Misperception and cognition in markets

Young, Benjamin Article Misperception and cognition in markets Games Provided in Cooperation with: MDPI – Multidisciplinary Digital Publishing Institute, Basel Suggested Citation: Young, Benjamin (2022) : Misperception and cognition in markets, Games, ISSN 2073-4336, MDPI, Basel, Vol. 13, Iss. 6, pp. 1-15, https://doi.org/10.3390/g13060071 This Version is available at: https://hdl.handle.net/10419/329982 Standard-Nutzungsbedingungen: Die Dokumente auf EconStor dürfen zu eigenen wissenschaftlichen Zwecken und zum Privatgebrauch gespeichert und kopiert werden. Sie dürfen die Dokumente nicht für öffentliche oder kommerzielle Zwecke vervielfältigen, öffentlich ausstellen, öffentlich zugänglich machen, vertreiben oder anderweitig nutzen. Sofern die Verfasser die Dokumente unter Open-Content-Lizenzen (insbesondere CC-Lizenzen) zur Verfügung gestellt haben sollten, gelten abweichend von diesen Nutzungsbedingungen die in der dort genannten Lizenz gewährten Nutzungsrechte. Terms of use: Documents in EconStor may be saved and copied for your personal and scholarly purposes. You are not to copy documents for public or commercial purposes, to exhibit the documents publicly, to make them publicly available on the internet, or to distribute or otherwise use the documents in public. If the documents have been made available under an Open Content Licence (especially Creative Commons Licences), you may exercise further usage rights as specified in the indicated licence. https://creativecommons.org/licenses/by/4.0/ Citation: Young, B. Misperception and Cognition in Markets. Games 2022,13, 71. https://doi.org/ 10.3390/g13060071 Academic Editors: Xiaojian Zhao and Ulrich Berger Received: 27 September 2022 Accepted: 21 October 2022 Published: 28 October 2022 Publisher’s Note: MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affiliations. Copyright: © 2022 by the author. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https:// creativecommons.org/licenses/by/ 4.0/). games Article Misperception and Cognition in Markets Benjamin Young Economics Discipline Group, UTS Business School, University of Technology Sydney, Sydney 2007, Australia; [email protected] Abstract: We consider a market setting where a consumer holds either a naive or sophisticated perception of their preference over products. We introduce the concept of a cognitive equilibrium, in which the consumer can transition between the cognitive states of naiveté and sophistication depending on the degree of exploitation in the market. We compare market outcomes under monopoly and competition. While competition unambiguously improves market outcomes when the consumer’s cognitive state is exogenous, it can strictly lower gains from trade when cognitive states are endogenously determined. Keywords: naiveté; sophistication; cognition; markets 1. Introduction There often exists a wedge between the preferences of an individual and their perception of these preferences. The economics literature has successfully incorporated such naive individuals into their models in order to explain empirical observations in a broad range of contexts 1 . Alternatively, individuals may have a sophisticated understanding of their true preferences. Whether an individual is naive or sophisticated is usually modeled as a fixed characteristic or ‘type’ of the individual in question 2 . There is psychological evidence, however, that suggests that movement between these cognitive states is more fluid 3 . In line with this idea, we provide a novel equilibrium concept called cognitive equilibrium in which a consumer can transition between naiveté and sophistication depending on the benefits and costs associated with doing so. We show that allowing for endogenous determination of the consumer’s cognitive state leads to novel findings in terms of how the degree of competition affects market outcomes relative to the exogenous case. Consider the following motivating example. A consumer seeks to enter into a new phone plan. Each plan has a number of features, such as the number of minutes for making calls, number of texts that can be sent, the amount of data included, etc. The value of each plan to the consumer will depend on the extent to which she will use each of these features (e.g., how often she will make phone calls, send text messages, and watch videos). At the time of selecting the plan, she expects to utilize each feature more than she actually will (i.e., is naive). The phone provider wants to take advantage of this misperception by offering more and more seemingly attractive features, even if they are not of use to the consumer. At the same time, however, the provider recognizes that, if they try to exploit the consumer too much, then they may become aware of their misperception and avoid entering the plan (i.e., become sophisticated). As such, the phone provider needs to take into account the consumer’s motivation to correct any preference misperception when designing plans. We develop an equilibrium theory that captures such potential for the consumer to endogenously transition from naiveté to sophistication. The model is as follows. A set of firms offer menus of products to a consumer. Products provide the consumer with a level of a provision of some good at a price. The consumer is either naive or sophisticated. If naive, they overvalue products relative to their sophisticated counterpart by perceiving that the good is more valuable to them than it actually is. Instead, if sophisticated, they correctly perceive their preference over products. The consumer starts Games 2022,13, 71. https://doi.org/10.3390/g13060071 https://www.mdpi.com/journal/games Games 2022,13, 71 2 of 15 out naive but can endogenously transition from naiveté to sophistication by investing in costly cognition. Specifically, the consumer makes this transition if the difference between the true utility received if they are sophisticated (and able to make an optimal decision) and that which they receives if naive (and make a sub-optimal decision) exceeds some cognitive cost. We introduce a solution concept called cognitive equilibrium in which firms take into account the effect that the products they design have on the consumer’s motivation to hold correct beliefs over preferences. We examine how the degree of competition affects market outcomes in our setting. We start by comparing monopoly to competition when the consumer’s cognitive state is exogenous (i.e., the consumer is deterministically sophisticated or naive). In this case, we find that competition is an unambiguous improvement over monopoly: competition increases consumer surplus at no cost to market efficiency. This is because competition erodes profits and transfers these rents to consumers, providing them with all gains from trade. Next, we compare cognitive equilibrium outcomes under monopoly and competition and show that the profit-reducing effect of competition can actually negatively impact on market outcomes. In particular, we derive the unique cognitive equilibrium outcome under monopoly and compare this to a natural cognitive equilibrium under competition. In this case, we show that total surplus is strictly higher under monopoly relative to competition; that is, monopoly can be strictly more efficient than competition when the consumer is able to transition from naiveté to sophistication. This occurs because competition erodes profits, which incentivizes competing firms to offer relatively inefficient products that are very attractive to a naive consumer in an attempt to earn profits. Instead, since the monopolist can earn profits, they are willing to offer a product more in-line with the consumer’s true preference. These findings have important policy implications. Since monopoly is more efficient than competition when the consumer’s cognitive state is endogenously determined, alternative policies, such as lump-sum redistributions from a monopolist to consumers, can improve on competitive outcomes. As such, competition is not the most effective tool at mitigating poor market outcomes resulting from consumer naiveté. Instead, if one makes the standard assumption that cognitive states are exogenously determined, one reaches the conclusion that competition unambiguously improves market outcomes. Hence, policymakers should display caution when proposing competition as a one-size-fits-all solution to the issues created by consumers exhibiting behavioral biases if such biases are endogenously determined. As such, the findings of this paper constitute important novel predictions that warrant further empirical investigation. The paper proceeds as follows. In Section 2 we discuss the related literature. In Section 3 we provide the details of the model. Section 4.1 investigates market outcomes when consumer’s cognitive state is exogenously determined. Section 4.2 allows for the consumer to endogenously transition from naiveté to sophistication and compares the cognitive equilibrium outcomes of monopoly to competition. Section 5concludes. All proofs are provided in Appendix A. 2. Related Literature There is a large literature investigating consumer naiveté in market settings. Some examples include work that looks at naiveté in self-control problems [ 8 ] and work in which consumers ignore hidden add-on prices [ 6 , 10 – 12 ]. A common finding in this literature is that competition does not correct the inefficiencies created by the presence of naive consumers, but often increases consumer surplus by transferring gains from trade from firms to consumers. This literature, however, assumes that naiveté is a fixed characteristic of the consumer. We contribute to this literature by showing that when the consumer’s cognitive state is endogenously determined, competition can strictly decrease efficiency. There is a selection of papers that find that competition can have negative consequences on market outcomes when consumers are naive. Herweg and Rosato argue that competition can increase inefficiency when a deceptive firm enters the market to compete Games 2022,13, 71 3 of 15 against a transparent retailer [ 13 ]. This relies on the fact that competing firms can create market segmentation by taking advantage of the consumer’s naiveté. In our paper, instead, all firms are identical and, as such, the environment is perfectly competitive. As such, we show that competition can be surplus-reducing even though all gains from trade are transferred from firms to the consumer. Jimenez-Gomez establishes that competition may reduce consumer welfare when firms can ‘phish’ consumers by introducing a wedge between their experienced utility and their decision utility [ 14 ]. This is because competition lowers prices, which can induce consumers to make make purchases that they otherwise should avoid. Instead, the driving force in our paper is that competing firms create large distortions in an attempt to earn profits, while a profit-making monopolist is more willing to align provision with true fundamentals. This paper also relates to the literature on contract design with either naive consumers or other cognitive considerations. Salant and Siegel provide a model where a consumer is subject to framing effects that increase the attractiveness of products [ 15 ]. The authors consider a form of consumer protection that is ex-post in nature: the frame can wear off and the consumer can return the product. Instead, we consider a form of consumer protection that is ex-ante in nature: the consumer’s cognitive state is determined before they select a product and is determined as a function of the products offered. Moreover, we consider the role of competition while Salant and Siegel focus on the case of monopoly. Eliaz and Spiegler investigate how a monopolist would effectively screen agents that have private information regarding their degree of naiveté [ 16 ]. Instead, we focus on how market structure affects market outcomes in a complete information environment where the consumer can transition between naiveté and sophistication. Finally, Tirole provides a model where parties to a contract can choose to keep some default, potentially sub-optimal, contract or to invest in cognition to investigate whether the contract can be improved upon [ 17 ] . In their paper, the contracting parties hold a sophisticated understanding of the environment, so that no party is exploited in equilibrium. Instead, we speak to exploitation in markets as our consumer can endogenously end up misperceiving their preference. The paper also relates to other notions of naiveté applied in other economic contexts. For example, receivers may naively interpret signals in communication settings [ 18 , 19 ]. Again, these papers model naiveté as a type of the receiver, while we consider the role of endogenous naiveté. Finally, Martin considers a naive buyer who does not properly understand equilibrium strategies in a strategic pricing game, but can acquire information about this strategy at cost [ 20 ]. Our model is similar in that our consumer may misperceive how they should behave, but is less likely to behave in this way the greater the benefits to avoiding doing so. Our focus, however, is on both the efficiency of, and the degree of exploitation in, the market, while Martin focuses on the informativeness of equilibrium prices. Finally, the paper contributes to the literature on motivated beliefs (see [ 21 ] for a detailed review). This literature generally seeks to rationalize the underlying motivations for why an individual may hold biased, self-serving beliefs [ 22 – 24 ]. This paper, on the other hand, takes biased beliefs as given and instead focuses on whether the individual is motivated to correct this misperception. This allows us to explore how firms manipulate a consumer’s motivation to be sophisticated. 3. The Model 3.1. Formal Details The Environment: There are N≥ 1 profit-maximizing firms that each provide a finite menu of products to a consumer. A product is a bundle x≡(q , t) , where q≥ 0 denotes the quality or level of provision of some good and t∈R is the price. Providing a product at quality q costs firm i C(q) , i= 1, . . . , N , where C is strictly increasing, strictly convex, differentiable, and C( 0 ) = C0( 0 ) = 0. Let h denote the inverse of C0 (the derivative of C ), which exists since Cis strictly convex4. If the consumer chooses product x= (q , t) from firm i , firm i earns profits equal to t−C(q) . Else, if the consumer does not purchase from firm i , firm i earns zero profits. Let Games 2022,13, 71 4 of 15 Xi denote the finite product menu designed by firm i and let X=N S i=1 Xi denote the entire set of products that the consumer can choose from. We assume that the consumer chooses only one product from X5. Consumer Preferences: The consumer decides whether to select a product from X or not. If the consumer has preference parameter ˜ θ≥ 0, the consumer values product x= (q,t)according to U(x|˜ θ)≡˜ θq−t. (1) Instead, if the consumer does not purchase a product from X , they receive reservation utility of zero, which is equivalent to selecting the product ( 0, 0 ) for all ˜ θ . Hence, we assume without loss of generality that (0, 0)∈X. Sophistication and Naiveté: The consumer will be in one of two cognitive states: they will either be sophisticated or naive. More precisely, assume that the true preference parameter of the consumer is given by θ> 0. If the consumer is sophisticated (denoted by S ), they understand that their true preference parameter is θ and, as such, value product x according to VS(x)≡U(x|θ) . Instead, if the consumer is naive (denoted by N ), they misperceive their preference over products by overvaluing provision of the good relative to its price. Specifically, they perceive that their preference parameter is given by θ+∆ , ∆> 0, so that they value product x according to VN(x)≡U(x|θ+∆) . We call ∆ the consumer’s degree of naiveté as it captures the extent to which a naive consumer misperceives their true preference. Notice that VN(x)>VS(x) for any product x= (q , t) with q> 0. Hence, the naive consumer over-values products relative to their sophisticated counterpart. Endogenous Naiveté : The consumer is initially naive but can endogenously transition from naiveté to sophistication through an investment in cognition. If they do not choose to make this investment, then the consumer utilizes the misperceived preference θ+∆ to make decisions. Formally, let a∈ {S , N} denote the cognitive investment decision of the consumer, where a=S implies an investment in cognition (i.e., a decision to become sophistication). We assume that such an investment is costly: the consumer incurs a cognitive cost of κ>0 if and only if they choose a=S. Suppose that the consumer faces menu of products X . Let xk denote a product that the consumer in cognitive state k∈ {S , N} selects; that is, xk∈arg max x∈XVk(x) . Then, the consumer’s true utility from choosing to be in cognitive state a∈ {S , N} when facing menu Xis given by V(a|X)≡VS(xa)−κ×I[a=S], (2) where I(·) is an indicator function. It is optimal for the consumer to invest in cognition if V(S|X)≥ V(N|X) and it is optimal for the consumer not to invest in cognition when V(S|X)≤ V(N|X). Cognitive Equilibria: We now define the relevant equilibrium concept that we use to make predictions, which we call a cognitive equilibrium. To do so, let σk:X→X denote a selection rule for the consumer in cognitive state k∈ {S , N} and α:X→ {S , N} a cognitive-investment strategy from an arbitrary menu X . Let πi(Xi , X−i , σS , σN , α) denote the expected profit of firm i when offering menu Xi , given that the rest of the firms offer jointly menu X−i≡S j6=i Xj and the consumer’s selection rules are given by σk , k∈ {S , N} and cognitive strategy is α . A notion of equilibrium in which the consumer’s cognitive state is endogenously determined is now defined. Definition 1. The tuple (X∗ 1 , . . . , X∗ N , σ∗ S , σ∗ N , α∗) is a cognitive equilibrium if, for all finite menus of products X, (i) σ∗ k(X)(x)>0only if x ∈arg max ˜ x∈XVk(˜ x)for k ∈ {S,N}; and (ii) α∗(X)(a0)>0only if a0∈arg max a∈{S,N}V(a|X), and X∗ imaximizes πi(Xi,X∗ −i,σ∗ S,σ∗ N,α∗)for each firm i ∈ {1, . . . , N}. Games 2022,13, 71 5 of 15 Essentially, a cognitive equilibrium is a subgame perfect equilibrium of a market game in which, first, each firm simultaneously designs a menu of products and, given the menu of products faced, the consumer decides whether or not to invest in cognition and makes a product selection given the realized cognitive state. Condition (i) of Definition 1requires the consumer to only select products that are optimal from the perspective afforded by each cognitive state that may realize. Instead, condition (ii) requires that the cognitiveinvestment decision is determined optimally given how the consumer would behave in each cognitive state. The final requirement is that the product-menu provided by a given firm constitutes a best response to other firms’ menus given the consumer’s decision rules. 3.2. Efficient and Exploitative Good Provision We now define the market outcome variables that are of interest in this paper. Let W˜ θ denote the maximal achievable total surplus when the preference parameter of the consumer is given by ˜ θ ; that is, W˜ θ≡max q≥0 ˜ θq−C(q) . The maximizer of this problem is h(˜ θ) , so that h(˜ θ) is the efficient level of provision of the good if the consumer has preference parameter ˜ θ . Since the true preference parameter of the consumer is θ , we have the following natural definitions of both an efficient product and an efficient market. Definition 2. A product x= (q , t) is efficient if q=h(θ) . A market is efficient if it generates total surplus equal to Wθ. Note that there always exists an efficient product that the consumer is willing to accept even if they are naive 6 . As such, the market can be efficient only if the consumer does not invest in cognition. Instead, if the consumer invests in cognition, the highest surplus that such a market can attain is Wθ−κ. Next, we define what is meant by consumer exploitation in our framework. Recall that the consumer can guarantee herself a reservation utility of at least zero by choosing the outside option. This leads to the following natural definition of consumer exploitation in the market. Definition 3. The market is exploitative if the true utility of the consumer is strictly lower than zero. Definition 3states the market is exploitative if the consumer earns true utility lower than their reservation level. This can happen either because the consumer naively chooses an exploitative product (i.e., one in which they overpay for provision) or because the consumer incurs cognitive costs that exceed the benefit of whichever product is selected under sophistication. 3.3. Discussion of Key Modeling Assumptions We have modeled the consumer’s misperception in a relatively general way: the consumer simply over-values provision of the good relative to their true preference. This could be for a number of reasons. For example, the consumer may be subject to framing or advertising effects that make them over-value the product [ 15 ], the consumer may not be paying attention to certain factors that affect their preference for the good [ 25 ], or may have alternative self-serving motivations to hold biased beliefs (e.g., thinking that they are more likely to go to the gym as it allows them to feel motivated to get fit). We do not take a stand on precisely the reason for this misperception and instead focus on the consumer’s incentives to correct this bias through costly cognitive investment. Regarding cognition, we have assumed the consumer has access to a very simple cognitive technology: if they invest in cognition they learn their true preference, θ , (i.e., become sophisticated), while if they do not make this investment they make decisions using the misperception, θ+∆ (i.e., stay naive). We do this to starkly illustrate how cognitive investment can serve as a bridge between naive decision-making and its sophisticated counterpart. Games 2022,13, 71 6 of 15 4. Results 4.1. Exogenous Cognitive States We first present results for a benchmark model in which the consumer’s cognitive state is determined exogenously; that is, the consumer’s cognitive state is fixed at either sophistication or naiveté and the firms know the realization of this cognitive state. Equivalently, we solve for a cognitive market equilibrium where the consumer’s cognitive cost is either κ= 0 (i.e., the consumer is always sophisticated) or κ= +∞ (the consumer is always naive). The first proposition describes the outcome under monopoly. Proposition 1. Under monopoly, a consumer in exogenously-known cognitive state selects (a) the efficient product xm S≡(h(θ),θh(θ)) if sophisticated; and (b) the inefficient product xm N≡(h(θ+∆),(θ+∆)h(θ+∆)) if naive. According to Proposition 1, the consumer is provided an efficient level of the good when sophisticated and is over-provided the good when naive. This is intuitive since the consumer over-values the good when naive and so the monopolist increases provision in order to satiate this preference and extract extra rents. This results in exploitation of the naive consumer. We now compare this outcome under monopoly to the competitive outcome. First, we derive the products selected by the consumer in each cognitive state in a competitive environment. To this end, suppose that there are N≥ 2 firms that all know the consumer’s cognitive state. The following proposition describes the equilibrium outcome in this competitive environment. Proposition 2. Under competition, a consumer in exogenously-known cognitive state selects (a) the efficient product xc S≡(h(θ),C(h(θ)) if sophisticated; and (b) the inefficient product xc N≡(h(θ+∆),C(h(θ+∆))) if naive. Proposition 2states that, similar to the case of monopoly, provision is efficient to a sophisticated consumer but the good is inefficiently over-provided to a naive consumer. Thus, competition does not correct the inefficiency created by naive consumers. It does, however, affect the price that the consumer pays for provision. A full comparison of market outcomes under monopoly and competition are provided in the following proposition. Proposition 3. If the consumer’s cognitive state is exogenously known, then (a) total surplus under monopoly and competition coincide; and (b) consumer surplus is strictly higher under competition relative to monopoly. Proposition 3establishes that the degree of competition has no effect on market efficiency when the consumer’s cognitive state is exogenously determined. Specifically, under both monopoly and competition, a sophisticated consumer selects a product with efficient provision of the good, h(θ) , while the naive consumer is over-provided the good at level h(θ+∆). The intuition for this result is as follows. Under both competition and monopoly, market forces incentivize the maximization of total surplus as if the consumer’s preference parameter was that which they actually perceive. In the case of sophistication, this results in efficient provision. Instead, when the consumer believes their preference parameter is greater than it is, the resulting provision in the market aligns with this misperception. The main difference between monopoly and competition is the transfer the consumer makes to firms for this common level of good provision. Specifically, part (b) of Proposition 3 states that the consumer is strictly better off under competition relative to monopoly. This is because competition drives down the price of provision which, consequently, increases the gains from trade realized by the consumer. This effect is present Games 2022,13, 71 7 of 15 regardless of whether the consumer is sophisticated or naive: all surplus (whether efficient or inefficient) accrues to the consumer under perfect competition. In summary, one can conclude that competition unambiguously improves market outcomes when the consumer’s cognitive state is common knowledge. This is broadly consistent with findings in the behavioral economics literature that suggests that competition does not necessarily increase market efficiency in the presence of exogenously naive agents but, rather, serves to transfer (inefficient) surplus from firms to consumers [ 6 , 8 ]. As such, competition is an effective policy tool to mitigate naiveté-based exploitation in this case. When the consumer’s cognitive state is endogenously determined, however, we will see that such a clear dominance relation cannot be established. We explore this in the next section. 4.2. Cognitive Equilibrium In this section, we consider the model described in Section 3in which the consumer’s cognitive state is endogenously determined. We first derive the unique cognitive equilibrium outcome under monopoly. Then, we derive a natural cognitive equilibrium under competition and, finally, compare monopoly and competitive cognitive equilibrium outcomes. 4.2.1. Monopoly We first consider the case where there is monopolistic provision of the good to the consumer. Since the consumer is only ever in one of two cognitive states, the monopolist needs to design only two products: xS which the consumer selects as a sophisticate, and xN which the consumer selects as a naif. Moreover, it is without loss of generality to assume that the consumer does not invest in cognition; that is, the consumer is incentivized by the monopolist to remain naive and select product xN . This is because the optimal product to offer a sophisticate is (h(θ) , θh(θ)) (derived in Proposition 1), which also satisfies the participation constraint of the naive consumer 7 . As such, we can write the monopolist’s optimization problem as follows: max xS,xN tN−C(qN)(3) subject to (IC-S):VS(xS)≥max{VS(xN), 0} (IC-N):VN(xN)≥max{VN(xS), 0} (No-Think):VS(xS)−VS(xN)≤κ. Constraints (IC-S) and (IC-N) ensure that the sophisticated and naive consumers, respectively, are willing to select the products designed for them. Instead, constraint (NoThink) ensures that the consumer does not invest in cognition but, rather, makes their product choice holding biased beliefs. The following proposition describes the product that the consumer selects in the cognitive equilibrium under monopoly. Proposition 4. Let κm≡∆h(θ) and κm≡∆h(θ+∆) . The product selected by the consumer in the cognitive equilibrium under monopoly, xm N= (qm N,tm N), has qm N=     h(θ)if κ≤κm κ/∆if κ∈(κm,κm) h(θ+∆)if κ≥κm. (4) The consumer is exploited for all cognitive costs κ>0. Proposition 4describes the product that the consumer selects in the unique cognitive equilibrium outcome under monopoly. The left panel of Figure 1illustrates the level of Games 2022,13, 71 8 of 15 good provision, qm N , to the consumer as a function of cognitive costs, κ . When κ is large, the monopolist provides the same product that is provided to an exogenously naive consumer (Proposition 1). Instead, when κ is small, the consumer is provided with an efficient level of the good, h(θ) . Finally, for intermediate levels of κ , provision in the monopolist’s optimal product is strictly in-between the levels provided to exogenously naive (i.e., h(θ+∆) ) and exogenously sophisticated (i.e., h(θ) ) consumers. Thus, provision to an endogenously naive consumer is always closer to the efficient level than to their exogenously naive counterpart. The intuition for this result is as follows. There are two main constraints that the monopolist must consider when designing the optimal product: (1) the participation constraint of the naive consumer and (2) the constraint that ensures the consumer does not invest in cognition. The participation constraint pushes provision towards the naif-optimal level, h(θ+∆) , as it utilizes the consumer’s misperception, θ+∆ . Instead, the (No-Think) constraint pushes provision towards the efficient level, h(θ) , as it is determined by the consumer’s true preference parameter, θ . When κ is sufficiently large, the former dominates and the consumer is over-provided the good. Instead, when κ is sufficiently small, the latter dominates and provision is efficient. Finally, for intermediate κ both constraints bind and optimal provision falls between these two levels. Proposition 4also states the consumer is exploited as long as the cost of becoming sophisticated is strictly positive. Thus, the monopolist is still able to take advantage of the fact that the consumer is bounded in their rationality. At the same time, the fact that the consumer can transition to sophistication does reduce exploitation, at least when the cost of cognition is small. This is because the threat of becoming sophisticated forces the monopolist to decrease exploitation in order to depress incentives to invest in cognition. The right panel in Figure 1illustrates consumer surplus in the cognitive equilibrium outcome under monopoly as a function of κ. κ qm N κmκm h(θ) h(θ+∆) (a) 0κ CSm κm −κm (b) Figure 1. Graphical depiction of the cognitive equilibrium outcome under monopoly. Panel (a) displays how good provision, qm N , varies with cognitive cost κ , while panel ( b ) displays how consumer surplus, CSm, varies with cognitive cost κ. 4.2.2. Competition We now consider the case where N≥ 2 firms compete for the consumer’s attention. We focus on a natural cognitive equilibrium under competition, which we call the (∗) - cognitive equilibrium. Recall the products xc S and xc N that are offered to sophisticates and naifs, respectively, in a competitive environment with exogenously known cognitive states (Proposition 2). The following proposition establishes that it is also a cognitive equilibrium under competition for each firm to offer precisely the menu of products {xc S,xc N}. Games 2022,13, 71 15 of 15 14. Jimenez-Gomez, D. Nudging and Phishing: A Theory of Behavioral Welfare Economics. 2018. Available online: https: //ssrn.com/abstract=3248503 (accessed on 20 October 2022). [CrossRef] 15. Salant, Y.; Siegel, R. Contracts with Framing. Am. Econ. J. Microecon. 2018,10, 315–346. [CrossRef] 16. Eliaz, K.; Spiegler, R. Contracting with Diversely Naive Agents. Rev. Econ. Stud. 2006,73, 689–714. [CrossRef] 17. Tirole, J. 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