Control delegation, information and beliefs in evolutionary oligopolies

Abstract

In an evolutionary delegation game, we investigate the effects on market outputs of different levels of information about the way managers are compensated. When managers are informed about their opponents, the long–run configuration of the industry depends on market conditions. When managers are informed only of the current composition of the population, only profit maximizing firms survive, no matter what market condition prevails. However, if we further lower the level of information –by hiding the current composition of the industry– then we show how managers’ beliefs affect the long run equilibrium.

Appendices

A Proofs

1.1 A.1 Proof of Lemma 1

The lemma follows from a standard local stability analysis on the map (7). In fact, a fixed point r ^∗ ∈ (0, 1) is locally asymptotically stable provided that $\left \vert \frac {dr(t+1)} {dr\left ( t\right ) }_{|r(t)=r^{\ast }}\right \vert <1$. By Assumption 4. on the Monotone Selection dynamics, r ^∗ is stable whenever

$$ \frac{dr(t+1)}{dr\left( t\right) }_{|r(t)=r^{\ast}}=1+r^{\ast}\left( \frac{\partial F\left( r^{\ast},0\right) }{\partial G}\frac{dG(r^{\ast} )}{dr}\right) \in\left( -1,1\right) $$

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which is equivalent to Eq. 8; a flip bifurcation occurs whenever $\frac {dr(t+1)}{dr\left ( t\right ) }_{|r(t)=r^{\ast }}=-1$, i.e. at $\frac {dG(r^{\ast })}{dr}=-\frac {2}{r^{\ast }\frac {\partial F\left ( r^{\ast },0\right ) }{\partial G}}$. It is straightforward to see from Eq. 20 that an inner steady state r ^∗ with $\frac {dG(r^{\ast })}{dr}>0$ is always unstable because of Assumption 1. QED

1.2 A.2 Proof of Proposition 4

To prove the claim, it is sufficient to show that G(w, r) in Eq. 18 is always positive if γ(N − 1) < 1. This indeed happens since |w − r| < 1.

1.3 A.3 Proof of Proposition 5

Consider the function g(r) = (w(r) − r)γ(N − 1) + 1, with w(r) defined in Eq. 19,

$$ g(r)=1+\gamma(N-1)\left\{ \begin{array}{ll} \delta^{1-\nu}r^{\nu} - r & 0\leq r\leq\delta\\ 1-(1-\delta)^{(1-\nu)}(1-r)^{\nu} - r & \delta<r\leq1 \end{array}\right. $$

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with 0 ≤ δ ≤ 1 and ν ∈ (0, 1)∪(1, + ∞). The roots of equation g(r) = 0 are also roots of G(w(r),r) = 0 in Eq. 18 and, by Assumption 3. on the evolutionary process, fixed points for Eq. 7.

Observe that g(0) = g(1) = 1 and that g is smooth in [0, 1], with derivative

$$ g^{\prime}(r)=\gamma(N-1)\left\{ \begin{array}{ll} \delta^{1-\nu}r^{\nu-1}\nu-1 & 0\leq r\leq\delta\\ (1-\delta)^{(1-\nu)}(1-r)^{\nu-1}\nu-1 & \delta<r\leq1 \end{array}\right. $$

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Define:

$$h(\nu,\delta)=(1-\delta)\nu^{\frac{1}{1-\nu}} - (1-\delta)\nu^{\frac{\nu }{1-\nu}} $$

and

$$l(\nu,\delta)=\delta\nu^{\frac{\nu}{1-\nu}}- \delta\nu^{\frac{1}{1-\nu}} $$

Observe that:

• h(ν, δ) = 0 for δ = 1, while for δ ∈ [0, 1), it is −1 < h(ν, δ) < 0 whenever 0 < ν < 1, and 0 < h(ν, δ) <1 for ν > 1;

• l(ν, δ) = 0 for δ = 0, while for δ ∈ (0,1], it is −1 < l(ν, δ)<0 whenever ν > 1, and 0 < l(ν, δ)<1 for 0 < ν < 1.

Now, proceed with the statements. We prove the first statement in the case δ = 0 and ν > 1 (the case δ = 1 and 0 < ν < 1 is similar).

Suppose δ = 0. The function g reduces to g(r) = 1+(1−(1−r)^ν−r)γ(N − 1). Given ν > 1, it is g ^′(r) > 0 when $r<\bar {r}_{1}$ and g ^′(r)<0 when $r>\bar {r}_{1}$. Thus g(r)≥g(0) = g(1) = 1. This proves that g(r) > 0 for each r ∈ [0, 1].

Finally we prove the second statement (the third is similar). Suppose ν > 1 and observe that $\bar {r}_{2}=\arg \min g(r)$, since g ^′(r) < 0 for $r<\bar {r}_{2}$ and g ^′(r) > 0 when $r>\bar {r}_{2}$. The minimum value of g is thus $g(\bar {r}_{2})=1+\gamma (N-1)l(\nu ,\delta )$. Define the following sets:

• ${E_{1}^{l}}(\gamma ,N)=\left \{ (\nu ,\delta )\text { s.t. }0<\delta \leq 1,l(\nu ,\delta )>\frac {-1}{\gamma (N-1)}\right \} ;$

• ${E_{2}^{l}}(\gamma ,N)=\left \{ (\nu ,\delta )\text { s.t. }0<\delta \leq 1,l(\nu ,\delta )=\frac {-1}{\gamma (N-1)}\right \} ;$

• ${E_{3}^{l}}(\gamma ,N)=\left \{ (\nu ,\delta )\text { s.t. }0<\delta \leq 1,l(\nu ,\delta )<\frac {-1}{\gamma (N-1)}\right \}$ ,

which are nonempty provided that the necessary condition γ(N − 1) ≥ 1 holds; these sets define the conditions on the plane (ν, δ) for the sign of $g(\bar {r}_{2})$:

When $(\nu ,\delta )\in {E_{1}^{l}}(\gamma ,N)$, then $g(\bar {r}_{2})>0$ and g(r) > 0 for each r ∈ [0,1].

When $(\nu ,\delta )\in {E_{2}^{l}}(\gamma ,N)$, then $g(\bar {r}_{2})=0$ and g(r) > 0 for each $r\in [0,1]\setminus \{\bar {r}_{2}\}$, while $\bar {r}_{2}$ is a fixed point.

When $(\nu ,\delta )\in {E_{3}^{l}}(\gamma ,N)$, then $g(\bar {r}_{2})<0$ and equation g(r) = 0 has two roots $r_{1}^{*}<\bar {r}_{2}$ and $r_{2}^{*}>\bar {r}_{2}$.

The stability analysis of these steady states for Eq. 7 follows from the proof of Proposition 1. The couple of fixed points (possibly stable and unstable) are created through a fold bifurcation for Eq. 7 on the bifurcation curves ${E_{2}^{l}}(\gamma ,N)$ (when ν > 1) and ${E_{2}^{h}} (\gamma ,N)$ (when 0 < ν < 1) in the parameter space (ν, δ).

B Derivation of uninformed managers’ expected utility

Consider, as in the case of informed managers, the (inverse) demand system (1), which can be written, for a given number K of P-firms, as follows

$$\begin{array}{@{}rcl@{}} p_{P,i}\left( K,x_{P,i}\right) &=& A-x_{P,i}-\gamma\left( \sum\limits_{h=1,h\neq i}^{K} x_{P,h}+\sum\limits_{h=1}^{N-K} x_{R,h}\right)\\ p_{R,j}\left( K,x_{R,j}\right) &=& A-x_{R,j}-\gamma\left( \sum\limits_{h=1}^{K} x_{P,h}+ \sum\limits_{h=1,h\neq j}^{N-K} x_{R,h}\right) \end{array} $$

where, by symmetry, it is

$$\begin{array}{cc} \sum\limits_{h=1,h\neq i}^{K} x_{P,h}=\left( K-1\right) \overline{x}_{P,i} & \sum\limits_{h=1,h\neq j}^{N-K} x_{R,h}=\left( N-K-1\right) \overline{x}_{R,j}\\ \sum\limits_{h=1}^{K} x_{P,h}=K\overline{x}_{P} & \sum\limits_{h=1}^{N-K} x_{R,h}=\left( N-K\right) \overline{x}_{R} \end{array} $$

Given K, the objective functions of managers in the i-th P-firm and j-th R-firm are, respectively,

$$\begin{array}{@{}rcl@{}} U_{P}(K,x_{P,i}) &=& x_{P,i}\left[ p_{P,i}\left( K,x_{P,i}\right) -c\right]\\ &=&-x_{P,i}^{2}+x_{P,i}\left[ A-c-\gamma\left( \left( K-1\right) \overline{x}_{P,i}+\left( N-K\right) \overline{x}_{R}\right) \right] \\ U_{R}(K,x_{R,j}) &=& x_{R,j}\left[ p_{R,j}\left( K,x_{R,j}\right) -\alpha c\right]\\ &=&-x_{R,j}^{2}+x_{R,j}\left[ A-\alpha c-\gamma\left( K\overline {x}_{P}+\left( N-K-1\right) \overline{x}_{R,j}\right) \right] \end{array} $$

At a given time, managers have beliefs w ∈ [0,1] that a P-firm is extracted from the population. To obtain the expected utility u _R (r, x _{R, j} ) of a R-manager we calculate

$$ \begin{array}{ll} u_{R}(w,x_{R,j}) & =\sum\limits_{i=0}^{N-1}U_{R}(N-1-i,x_{R,j})\left( \begin{array}{c} N-1\\ i\end{array}\right) w^{N-1-i}(1-w)^{i}\\ & =-x_{R,j}^{2}+x_{R,j}(A-\alpha c)\\ &\quad-\gamma x_{R,j}\sum\limits_{i=0}^{N-1}\left[ (N-1-i)\overline{x}_{P}+i\overline{x}_{R,j}\right] \ast\\ \ast\left( \begin{array}{c} N-1\\ i\end{array}\right) w^{N-1-i}(1-w)^{i} & =\\ & =-x_{R,j}^{2}+x_{R,j}\left( A-\alpha c\right)\\ &\quad-\gamma x_{R,j}\left[ \left( N-1\right) \left( w\overline{x}_{P}+\left( 1-w\right) \overline {x}_{R,j}\right) \right] \end{array} $$

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The expected utility u _P (w, x _{P, i} ) of a P-manager is calculated analogously.