CEO confidence bias and strategic choice: a general framework

ABSTRACT An owner of a firm may choose to hire an unbiased CEO or one with confidence bias. We develop a model that demonstrates that the owner’s optimal choice depends on whether the firm and rival choice variables are strategic substitutes or strategic complements. When choice variables are strategic substitutes or strategic complements for both firms, owners optimize by hiring overconfident CEOs. When choice variables are substitutes for one firm and complements for the rival firm, each firm optimizes by hiring an underconfident CEO. We show that the model applies to price and output competition, advertising, research and development spending, and product design.


Introduction
Seminal studies by Malmendier andTate (2005, 2008) show that many large corporations hire chief executive officers (CEOs) who are overconfident.Research further indicates that overconfident CEOs overpredict the likelihood of success, engage in more aggressive merger and borrowing behavior, and inflate estimates of future earnings. 1Despite the attention paid to CEO overconfidence, evidence of underconfidence exists as well (Ben-David, 2013;Moore & Cain, 2007). 2 For example, Chen, Sau Leung, Song, and Goergen (2019) identify industry-specific CEO confidence levels and find that public utility and telecommunication industries hire relatively underconfident CEOs.
Early theoretical studies were designed to rationalize the presence of overconfident CEOs.For example, Goel and Thaker (2008) and Campbell, Gallmeyer, Johnson, Rutherford, and Stanley (2011) show how moderate CEO overconfidence can attenuate corporate risk.Englmaier (2010Englmaier ( , 2011) ) and Englmaier and Reisinger (2014) investigate duopoly games that allow firm owners to hire CEOs who overestimate the effectiveness of various actions, such as spending on research and development, to increase firm demand.These studies demonstrate that an owner can increase profit by hiring an overconfident CEO in both Cournot and Bertrand product market games.This can occur when the

DEVELOPMENT ECONOMICS
action of an overconfident CEO profitably changes the behavior of competitors.More recent work proves that it can be profitable for owners to hire underconfident CEOs when there are strategic asymmetries (Schroeder et al., 2021a;2021b).
As Malmendier and Tate (2015) point out, the overconfidence literature provides a different reason for the agency problem in which CEOs fail to maximize simple profits.In the classic principal-agent problem, this failure is due to the fact that CEO compensation is not aligned with profits.In the overconfidence literature, however, the source of failure is CEO confidence bias.In this case, CEOs can fail to maximize simple profits even when they are financially incentivized to do so.Thus, a contract that aligns CEO compensation with profit need not affect CEO behavior or benefit owners. 3he purpose of this study is to develop a general model that encompasses previous and new results regarding confidence bias for a broad class of strategic variables.The model identifies conditions under which it is optimal for owners to hire perfectly rational, overconfident, or underconfident CEOs.One determining factor is whether firms compete in a strategic or non-strategic setting.Another is whether the relevant choice variables are strategic substitutes or strategic complements.We develop the model in section 2. In section 3, we show how our model subsumes a wide range of economic applications that are relevant to CEO confidence bias.The study concludes in section 4.

Confidence bias and strategic behavior
In this section, we develop a general framework for identifying conditions that support an owner's decision to purposefully hire a CEO with confidence bias.In our model, costs are sufficiently low to assure firm participation.Profit functions are strictly concave and twice-continuously differentiable, such that the first-and second-order conditions of profit maximization are met for all choice variables.
Consider a market where two firms (1 and 2) compete in a sequential game.Subscript i will identify one firm, and subscript j will represent firm i's competitor.In stage I, owners or boards of directors decide whether to hire CEOs who are perfectly rational, overconfident, or underconfident.The goal of each owner is to maximize the company's true profit, π.Once hired, all corporate decisions are delegated to CEOs. 4 In stage II, the CEO of firm i chooses the value of the strategic variable s i 2 S, where S � R is the feasible choice set.s i could represent a variety of choice variables, including output, price, product design, or research and development.Bias occurs when a CEO overestimates or underestimates the effectiveness of s i to increase profit.The CEO of firm i maximizes expected profit, which is based on potentially biased beliefs about the effectiveness of strategic variables.In each stage, decisions are made simultaneously.Information is perfect and complete.
Potential CEO bias is indexed by parameter ϕ.For the perfectly rational CEO who has an accurate assessment of the effectiveness of s i , ϕ i ¼ 0. When overconfident, the CEO of firm i overestimates the effectiveness of s i by ϕ i 2 0; 1 ð Þ.An underconfident CEO underestimates the effectiveness of s i , such that ϕ i 2 À 1; 0 ð Þ.In our model, Þ and the level of overconfidence (underconfidence) increases (decreases) with ϕ i .Owner search reveals the confidence level of each CEO candidate before hire.With this notation, firm i's profit is π i ϕ i ; ϕ j ; s i ; s j � � .Our main results depend upon the presence of strategic effects and whether s i and s j are strategic substitutes or complements (Bulow et al., 1985).At stage II, each firm has a unique best reply, denoted by s BR i ¼ argmax s i 2S π i s i ; s j À � .When s i and s j are strategic substitutes, the choice variables are combative -an increase in s j imposes a negative externality on firm i by lowering its marginal profit i:e:; @ 2 π i @s i @s j < 0 � � , and firm i's bestreply function has a negative slope.When strategies are complements, the choice variables are constructive -an increase in s j produces a positive externality for firm i by raising its marginal profit i:e:; @ 2 π i @s i @s j > 0 � � , and firm i's best-reply function has a positive slope.
Backward induction is used to identify the subgame-perfect Nash equilibrium (SPNE).In stage II, CEOs simultaneously choose s, based on the degree of confidence bias and given owner decisions in stage I. Firm i's Nash equilibrium value in this subgame is . In stage I, firm i's profit depends on first-stage choices, ϕ i and ϕ j , and the anticipated actions in stage II.Thus, owner i maximizes with respect to ϕ i .If deviation from perfect rationality (ϕ i ¼ ϕ j ¼ 0) raises owner profit, then it is optimal for an owner to hire a biased CEO in stage I.When evaluated at ϕ i ¼ ϕ j ¼ 0, the total effect is: The first set of terms on the right-hand side of the equality is the direct effect on firm i's profit of a marginal increase in ϕ i (i.e., greater overconfidence).The second set of terms identifies the indirect or strategic effect that results from a change in the action of firm j.
If Equation ( 1) is zero, then it pays to hire a perfectly rational CEO.If it is positive (negative), however, it is optimal to hire an overconfident (underconfident) CEO.Equation (1) demonstrates that an owner will hire a perfectly rational CEO when strategic effects are absent.From the first-order condition of profit maximization, @π i @s i ¼ 0. For a monopolist, @s j =@s i ¼ 0 because the firm has no competitors.In perfect competition, @s j =@s i ¼ 0 because each firm is infinitesimally small and its action has no effect on rival demand or cost conditions.Thus, in the absence of strategic effects, dπ i dϕ i ¼ @π i @s i ¼ 0 and an owner maximizes profit by hiring a perfectly rational CEO. 5n a strategic setting, an owner's commitment to hiring a biased CEO may change rival behavior and increase firm profit. 6The next three propositions identify conditions under which it is optimal for an owner to hire a CEO with confidence bias in a duopoly market.
The first two propositions consider the symmetric cases where the choice variables are either strategic substitutes or strategic complements for both firms.The third proposition treats the asymmetric case where the choice variables are strategic substitutes for one firm and strategic complements for the other firm.
Proposition 1: In the duopoly game where the choice variables, s i and s j , are strategic substitutes for both firms, it is optimal for each owner to hire an overconfident CEO (ϕ i > 0).Proof: From Equation (1), the direct effect is zero because @π i ϕ 1 ¼ϕ 2 ¼0 ð Þ @s i ¼ 0 from the firstorder condition of profit maximization.For strategic substitutes where s i and s j are combative, @s j @s i < 0 (i.e, firm j's best-reply function has a negative slope) and an increase in s j lowers firm i's profit Because overconfidence leads to an overestimation of the effectiveness of s i , @s i @ϕ i > 0. Thus, > 0, and it is optimal for the owner of firm i to hire an overconfident CEO (ϕ i > 0).7 Q.E.D.
Proposition 2: In the duopoly game where the choice variables, s i and s j , are strategic complements for both firms, it is optimal for each owner to hire an overconfident CEO (ϕ i > 0).Proof: From Equation (1), the direct effect is zero because @π i ϕ 1 ¼ϕ 2 ¼0 ð Þ @s i ¼ 0 from the firstorder condition of profit maximization.For strategic complements, where s i and s j are constructive, @s j @s i > 0 (i.e, firm j's best-reply function has a positive slope) and an increase in s j raises firm i's profit Because overconfidence leads to an overestimation of the effectiveness of s i , @s i @ϕ i > 0. Thus, > 0, and it is optimal for the owner of firm i to hire an overconfident CEO (ϕ i > 0).8 Q.E.D.
The next proposition considers the asymmetric case where the choice variables are strategic substitutes for one firm but strategic complements for the other firm.For concreteness, let s 1 and s 2 be strategic complements for firm 1 @ 2 π 1 @s 1 @s 2 > 0 � � and strategic substitutes for firm 2 @ 2 π 1 @s 1 @s 2 < 0 � � .That is, s 1 is combative and s 2 is constructive, such that best-reply functions are positively sloped for firm 1 and negatively sloped for firm 2.9 Examples of this case will be discussed in the next section.
Proposition 3: In the duopoly game where the choice variables, s i and s j , are strategic complements for firm 1 and strategic substitutes for firm 2, it is optimal for each owner to hire an underconfident CEO (ϕ i < 0).
These results indicate that an owner's decision to hire a biased CEO requires there to be strategic effects and depends on whether the choice variables are strategic substitutes or complements.In the absence of strategic effects, as with perfect competition and monopoly, it is optimal for owners to hire perfectly rational CEOs.In strategic settings where both choice variables are either strategic complements or strategic substitutes, it is optimal for owners to hire overconfident CEOs.In the asymmetric setting where the choice variables are strategic substitutes for one firm but strategic complements for the other firm, it is optimal for owners to hire underconfident CEOs.

Common economic applications
In this section, we analyze several applications of CEO confidence bias in strategic settings.We consider specific choice variables and show how our model serves as an umbrella for several established and new results.Famously, the first duopoly models of non-cooperative oligopoly were developed by Cournot (1838) and Bertrand (1883).In the Cournot model, the choice variable is output (q), and two firms simultaneously choose output levels: s 1 ¼ q 1 and s 2 ¼ q 2 .In the Bertrand model, the choice variable is price (p), and two firms simultaneously choose price levels: s 1 ¼ p 1 and s 2 ¼ p 2 .In both cases, owners/managers are assumed to be profit maximizers who are unbiased (i.e., perfectly rational).
Different results may emerge, however, when owners (or boards of directors) are different from managers.In this case, owners may commit to hiring managers who overor under-estimate demand.In the Cournot model, the choice variables are strategic substitutes for both firms, and the choice variables are strategic complements for both firms in the Bertrand model.Thus, Propositions 1 and 2 indicate that the SPNE is for owners to hire overconfident CEOs.Several authors verify that it is profit maximizing for owners to hire overconfident CEOs in Cournot and Bertrand settings for linear demand and cost functions. 12 Schroeder et al. (2021b) prove this with more general demand and cost conditions.
A related model that assumes strategic asymmetry has also emerged in the literature, the Cournot-Bertrand model.This is a hybrid model where one firm competes in output, as in Cournot, and the other firm competes in price, as in Bertrand (Bylka and Komar, 1976;Tremblay & Tremblay, 2019).In this case, the choice variables, s i ¼ q i and s j ¼ p j , are strategic substitutes for the Bertrand-type firm and are strategic complements for the Cournot-type firm.Consistent with Proposition 3, Schroeder et al. (2021b) prove that it is optimal for owners to hire underconfident CEOs with Cournot-Bertrand competition.
Real firms compete in a variety of other variables, including research and development (R&D), advertising, and product design.Models with these strategic options require an additional stage in the sequential game.With R&D, for example, owners choose the level of CEO confidence bias in stage I, firms compete in R&D in stage II, and firms compete in output (or price) in stage III.Confidence bias enters the model in stage II, where an overconfident (underconfident) CEO overestimates (underestimates) the effectiveness of R&D in increasing demand or decreasing costs.R&D can generate product innovations (i.e., create an improved product) or process innovations (i.e., reduce the cost of producing an existing product).R&D is combative when firm i's product innovation steals customers from firm j.When this is true for both firms, R&D i and R&D j are strategic substitutes.In contrast, R&D is constructive if information about firm i's process innovation (partially) spills over to firm j, thus benefiting firm j as well as firm i.In this case, R&D i and R&D j are strategic complements.Strategic asymmetry is also possible, where one firm focuses on product innovations and the other focuses on process innovations.In this instance, R&D i and R&D j would be strategic substitutes for one firm and strategic complements for the other firm.
As in the previous section, backward induction is used to identify the SPNE.In stage III, firm i's Nash equilibrium output is q NE i ϕ i ; ϕ j ; R&D i ; R&D j � � .In stage II, CEOs simultaneously make R&D decisions, given stage I realizations (ϕ i and ϕ j ) and the anticipated best-replies in stage III, q NE i and q NE j .At this stage, firm i's Nash equilibrium would be R&D NE i ϕ i ; ϕ j � � .Thus, in stage I, owners simultaneously make CEO decisions, given q NE i , q NE j , R&D NE i , and R&D NE j .In these games, Propositions 1 and 2 indicate that owners benefit from hiring overconfident CEOs when R&D i and R&D j are either strategic substitutes or strategic complements for both firms.Englmaier and Reisinger (2014) and Tondji (2021) consider such models of competition in R&D.They assume linear demand and production cost functions and that both firms engage in process enhancing R&D with positive spillovers (strategic complements).Their results are consistent with Proposition 2: owners benefit from hiring overconfident CEOs.To date, no one has investigated the cases where R&D i and R&D j are strategic substitutes for both firms, which would support overconfidence (Proposition 1), or are strategic substitutes for one firm but strategic complements for the other firm, which would support underconfidence (Proposition 3).
Another important choice variable is advertising.In this case, bias enters the game in stage II when CEOs have a potentially biased view of advertising effectiveness.As discussed by Marshall (1890) and Friedman (1983), advertising can be expansionary (constructive) when it increases market demand or predatory (combative) when one firm's advertising steals customers from its competitor.When both firms engage in combative (constructive) advertising, these choice variables are strategic substitutes (complements) and owners benefit from hiring overconfident CEOs (Propositions 1 and 2).There also are realworld examples where a maverick firm engages in combative advertising and competes with a dominant firm that chooses more constructive advertising campaigns (Tremblay and Tremblay, 2012).In this case, Proposition 3 implies that owners will prefer to hire underconfident CEOs.Schroeder et al. (2021a) analyze advertising games with these characteristics and find results that are consistent with Propositions 1, 2, and 3.
As a final example, consider the case where firms compete in product design, as discussed in Johnson and Myatt (2006).Here, firms have the option of developing a product that appeals to the masses or to a niche set of consumers.For example, an economy sedan like a Honda Civic might have mass market appeal, while a performance sedan like a BMW 330 might appeal to a smaller set of performance enthusiasts.In this application, overconfidence (underconfidence) occurs when a CEO overestimates (underestimates) the appeal of a particular design to consumers.If firms 1 and 2 both compete in either economy sedans or performance sedans, these choices are combative.As a result, owners benefit from hiring overconfident CEOs (Proposition 1).On the other hand, if firm 1 chooses to produce an economy sedan and firm 2 chooses to produce a performance sedan, an asymmetry can result.The introduction of the economy sedan by firm 1 may attract a new set of potential consumers to the market and benefit firm 2, while firm 2ʹs introduction of the performance sedan may steal sales from firm 1.In this case, the choice variables are strategic substitutes for firm 1 but strategic complements for firm 2. As a result, both owners benefit from hiring underconfident CEOs (Proposition 3).The issue of CEO confidence bias and product design has not been considered in the literature.

Concluding remarks
We develop a general duopoly model of CEO confidence and strategic choice that provides a unifying framework for the existing literature, as well as suggesting some new results.In the first stage of the model, owners or boards of directors choose the level of CEO confidence bias when hiring a CEO.In the second stage, this CEO chooses the value of the strategic variable, maximizing profit according to potentially biased beliefs.This model flexibly encompasses a wide range of specific models, given its general demand and cost conditions and for a variety of possible strategic choice variables, including output, price, R&D, advertising, and product design.The model identifies conditions under which it is optimal for owners to hire CEOs who are overconfident, underconfident, or perfectly rational.
In perfect competition or monopoly, the absence of strategic effects leads to the conclusion that it is optimal for owners to hire perfectly rational CEOs.In strategic settings, the result depends on whether the choice variables are strategic substitutes or strategic complements.When the game is symmetric, in that the choice variables are either strategic substitutes for both firms or strategic complements for both firms, it is optimal for each owner to hire an overconfident CEO.When the choice variables are strategic complements for one firm and strategic substitutes for the other firm, however, it is optimal for both firms to hire underconfident CEOs.
These results shed light on a variety of findings in the theoretical and empirical literatures.Empirically, we have seen that both CEO overconfidence and underconfidence exist in the real world.Similarly, the theoretical literature provides examples of duopoly models in which overconfidence or underconfidence is optimal.Our general framework highlights the key feature of these models that are driving the differing results: whether the choice variables are strategic substitutes or complements.
Although there are advantages to a general model, welfare analysis is infeasible.The studies found in Section 3 that used specific functional forms demonstrate that the effect of confidence bias on welfare varies across economic settings.For example, product market studies found that the presence of confidence bias increases welfare (i.e., consumer and total surplus) with Cournot competition, decreases welfare with Bertrand competition, and has an indeterminate effect on welfare with Cournot-Bertrand competition.When the choice variable is advertising, however, the welfare effect of confidence bias depends upon whether advertising is informative or persuasive, not on whether advertising is a strategic substitute or complement alone.These results further illustrate the inherent complexity of policy analysis regarding imperfectly competitive markets.An optimal policy would need to be case-specific.
This paper suggests several avenues for future research.First, analysis of alternative choice variables, using both general and specific models, may reveal additional instances in which confidence bias benefits owners and/or improves welfare.In addition, a future study might incorporate the classic principal-agent problem with confidence bias and allow CEO bias to influence the contract between owners and CEOs.Such a model would provide a more complete picture of agency problems and CEO bias.

Disclosure statement
No potential conflict of interest was reported by the author(s).

Notes on contributors
Elizabeth Schroeder is an Associate Professor of Economics at Oregon State University.She conducts research in applied microeconomics and microeconometrics.Research topics include evaluating the impact of education and credit policies in developing countries; studying partisan media and political speech; and the behavioral effects of nudges.She has published in journals such as Journal of Public Economics, Economics of Education Review, and Journal of Development Studies.Carol Horton Tremblay is Professor Emeritus of Economics at Oregon State University.Her fields of interest are applied econometrics, labor economics, industrial organization, and behavioral economics.Recent work includes research on neuroeconomics, brand names and advertising; mental illness and labor market outcomes; AIDS education; and the cigarette and brewing industries.Dr. C.H. Tremblay has published numerous articles in journals such as Economics Letters, Health Economics, Journal of Human Resources, Journal of Industrial Economics, and Review of Economics and Statistics.Victor J. Tremblay is Professor Emeritus of Economics at Oregon State University.His research interests are in industrial organization, microeconomics, and game theory.Recent work has focused on Cournot-Bertrand models, behavioral economics, the economics of the brewing industry, and the economics of advertising.Dr. V. J. Tremblay is on the editorial board of Games and of Theoretical Economics Letters.His work has been published in many economics journals, including Economics Letters, Journal of Industrial Economics, International Journal of Industrial Organization, Review of Industrial Organization, and Antitrust Bulletin.