Social Sciences & Management Studies - Juniper Publishers Abstract In this paper, we study the individual payoff effects of over confident self-perception in teams. In particular, we demonstrate that the welfare of an overconfident agent who works in a team with a rational agent or in a team with an overconfident agent can be higher than the welfare of the members of a team of two rational agents. This result holds irrespective of the assumption about the agents’ awareness of their colleague’s bias. Moreover, we show that an overconfident agent is always better off when he is unaware of a potential bias of his colleague. Thus, our results provide a potential rationale for the widespread dissemination of overconfidence. Keywords: Overconfidence; Team Production; Unawareness; Self-perception; Synergy effects JEL classification: D21, D62, L23 Go to Introduction Considerable evidence from psychology suggests that individuals tend to overestimate their own skills (e.g. [1-4] for recent reviews see [5-7]).1 Given the apparent relevance of the phenomenon for many economic contexts, the effects of overconfidence have also received considerable attention in the economic literature2. One prevalent effect of overconfidence seems to be that individuals who overestimate their own skill tend to work harder than individuals assessing their ability correctly (see, e.g., [20-22] and more recently [23]). Interestingly, the effort-increasing effect of overconfidence implies that the bias of some agents can affect the actions or payoffs of other agents. For example, the bias may change the incentive structure of the others if the agents’ payoffs depend not only on their own effort but also on the effort of others, e.g. in a teamwork setting, or the bias may change the payoff and/or the optimal incentive scheme from the perspective of a principal. Yet, such changes may crucially depend on the information agents possess about the actions and/or biases of the (overconfident) agents since agents can only react to what they observe or believe. The importance of the information structure is, for example, demonstrated by Santos-Pinto [24] in a principal- agent setting. Focusing on the principal, Santos-Pinto considers a situation where the principal can condition wages on each agent’s output. He shows that overconfidence is beneficial for the principal if effort is observable, while it need not be beneficial in the presence of moral hazard. In the present paper, we take up the discussion about the effects of overconfidence and analyze (unlike Santos-Pinto) a model of a teamwork situation with effort complementarities (see also Hakens and Katolnik [25] who study optimal team size in teams of overconfident agents). We first consider the potential advantage overconfident agents may have in an environment of mainly rational agents. The argument is related to works by De la Rosa [26], Gervais and Goldstein [27] or Hvide [28]. De la Rosa [26], for example, analyses welfare effects of overconfidence in a setting in which firms compete for an overconfident and risk-averse agent; he finds that the agent benefits when his bias is moderate. Along the same lines, Gervais and Goldstein [27] analyze a model of team production with effort complementarities. They show how overconfidence reduces free-riding, how it might increase both a rational as well as an overconfident agent’s welfare and give rise to a Pareto-improvement. Hvide [28], in turn, considers a case where the agent can actually choose the beliefs about his ability and shows that biased beliefs can be beneficial to the agent - as they may improve his outside option - while they are detrimental for the firm. 1 Note that the notion of overconfidence in general is not uncontested [8-11]. A recent meta-study by Koehler et al. (2002), however, describes overconfidence as a prevalent phenomenon. 2 For example, effects of overconfidence on decisions by managers and stock-traders have been analysed by Grinblatt and Keloharju [13], Malmendier and Tate [14,15], Heaton [16], Hirshleifer and Luo [16], and Kyle and Wang [18]. The effects on employee turnover and firm profits are analyzed by Hoffman and Burks [19]. In a second step, we then ask how individual payoffs are affected if both team members are biased and how awareness of the biases of others impacts on the agents’ payoffs. Whether or not people are actually aware of the bias of others, of course, remains an empirical issue which so far has received little attention. However, the findings by Ludwig and Nafziger [29] indicate that overconfident people tend to be unaware of the biases of others. Also, Bruhin et al. [30] find that subjects in an experiment do not appear to strategically respond to overconfidence of another team member. In the subsequent analysis, we show that overconfidence may not only enhance the team’s productivity (due to increased efforts), i.e. benefit the firm, but may also increase the welfare of the biased agent himself. And this holds in a team of one overconfident and one rational agent as well as in a team of two overconfident agents. Moreover, the result is particularly strong if the considered agent’s overconfidence is combined with unawareness of other people’s biases (despite the fact that being aware of the other’s bias is closer to the true state of the world). Thus, our results not only provide a potential rationale for the wide dissemination of overconfidence suggested by the studies cited above. They also provide a potential rationale for the empirical finding that overconfident people appear to be unaware of the biases of others [29,30]. The intuition behind these results is rather straightforward: Due to the effects of synergy, overconfidence of another team member increases the optimal effort level for any agent who is aware of this bias. However, if an agent is overconfident himself, his effort level is already above the individual optimum – because of his own bias which he is unaware of. Awareness of a colleague’s bias, then, leads to a further (suboptimal) increase in his effort. By contrast, lack of such awareness keeps the expectation about the colleague’s effort and, hence, the agent’s extra effort, which he exerts in order to exploit effort complementarities, low. In combination with the increase in the agent’s effort due to his own overconfidence, the agent’s effort choice gets closer to the overall individual optimum than if he were aware of the other’s bias. In a sense, all necessary upward-adjustments in the agent’s effort (in order to exploit the synergies from the colleague’s overconfidence) are already accounted for in the agent’s effort choice - although for a different reason, namely the agent’s own overconfidence (which he is unaware of). And this intuition essentially covers both cases, i.e. a team with one biased and one rational agent and a team with two biased agents. The rest of the paper is structured as follows: Section 2 presents our baseline model of a teamwork situation with effort complementarities. Section 3 introduces overconfidence in a team of one overconfident and one rational agent. Moving to teams of two overconfident agents, Section 4 consider the effects of changes in the information structure in such instances. Section 5, then, compares teams of two overconfident agents with teams of two rational agents and summarizes the main points of the analysis. Section 6 concludes. Go to The Baseline Model Consider a firm whose output generates from a single oneperiod project which is carried out by two risk neutral agents, i = 1, 2, where teamwork is implemented in order to create positive externalities 3 . The value of the project is the value of its expected cash flow which depends on the agents’ efforts, e i , and their abilities, a i ; for the sake of argument, we assume a 1 = a 2 = a. 4 Moreover, we assume that agent i‘s expected return from the project, denoted by R i (e i , e -i ) is increasing in effort and ability and that the marginal return to effort is increasing in ability, i.e. d 2 R i /d e i da i > 0. 5 The agents’ cost of effort is denoted by c(e i ) with c (0) = 0, c ' > 0 and c " > 0. Finally, in order to make the subsequent discussion meaningful, we follow, for example, Gervais and Goldstein [15] and assume that the agents’ efforts are strategic complements, i.e.: 6 3 On positive externalities through teamwork see e.g. Alchian and Demsetz [32], Grossmann and Hart [33], Alchian and Woodward [34], Aghion and Tirole [35], Jensen and Meckling [36], or Holmström and Roberts [37]. 4 Note that assuming equal ability is not restrictive for the present argument. In particular, the focus of the analysis is on the individual effects of overconfidence and information about such biases of other team members. And, as such, the discussion is essentially confined to the consequences of changes in these parameters for one of the two agents. In fact, actual ability is not explicitly accounted for as we will treat it as fixed throughout the analysis. 5 The complementarity assumption between ability and the value of effort is reasonable in many settings since it is often the case that a more able agent needs less time to carry out a certain task. 6 Efforts being strategic complements corresponds to the slope of the best reply being positive, i.e. Under the above assumptions, the maximization problem of agent i can be written as follows: with first-order condition (FOC): Moreover, the corresponding second-order condition (SOC) is satisfied if: which we assume to hold in the following. Substituting the corresponding equilibrium efforts, denoted by with i = 1, 2, into the agents’ payoff functions, we obtain the following general expression for the agents’ equilibrium payoffs in the case without overconfidence: These payoffs will serve as our benchmark for later comparisons. Go to Overconfidence In order to analyze the effects of overconfidence, we first consider a team in which one agent, say agent 2, is overconfident, while the other agent, agent 1, is rational and aware of agent 2’s bias. In particular, we assume that agent 2 overrates his own skill by b 2 > 0, i.e. his perceived ability is a ':= a + b 2 . 7 Moreover, we assume that agent 2 is not aware of his own bias so that agent 2’s maximization problem can be written as follows:8 where denotes the expected return to the project as (wrongly) perceived by the biased agent . The resulting FOC is given by: with SOC: Note that, compared to a situation without overconfidence (b 2 ) = 0 the effort of agent 2 increases for a given effort level of agent 1, as the marginal return to effort of agent 2 is increasing in b 2 : recall that, by construction, the nominator of (9) is positive due to the assumed positive effect of the agents’ ability on marginal productivity - and the denominator is negative which follows from the SOC (see (4) and (8)). The maximization problem of agent 1, in turn, is the same as described in the baseline model of a fully rational team except that agent 1 now takes the bias b2 of agent 2 into account; i.e. agent 1 knows that agent 2’s effort changes due to his overconfidence and accounts for this. Thus, agent 1 knows that agent 2 is biased and while agent 2 knows this, he disagrees with agent 1, i.e. the agents agree to disagree as, for example, in Morris [39] and Squintani [40].9 Accordingly, optimal efforts are derived as follows: Agent 2 maximises his incorrectly perceived payoff (correctly) anticipating that agent 1 is rational (and that agent 1 believes that agent 2 is biased); and agent 1 maximizes his actual payoff (correctly) anticipating that agent 2 is biased and thus maximizes his perceived payoff. Denoting the resulting efforts with and , the agents’ individual payoffs based on actual and not on perceived abilities (and thus on actual rewards) are: The qualitative effect of changes in agent 2’s perceived ability on agent 1’s expected payoff, then, can be summarized as follows: for any: ,where denotes some upper bound on agent 2’s bias (possibly ), it holds that 7 Note that we consider overconfidence in the form of overestimation of one’s absolute ability (see, e.g., [27], for a similar approach). In general, overconfidence can arise in other forms like overestimation of relative abilities (“better-than- average effect”, e.g. [3]) or personal control (“illusion of control”, e.g., [38]); as well as unrealistic optimism about the future (e.g. Weinstein, N.D. (1980), ”Unrealistic Optimism about Future Life Events,” Journal of Personality and Social Psychology, 39, 806-820). 8 Note that overconfidence would have no behavioral effect if agents were aware of their bias (and otherwise rational, i.e. expected utility maximisers). 9 Note that beliefs in this type of argument are used essentially to motivate behavior but are not themselves part of the equilibrium in that they have to be correct. This is somewhat similar to models of level-k thinking used to analyze initial responses in normal form games see, for example, [41-43]. In view of applications, such an implicit exclusion of the consistency condition regarding beliefs appears to be a justifiable simplification, for example, in settings where there are few opportunities for learning (e.g. due to a low frequency of repetition) or where the common restrictions of the agents’ mental capacities are binding (e.g. due to time constraints or some other details of the job the agents have to carry out). 10 Since both agents are unaware of each other’s bias, both believe that the colleague is unbiased. Moreover, each agent is unaware of the own bias. Thus, the agents’ beliefs are effectively inconsistent with actual strategies (as they are unaware of the biases); see also footnote 9. However, as soon as we deal with biased agents, consistency of beliefs is always an issue as biased agents, by definition, are at least unaware of their own bias. As the first term is zero by the envelope theorem, the impact of agent 2’s overconfidence on agent 1’s payoff depends on the sign of the strategic effect, which is positive. Hence, agent 1’s expected payoff increases in agent 2’s overconfidence. Furthermore, the impact of b2 on agent 2’s own expected payoff is given by: The first term again reflects the strategic effect, which is positive as (1) efforts are strategic complements, i.e. by assumption and (2) agent 2’s effort is increasing in his bias b2 i.e. as the marginal return to effort is increasing in ability. By contrast, the second term, which reflects the payoff effect of agent 2’s mistaken belief about his own ability, is, of course, negative as the mistaken belief induces agent 2 to exert too much effort, i.e. which in turn implies . Eventually, the overall effect on agent 2’s expected payoff is determined by the trade-off between the strategic effect and the effect of agent 2’s mistaken belief. In particular, if synergy effects are large, the strategic effect dominates and agent 2’s payoff increases in 2 b This also holds if both synergy effects and agent 2’s bias are small as a small bias results in a moderate increase in agent 2’s effort and thus the mistaken belief effect is negligible. If synergies are small while the bias is comparably large, though, the overall effect on agent 2’s utility is negative. The overall effect of agent 2’s overconfidence on agent 1’s expected payoff, by contrast, depends only on the sign of the strategic effect, which is positive. Accordingly, agent 1’s payoff always increases in agent 2’s overconfidence. Summing up, both agents’ efforts increase in b2 if efforts are strategic complements and the marginal return to effort of agent 2 is increasing in b2 - as assumed for the present discussion. Moreover, such an increase in efforts does not only lead to a higher team productivity (i.e. a higher firm value) and a higher expected payoff of agent 1 (which is increasing in b2). It also increases the expected payoff of the overconfident agent 2, provided that either synergies are large or, if they are small, also the bias, b2, itself is sufficiently small. Intuitively, the latter effect is due to the fact that agent 2 benefits from the positive externalities of the increased effort of agent 1. Even if these externalities are rather small, this effect outweighs the decrease in expected payoff resulting from agent 2’s increased effort as long as the extent of overconfidence is moderate. Thus, we conclude: Lemma 1 Within the considered model of team production, being overconfident (and paired with a rational agent) increases the payoff of the overconfident agent if either synergy effects are sufficiently large or if both synergy effects and the agent’s bias are small. Go to Bias-Awareness In a next step, we turn to the discussion of teams which consist of two overconfident agents. We address the question whether it is optimal for either agent to be informed or ignorant of his colleague’s bias. In doing so, we distinguish three settings: (Case 1) both agents are unaware of each other’s biases; (Case 2) one agent is aware of the other’s bias while the other agent is unaware of the colleague’s bias; (Case 3) both agents are aware of each other’s bias. As we will see, it is always better for agent 2 to be unaware of his colleague’s overconfidence - irrespective of whether agent 1 is aware or unaware of agent 2’s bias. The section concludes with some brief statements about the effect of partial awareness. Case 1: Both agents are unaware of each other’s bias. If both agents are overconfident but unaware of their colleague’s bias, each agent’s decision situation is basically analogous to the situation of agent 2 considered in Section 3, i.e. the situation where an overconfident agent 2 is paired with a rational agent 1. Accordingly, each agent maximizes his (incorrectly) perceived payoff (incorrectly) anticipating that the other agent behaves rationally. Thus, the derivation of the maximization problems and the optimal efforts for both agents is analogous to that for agent 2 in the previous section.10 Accordingly, agent 2’s decision in the present setting is identical to the one discussed in Section 3: Agent 1, in turn, now acts in the same way as agent 2; i.e. he also increases his effort compared to the individually rational level, , because of his own overconfidence (but no longer, as he did before, because of - the knowledge of - his colleague’s bias). Thus, agent 1’s maximization problem is given by: Note that the first term of this expression derives from agent i’s mistaken belief and is negative as (recall that the marginal return to effort increases in ability). Moreover, the strategic effect is zero as both agents are unaware of the other’s bias, i.e. Thus, we conclude: Lemma 2 Being overconfident reduces agent i’s payoff if agent -i is unaware of this bias. Case 2: One agent is aware, one unaware of the other’s bias Suppose agent 2 is aware of the bias of agent 1 but agent 1 is still unaware of his colleague’s bias. 13 Then, agent 1 maximizes his (incorrectly) perceived payoff (incorrectly) anticipating that agent 2 behaves rationally; and agent 1 disagrees with agent 2’s belief that agent 1 is overconfident. Thus, the maximization problem and the corresponding optimal effort of agent 1 remain the same as in Case 1. Thus, we have: 14 For agent 2, however, things are different. In particular, agent 2 again maximizes his (incorrectly) perceived payoff but now accounts for agent 1’s overconfidence. Thus, as efforts are strategic complements, agent 2’s effort increases in b1 (because agent 1’s marginal return to effort increases in 1 b ): Note that there are now two reasons for agent 2 to increase his effort: (1) the biased perception of his own ability (which he is not aware of), and (2) the awareness of the colleague’s overconfidence. Thus, agent 2’s effort is not only higher than in the fully rational team, but also higher than his effort in