commitment: Proposition 1.
Let G be an underlying game satisfying Assumptions 2–5. All agents:
1. over reply in any naive analytics equilibrium if G has a beneficial upward commitment;2. under reply in any NAE if G has a beneficial downward commitment. Sketch of proof.
If any player i differs from the (unbiased) best replying in the directionopposite of beneficial commitment, then this cannot be a naive analytics equilibrium. Thisis so because a deviation of player i to bias α i slightly closer to 1 must increase the deviator’spayoff (contradicting the profile being a NAE) because the deviation yields both a directadvantage (the new strategy is closer to best-replying) and a strategic advantage (the newstrategy has shifted in the direction that yields a beneficial commitment).Next, we characterize the direction of beneficial commitment in terms of the number ofpositive derivatives out of the three monotone derivatives of Assumptions 4–5. Specifically,the direction of beneficial commitment is upwards if the number of positive derivatives iseven, while it is downwards if the number of positive derivatives is odd: Claim . Let G be an underlying game that satisfies Assumptions 2 and 4–5. Then G :1. has a beneficial upward commitment iff the number of positive derivatives is even;18. has a beneficial downward commitment iff the number of positive derivatives is odd. Proof sketch for the motivating example ( d ( RIDE i ) dx j < < d ( RIDE i ) dx i , d ( π i ) dx j ); proof is in Appx. A.6. If a player increases her strategy, it decreases the opponents’ RIDEs (due to negative RIDEexternalities). Due to increasing RIDE derivative, it implies that all opponents j = i haveto increase their own strategies in order to maintain RIDE j = α j in the new naive ana-lytics equilibrium. The change of the opponents’ strategies increases player i ’s payoff dueto having positive externalities, which implies that the game has a beneficial downwardcommitment. Next we characterize the condition for analysts to either over estimate or under estimate thesensitivity of demand in any naive analytics equilibrium. Specifically, we show that1. All agents overestimate the sensitivity of demand (i.e., α i > i ) iffboth the RIDE externality and payoff externality have the same sign.2. All agents underestimate the sensitivity of demand (i.e., α i < i ) iffthe RIDE externality and payoff externality have different signs. Proposition 2.
Let G be an underlying game that satisfies Assumptions 2–5. Let ( α ∗ , x ∗ ) be a naive analytics equilibrium, and let i ∈ N . Then:1. α ∗ i > if d ( RIDE j ) dx i · dπ j dx i > .2. α ∗ i < if d ( RIDE j ) dx i · dπ j dx i < .Sketch of proof for the motivating example (in which d ( RIDE i ) dx j < < d ( RIDE i ) dx i , d ( π i ) dx j ). By Claim 3 the game has a beneficial commitment advantage. By Proposition 1 all agentsover reply in any naive analytics equilibrium ( α ∗ , x ∗ ) (i.e., x ∗ i > BR (cid:16) x ∗− i (cid:17) ). Due to the RIDEderivative being positive ( d (RIDE i ) dx i > α ∗ i = RIDE i ( x ∗ ) > ⇒ α ∗ i < Definition 5.
Strategy profile x is symmetric if x i = x j for any pair of players i, j ∈ N . Proposition 3.
Let G be an underlying game that satisfies Assumptions 2–5. Let ( α ∗ , x ∗ ) be a naive analytics equilibrium.
1. If G has strategic complements, then x ∗ i > x NEi and π i ( x ∗ ) > π i (cid:16) x NE (cid:17) for each i ∈ N .2. If G has strategic substitutes, and x ∗ and x NE are symmetric profiles, then x ∗ i < x NEi and π i ( x ∗ ) < π i (cid:16) x NE (cid:17) for each player i ∈ N . Sketch of proof for the motivating example (in which d ( RIDE i ) dx j < < d ( RIDE i ) dx i , d ( π i ) dx j ). The fact that ( α ∗ , x ∗ ) is a naive analytics equilibrium implies that π i ( x ∗ ) = π i ( x ( α ∗ )) ≥ π i (cid:16) x (cid:16) , α ∗− i (cid:17)(cid:17) . The fact that x i (cid:16) , α ∗− i (cid:17) is the unbiased best reply of player i implies that π i (cid:16) x (cid:16) , α ∗− i (cid:17)(cid:17) > π i (cid:16) x NEi , x − i (cid:16) , α ∗− i (cid:17)(cid:17) . Due to Proposition 1 all players over reply in ( α ∗ , x ∗ ) (i.e., x ∗ i > BR i (cid:16) x ∗− i (cid:17) ). This impliesthat all players j = i over reply also in x (cid:16) , α ∗− i (cid:17) (because they have the same values of α ∗ j in both profiles). In games with strategic complements this observation implies that x j (cid:16) , α ∗− i (cid:17) > x NEj for each player j ∈ N (as formalized in Lemma 3 in Appendix A.8). Finally, The fact that the game has positive payoff externalities implies that π i (cid:16) x NEi , x − i (cid:16) , α ∗− i (cid:17)(cid:17) > π i (cid:16) x NE (cid:17) . Combining the three inequalities we obtain π i ( x ∗ ) > π i (cid:16) x NE (cid:17) .Table 2 summarizes the results presented so far in Section 4. Games with strategic substitutes do not have a property analogous to Lemma 3. Due to this ourstatement of Pareto domination for games with strategic substitutes holds only for symmetric profiles. d Π i dx j RIDEextr. d ( RIDE i ) dx j RIDEderiv. d ( RIDE i ) dx i Direction ofbeneficialcommit-ment Under /Overestimation Strategicsubstitutes/complementsPrice competitionw. subs. goods(motiv. example) + - +
Beneficialupward α ∗ i < x ∗ i > x NEi ∀ i Advertising withpositive exter. &team production + + - commitment α ∗ i > - + + over replyin any α ∗ i < - - - NAE. α ∗ i > An interesting representation of the NAE, which will prove useful in the applications inthe next section, is to characterize it as an α -equilibrium in which each player plays her(unbiased) optimal Stackelberg-Leader strategy. The representation holds in a general setup,without relying on the assumptions of monotone derivatives of Section 4.1.We begin by defining an α − i -equilibrium given a fixed action of player i . Definition 6.
Fix player i ∈ N , strategy x i ∈ X i , and a bias profile of the remainingplayers α − i ∈ A n − . A profile of the remaining players x − i ∈ X − i is an α − i -equilibrium ifRIDE j ( x ) = α j for each player j = i .We assume that each strategy x i induces a unique α − i -equilibrium. Assumption 3’. (Adapted uniqueness)
For each player i ∈ N , strategy x i ∈ X i , and biasprofile α − i ∈ A n − , there exists a unique α − i -equilibrium, denoted by x − i ( x i , α − i ) ∈ X − i .Next we define X SLi ( α − i ) as the set of optimal strategies of an (unbiased) Stackelberg-leader player i who faces opponents with bias profile α − i (when the set of biases is restricted,i.e., A = R ++ , we restrict the feasible Stackelberg-leader strategies to those for which themultiplicative inverse of the induced RIDE is in A ). Definition 7.
Let G be an underlying game satisfying assumptions 1, 2 and 3’ with set of21f feasible biases A . Player i ’s Stackelberg-leader strategy against bias profile α − i ∈ A n − is: X SLi ( α − i ) = argmax { x i ∈ X i | RIDE − i ( x i ,x − i ( x i ,α − i )) ∈ A } π i ( x i , x − i ( x i , α − i )) . Next we characterize a naive analytics equilibrium as an α -equilibrium in which everyoneplays Stackelberg-leader strategies. Claim . Let G be an underlying game that satisfies Assumptions 1–3 and 3’. The pair( α ∗ , x ( α ∗ )) is a naive analytics equilibrium iff x i ( α ∗ ) ∈ X SLi ( α − i ) for each player i ∈ N .Moreover, if x i ( α ∗ ) ∈ Int ( X i ) and α i ∈ Int ( A i ), then α ∗ i − X j = i d x j (cid:16) x i , α ∗ j (cid:17) dx i · ∂q i ∂x j ∂q i ∂x i . Sketch of proof; proof in Appx. A.1. If x i ( α ∗ ) is (resp., is not) a Stackelberg-leader strategyin ( α ∗ , x ( α ∗ )), then there does not (resp., does) exist a bias α ′ i that induces an ( α ′ i , α − i )-equilibrium where player i plays a Stackelberg-leader strategy and gains a payoff higherthan in x ( α ∗ ). The “moreover” part is implied by substituting the FOC in the definition of α ∗ -equilibrium (namely, 0 = ∂π i ∂x i + ∂π i ∂q i · α i · ∂q i ∂x i ) in the FOC of a Stackelberg-leader strategy0 = dπ i (cid:16) x i , x − i (cid:16) x i , α ∗ j (cid:17)(cid:17) dx i = ∂π i ∂x i + ∂π i ∂q i · ∂q i ∂x i + X j = i d x j (cid:16) x i , α ∗ j (cid:17) dx i · ∂q i ∂x j · ∂π i ∂q i . We present three applications of our model and solution concept: price competition, ad-vertising competition, and team production. For each application we derive the predicteddeviation of the analytics bias and the predicted levels of strategies and payoffs comparedto the Nash equilibrium. We further provide micro-foundations for the processes that causethe predicted bias in analytics. 22 .1 Price Competition
Our first application generalizes the motivating example of Section 2.
Underlying Game
The underlying game G p = ( N = { , } , X, q, π ) is a price competitionbetween two firms. The demand of each firm i ∈ { , } is the following linear function: q i ( x ) = a i − b i x i + c i · x − i , where c i · c − i , a i , b i >
0, and | c i | < b i for each player i . The sign of c i (which coincides withsign of c − i ) determines whether the sold goods are substitutes ( c i > c i < | c i | < b i constrains thecross-elasticity parameters to be sufficiently small relative to the own-elasticity parameters.If c i < | c i | < a i a − i b − i ∀ i ∈ { , } . (5.1)Each seller i sets a price x i ∈ X i , where X i = R + if c i > X i = h , a i b i i if c i < a i b i is without loss of generality because setting a higher priceimplies that the seller’s demand cannot be positive. Finally, the profit of each firm is givenby π i ( x i , q i ) = q i ( x ) · x i . This profit function corresponds to constant marginal costs, whichhave been normalized to zero.The game G p has strategic complements if c i > c i <
0. Thiscan be observed directly from a simple analysis of the payoff function, and is immediatelyimplied by Proposition 5 and Claim 2. All of our results remain the same if one adapts the demand function to be non-negative (as is commonlydone in models of price competitions), i.e., q i ( x ) = max ( a i − b i x i + c i · x − i , aive Analytics Equilibrium It is simple to show that G p satisfies all the assumptionsof the general model. Claim . Price competition game G p satisfies Assumptions 2–6 and 3’ (WRT an unrestrictedset of biases A = R ++ ). Moreover, the RIDE derivative is always positive, and the sign ofthe payoff (resp., RIDE) externalities is the same as (resp., opposite of) the sign of c i .Our next result shows that price competition admits a unique naive analytics equilibriumin which both players under-estimate the elasticity of demand in the same way (despite thegame being asymmetric). The prices in this NAE are higher than in the Nash equilibrium.The equilibrium Pareto dominates the Nash equilibrium if the game has strategic comple-ments ( c i > c i < Corollary 1. G p admits a unique naive analytics equilibrium ( α ∗ , x ∗ ) satisfying:1. Symmetric under-estimation of elasticity of demand: α ∗ = α ∗ = q − c i c − i b i b − i ∈ (0 , .
2. Prices are higher than the Nash equilibrium prices: x i ( α ∗ ) > x i (cid:16) −→ (cid:17) .
3. Pareto dominance relative to the Nash equilibrium: c i > ⇒ ˜ π i ( α ∗ ) > ˜ π i (cid:16) −→ (cid:17) , and c i < ⇒ ˜ π i ( α ∗ ) < ˜ π i (cid:16) −→ (cid:17) .Sketch of proof; proof of part (1) in Appendix A.10. It is simple to see that dx j ( x i ,α ∗ j ) dx i = c j b j ( α ∗ j ) , ∂q i ∂x j = c i , ∂q i ∂x j = − b i . Claim 4 implies that x ∗ i must satisfy α ∗ i − x j (cid:16) x i , α ∗ j (cid:17) dx i · ∂q i ∂x j ∂q i ∂x i = c i · c j b i · b j (cid:16) α ∗ j (cid:17) ⇔ (cid:16) α ∗ j (cid:17) (1 − α ∗ i ) = c i · c j b i · b j . (5.2)Observe that the RHS of (5.2) remains the same when swapping i and j . This implies that α ∗ j and α ∗ i must be equal, and that α ∗ = α ∗ = q − c i c − i b i b − i , which proves part (1). Parts(2)–(3) are immediately implied by α ∗ = α ∗ , Claim 5, and Proposition 3. Remark . Our results can be extended from duopoly to oligopoly ( n > α ∗ , x ∗ ) with symmetric biases exist, and it has similar qualitativeproperties as in Proposition 1. 24 icro-Foundations for α i < Implications
The results of Claim 1 fit the direction of demand elasticity bias whennot controlling for price endogeneity, as in, e.g., Table 1 of Berry (1994) and Table 2 ofVillas-Boas and Winer (1999). It is commonly assumed in empirical research that firmsslowly converge to the correct optimal (best-response) pricing, and that inconsistencies withempirical results, such as the appearance that firms are pricing on the inelastic portion ofthe demand curve are due to an incorrect econometric analysis by the researcher. Our re-sults provide an alternative explanation to such an assumption—firms in a naive analytics25quilibrium would believe that they are pricing optimally, and having all firms price on theinelastic portion of the demand curves would be to their benefit.
In this subsection we present the second application of our model, advertising competition.Research that estimates the effectiveness of advertising often showed that extra care is re-quired to arrive at non-biased estimates, and that not correcting for these biases often resultsin overestimating advertising effectiveness (Lodish, Abraham, Kalmenson, Livelsberger, Lubetkin, Richardson, and Stevens,1995; Blake, Nosko, and Tadelis, 2015; Gordon, Zettelmeyer, Bhargava, and Chapsky, 2019;Shapiro, Hitsch, and Tuchman, 2019). In the second application of our model we analyzea duopoly competition in advertising to understand why firms might benefit from naiveanalytics that overestimates the effectiveness of advertising.
Underlying Game
The underlying game G a = ( N = { , } , X, q, π ) describes an adver-tising competition among two firms. Firm i ∈ { , } sells a product with exogenous profitmargins p i > x i ∈ X i of both firms: q i = a i + b i · √ x i + c i · √ x i · x − i , (5.3)where a i , b i , c i · c − i >
0. When c i > X i = R + ).When c i <
0, we restrict the maximum budget to be M i = ( p i · b i ) (i.e., X i = [0 , M i ]). Thisrestriction is without loss of generality as it can be shown that no firm will select x i > M i ,since for budgets above M i the marginal revenue from increasing advertising is below itsmarginal cost, regardless of the opponent’s strategy.In this market a firm’s own advertising increases demand for its own product ( b i ispositive), but the competitor’s advertising may affect the level of the increase. When c i > c i < c i and c − i to coincide. Positive c i -s might correspond to a newcategory of goods, in which advertising attracts attention to the category. Negative c i -smight correspond to a mature category in which advertising mainly causes consumers toswitch among competing goods. The payoff of player i is given by π i ( x i , x − i ) = q i ( x i , x − i ) · p i − x i = p i · ( a i + b i · √ x i + c i · √ x i · x − i ) − x i . (5.4)We require the advertising externalities to be sufficiently small: | c i | ∈ (cid:16) , p i (cid:17) , which implieswell-behaved interior Nash equilibrium. Further, if c i < | c i | < b i b − i · p − i . (5.5) Naive Analytics Equilibrium
Observe that the RIDE in advertising competition is alinear transformation of the return on investment (ROI), which is often used as a measureof advertising effectiveness (Blake, Nosko, and Tadelis, 2015; Lewis and Rao, 2015):RIDE i ( x ) = − ∂π i ∂q i ( x ) · ∂q i ∂x i ∂π i ∂x i = p i · ∂q i ∂x i = p i √ x i ( b i + c i √ x − i ) = 12 π i ( x i , x − i ) − π i (0 , x − i ) x i | {z } ROI +1 . (5.6)It is simple to show that G a satisfies all the assumptions of the general model. Claim . The price competition game G a satisfies Assumptions 2–5 and Assumption 3’ withrespect to the feasible set of biases A = (0 , c i ’s sign.Our next result shows that advertising competition admits a unique naive analytics equi-librium in which both players over-estimate the effectiveness of advertising in the same way(despite the game being asymmetric). Both firms spend more on advertising than in theNash equilibrium. The NAE Pareto dominates the Nash equilibrium if advertising expendi-tures are strategic complements ( c i > Corollary 2. G a admits a unique naive analytics equilibrium ( α ∗ , x ∗ ) satisfying: . Symmetric over-estimation of effectiveness: α ∗ = α ∗ = √ − c c p p ∈ (1 , .2. Advertising budgets are higher than the Nash equilibrium budgets: x i ( α ∗ ) > x i (cid:16) −→ (cid:17) .
3. Pareto dominance relative to the Nash equilibrium: c i > ⇒ ˜ π i ( α ∗ ) > ˜ π i (cid:16) −→ (cid:17) , and c i < ⇒ ˜ π i ( α ∗ ) < ˜ π i (cid:16) −→ (cid:17) .Sketch of proof; proof of part (1) in Appendix. A.12. It is simple to see that dx j ( x i ,α ∗ j ) dx i = √ x j √ x i α ∗ j p j c j , ∂q i ∂x j = c i √ x i √ x j , ∂q i ∂x j = α ∗ i pi . Claim 4 implies that x ∗ i must satisfy α ∗ i − √ x j √ x i α ∗ j p j c j c i √ x i α ∗ i p i √ x j = α ∗ j α ∗ i p i p j c j c i . (5.7)Observe that the RHS of (5.7) remains the same when swapping i and j . This implies that α ∗ j and α ∗ i must be equal. The resulting one-variable quadratic equation has a unique solutionsatisfying α ∗ i <
2, which is α ∗ = α ∗ = √ − c c p p , proving part (1). Parts (2)–(3) areimmediate implications of α ∗ = α ∗ , Claim 6, and Proposition 3. Micro-Foundations for α i > Thus far we have interpreted q ( x ) as market demand and α i = 1 as bias due to naiveanalytics. We now demonstrate that our model applies in more general settings. Specifi-cally, we apply the model to an underlying game of team production with strategic com-plementarity. Team production is common in partnerships and other input games (see,e.g., Holmstrom, 1982; Cooper and John, 1988; Heller and Sturrock, 2020). Examples in-clude sales force members who are compensated based on the performance of the joint salesof a team, and entrepreneurs who receive a share of the exit value of a startup. It isoften observed that entrepreneurial firms are founded by teams of overconfident founders(Astebro, Herz, Nanda, and Weber, 2014; Hayward, Shepherd, and Griffin, 2006). Takingthis perspective, we interpret x i as the contribution of each team member, and q ( x ) as thevalue created by the team. This analogy directly leads to interpreting α i = 1 as a biasplayer i has when evaluating their contribution to the value created by the team, whichcan be seen as a measure of confidence . We show that in any naive analytics equilibriumall agents are overconfident in the sense of overestimating their ability to contribute to theteam’s output (i.e., having α i > Our results provide a novel foundation for the tendency of people (and, in particular,entrepreneurs) to be overconfident in the sense of overestimating one’s ability. We show thatwhen skills are complementary, overconfidence contributes to increased team efficiency, andis a response to the internal firm environment. The results provide a novel explanation to See also Heller (2014) who demonstrates that overconfidence of entrepreneurs can help an investor indiversifying aggregate risk.
Underlying Game
We describe a team production game G t with strategic complements.Consider two players ( N = { , } ), each choosing how much effort x i ∈ X i ≡ R + to exert ina joint project. The project yields all agents a value of q ( x ), where q is twice continuouslydifferentiable in R , strictly increasing, strictly concave, and supermodular (i.e., satisfiesstrategic complementarity) with respect to its two parameters (i.e., ∂q ( x ) ∂x i > ∂ q ( x ) ∂x i < ∂ q ( x ) ∂x x > x ∈ X and any i ∈ { , } ). The payoff of each player i is equal to theproject’s value minus her effort: π i ( x ) = q ( x ) − x i . We assume that the marginal contribution of effort is sufficiently large if efforts are small,and it is sufficiently small if efforts are large. Formally:
Assumption 7.
For each α ∈ R ++ , there exist strategy profiles x ≤ x , such that ∂q ( x ) ∂x i ≤ α ≤ ∂q ( x ) ∂x i for each player i . Finally, we assume that the Hessian determinant of q ( x ) never changes its sign (i.e., itis never equal to zero). Formally: Assumption 8.
Monotone Hessian determinant : ∂ q ( x ) ∂x ∂ q ( x ) ∂x = (cid:16) ∂ q ( x ) ∂x x (cid:17) ∀ x ∈ X . We interpret Assumption 8 as having a monotone relation between concavity and su-permodularity, i.e., either the amount of concavity is always larger than the amount ofsupermodularity (i.e., ∂ q ( x ) ∂x ∂ q ( x ) ∂x > (cid:16) ∂ q ( x ) ∂x x (cid:17) for every x ∈ X ), or the amount of concavity isalways smaller than the amount of supermodularity (i.e., ∂ q ( x ) ∂x ∂ q ( x ) ∂x < (cid:16) ∂ q ( x ) ∂x x (cid:17) ∀ x ∈ X ). Example 2 (Cobb-Douglas production) . The Cobb-Douglas production function q ( x ) = x β x β satisfies Assumptions 7 and 8 if β + β < ∂q (˜ x, ˜ x ) ∂x i = β i ˜ x − β i − β − i ⇒ lim ˜ x → ∂q (˜ x, ˜ x ) ∂x i = ∞ , lim ˜ x →∞ ∂q (˜ x, ˜ x ) ∂x i = 0 ,∂ q ( x ) ∂x i = − (1 − β i ) β i x β − i − i x − β i i < , ∂ q ( x ) ∂x x = β β x − β x − β > . aive Analytics Equilibrium The RIDE is equal to the agent’s marginal contributionto the project: RIDE i ( x ) = − ∂π i ∂q ( x ) · ∂q∂x i ∂π i ∂x i = − · ∂q∂x i − ∂q∂x i . (5.8)Further, observe that the underlying team-production game satisfies Assumptions 1–5. Claim . The team-production game G t satisfies Assumptions 1–5 with respect to an unre-stricted set of biases A = R ++ . Moreover, the payoff and RIDE externalities are positive,while the RIDE self-derivative is negative.The results of Section 4 and Claim 7 immediately imply that in any naive analyticsequilibrium both players: experience overconfidence (i.e., α ∗ i > Corollary 3.
In any naive analytics equilibrium ( α ∗ , x ∗ ) of G t :1. Both players over-estimate their influence on the joint project, i.e., α ∗ i > ∀ i .2. Both players exert more efforts than in the Nash equilibrium, i.e., x ∗ i > x NEi ∀ i .3. The NAE Pareto dominates the Nash equilibrium. Naive analytics equilibrium can be used to analyze such games where players have uncertaintyabout the indirect impact of their actions on their payoffs, and allows players to use biaseddata analytics to estimate this impact. This scenario is common in economic applicationssuch as price competition, advertising competition and team production.The predictions of our results are consistent with commonly observed behaviors of firmsand teams. In equilibrium, players are predicted to converge to biased estimates in the direc-tion that causes their opponents to respond in a beneficial manner. In pricing competition,players are better off if they perceive consumers to be less price elastic than they actuallyare, which is a possible interpretation of observed firm pricing if they do not correct for priceendogeneity in their econometric analysis. In advertising competition, it is observed that31rms often overestimate the response to their advertising and over-advertise, as predictedby our results. These deviations from unbiased estimates cause deviations from the Nashequilibrium that can be beneficial or detrimental to players. When games have strategiccomplements, players will choose strategies that deviate from the Nash equilibrium in thedirection that benefits the opponents, and their equilibrium payoffs will dominate those ofthe Nash equilibrium. The converse is true for games with strategic substitutes.The results of our analysis provide testable empirical predictions about the direction andmagnitudes of the biases. In particular, the analysis predicts that different firms within amarket will have similar level of biasedness, while the level of biasedness will differ across mar-kets. Initial evidence for this phenomenon is observed in Table 2 of Villas-Boas and Winer(1999). Another prediction is that biasedness away from α ∗ = 1 will disappear as the numberof players grows, or in a monopolistic market. These predictions could be tested in empiricaldata as well as serve as a basis for analysis about the adoption and sophistication of analyticsin various industries. Further, our results may bring to question some of the assumptionsused in practice when performing counterfactual analysis to estimate welfare and assess theimpact of regulatory policy. In these analyses, it is often assumed that firms correctly per-ceive their economic environment and that any observed inconsistency with this assumptionmay be due to unobserved factors by researchers. However, as we have shown, in a naiveanalytics equilibrium firms will be profit maximizing if they misperceive their environment.One would expect the conclusions from a counterfactual analysis that utilizes the standardassumptions to be biased if firms are indeed playing an analytics game.A second implication of our results is for research that focuses on biases in decision mak-ing from non-causal inferential methods. The research implicitly assumes that focusing oncausality and more precise estimates are better for firm performance, which often translatesto normative recommendation about firm practices (see, e.g., Siroker and Koomen (2013)and Thomke (2020) on A/B testing). Our results point to the conclusion that firms maybe better off with opting for more naive heuristics, which are indeed quite popular becausethey are easy to implement. This may suggest that normative recommendations for deploy-ing more sophisticated analytics capabilities should be made with caution in competitiveenvironments. 32inally, there are two natural extensions to our work which we leave for the future. First,our analysis focused on the case in which the underlying game admits a unique α -equilibriumfor each bias profile α , while extending the results to games with multiple equilibria is desired.Second, the model we derived assumed that the direction of monotone derivatives of playersare the same, i.e., one type of derivative is either increasing or decreasing for all players.A natural question to ask is under what conditions the results of the characteristics of thenaive analytics equilibrium we described extend to games where the monotone derivativesare mixed in directions among players. References
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A.1 Proof of Claim 4 (Stackelberg-Leader Representation)
Assume to the contrary that ( α ∗ , x ( α ∗ )) is a naive analytics equilibrium and x i ( α ∗ ) / ∈ X SLi ( α − i ) for player i . Let x SLi ∈ X SLi be an optimal Stackelberg-leader strategy of player i . Let α ′ i = i ( x SLi ,x − i ( x i ,α ∗− i )) . Then:˜ π i (cid:16) α ′ i , α ∗− i (cid:17) = π i (cid:16)(cid:16) x SLi , x − i (cid:16) x i , α ∗− i (cid:17)(cid:17)(cid:17) > π i ( x i ( α ∗ ) , x − i ( α ∗ )) = π i ( α ∗ ) , and we get a contradiction to ( α ∗ , x ( α ∗ )) being a NAE (where the first equality is due toAssumption 3 (uniqueness) and the inequality is due to the definition of X SLi ( α − i )).Next assume to the contrary that x i ( α ∗ ) ∈ X SLi ( α − i ) for each player i ∈ N and( α ∗ , x ( α ∗ )) is not a naive analytics equilibrium. This implies that there is player i andbias α ′ i such that ˜ π i (cid:16) α ′ i , α ∗− i (cid:17) > ˜ π i ( α ∗ ). Let ˜ x i = x i (cid:16) α ′ i , α ∗− i (cid:17) . Observe that π i (cid:16) ˜ x i , x − i (cid:16) ˜ x i , α ∗− i (cid:17)(cid:17) = ˜ π i (cid:16) α ′ i , α ∗− i (cid:17) > ˜ π i ( α ∗ ) = π i (cid:16) x i ( α ∗ ) , x − i (cid:16) x i ( α ∗ ) , α ∗− i (cid:17)(cid:17) , which contradicts the assumption that x i ( α ∗ ) ∈ X SLi ( α − i ).Next we prove the “moreover” part. The fact that x i ( α ∗ ) ∈ Int ( X i ) is an optimalStackelberg-leader strategy implies that it satisfies the following first order condition:0 = dπ i (cid:16) x i , x − i (cid:16) x i , α ∗ j (cid:17)(cid:17) dx i = ∂π i ∂x i + ∂π i ∂q i · ∂q i ∂x i + X j = i d x j (cid:16) x i , α ∗ j (cid:17) dx i · ∂q i ∂x j · ∂π i ∂q i . (A.1)Substituting 0 = ∂π i ∂x i + ∂π i ∂q i · α i · ∂q i ∂x i (implied by x ( α ∗ ) being an α ∗ -equilibrium) in A.1 yields:0 = (1 − α ∗ i ) · ∂π i ∂q i · ∂q i ∂x i + X j = i d x j (cid:16) x i , α ∗ j (cid:17) dx i · ∂q i ∂x j · ∂π i ∂q i ⇔ α ∗ i − X j = i d x j (cid:16) x i , α ∗ j (cid:17) dx i · ∂q i ∂x j ∂q i ∂x i . (A.2)1 .2 Proof of Claim 1 (Unimodal Monotone RIDE) Fix player i and profile x − i . Let x ∗ i ∈ X i be the strategy satisfying RIDE i ( x ∗ i , x − i ) = 1.Assume first that ∂π i ∂x i >
0. Observe that dπ i dx i = 0 (resp., dπ i dx i < dπ i dx i >
0) iff ∂π i ∂x i = (cid:12)(cid:12)(cid:12) ∂π i ∂q i · ∂q i ∂x i (cid:12)(cid:12)(cid:12) (resp., ∂π i ∂x i < (cid:12)(cid:12)(cid:12) ∂π i ∂q i · ∂q i ∂x i (cid:12)(cid:12)(cid:12) , ∂π i ∂x i > (cid:12)(cid:12)(cid:12) ∂π i ∂q i · ∂q i ∂x i (cid:12)(cid:12)(cid:12) ), which holds iff RIDE i ( x ) = 1 (resp., RIDE i ( x ) > RIDE i ( x ) < d ( RIDE i ( x )) dx i > x i = x ∗ i (resp., x i > x ∗ i , x i < x ∗ i ), which implies unimodality. If d ( RIDE i ( x )) dx i <
0, then
RIDE i ( x ) = 1 (resp., RIDE i ( x ) > RIDE i ( x ) <
1) holds iff x i = x ∗ i (resp., x i < x ∗ i , x i > x ∗ i ), which violates unimodality.Next assume first that ∂π i ∂x i <
0. Observe that dπ i dx i = 0 (resp., dπ i dx i < dπ i dx i >
0) iff (cid:12)(cid:12)(cid:12) ∂π i ∂x i (cid:12)(cid:12)(cid:12) = ∂π i ∂q i · ∂q i ∂x i (resp., (cid:12)(cid:12)(cid:12) ∂π i ∂x i (cid:12)(cid:12)(cid:12) > ∂π i ∂q i · ∂q i ∂x i , (cid:12)(cid:12)(cid:12) ∂π i ∂x i (cid:12)(cid:12)(cid:12) < ∂π i ∂q i · ∂q i ∂x i ), which holds iff RIDE i ( x ) = 1 (resp., RIDE i ( x ) < RIDE i ( x ) > d ( RIDE i ( x )) dx i < x i = x ∗ i (resp., x i > x ∗ i , x i < x ∗ i ), which implies unimodality. If d ( RIDE i ( x )) dx i > RIDE i ( x ) = 1 (resp., RIDE i ( x ) > RIDE i ( x ) <
1) holds iff x i = x ∗ i (resp., x i < x ∗ i , x i > x ∗ i ), which violates unimodality. A.3 Fact on Relabeling Strategies ( x i → − x i ) Fact 1.
Let G = ( N, X, q, π ) be an underlying game that satisfies Assumptions 1–5. Let G ′ = ( N, X ′ , q ′ , π ′ ) be the same game after relabeling the strategies x i → − x i , i.e., X ′ i = − X i = { x ′ i ∈ R | − x ′ i ∈ X i } , q ′ i ( x , ..., x n ) = q i ( − x , ..., − x n ) and π ′ i ( x i , q i ) = π i ( − x i , q i ) .Then G ′ satisfies Assumptions 1–5 and the sign of each of its three monotone derivatives isthe opposite of the respective sign in G , i.e., for each players i = jdπ i ( x ) dx j > ⇔ dπ ′ i ( x ) dx j < , d ( RIDE i ( x )) dx i > ⇔ d ( RIDE ′ i ( x )) dx i < , AND d ( RIDE i ( x )) dx j > ⇔ d ( RIDE ′ i ( x )) dx j < . A.4 Proof of Claim 2 (Strategic Substitutes/Complements)
In what follows we prove the “if” parts in both parts of the claim. The “only if” parts are im-mediately implied from the “if” parts due to d ( RIDE j ) dx j · d ( RIDE j ) dx i < d ( RIDE j ) dx j · d ( RIDE j ) dx i > i, j ∈ N , strategy profile x and strategy x ′ i > x i . Assume first that d ( RIDE j ) dx i >
0. This im-plies that
RIDE j ( x ′ i , x − ij ) > RIDE j ( x i , x − ij ). This, in turn, implies that BR j ( x ′ i , x − ij ) >BR j ( x i , x − ij ) (resp., BR j ( x ′ i , x − ij ) < BR j ( x i , x − ij )) if d ( RIDE j ) dx j < d ( RIDE j ) dx j > d ( RIDE j ) dx i <
0. This implies that
RIDE j ( x ′ i , x − ij ) < RIDE j ( x i , x − ij ). This, inturn, implies that BR j ( x ′ i , x − ij ) > BR j ( x i , x − ij ) (resp., BR j ( x ′ i , x − ij ) < BR j ( x i , x − ij )) if d ( RIDE j ) dx j > d ( RIDE j ) dx j < A.5 Proof of Proposition 1(Over/Under Reply)
Let ( α ∗ , x ∗ ) be a naive analytics equilibrium in a game with beneficial upward commit-ment (the proof for the case of beneficial downward commitment is analogous). Assumeto the contrary that there is player i for which x ∗ i < BR i (cid:16) x ∗− i (cid:17) . Consider a first-stagedeviation of player i to bias α ′ i sufficiently close to α ∗ i that induces him to increase itssecond-stage (cid:16) α ′ i , α ∗− i (cid:17) -equilibrium strategy from x ∗ i to x ′ i such that x ∗ i < x ′ i < BR i ( x ∗− i ).That is, α ′ i = α ∗ i + ǫ if the RIDE derivative is negative and α ′ i = α ∗ i − ǫ if the RIDE deriva-tive is positive for a sufficiently small ǫ >
0. Then in the new naive analytics equilibrium( α ′ , x ′ ) ≡ (cid:16)(cid:16) α ′ i , α ∗− i (cid:17) , x ′ (cid:17) , all other players adjust their strategies, x ′ j in the direction thatbenefits player i due to the game having a beneficial upward commitment (i.e., when thepayoff externalities are positive x ′ j > x ∗ j and when negative x ′ j < x ∗ j ). This, in turn, impliesthat π i ( α ′ , x ′ ) > π i ( α ∗ , x ∗ ) because player i gains both from increasing his own x i closer tohis best-reply and from the others changing their x ′ j in the direction in its favor due to theunderlying game having a beneficial upward commitment. A.6 Proof of Claim 3 (Beneficial Commitment)
The following simple lemma will be helpful in the proof of Claim 3.
Lemma 1.
Fix i = j ∈ N , x ∈ R n + . Assume that RIDE j ( x i , x j , x − ij ) = RIDE j (cid:16) x ′ i , x ′ j , x − ij (cid:17) with x i = x ′ i and x j = x ′ j . Then ( x i − x ′ i ) · (cid:16) x j − x ′ j (cid:17) < iff the signs of the RIDE derivative nd RIDE externalities coincide.Proof. We prove the lemma for one of the four possible cases in which the signs of theRIDE derivative and RIDE externalities are both positive (the analogous arguments in theremaining three cases is omitted for brevity). If ( x i − x ′ i ) · (cid:16) x j − x ′ j (cid:17) >
0, then either:(I) x ′ i > x i and x ′ j > x j implying that RIDE j ( x i , x j , x − ij ) < RIDE j (cid:16) x ′ i , x ′ j , x − ij (cid:17) , or(II) x ′ i < x i and x ′ j < x j implying that RIDE j ( x i , x j , x − ij ) > RIDE j (cid:16) x ′ i , x ′ j , x − ij (cid:17) .Thus, RIDE j ( x i , x j , x − ij ) = RIDE j (cid:16) x ′ i , x ′ j , x − ij (cid:17) implies that ( x i − x ′ i ) · (cid:16) x j − x ′ j (cid:17) < RIDE j ( x i , BR j ( x i , x − ij ) , x − ij ) = RIDE j ( x ′ i , BR j ( x ′ i , x − ij ) , x − ij ) = 1implies that ( x i − x ′ i ) · ( BR j ( x i , x − ij ) − BR j ( x ′ i , x − ij )) < i would benefit froma best-replying player j increasing her play, which would happen when player i decreases(resp., increases) her strategy when the signs of the RIDE derivative and RIDE externalitiescoincide (resp., are different). This implies that the game has beneficial upward commitmentiff exactly one of the signs of the RIDE derivative/externalities is negative.Next, assume that the game has negative payoff externalities. Then, player i would ben-efit from a best-replying player j decreasing her play, which would happen when player i in-creases (resp., decreases) her strategy when the signs of the RIDE derivative and RIDE exter-nalities coincide (resp., are different). This implies that the game has beneficial upward com-mitment iff either both or none of the signs of the RIDE derivative/externalities is negative.Combining the above argument implies that the game has beneficial upward commitmentiff the number of positive derivatives (among all three monotone derivative/externalities) iseven. 4 .7 Proof of Proposition 2 ( α ∗ i > or α ∗ i < ) Assume first that the game has a beneficial upward commitment. By Proposition 1 x ∗ i >BR (cid:16) x ∗− i (cid:17) . This implies that RIDE ( x ∗ i ) > d ( RIDE i ) dx i >
0. The fact that x ∗ is an α ∗ -equilibrium implies that RIDE ( x ∗ i ) = α ∗ i , which, in turn, implies that α ∗ i < d ( RIDE i ) dx i > d ( RIDE i ) dx i > d ( RIDE i ) dx j · dπ i dx j < α ∗ i < d ( RIDE i ) dx j · dπ i dx j < x ∗ i
0. The fact that x ∗ is an α ∗ -equilibrium implies that RIDE ( x ∗ i ) = α ∗ i , which, in turn, implies that α ∗ i > d ( RIDE i ) dx i > d ( RIDE i ) dx i > d ( RIDE i ) dx j · dπ i dx j > α ∗ i > d ( RIDE i ) dx j · dπ i dx j > A.8 Proof of Proposition 3 (Pareto domination of NAE and NE)
The following two lemmas will be helpful in the proof of Proposition 3. Lemma 2 shows thatwhether player i is over replying in an α -equilibrium depends only on her own biasednessparameter α i . Lemma 2.
Let G be a game satisfying Assumptions 1–5. Player i over replies in an α -equilibrium iff she over replies in an (cid:16) α i , α ′− i (cid:17) -equilibrium for any α ∈ R n + and α ′− i ∈ R n + .Proof. Observe that
RIDE i ( x ( α )) = RIDE i (cid:16) x (cid:16)(cid:16) α i , α ′− i (cid:17)(cid:17)(cid:17) = 1 α i , and RIDE i ( BR i ( x − i ( α − i )) , x − i ( α )) = RIDE i (cid:16) BR i (cid:16) x − i (cid:16) α i , α ′− i (cid:17)(cid:17) , x − i (cid:16) α i , α ′− i (cid:17)(cid:17) = 1 . Player i can over reply only if α i = 1. There are four exhaustive cases, in all of which player i over-replying behavior is the same in the α -equilibrium and in the (cid:16) α i , α ′− i (cid:17) -equilibrium:1. If dRIDE i ( x ) dx i > α i <
1, then
RIDE i ( x ( α )) = RIDE i (cid:16) x (cid:16)(cid:16) α i , α ′− i (cid:17)(cid:17)(cid:17) = α i > x i ( α ) > BR i ( x − i ( α − i )) and x i (cid:16) α i , α ′− i (cid:17) > BR i (cid:16) x − i (cid:16) α i , α ′− i (cid:17)(cid:17) , i.e.,5layer i over replies in both the α -equilibrium and the (cid:16) α i , α ′− i (cid:17) -equilibrium.By an analogous argument:2. If dRIDE i ( x ) dx i < α i >
1, then Player i over replies in both biased equilibria.3. If dRIDE i ( x ) dx i < α i <
1, then Player i under replies in both biased equilibria.4. If dRIDE i ( x ) dx i > α i >
1, then Player i under replies in both biased equilibria.Lemma 3 (which is a standard result) shows that in games with strategic complementsif all agents over (resp., under) reply to each other, then they all must play strategies above(resp., below) their Nash equilibrium strategies. Formally, Lemma 3.
Let G be a game with strategic complements, concave payoffs (Assumption 1)and a unique Nash equilibrium x NE . Let x ∗ be a strategy profile.1. If x ∗ i ≥ BR i (cid:16) x ∗− i (cid:17) for each player i ∈ N with strict inequality for at least one player,then x ∗ i > x NEi for each player i ∈ N .2. If x ∗ i ≤ BR i (cid:16) x ∗− i (cid:17) for each player i ∈ N with strict inequality for at least one player,then x ∗ i < x NEi for each player i ∈ N .Proof.
1. We begin by showing the weak inequality x ∗ i ≥ x NEi for each player i ∈ N . Assume tothe contrary that there exists player j for which x ∗ j < x NEj . Consider an auxiliary game G R similar to G except that each player i is restricted to choose a strategy up to x ∗ i .Due to the concavity, the game G R admits a pure Nash equilibrium, which we denoteby x RE . Note that x RE = x NE because x REj ≤ x ∗ j < x NEj . The fact that x NE is aunique equilibrium in G implies that x RE cannot be an equilibrium of G . This implies(due to the concave payoffs) that there must exist player k for which x REk = x ∗ k and x ∗ k < BR k (cid:16) x RE − k (cid:17) . The fact that x REi ≤ x ∗ i for each player i and the assumption thatthe game has strategic complements jointly imply that x ∗ k ≥ BR k (cid:16) x ∗− k (cid:17) ≥ BR k (cid:16) x RE − k (cid:17) and we get a contradiction. 6ext, we want to show the strict inequality x ∗ i > x NEi for each player i ∈ N . Observethat x ∗ = x NE because there exists a player who (strictly) over replies. This impliesthat there exists player j for which x ∗ j > x NEj . The fact that the game has strategiccomplements imply that x ∗ i ≥ BR i (cid:16) x ∗− i (cid:17) > BR i (cid:16) x NE − i (cid:17) = x NEi , which completes the proof.2. The proof of part (2) is analogous to part (1), and is omitted for brevity.Next we prove Proposition 3 by relying on the above lemmas.1. Assume that the game has a beneficial upward commitment. Due to Proposition 1 allplayers over reply in x ∗ (i.e., x ∗ i > BR i (cid:16) x ∗− i (cid:17) for each player i ∈ N ). The fact that( α ∗ , x ∗ ) is a naive analytics equilibrium implies that π i ( x ∗ ) = π i ( x ( α ∗ )) ≥ π i (cid:16) x (cid:16) , α ∗− i (cid:17)(cid:17) . Next observe that because x i (cid:16) , α ∗− i (cid:17) = BR i (cid:16) x − i (cid:16) , α ∗− i (cid:17)(cid:17) , then π i (cid:16) x (cid:16) , α ∗− i (cid:17)(cid:17) > π i (cid:16) x NEi , x − i (cid:16) , α ∗− i (cid:17)(cid:17) Further observe that player i plays a best reply in x (cid:16) , α ∗− i (cid:17) (because he is unbiased),while each other player j = i over replies in x (cid:16) , α ∗− i (cid:17) because she has over repliedin x ( α ∗ ) and she has the same value of α ∗ j in both naive analytics equilibria (thisobservation is formalized in Lemma 2). In games with strategic complements thisobservation implies that x j (cid:16) , α ∗− i (cid:17) > x NEj for each player j ∈ N (as proven in Lemma3). Due to Claim 2 exactly one of the RIDE derivatives is positive. As Claim 3 impliesthat the total number of positive derivatives is even, it implies that the remainingderivative is positive, i.e., that the game has positive payoff externalities, which implies7hat π i (cid:16) x NEi , x − i (cid:16) , α ∗− i (cid:17)(cid:17) > π i (cid:16) x NE (cid:17) Combining the three inequalities we obtain π i ( x ∗ ) > π i (cid:16) x NE (cid:17) .The proof for the case in which the game has beneficial downward commitment isanalogous and omitted for brevity (here and in part 2 below).2. Assume that the game has upward commitment benefit. Due to Proposition 1 allplayers over reply in x ∗ (i.e., x ∗ i > BR i (cid:16) x ∗− i (cid:17) ). In symmetric profiles x ∗ i > BR i (cid:16) x ∗− i (cid:17) iff x ∗ i > x NEi because x ∗ i > x NEi ⇔ x ∗− i > x NE − i ⇔ BR i (cid:16) x ∗− i (cid:17) < BR i (cid:16) x NE − i (cid:17) = x NEi < x ∗ i , where the the first iff is due to the strategy profile being symmetric, and the second iffis due to the game having strategic substitutes. Due to Claim 2 either zero or two theRIDE derivatives are positive. As Claim 3 implies that the total number of positivederivatives is even, it implies that the remaining derivative is negative, i.e., that thegame has negative payoff externalities, which implies that π i ( x ∗ ) < π i (cid:16) x ∗ i , x NE − i (cid:17) < π i (cid:16) x NE (cid:17) . A.9 Proof of Claim 5 (Price Competition ⇒ Assumptions 2–6)
We begin by showing that the RIDE coincides with the elasticity of demand:RIDE i ( x ) = − ∂π i ∂q i · ∂q i ∂x i ∂π i ∂x i = − x i · ∂q i ∂x i q i ( x ) = | η x i ,q i | = x i · b i q i ( x ) (A.3)Next we show that G p satisfies Assumptions 2–6.• Assumption 2 (opposing payoffs): ∂π i ∂x i · ∂q i ∂x i = q i · ( − b i ) < α -equilibrium): Strategy profile x is an an α -Equilibrium iff for8ach player i α i = RIDE i ( x ) = x i · b i q i ( x ) = x i · b i a i − b i x i + c i · x − i ⇔ x i = a i + c i · x − i b i (1 + α i ) . Substituting x − i = a − i + c − i · x i b − i (1+ α − i ) and rearranging yields the unique α -Equilibrium x ( α ): x i ( α ) = a i b − i (1 + α − i ) + c i a − i b i b − i (1 + α i ) (1 + α − i ) − c i c − i . (A.4)Observe that the numerator of A.4 is positive due to (5.1) and the denominator ispositive due to the assumption of | c i | < b i . This implies that x ( α ) is a well-definedpositive price profile.• Assumption 3’:• Assumption 4 (monotone payoff externalities with the same sign as c i ): dπ i ( x ) dx − i = ddx − i ( x i · q i ( x )) = ddx − i ( x i · ( a i − b i x i + c i · x − i )) = c i . • Assumption 5 (monotone RIDE externalities with the opposite sign of c i ): d RIDE i ( x ) dx − i = ddx − i x i · b i q i ( x ) ! = ddx − i x i · b i a i − b i x i + c i · x − i ! = − x i · b i ( a i − b i x i + c i · x − i ) · c i . • Assumption 6 (increasing RIDE, which has the same sign as ∂π i ∂x i = q i > d RIDE i ( x ) dx i = ddx i x i · b i q i ( x ) ! = ddx i x i · b i a i − b i x i + c i · x − i ! = b i · ( a i + c i x − i )( q i ( x )) > , where the last inequality is immediate if c i >
0, and it is implied by x − i ≤ M − i = a − i b − i < a i | c i | (where the last inequality is due to (5.1)) if c i < .10 Proof of Part (1) of Claim 1 (Price Competition) It is simple to see that the biased best-reply of player j to strategy x i is x j (cid:16) x i , α ∗ j (cid:17) = a j + c j x i b j (cid:16) α ∗ j (cid:17) ⇒ dx j ( x i , α ∗ i ) dx i = c j b j (cid:16) α ∗ j (cid:17) . Clearly any NAE ( α ∗ , x ∗ ) must have positive prices, which implies that x ∗ i ∈ Int ( X i ) foreach player i. Claim 4 implies that x ∗ i must satisfy α ∗ i − x j (cid:16) x i , α ∗ j (cid:17) dx i · ∂q i ∂x j ∂q i ∂x i . (A.5)Substituting dx j ( x i ,α ∗ i ) dx i = c j b j ( α ∗ j ) , ∂q i ∂x j = c i , ∂q i ∂x i = − b i yields:1 − α ∗ i = c i · c j b i · b j (cid:16) α ∗ j (cid:17) ⇔ (cid:16) α ∗ j (cid:17) (1 − α ∗ i ) = c i · c j b i · b j . (A.6)Observe that the RHS of (A.6) remains the same when swapping i and j . This implies that α ∗ j and α ∗ i must be equal, which, in turn, implies that1 − ( α ∗ i ) = c i · c j b i · b j ⇒ α ∗ = α ∗ = s − c i c − i b i b − i . A.11 Proof of Claim 6 (Advertising Competition ⇒ Asm. 2–6) • Assumption 1 (concave payoffs): dπ i ( x ) dx i = ∂π i ∂q i ( x ) · ∂q i ∂x i + ∂π i ∂x i = p i √ x i ( b i + c i √ x − i ) − d π i ( x ) dx i = − p i x / i ( b i + c i √ x − i ) < , where the last inequality is immediate if c i >
0, and it is implied by the assumptionsthat √ x − i ≤ p − i b − i and | c i | < b i b − i .p − i if c i < α -equilibrium): Strategy profile x is an α -Equilibrium iff foreach player i α i = RIDE i ( x ) = p i √ x i ( b i + c i √ x − i ) ⇔ √ x i = α i p i b i + c i √ x − i ) , where if c i < b i + c i √ x − i > | c i | < b i b − i .p − i and x − i ≤ ( p − i b − i ) , which implies that x i ( x − i ) is well-defined and positive for any x − i . Substi-tuting √ x − i = α − i p − i (cid:16) b − i + c − i √ x i (cid:17) and rearranging yields the unique α -Equilibrium: q x i ( α ) = α i p i (2 b i + α − i b − i c i p − i )4 − α i α − i c i c − i p i p − i ! , (A.7)where the numerator is positive when c i < | c i | < b i b − i .p − i and α − i ≤
2, and the denominator is positive due to the assumptions that α i ≤ c i < p i . This implies that x ( α ) is a well-defined positive advertising budget profile.• Assumption 3’:• Assumption 4 (monotone payoff externalities with the same sign as c i ): dπ i ( x ) dx − i = p i · dq i ( x ) dx − i = p i · √ x i · √ x − i c i . • Assumption 6 (negative RIDE): d RIDE i ( x ) dx i = ddx i p i √ x i ( b i + c i √ x − i ) ! = − p i x / i ( b i + c i √ x − i ) < c i < b i + c i √ x − i > | c i | < b i b − i .p − i and x − i ≤ ( p − i b − i ) .• Assumption 4 (monotone RIDE externalities with the same sign as c i ): d RIDE i ( x ) dx − i = ddx − i p i √ x i ( b i + c i √ x − i ) ! = p i √ x i x − i · c i . .12 Proof of Part (1) Claim 2 (Adverting Competition) In the proof of Claim 6 we have shown that √ x j = α ∗ j p j (cid:16) b j + c j √ x i (cid:17) , which implies: x j (cid:16) x i , α ∗ j (cid:17) = (cid:18) α j p j b j + c j √ x i ) (cid:19) ⇒ dx j ( x i , α ∗ i ) dx i = √ x j √ x i α j p j c j Clearly any NAE ( α ∗ , x ∗ ) must have positive advertising budgets (as otherwise the firmmakes no profit), which implies that x ∗ i ∈ Int ( X i ) for each player i. Claim 4 implies that x ∗ i must satisfy α ∗ i − x j (cid:16) x i , α ∗ j (cid:17) dx i · ∂q i ∂x j ∂q i ∂x i . (A.8)Substituting dx j ( x i ,α ∗ i ) dx i = √ x j √ x i α ∗ j p j c j , ∂q i ∂x j = c i √ x i √ x j , ∂q i ∂x i = b i + c i √ x j √ x i = α ∗ i pi yields:1 − α ∗ i = √ x j √ x i α ∗ j p j c j c i √ x i α ∗ i p i √ x j = α ∗ j α ∗ i p i p j c j c i . (A.9)Observe that the RHS of (A.9) remains the same when swapping i and j . This implies that α ∗ j and α ∗ i must be equal, which, in turn, implies that1 − α ∗ i ( α ∗ i ) = p i p j c j c i ⇒ α ∗ = α ∗ = 21 + √ − c c p p or 21 − √ − c c p p . Because α i ∈ (0 , α ∗ = α ∗ = √ − c c p p . A.13 Proof of Claim 7 (Team Production)
1. Assumption 1 (concave payoffs): d π i dx i = d dx i ( q ( x ) − x i ) = d q ( x ) dx i < q being concave..2. Assumption 3 (unique α -equilibrium): Strategy profile x is an an α -Equilibrium iff for12ach player i α i = RIDE i ( x ) = ∂q∂x i . Let ǫ = max (cid:16) α , α (cid:17) TBD : (1) Supermodularity implies that NAE exists. (2) theother two assumptions imply an interior solution. (3) showing that monotone Hessiandeterminant implies uniqueness.3. Assumption 4 (positive payoff externalities): dπ i ( x ) dx − i = ddx − i ( q ( x ) − x i ) = ddx − i ( q ( x )) > q being increasing in x − i .4. Assumption 6 (negative RIDE self-derivative): d RIDE i ( x ) dx i = ddx i ∂q∂x i ! = d qdx i < q being concave in x i .5. Assumption 4 (positive RIDE externalities): d RIDE i ( x ) dx − i = ddx i − ∂q∂x i ! > q being supermodular. B Conditions for Unique α -equilibrium (Asm. 3) In this appendix we present conditions on the RIDE that imply existence and uniqueness ofan α -equilibrium for any α (Assumption 3).Our first result presents a necessary and sufficient condition for the existence of α -equilibrium. The condition requires that there are two strategy profiles , where one profile isweakly higher than the other, such that for each of these profiles, each player has a strategythat yields him a RIDE of α i . The standard proof relies on applying Brouwer fixed-point13heorem. Claim . Let G be an underlying game that satisfies Assumptions 4–5. Let α ∈ A n . Thenthe game admits an α -equilibrium iff there exist strategy profiles x ≤ x (i.e., x i ≤ x i for eachplayer i ), such that for each player i , there exist strategies x ′ i , x ′ i ∈ [ x i , x i ] that satisfy RIDE i ( x ′ i , x − i ) = RIDE i ( x ′ i , x − i ) = α i . Proof. “If side”: Let ¯ X = { x ∈ X | x ≤ x ≤ x } be the compact and convex subset of profilesbetween x and x . Define g : ¯ X → ¯ X as follows g i ( x ) = n x ′ i ∈ ¯ X | RID E i ( x ′ i , x − i ) = α i o . Assumptions 4–5 imply that there exists a unique x ′ i ∈ [ x ′ i , x ′ i ] such that RID E i ( x ′ i , x − i ) = α i , g i ( x ) ,which implies that g is a well-defined function. The fact that π and q are both twice con-tinuously differentiable implies that g is continuous. Brouwer fixed-point theorem implies g admits a fixed point, which must be an α -equilibrium.“Only if side”: If there exist an α -equilibrium x ∗ , then taking x = x = x ∗ satisfies thecondition of the claim.Next we present a sufficient condition for the uniqueness of α -equilibrium. For each x ∈ X , let J ( x ) be the n × n Jacobian matrix of partial derivatives of
RIDE at x : J ij ( x ) = d ( RIDE i ) dx j . Definition 8. J ( x ) is uniformly directional if for each v ∈ R n , there exists r ∈ R n such that v · J ( x ) · r > for any x ∈ X .Observe that if either J ( x ) is positive-definite for all x or negative-definite for all x , thenit is uniformly directional (where r = v for the positive-definite case, and r = − v for thenegative-definite case). Our next result shows that uniform directionality implies uniquenessof α -equilbirium. Claim . Let G be an underlying game that satisfies Assumptions 4–5. Assume that theJacobian J ( x ) is uniformly directional. Let α ∈ A n be a bias profile. Then there exists atmost one α -equilibrium. 14 roof. Assume to the contrary that there exists x ′ = x ′′ ∈ X such that RIDE ( x ′ ) = RIDE ( x ′′ ) = α . Let v = x ′′ − x ′ . By uniform directionality there exists r ∈ R n such that v · J ( x ) · r > for any x ∈ X . Let ˆ X = { x ∈ X | x ′ i ≤ x i ≤ x ′′ i or x ′′ i ≤ x i ≤ x ′ i } be the subsetof X between x ′ and x ′′ . By the compactness of ˆ X , there exist δ > such that v · J ( x ) · r > δ for any x ∈ ˆ X . The fact that π and q are twice continuously differentiable implies that forany ǫ > there exists n such that (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) RIDE ( x ′′ ) − RIDE ( x ′′ ) + n X k =1 x ′′ − x ′ n ! · J n − kn x ′ + kn x ′′ !!! · r (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) < ǫ. The fact that
RIDE ( x ′ ) = RIDE ( x ′′ ) implies that ǫ > n (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) n X k =1 ( x ′′ − x ′ ) · J n − kn x ′ + kn x ′′ ! · r (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) = 1 n (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) n X k =1 δ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) = δ, and we get a contradiction for a sufficiently small ǫ .In the two-player case we present a simpler sufficient condition for uniqueness, namelythat the determinant of the Jacobian never changes its sign, which is equivalent to requiringthat the product of the cross-RIDE derivatives is either always larger, or always smaller,than the product of the self-RIDE derivatives. Claim . Let G be an underlying two-player game that satisfies Assumptions 4–5. Assumethat | J ( x ) | ≡ dRIDE ( x ) dx · dRIDE ( x ) dx − dRIDE ( x ) dx · dRIDE ( x ) dx = 0 for any x ∈ X . Then there exists at most one α -equilibrium. Proof.
Assume to the contrary that there exist x ′ , x ′′ such that RIDE ( x ′ ) = RIDE ( x ′′ ) .Assume WLOG that x ′ < x ′′ . For each x ∈ [ x ′ , x ′′ ] , let f ( x ) ∈ X be the unique profile thatsatisfies RIDE ( x , f ( x )) (uniqueness is implied by Assumption 6 of RIDE monotonicity).In particular it must be that x ′ = f ( x ′ ) and x ′′ = f ( x ′′ ) . In what follows we show that RIDE ( x , f ( x )) is strictly monotone in x , which contradicts RIDE ( x ′ ) = RIDE ( x ′′ ) .15bserve that the definition of f ( x ) implies that dRIDE ( x , f ( x )) dx = J ( x , f ( x ))+ J ( x , f ( x )) f ′ ( x ) ⇔ f ′ ( x ) = − J ( x , f ( x )) J ( x , f ( x )) .Next we calculate the derivative dRIDE ( x , f ( x )) dx = J ( x , f ( x )) + J ( x , f ( x )) f ′ ( x ) = J ( x , f ( x )) − J ( x , f ( x )) J ( x , f ( x )) J ( x , f ( x )) = − | J ( x , f ( x )) | J ( x , f ( x )) .J ( x ) never changes its sign due to Assumption 5. This implies that if the determinant | J ( x ) | never changes its sign, then RIDE ( x , f ( x )) is strictly monotone in x . C Microfoundations for biased estimation
C.1 Biased Price Competition Elasticity Estimates res α i < Suppose the analysts hired by each of the two firms decide to experiment with prices to findthe price elasticity of demand by alternating between a high price ( p H ) and a low price ( p L ),setting a low price (discount) µ L -share of the time. The experiment can be characterized bya level of sloppiness γ i ∈ [0 , . In a fraction γ i of the time, the analyst doesn’t monitor thefirm’s employees and does not carefully supervise that the employees choose the discounttimes uniformly at random. Hence, it is possible, for example, that the firm’s employeeswill implement discounts on days of low demand, possibly due to the employees having morefree time in these days to deal with posting different prices. In the rest of the time ( − γ i fraction), the analyst verifies that the prices are set randomly. Consequently, when eitheranalyst sets prices uniformly at random, there will not be correlation between the firm’sprices. This happens − γ γ fraction of the time. In the remaining γ γ fraction of thetime, there might be correlation between the firm’s prices, which we denote by ρ . The jointdistribution of prices conditional on the correlation ρ and the fractions γ , γ is described inTable 3. 16 L p H p L µ LL = µ L + µ L (1 − µ L ) γ γ ρ µ LH = µ L (1 − µ L )(1 − γ γ ρ ) p H µ HL = µ L (1 − µ L )(1 − γ γ ρ ) µ HH = (1 − µ L ) + µ L (1 − µ L ) γ γ ρ Table 3: Joint distribution of prices with correlation ρ and the fractions γ , γ When calculating the price elasticity of demand to decide how to change prices, theanalyst calculates: η i = − ∆ Q i Q i ∆ P i P i (C.1)Where ∆ Q i is the difference in average demand between high priced and low priced periods, Q i is the average realized demand, ∆ P i = p H − p L is the difference in price between highand low price periods, and P i = µ L p L + (1 − µ L ) p H is the average price set by the firm.The demand observed by firm i when setting price p i and when its competitor sets aprice p − i is Q i ( p i , p − i ) = a i − b i p i + c i p − i .Using the joint probabilities in Table 3, we find that Q i = a − ( b − c ) ( µ L p L + (1 − µ L ) p H ) and ∆ Q i = − ( p H − p L ) ( b − cγ γ ρ ) .Plugging into (C.1), firm i will estimate its price elasticity as: η i = ( b − cγ γ ρ ) ( µ L p L + p H (1 − µ L )) a − ( b − c ) ( µ L p L + p H (1 − µ L )) , (C.2)while the true elasticity is η Ti = b ( µ L p L + p H (1 − µ L )) a − ( b + c )( µ L p L + p H (1 − µ L )) . Hence the analyst will estimate thefirm’s price elasticity as being lower than η Ti when c > and ρ > . C.2 Biased Advertising Effectiveness Estimates Result in α i > Assume the firm’s sales at time t behave according to the linear model sales t = µ + x t + ǫ t where µ is the average sales, x t is the level of advertising, that can be x L or x H with x H > x L ≥ , and ǫ t is demand shock which is distributed i.i.d N (0 , . In this model, thetrue effect of advertising, d ( sales t ) dx t equals 1.The firm has a sales target µ and its advertising strategy is to increase advertising to x t +1 = x H if sales fall below µ at time t , i.e., if sales t < µ , and otherwise set x t +1 = x L .To estimate the effect of advertising, the firm looks at the difference in sales when ad-17ertising is increased or decreased (otherwise the change cannot be attributed to changes inadvertising) and takes the average to calculate E [∆ sales ]∆ x = E [ sales t +1 − sales t | x t +1 = x H ,x t = x L ] x H − x L + E [ sales t +1 − sales t | x t +1 = x L ,x t = x H ] x L − x H (C.3)More sophisticated approaches can take into account a weighted average of these estimatesand also take into account the baseline sales when advertising does not change.The lefthand part of the summand equals: E [ sales t +1 − sales t | x t +1 = x H , x t = x L ] x H − x L = µ + x H + E [ ǫ t +1 ] − ( µ + x L + E [ ǫ t | sales t < µ ]) x H − x L = = µ + x H − (cid:16) µ + x L − φ ( − x L )Φ( − x L ) (cid:17) x H − x L = 1 + φ ( − x L )Φ( − x L ) x H − x L > where φ ( · ) is the standard Normal pdf and Φ( · ) its cdf. The righthand part equals: E [ sales t +1 − sales t | x t +1 = x L , x t = x H ] x L − x H = µ + x L + E [ ǫ t +1 ] − ( µ + x H + E [ ǫ t | sales t ≥ µ ]) x L − x H = µ + x L − (cid:16) µ + x H − φ ( − x H )1 − Φ( − x H ) (cid:17) x L − x H = 1 − φ ( x H )Φ( x H ) x H − x L < Because φ ( x )Φ( x ) is decreasing in xx