Frequency of Visiting a Doctor: A right Truncated Count Regression Model with Excess Zeros- Juniper Publishers

 

Biostatistics and Biometrics - Juniper Publishers

Abstract

Count response variables are frequently encountered in medical data, which calls for the use of count regression models. In this study, we introduce the hurdle Conway-Maxwell Poisson (HCMP) regression model where the outcome variable is the number of doctor visits, complicated by excess zeros and over-dispersion from troublesome extreme values. A truncation approach is proposed to handle extreme values, leading to the definition of a truncated HCMP (THCMP) model. Parameter estimates are derived using maximum likelihood. Results of a case study on a RWM dataset investigated effects of response truncation at 6.65, 3.08 and 1.75% for the THCMP and truncated hurdle Poisson (THP) models. In a simulation study, responses were generated from a mixture of HCMP (50%) and HP (50%) probability models. THCMP and THP model performance was compared with respect to parameter estimation bias, goodness-of-fit and outcome estimates for truncation levels of 5 and 10%. As measured by AIC, the THCMP model exhibited better goodness-of-fit at all truncation levels compared to the THP model. Estimation bias increased with higher truncation levels for both models, but to a lesser degree for the THCMP model.

Keywords:Hurdle model; Conway-maxwell poisson; Over-dispersion, Parameter estimation; Model selection

Abbreviations: HCMP: Hurdle Conway-Maxwell Poisson; THP: Truncated Hurdle Poisson; CMP: Conway-Maxwell Poisson; GSOEP: German Socioeconomic Panel; LL: Log-Likelihood; AIC: Akaike’s Information Criterion; BIC: Bayesian Information Criterion; TP: Truncated Poisson; TCMP: Right-Truncated CMP; HNB: Hurdle Negative Binomial; HGP: Hurdle Generalized Poisson

Introduction

Health care is one of the most important factors in human life. Good health care is a major contributor to quality of life, and ready access to a physician is an important component of a good health care system. The number of doctor visits for a household over a fixed interval is a useful metric for studying factors that affect physician accessibility. In this study, we introduce a new regression model for studying count data occurring in medical studies. The count variable in this study is number of doctor visits over a fixed time period.

There are numerous publications describing applications of count models to healthcare demand data, and applications of negative binomial models in particular. The negative binomial model has been applied to cross-sectional data, and in econometric models to analyze cross-sectional data with multiple outcomes per observation [ 1]. Modelling count data using a random effects negative binomial regression model is discussed in [ 2], and application of the negative binomial hurdle model to physician visit data is demonstrated in [ 3].

The Conway-Maxwell Poisson (CMP) distribution-a generalization of the Poisson-was introduced in [ 4] with applications to queues and service rates. The CMP distribution belongs to the exponential family and the two-parameter power series family of distributions. In the 50 plus years since its introduction, the CMP model has not been widely employed; however, a revival has arisen of late from a recognition of its utility in fitting discrete data [ 5]. The CMP distribution has two parameters and can handle both under- and over-dispersed data. This is in contrast to the commonly used negative binomial model which can only handle overdispersion.

We found several studies describing the properties of the CMP model with a variety of applications. The CMP distribution was applied to model timing of bid placement and the extent of multiple bidding in online auctions [ 6]. A Bayesian analysis of the CMP distribution is discussed in [ 7] and conjugate priors for the distribution parameters are derived. A flexible cure rate survival model is expanded to follow the CMP distribution in [ 8]. The joint generalized quasi-likelihood estimating equations are compared to the marginal equations in a CMP generalized linear model describing the number of car breakdowns in [ 9]. The structural properties of the CMP distribution, including moments and probability generating function are derived in [ 10]. Notwithstanding the many published applications, we have yet to find the CMP distribution used in a medical context-which is the motivation for this paper.

In many real world applications, the problem of excess zeros is encountered-the actual zero frequency outcome is higher than that predicted by the theoretical model. When this is the case, a hurdle model may be used that models the zero outcome separately from the non-zero outcomes [ 11]. In a hurdle model two densities are used, one that generates the zeroes, and another called the zero-truncated density that generates the positive values. The finite mixture distribution generated by combining two densities is discussed in [ 12]. The mechanisms by which excess zero frequencies occur for various types of count data, and how the zero-inflated Poisson model has application to such data in a medical context are described in [ 13]. Excess zeros in count data with application to public health employing a likelihood ratio test is addressed in [ 14].

Another aspect of our paper focuses on extreme values in the context of the CMP model and the adverse effect of troublesome ‘outliers’ on estimates of the CMP distribution mean and variance. The effect of outliers is to inflate the variance to make it larger than the mean, which is the definition of over-dispersion in the CMP model. One approach for reducing over-dispersion is right truncation, where values greater than a fixed constant are removed from the sample. A right-truncated Poisson regression model for handling over-dispersion is discussed in [ 15]. Applications of hurdle models with right censoring are the hurdle generalized Poisson regression model and the hurdle negative binomial regression model applied to the number of fish caught by fishermen at a state park, where the response was right censored [ 16,17].

The main focus of this study is on regression analysis based on CMP distribution. CMP has two parameters and this feature makes the distribution more flexible compared to Poisson model. Negative binomial (NB) model is a competitive model for CMP, however the dispersion parameter in NB model can only deal with over-dispersed data. CMP model is more flexible in that sense and can handle both over- and under-dispersion scenarios. The application part of this study (including read data example and simulation study) illustrates the performance of CMP model over alternative models under both under- and over-dispersed data.

The novel contribution of our paper is the introduction of a hurdle model based on the CMP distribution, the truncated hurdle Conway-Maxwell Poisson (THCMP) distribution, which can handle excess zeros in right-truncated count data. We illustrate the THCMP model in an application involving an analysis of the number of doctor visits over a fixed interval and compare it with the truncated hurdle Poisson (THP) model—a less suitable, but possible choice among presently available alternatives. In section 2, we describe the health care data set and variables for a case study analysis. Inasmuch as we have found no published study applying the CMP distribution to an outcome in medicine or public health, our paper is unique in this regard. In section 3, the THCMP regression model for right truncated data is introduced and parameter estimates derived. A case study analysis of the THCMP model is presented in section 4 along with a description of the methodology for a simulation study in which the THCMP model is compared to the truncated hurdle Poisson (THP) model on both over- and under-dispersed data. In section 5, we discuss results of a simulation study and evaluate the performance of the THCMP regression model versus the THP model.

Description of RWM health care data

The RWM data set [ 18] used in this study is taken from the German Socioeconomic Panel (GSOEP). The GSOEP, conducted by the German Institute for Economic Research in Berlin, surveys a representative sample of East and West German households. Researchers have recently used this cross-sectional data set to evaluate performance of their proposed count regression models [ 18,19]. The RWM data set is an unbalanced panel survey of health care utilization of 27,326 German individuals. The frequency table for the number of visits to the doctor is given in Table 2.

The dependent variable in our analyses, DocVis, is a count variable—the number of visits to a doctor (including dentists) during a fixed time interval. The RWM data set is viewed as a cross-sectional dataset in our analysis [ 18,19]-the outcome is not time varying, as contrasted with [ 18]-and we assume that counts among study subjects are independent. The particular irregularities/anomalies of this dataset make it a good candidate for illustrating the unique features of the THCMP model. In Table 1, the mean and variance of DocVis are 3.18 and 32.37, respectively, which indicates substantial overdispersion in the data; minimum and maximum values are 0 and 121, respectively. In addition, the frequency of the zero response in DocVis is higher than expected (median=1, mode=0) (Table 2).

The explanatory variables consist of socioeconomic characteristics and demographic variables. All count regression models on DocVis were fitted as functions of sex (1=female, 0=male), age (years), education (years of schooling), marital status (1=married, 0=single) and children in the household (1=children present). Independent variables are summarized in Table 1 which shows that 48% of visits are by females, average age is 43.5, children are present in 40% of households, average years of schooling is 11.3, and 24.1% of respondents are single. The base case count model used in the analysis included the following variables in addition to the constant term:

The frequency distribution of DocVis is shown in Table 2. According to the percentage of zeros in the response variable (37.1%), there is an excess of zeros. In addition, the 95th percentile is 12 which means that there are some extreme values in the sample. It is apparent from the histogram in Figure 1 that the zero count frequency of DocVis exceeds that expected in a Poisson distribution.

Methodology

In this section, the right-truncated hurdle Conway-Maxwell Poisson (THCMP) regression model is introduced for handling count data with excess zeros and right-tail data truncation. Parameter estimation and the goodness-of-fit statistics are discussed.

1.1. The model

Let the response variable *,1,,iYin= be the number of visits to a doctor over a fixed time period. The HCMP regression model ()*,,,iifvyλ is given by

Where ()*iiEYλ= of a Poisson distribution associated with observation , and 0v≥ is the dispersion parameter. The CMP regression model can handle both over-dispersion ()1v< and under-dispersion ()1,v> and when 1,v=the probability function (1) reduces to a Poisson model. A geometric distribution is obtained from (1) when0v= and 1.iλ< When v→∞ in (1) with probability ,1iiλλ+ the result is a Bernoulli distribution.

In many practical applications, it is common to assume that the parameter iλ depends on a vector of explanatory variables

variables are commonly incorporated in the context of a log-linear model, where log indicates the base e or natural logarithm,

The'jsβ are coefficients of the explanatory variables in the regression model and m is the number of explanatory variables.

vector of unknown parameters. In this model set up, the non-negative function 0w is modeled using a logit link function.

The moments of the HCMP distribution are obtained as

Now, we can define the right truncated hurdle Conway-Maxwell Poisson regression model as

Where t is the truncation point for .iy This means that we truncate the response variable when ,iyt> leading to the definition of B as

Thus, the log-likelihood function for the HCMP model with right truncation can be written as

Where k is the number of observations after truncation.

Parameter estimation

In this section we obtain parameters estimates using maximum likelihood. The likelihood equations for estimating ,rtβδ and v are obtained by taking the partial derivatives of (4) and setting them equal to zero yielding

These partial derivative equations cannot be further simplified. Calculating the Hessian matrix directly is computationally laborious, and so the Conjugate Gradient Optimization method implemented in SAS was used to numerically obtain the Hessian variance-covariance matrix. In approximating standard errors of the parameter estimates, the Hessian matrix must be computed at least once, regardless of the optimization technique.

The Fisher information matrix for the THCMP regression model is obtained as

The elements of the Fisher information matrix are available in the Appendix.

Model selection and test for dispersion

Goodness-of-fit statistics for the THCMP model are based on the deviance statistic, defined as

the model likelihood function evaluated at μ and ,y respectively. The log-likelihood function is defined in equation (4).

The deviance statistic can be approximated by a chi-square distribution when 'sμ is large. In the application section, we use 2,LLAIC− and BIC to compare the different regression models in terms of goodness-of-fit. For all of these statistics, a smaller value indicates a better fit.

From section 3.1, it is obvious that the THCMP model reduces to THP model when 1.v= To assess the adequacy of the THCMP model over the truncated hurdle Poisson model, we test the hypothesis

The purpose of (5) is to evaluate the significance of the dispersion parameter. It follows that the THCMP model should be used instead of the THP model whenever 0H is rejected. To test the null hypothesis 0H in (5), the likelihood ratio statistic can be used. An alternative statistic for the parameter v is the asymptotic Wald statistic where the dispersion parameter is calculated after fitting the THCMP regression model.

Results

Case study

In this case study using the RWM data set (n=27,326), the THCMP regression model is used to model the number of doctor visits (DocVis) per patient over a period of three months as a function of the independent variables sex, age, children, education and married. The THP model is also considered as an alternative model in the analysis of the RWM. The THCMP model will be compared to the THP model relative to parameter estimates, standard errors, goodness-of-fit statistics and accuracy in modelling the response.

Five independent variables are used in the model and all are incorporated into both the logit and non-logit parts of the model. Therefore, the link functions can be written as

plus the dispersion parameter (when used) will be estimated using the ML method.

Three truncation points, t=10, t=15, and t=20 are employed in comparing the effects of truncation, and correspond to truncation percentages of 6.65, 3.08 and 1.75, respectively.

Parameter estimates for the THP and the THCMP model were obtained for the specified truncation points and are summarized in Table 3. Link functions are obtainable from the estimates shown in Table 3. For example, the respective log and logit link functions for the THCMP regression model for truncation point 110t= is

Using the log link function coefficients in Table 3, one can see the estimated change in DocVis per unit change in each independent variable, all others being held constant. To illustrate, the positive 1β coefficients corresponding the variable sex in the HP model in Table 3 for the three truncation times are 0.0997, 0.11 and 0.0935, which indicate higher values of ()logDocVis for females compared to males for all truncation points. For the THCMP model, decreases in ()logDocVis of 0.0158 and 0.003 for females versus males are predicted for 215t= and 320,t= respectively. For a one-unit increase in age, expected ()logDocVis is estimated to increase by ~0.01 on average using the THP model and ~0.003 using THCMP model for all truncation points. Thus, older patients are predicted to have a higher number of doctor visits per unit time. ()logDocVis is estimated to be lower for households with children relative to those with no children for both THP and THCMP models at all truncation points—with the exception of THCMP when 215.t= For a one-unit increase in the number of years of schooling, the estimated change in the number of doctor visits decreased for all truncation points. ()logDocVis showed more frequent doctor visits for married versus single individuals for all truncation points.

The THCMP regression model indicated overdispersion in the RWM data. Dispersion parameter estimates corresponding to truncation percentages of 6.65,3.08 and 1.75 were 0.108,v= 0.069v= and 0.0621v= respectively, where 1v< indicates overdispersion.

THP and THCMP regression model coefficients for the RWM analysis based on the logit link function are also given in Table 3. These coefficients correspond to the excess zeros component of the models. Both models show a negative effect of sex for all truncation percentages, indicating a higher rate of zero doctor visits in males than in females. The log odds of excess zeros increases for each unit decrease in age in both the THP and THCMP models at all truncation points. This means that zero doctor visits were increasingly more likely with aging. A positive coefficient for children in both THP and THCMP models indicated that households with children exhibited a higher rate of zero visits to a doctor compared to households without children for all truncation percentages. The log odds of excess zeros increased for each unit increase in the number of years of schooling for all truncation points for both regression models. This indicates that fewer years of schooling were associated with higher odds of zero visits to a doctor. Both THP and THCMP models resulted in negative coefficients for the married variable for all truncation points with the exception of the 320t= truncation point for the THCMP model. So, generally, single individuals exhibited higher rates of zero doctor visits than married individuals.

Table 4 compares the THP and THCMP regression models on three goodness-of-fit measures: log-likelihood (LL), Akaike’s Information Criterion (AIC) and the Bayesian Information Criterion (BIC). The right-truncated Poisson (TP) and right-truncated CMP (TCMP) regression models are included in Table 4 as special cases to investigate whether a zero frequency of 37.1% should be considered an excess zero scenario. We were also able to investigate whether the TP and TCMP regression models (without excess zero scenario) were able to fit the data as well as the hurdle regression models. The THP and THCMP models demonstrated better goodness-of-fit than the TP and TCMP models on all measures (smaller is better). The THCMP regression model exhibited superior goodness-of-fit compared to the THP regression model for all truncation levels in the RWM data analysis.

Table 5 shows that the THCMP regression model exhibited a better fit to the RWM data versus the THP model based on the predicted versus observed frequency count criterion for all truncation points. The RWM dataset observed zero frequency was 10135. Predicted zero frequencies for THCMP/THP at the truncation points were 1:10118/10012;t 2:10319/9971;t and 3:9626/9008.

A subset of RWM data where under-dispersion is present was considered. We have focused on the individuals with low health satisfaction score (<4 out of 10) and split up the data set by marriage status to come up with two under-dispersed scenarios. The married covariate was excluded from the list of independent variables and other covariates were kept in. TP and THCMP models with a right truncation point of 4 applied to the data. The goodness-of-fit statistics (-2LL and AIC) for THCMP/THP were 653.2/686 and 675.2/706 for unmarried individuals, and 1847.5/1908.9 and 1869.5/1928.9 for married individuals. The dispersion parameter of THCMP model was significant in both scenarios (4.12 (3.02, 5.33) and 3.42 (2.77, 4.07), p-value<0.001).

A Simulation study

We conducted a simulation study to assess and compare THP and THCMP regression model performance. Goodness-of-fit was measured using the Akaike information criterion (AIC). Data were simulated with 50% of responses generated by the HCMP regression model and 50% by the HP model. Simulation models

erated from a uniform distribution on []0,1. A sample size of 200n= was used in conjunction with varying proportions of zero outcomes and 0w set to values of 0.2, 0.3 and 0.4. Truncation percentages were set at 5% and 10% of the simulated distribution tail, although actual simulated truncation percentages were not always exactly equal to 5% and 10%.

The six working simulation models, differing in their coefficient parameters, are shown in Table 6. Three are mixtures of under-dispersed (υ>1) HCMP regression models with the HP regression model. These models generate count data outcomes of 0, 1, 2, and 3 for the most part, resulting in short-tailed count frequency distributions. Conversely, the remaining three models are mixtures of over-dispersed (υ<1) HCMP regression models with the HP regression model, which generate more extended right tails in the response distribution. Using the three combinations of coefficient parameters, the model with 01,β= 11.5,β= 20.9,β= 01a= generates long right-tailed count distributions with high count outcomes; the model with 01,β= 11,β= 21,β= 02a= generates distributions with tails of intermediate length; and the model with 01,β= 10.8,β= 20.8,β= 00.3a= distributions with relatively short tails.

The number of replications was set at 1,000, which was sufficient for our purposes. Additional replications would have unnecessarily increased the computational burden. The simulation was programmed in FORTRAN. Maximum likelihood estimates were obtained via numerical maximization using the simulated annealing algorithm [ 20].

Simulation results are summarized in Table 7(i) and (ii) where the values given are the average values of 1,000 replications. From Table 7(i), the average AIC for the THCMP regression model was significantly lower than that for the THP regression model for simulation models (a) to (c). When the truncation percentage was increased to 10% as in Table 7(ii), the average AIC difference was smaller as compared to Table 7(i). Nevertheless, the THCMP regression model performed slightly better for model (a) and definitely outperformed models (b) and (c). For model (d), the average AIC difference between the THP and THCMP regression models was less than 2, regardless of the proportion of zeros and the percentage of truncation. For models (e) and (f), the THCMP regression model was slightly better than the THP model with 5% truncation in the tail, where the AIC average difference was greater than 2.

Discussion

In this paper, we introduce the THCMP regression model and illustrate its application in an analysis of the RWM data in which the outcome variable of interest is the number of doctor visits occurring in a fixed interval. We show how the THCMP model can be used to handle dual data anomalies of excess zeros and extreme values in a count response variable. Parameter estimates and standard errors for the TCMP model—and the alternative THP model-were obtained for selected data truncation levels using ML estimation. Covariate effects were interpreted in the context of model link functions.

A comparison of goodness-of-fit of the THCMP and THP models to the RWM data, as assessed by -2LL, AIC and BIC, showed better performance for the THCMP model at the three truncation levels studied (6.65%, 3.08% and 1.75%). The percentage of zeros in the response variable of the RWM case study was 37.1%—which is cited in the literature as a threshold for excess zeros [ 18, 19]. The results showed that a right truncated hurdle model can fit these data with inflation at zero better than a simple right-truncated model where excess zero part of the model is not taken into account. We also examined goodness-of-fit for the right truncated Poisson and CMP models as well as the excess zero models in the RWM case study analysis.

In the RWM analysis, the THCMP model exhibited better agreement between observed and predicted zero frequency counts than the THP model for all three truncation points. However, greater truncation levels resulted in greater bias in estimating the zero frequency for both models, although to a lesser degree for the THCMP model. In addition, the THCMP model generally exhibited better goodness-of-fit in modelling counts greater than zero. Exceptions favoring the THP model were a few cases involving frequency estimates for 2 and 7 at 6.65% truncation, and 8 and 9 at 3.08% and 1.75% truncation. The case study analysis results suggest advantages of the THCMP regression model compared to the THP model for analyzing ‘real life’ count data when there are both an excess of zeros and extreme values in the observed response.

In the under-dispersed subset of RWM data sets, we have tried hurdle negative binomial (HNB) and hurdle generalized Poisson (HGP) models with right truncation approach to investigate the performance of HCMP over HNB and HGP. HNB model was not converged because the final Hessian matrix, though full rank, had at least one negative eigenvalue, and therefore the second-order optimality condition violated. This is expected as NB model can handle over-dispersion scenario not under-dispersion. HGP model also was not converged as the final Hessian matrix was not positive definite and therefore the estimated covariance matrix was not full rank and may not be reliable. Hence, the HCMP model outperform HNB and HGP when the under-dispersed right truncated outcome has excess zeros.

In the simulation study, mean AIC for the THCMP model was significantly lower than mean AIC for the THP model (AIC mean difference >11) in under-dispersed scenarios with 5% tail truncation, indicating substantial improvement in fitting the data [ 21]. At the 5% truncation level, a clear lack-of-fit of THP is exhibited for simulation models (a) to (c), indicating under-dispersed scenarios, compared to THCMP. However, at 10% truncation, the mean AIC difference between models was smaller in under-dispersed scenarios. In the simulation study, the THCMP regression model performed slightly better than the THP model in the short-tailed data scenario (model (a)) and clearly outperformed THP in the medium- and long-tailed data scenarios (models (b) and (c)). In an over-dispersed scenario with short tail (model (d)), the average AIC difference for the THP and THCMP models was less than 2 regardless of the proportion of zeros or the truncation percentage, implying similar performance in this case. For over-dispersed scenarios involving medium and long tails (model (e) and (f)), and less than 5% truncation, the THCMP regression model performed slightly better than the THP model (AIC mean difference >2). In short, the comparison of average AIC for the THCMP and THP regression models indicated superiority of the THCMP to the THP model for outcome data with lower than expected proportions of zeros, lower percentages of tail truncation, and consists of mostly low values of the response variable.

A strong point of the simulation study is that the CMP model, unlike more frequently used models such as the negative binomial model, is more flexible and able to handle under-dispersion as well as over-dispersion. The negative binomial model was not entertained as an alternative model in this study because the NB model cannot accommodate under-dispersed data. The THCMP model clearly outperformed THP model for under-dispersed scenarios. Therefore, the THCMP regression model would be expected to provide better outcomes in terms of the parameter estimates and goodness-of-fit statistics when data are under-dispersed—even when compared to some alternative models such as hurdle negative binomial model.

In summary, we introduced the truncated hurdle Conway-Maxwell Poisson regression model. We carried out a simulation study, based on data generated from a mixture of HCMP (50%) and HP (50%) probability models, which showed the THCMP regression model accommodated various degrees of truncation and anomalous zero frequencies better than the THP regression model. We recommend the THCMP regression model as a flexible distribution for analyzing count data exhibiting the dual anomalies involving zero frequencies and a low level of tail truncation in handling over- and under-dispersion.

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On the Positive Effects of Overconfident Self-Perception in Teams - Juniper Publishers

 

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

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.

1Note 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.

2For 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.

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 externalities3. The value of the project is the value of its expected cash flow which depends on the agents’ efforts, ei , and their abilities, ai ; for the sake of argument, we assume a1 = a2 = a.4 Moreover, we assume that agent i‘s expected return from the project, denoted by Ri(ei, e-i) is increasing in effort and ability and that the marginal return to effort is increasing in ability, i.e. d2 Ri /d ei dai > 0.5 The agents’ cost of effort is denoted by c(ei) 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

3On 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].

4Note 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.

5The 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.

6Efforts 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.

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 b2 > 0, i.e. his perceived ability is a ':= a + b2 . 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 (b2) = 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 b2 :

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

7Note 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).

8Note that overconfidence would have no behavioral effect if agents were aware of their bias (and otherwise rational, i.e. expected utility maximisers).

9Note 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).

10Since 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.

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 the case where he is unaware of agent 1’s bias, i.e.:

This implies that the team’s productivity is increased compared to the fully rational team and the team with two overconfident agents who are both unaware of their colleague’s bias.15

Moreover, the corresponding optimal payoffs of the agents in this case are as follows:

Payoff comparison when one agent is unaware of the colleague’s bias.

A simple payoff comparison yields that if one agent, say agent 1, is unaware of agent 2’s bias, agent 2 is better off being unaware of the bias of agent 1:

11Here as below, the double digit in the exponent (“00” in this case) refers to the agents’ awareness of biases: the first digit refers to agent 1 and the second to agent 2 (“0” indicating unawareness of the respective other agent’s bias and “1” indicating awareness of it).

12Note that, as both agents are unaware of the other’s bias, optimal effort levels only depend on each agent’s own bias.

13Due to the symmetry of the problem, the case that agent 1 is aware of agent 2’s bias follows immediately from interchanging the agents.

14Note that “01” in the exponent now indicates that agent 1 is unaware of agent 2’s bias while agent 2 is aware of agent 1’s bias.

15It can also be shown that the team’s productivity increases compared to the team with only one overconfident agent.

16Recall that the “01” in the exponent refers to the case where both agents are overconfident but only agent 1 is aware of the bias of agent 2, which is analogous to Case 2 except that the information structure is reversed. Note further that 

Intuitively, accounting for agent 1’s overconfidence induces agent 2 to further increase his effort in an attempt to exploit effort complementarities. Yet, his effort is already above the individual optimum because of his own overconfidence and the further increase in effort is not complemented by agent 1. Thus, we conclude:

Lemma 3 If both agents are overconfident and agent 1 is unaware of the bias of agent 2, then agent 2, ceteris paribus, is better off if he is also unaware of agent 1’s bias than if he were aware of it.

Case 3: Both agents are aware of each other’s bias.

When both agents are aware of each other’s bias (but unaware of their own bias), both maximise their (incorrectly) perceived payoff (correctly) anticipating that the other agent is overconfident. Yet, agents disagree with the other agent’s belief that they are biased themselves. Hence, the situation is analogous to the situation of agent 2 in Case 2 where agent 2 is aware of agent 1’s bias; i.e. the optimal effort level of agent 2 is given by:

However, for agent 1, who now takes into account the bias of agent 2, the optimal effort level is increased compared to Case 2, i.e.:

as efforts are strategic complements and agent 2’s marginal return to effort is increasing in his bias. Note that under these conditions both agents increase their effort for two reasons: (1) their own overconfidence and (2) their attempt to complement their colleague’s increased effort.

Accordingly, the agents’ resulting payoffs in this case are given by:

Payoff comparison when one agent is aware of the colleague’s bias.

Next, we consider agent 2 and compare his payoff for the case where he is aware of agent 1’s bias with the case where he is not – assuming that agent 1 is aware of agent 2’s bias. A comparison of agent 2’s payoff in both instances shows that being unaware of agent 1’s bias is preferable for agent 2 if:16

which holds as  is concave and . 17 The intuition for this result is the same as before: Complementing agent 1’s additional effort is detrimental for agent 2 because agent 2’s effort is already above the optimum – due to his own bias – and because the further increase is not complemented by agent 1. Similar to the previous situation, we thus conclude:

Lemma 4 If both agents are overconfident and agent 1 is aware of the bias of agent 2, then agent 2, ceteris paribus, is better off if he is unaware of agent 1’s bias.

Consequences of Partial Awareness.

Finally, we want to briefly comment on the effects of partial awareness of biases; see Appendix A for a formal discussion. For the sake of argument, we assume that an agent who is “partially aware” of his colleague’s overconfidence assigns probability p ∈ [0, 1] to the case that his colleague has bias bi > 0, where bi is the true bias of agent i.18 As it turns out, partial awareness essentially reduces the strength of the effects discussed above while keeping the direction of changes. In particular, it holds (see Appendix A for a formal derivation):

17Irrespective of whether agent 2 is aware or unaware of agent 1’s bias, it is obviously better for agent 2 if agent 1 is aware of agent 2’s bias than if he is not, i.e.

18It is straightforward to generalize our analysis to more general cases of “partial aware- ness”, where an agent attaches different probabilities to different sizes of the bias.

19Here we consider only the case in which information about biases is optimal, i.e. biased agents are unaware of the biases of others. Similar results hold if one or both agents are (partially) aware of the bias of their colleague, albeit with slightly stricter restrictions on synergy effects and the size of the biases.

Lemma 5 An agent is best off being unaware of the colleague’s bias; and being partially aware is better than being fully aware. Moreover, for an over- confident agent it is optimal if his colleague is fully aware of the bias; and partial awareness is better than unawareness.

Comparison with Rational Team

In the previous sections, we have shown that within the proposed model of team production (1) overconfidence can be beneficial for the biased agent and (2) if an agent is overconfident, it is always best for him to be unaware of a potential bias of his colleague. In view of a general comparison between rational and overconfident agents, however, it is interesting to ask how individual payoffs in a team of two overconfident agents compare to those in a fully rational team. In the remainder of this section, we show that (un- der fairly weak conditions) individual payoffs in a team of two overconfident agents are higher than in a team of two rational agents.

Consider a situation in which both agents are overconfident but unaware of their colleague’s bias, i.e. a situation where overconfidence is present in its “individually optimal” form (i.e. it is combined with unawareness of the colleague’s bias). Then, both agents’ overconfidence is not complemented by a higher effort of the respective colleague through awareness of biases.

In order to obtain a clear picture of the individual payoff comparison for this scenario, let us first consider the case in which one agent, agent i, is biased and the other agent exerts his benchmark effort  (e.g. because he is rational but unaware of his colleague’s bias). For this case, the following holds:

In fact, the comparison remains positive also for small synergy effects if biases are moderate. Intuitively, this holds as a small bias of agent i induces only a moderate increase in agent i’s own effort. Hence, a smaller “synergetic feedback” through agent −i’s effort is required to “reimburse” the biased agent i.

Summing up, the above result in favor of overconfidence is rather intuitive as we have already seen that individual payoffs for a biased agent in a team of one overconfident and one rational agent are higher (cf. Section 3). The maximization problem of the overconfident agent, say agent 2, is the same in both the team with one and the team with two overconfident agents: He is biased himself (and unaware of his bias) and thinks his col- league, agent 1, is unbiased and, hence, will exert the same effort in both cases. Moreover, if the additional effort exerted by a rational agent 1 in order to complement agent 2’s additional effort (due to agent 2’s overconfidence) is enough to overcompensate agent 2 for his increased effort cost, then it is natural to expect that an overconfidence bias of agent 1 has a similar effect. Eventually, both the awareness of agent 2’s bias (of the rational agent 1) and the own overconfidence of agent 1 have a similar effort enhancing effect; and the higher effort of agent 1 (due to his overconfidence) is what compensates agent 2 for his additional cost.

Finally, it is interesting to note that the favorable comparison of individual payoffs in an overconfident team with those in a fully rational team does not depend on the overconfident agents’ unawareness of their colleague’s bias. In fact, even if one or both agents are (partially) aware of their col- league’s bias, individual payoffs are higher than those in a fully rational team if either synergy effects are comparably large, or if synergy effects are small and biases are moderate; see Appendix B for a more detailed argument. Proposition 1 below qualitatively summarizes the main points of the preceding discussion.

Proposition 1 For the above model of team production with synergy effects, the following results hold:

i. Individual payoffs in a team of one overconfident and one rational agent are higher than those in a team with two rational agents - provided that the rational agent is aware of his colleague’s bias and either synergy effects are sufficiently large, or synergy effects are small and the bias is moderate.

ii. The individual payoff of an overconfident agent whose colleague is also overconfident is always higher if he is not aware of his colleague’s bias (irrespective of whether the colleague is aware of the other agent’s bias).

iii. Individual payoffs in a team of two overconfident agents which are both unaware of the other’s bias are higher than those in a team of two rational agents - provided that either synergy effects are sufficiently large, or synergy effects are small and biases are moderate.19

Conclusion

In this paper, we have considered an intuitive model of team production with effort complementarities in order to emphasize the potentially positive effects of being overconfident. As we have shown, a more rational perspective on others, i.e. awareness of the overconfidence of others, is suboptimal for an agent who is overconfident himself. More specifically, within the considered model of team production, the payoff of an overconfident agent, whose col- league is also overconfident, is always higher if he is unaware of his colleague’s bias. Thus, although the empirical evidence on the matter is scarce, our results provide a possible rationale for why many people appear to be unaware of the overconfidence biases of others [29,30].

Moreover, we have shown that individual payoffs in both a team of a rational and an overconfident agent as well as in a team of two overconfident agents are higher than in a team of two rational agents whenever either synergy effects are sufficiently large or biases are moderate. Thus, the present analysis gives further support to the notion that being overconfident is beneficial not only in view of aggregate outcomes (as overconfidence seems to enhance effort and therefore team productivity) but also for the overconfident individuals themselves (see also [29]). In fact, the analysis also suggests that overconfident agents have no incentive to gather information about a colleague’s potentially biased selfperception (even if such information was costless). Thus, our results provide a possible rationale for why overconfidence may indeed be (and remain) as widespread a phenomenon as empirical and experimental research indicates.

Appendix

A. Partial Awareness

In order to model a situation in which agent i is uncertain of his colleague’s bias, we assume that agent i assigns probability p ∈ [0, 1] to the case that his colleague -i has bias b−i > 0 and otherwise is unbiased. For the sake of argument, suppose i =1. Thus, agent 1 believes that with probability p agent 2 follows strategy ~ e2 (where the tilde denotes that agent 2 is biased) and with probability 1− p strategy 2 e . Accordingly, agent 1 has to solve the following maximization problem:

Since (by assumption) an agent’s effort rises in his ability and, thus, in his bias, i.e. , we have  Moreover, as efforts are strategic complements, it follows:

Hence, the left hand side of the FOC must be increasing in p. For p = 1, agent 1 attaches probability one to the case that agent 2 has bias 2b . (which corresponds to the case that agent 1 is completely aware of agent 2’s bias). In this case, the FOC becomes:

Obviously, the left hand side of this FOC is larger than if agent 1 is aware of agent 2’s bias. Hence, also the right hand side must be larger.

Note that agent 2’s effort does not depend on agent 1’s awareness of agent 2’s bias but only on agent 2’s awareness of his colleague’s bias (see also the discussion of Case 1-3 in Section 4):

Thus, we conclude

Lemma A.1 An agent is best off being unaware of the colleague’s bias; and being partially aware is better than being fully aware.

For agent 2 it also holds that his effort is higher if he is partially aware than if he is unaware and highest if he is aware of agent 1’s bias - irrespective of agent 1’s awareness status x∈[0, 1] ,

Thus, we conclude

Lemma A.2 An overconfident agent is best off if the colleague is aware of the bias; and partial awareness is better than unawareness.

B. Comparison: 2 Overconfident vs. 2 Rational Agents

i. Both overconfident agents are aware of their colleague’s bias.

If both agents are overconfident and aware of their colleague’s bias, we have:

If both agents are overconfident and aware of their colleague’s bias, we have:

provided that synergy effects are sufficiently large. And this also holds if biases are moderate (as a small bias results in a moderate increase in effort and therefore a smaller “synergetic feedback” through agent −i’s effort is required).

ii. Only one overconfident agent is aware of his colleague’s bias.

Similar to the above argument individual payoffs again are higher than for a fully rational team if synergy effects are sufficiently large or biases are moderate:

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