Optimal Team Size and Overconfidence

Abstract

In a team formation model with endogenous team size, we show that overconfidence may dominate rationality by increasing agents’ individual payoffs in teams. If team members are overconfident in their own ability, effort levels increase and the free rider problem is partially resolved. Because each member believes himself to be more skilled than the other members, agents prefer larger-sized teams only if complementarities are sufficiently strong. From the perspective of individual welfare, overconfidence partially undermines the efficient formation of teams. Although team members can benefit from their overconfidence only if complementarities exist, team formation can even be advantageous if members’ inputs are substitutes as it prevents agents from overinvesting in effort. We consider different extensions, including asymmetric agents, repeated interactions and the roles of monitoring and budget breaking as possible remedies to free riding.

Appendix

Proof of Lemma 1

The first-best effort $e^{FB}$ maximizes $\sum _{i=1}^n \varPi _i=Y-\sum _{i=1}^n C(e_i)$, with Y consisting of the sum of n individual outputs $a_i\,e_i$ and of the sum of $n\,(n-1)/2$ interactions between team members’ outputs. Setting $a\,e, \forall i \in \{1,\ldots ,n\}$, in (2), the maximization problem is given by

$$\begin{aligned} \max _{e} n\, \varPi&=\max _{e} Y-n\, C(e)=\max _{e}\,n\,a\,e\,(1+({n-1})\,\kappa \,a\,e/2)- n\,c\, {e^{2}}/{2}. \end{aligned}$$

(15)

The first-order condition, $\partial (n\,\varPi )/\partial e=0$,

$$\begin{aligned} n\,(a\,(1+(n-1)\,\kappa \,a\,e^{FB})-c\,e^{FB})=0, \end{aligned}$$

(16)

yields $e^{FB}$ in (3), which applies for $c>(n-1)\,\kappa \,a^2$ if $n>1$; otherwise, $e^{FB}\rightarrow \infty$.

$\square$

Proof of Lemma 2

In a symmetric Nash equilibrium, each agent’s effort maximizes $\varPi _i=Y/n-C(e_i)$. From the perspective of agent $i=1$, Y is given by $a_1\,e_1+(n-1)\,a_j\,e_j, \forall j\ne 1$. Accordingly, agent 1 interacts with $n-1$ team members, while the $n-1$ members additionally interact with $(n-2)/2$ members. Using (4), agent 1’s maximization problem can be written as

$$\begin{aligned} \max _{e_1} \varPi _{1}&=\max _{e_1} \,{Y}/{n}-C(e_1)=\max _{e_1} \,{a}\, \big (e_1 + (n-1)\,e_j\nonumber \\&\quad +(n-1)\,\kappa \,a\,e_j\,\big (e_1+({n-2})\,e_j/2)\big )/n- c\, {e_1^{2}}/{2}. \end{aligned}$$

(17)

The first-order condition, $\partial \varPi _{1}/\partial e_1=0$,

$$\begin{aligned} {a}\,(1+(n-1)\,\kappa \,a\,e_j^*)/n-c\,e_1^*=0, \end{aligned}$$

(18)

yields $e_1^*$ in (5). With identical agents, $e_1^*=e_j^*=e^*$, which can be substituted into (18). This yields $e^*$ in (6), which applies for $c>\kappa \,a^2$ if $n>1$; otherwise, $e^*\rightarrow \infty$.

$\square$

Proof of Proposition 1

With the equilibrium effort $e^*$ from Lemma 1, the rational agents’ individual payoff $\varPi (e^*,n)$ is given by (7). Solving for the equilibrium team size, the first-order condition, $\partial \varPi (e^*,n)/\partial n=0$,

$$\begin{aligned} \kappa \,a^3-2\,a\,c\,\Big (1-\frac{c}{n^*\,c-(n^*-1)\,\kappa \,a^2}\Big )=0, \end{aligned}$$

(19)

yields $n^*$ in (8), which applies for $c>\kappa \,a^2$; otherwise, $n^*\rightarrow \infty$. Thereby, teams are preferred to individual work, $n^*>1$, only if $\kappa >0$; for $\kappa \le 0$, we have $n^*=1$. The second-order condition, $\partial ^2 \varPi (e^*,n)/\partial n^2<0$, requires that

$$\begin{aligned} (2\,n^*-3)\,c^2+\kappa \,a^2\,((n^*-1)\,\kappa \,a^2-(3\,n^*-2)\,c)<0 \end{aligned}$$

(20)

for $c>\kappa \,a^2$, which holds at the equilibrium $n^*$ for $\kappa \ge 0$. $\square$

Proof of Lemma 3

With overconfidence, agent 1’s perceived individual abilities are given by $P^1[a_1]=b\,a$ and $P^1[a_j]=a, \forall j\ne 1$. Using this in (17), agent 1’s maximization problem can be rewritten as

$$\begin{aligned} \max _{e_1} P^1[\varPi _{1}]&=\max _{e_1}\, {P^1[Y]}/{n}-C(e_1)= \max _{e_1}\,\big (P^1[a_1]\,e_1 + (n-1)\,P^1[a_j]\,e_j \nonumber \\&\quad + (n-1)\, \kappa \,P^1[a_j]\,e_j\,(P^1[a_1]\,e_1+(n-2)\,P^1[a_j]\,e_j/2)\big )/{n}- c \, {e_1^{2}}/{2}\nonumber \\&= \max _{e_1}{a} \,\big (b\,e_1 +(n-1)\,e_j+(n-1)\, \kappa \,a\,\,e_j\,(b\, e_1\nonumber \\&\quad +({n-2})\,e_j/2)\big )/n- c \, {e_1^{2}}/{2}. \end{aligned}$$

(21)

The first-order condition, $\partial P^1[\varPi _{1}]/\partial e_1=0$,

$$\begin{aligned} {b\,a}\,(1+(n-1)\,\kappa \,a\,e_j^B)/n-c\,e_1^B=0, \end{aligned}$$

(22)

yields $e_1^B$ in (9). As agents agree to disagree, $e_1^B=e_j^B=e^B$, and we can rearrange (22) to obtain $e^B$ in (10), which applies for $c>\kappa \,b\,a^2$ if $n>1$; otherwise, $e^B\rightarrow \infty$.

$\square$

Proof of Proposition 2

Using the equilibrium effort $e^B$ derived in Lemma 3, the overconfident agents’ perceived individual payoff $P[\varPi (e^B,n)]$ is given by (11). Solving for the equilibrium team size, the first-order condition, $\partial P[\varPi (e^B,n)]/\partial n=0$,

$$\begin{aligned} \frac{b\,a\,(2\,(n^B+b-2)\,c^2+\kappa \,b\,a^2\,((n^B-1)\kappa \,b\,a^2\,-(3\,n^B+2\,b-4)\,c))}{(n^B-1)\,\kappa \,b\,a^2-n^B\,c}=0, \end{aligned}$$

(23)

yields $n^B$ in (12), which applies for $c>\kappa \,b\,a^2$; otherwise, $n^B \rightarrow \infty$. Thereby, $n^B>1$, only if $\kappa>{\hat{\kappa }}_1=2\,(b-1)\,c/((2\,b-1)\,b\,a^2)>0$; otherwise, $n^B=1$. The second-order condition, $\partial ^2 P[\varPi (e^B,n)]/\partial n^2<0$, requires that

$$\begin{aligned} (2\,n^B+3\,b-6)\,c^2+\kappa \,b\,a^2\,((n^B-1)\,\kappa \,b\,a^2-(3\,n^B+3\,b-5)\,c)<0 \end{aligned}$$

(24)

for $c>\kappa \,b\,a^2$, which holds at the equilibrium $n^B$ for $\kappa \ge {\hat{\kappa }}_1$. $\square$

Proof of Proposition 3

Replacing $P^1[a_1]=b\,a$ by $a_1=a$ in (21), the overconfident agents’ actual individual payoff function is consistent with the rational agents’ payoff function given by (17). Incorporating the equilibrium effort $e^B$ from Lemma 3, yields the overconfident agents’ actual individual payoff $\varPi (e^B,n)$ in (13). Solving for the welfare-optimal team size, the first-order condition, $\partial \varPi (e^B,n)/\partial n=0$,

$$\begin{aligned} \frac{b\,a\,(2\,(n^W-b)\,c^2+\kappa \,b\,a^2\,((n^W-1)\, \kappa \,b\,a^2-(3\,n^W-2\,b)\,c)) }{(n^W-1)\,\kappa \,b\,a^2-n^W\,c}=0, \end{aligned}$$

(25)

yields $n^W$ in (14), which applies for $c>\kappa \,b\,a^2$; otherwise, $n^W \rightarrow \infty$. Thereby, $n^W>1$ if $b\ge 1.5$ or if $\kappa >{\hat{\kappa }}_2=2\,(b-1)\,c/((2\,b-3)\,b\,a^2)<0$; otherwise, $n^W=1$. The second-order condition, $\partial ^2 \varPi (e^B,n)/\partial n^2<0$, requires that

$$\begin{aligned} (2\,n^W-3\,b)\,c^2+\kappa \,b\,a^2\,((n^W-1)\,\kappa \,b\,a^2-(3\,n^W-3\,b+1)\,c)<0 \end{aligned}$$

(26)

for $c>\kappa \,b\,a^2$, which holds at the optimal $n^W$ for $b\ge 1.5$ or $\kappa \ge {\hat{\kappa }}_2$. $\square$

Proof of Proposition 4

Similar to the proposition, the proof consists of two parts. Part (i) proves that $\varPi (e^B, n^W)>\varPi (e^*, n^*)$ for $\kappa >0$, and part (ii) shows that $\varPi (e^B,n^B)>\varPi (e^*, n^*)$ for $\kappa>{\underline{\kappa }}> {\hat{\kappa }}_1>0$. Throughout, we require $c>\kappa \,b\,a^2$ to hold.

(i) First, $n^*, n^W>1$ for $\kappa >0$. Incorporating $n^*$ in (7) and $n^W$ in (13), yields

$$\begin{aligned} \varPi (e^B, n^W)>\varPi (e^*, n^*)\, \Leftrightarrow \, \frac{a^2\,(\kappa \,b\,a^2-2\,c)^2}{8\,c\,(c-\kappa \,(b-1)\,a^2)\,(c-\kappa \,b\,a^2)}>\frac{a^2\,(\kappa \,a^2-2\,c)^2}{8\,c^2\,(c-\kappa \,a^2)}. \end{aligned}$$

(27)

It is easy to see that for $\kappa >0$, the inequality holds for all values of $b>1$.

Second, $n^W>n^*=1$ for $\kappa \le 0$ if $b\ge 1.5$ or $\kappa > {\hat{\kappa }}_2$. Using this in (7) and (13) yields

$$\begin{aligned} \varPi (e^B,n^W)>\varPi (e^*,n^*=1) \,\Leftrightarrow \, \frac{a^2\,(\kappa \,b\,a^2-2\,c)^2}{8\,c\,(c-\kappa \,(b-1)\,a^2)\,(c-\kappa \,b\,a^2)}>\frac{a^2}{2\,c}. \end{aligned}$$

(28)

For $\kappa =0$, both sides of (28) coincide. Since the left-hand side strictly increases with $\kappa$, this condition does not hold.

Third, $n^*,n^W=1$ for $\kappa \le {\hat{\kappa }}_2<0$ and $b< 1.5$. Using this in (7) and (13), yields

$$\begin{aligned} \varPi (e^B,n^W=1)>\varPi (e^*,n^*=1)\, \Leftrightarrow \, {(2-b)\,b\,a^2}/{(2\,c)}>{a^2}/({2\,c}), \end{aligned}$$

(29)

which is not fulfilled because $b>1$. This completes the first part of the proof.

(ii) First, $n^*,n^B >1$ for $\kappa>{\hat{\kappa }}_1>0$. Incorporating $n^*$ in (7) and $n^B$ in (13), yields

$$\begin{aligned} \varPi (e^B,n^B)&>\varPi (e^*,n^*)\,\Leftrightarrow \, \frac{a^2\,(\kappa \,b\,a^2-2\,c)^2}{8\,c^2\,(c-\kappa \,b\,a^2)}\, \frac{b\,c\,(3\,\kappa \,(b-1)\,b\,a^2-(3\,b-4)\,c)}{((b-2)\,c-\kappa \,(b-1)\,b\,a^2)^2}\nonumber \\&>\frac{a^2\,(\kappa \,a^2-2\,c)^2}{8\,c^2\,(c-\kappa \,a^2)}. \end{aligned}$$

(30)

The first term on the left-hand side is positive and always greater than the right-hand side of (30). For $\kappa >\hat{\kappa }_1$, the second term on the left-hand side strictly increases with $\kappa$ and tends to $b>1$ for $\kappa \rightarrow c/(b\,a^2)$. There is then a threshold value, $\underline{\kappa }>{\hat{\kappa }}_1>0$, such that (30) holds for all $\kappa >{\underline{\kappa }}$.

Second, $n^*>n^B=1$ for $0<\kappa \le {\hat{\kappa }}_1$. Incorporating this in (7) and (13), gives

$$\begin{aligned} \varPi (e^B,n^B=1)>\varPi (e^*,n^*)&\, \Leftrightarrow \, \frac{(2-b)\,b\,a^2}{2\,c}>\frac{a^2\,(\kappa \,a^2-2\,c)^2}{8\,c^2\,(c-\kappa \,a^2)}. \end{aligned}$$

(31)

It is easy to see that this condition is not fulfilled because $b>1$.

Third, $n^*,n^B=1$ for $\kappa \le 0$. Using this in (7) and (13), we have $\varPi (e^B,n^B=1)>\varPi (e^*,n^*=1)$ if (29) holds. This condition is not fulfilled because $b>1$. This completes the second part of the proof. $\square$