The organization of project evaluation under competition

Abstract

This article examines the optimal organizational form of project evaluation under competition. The evaluations are carried out by two fallible screening units that sequentially assess projects. Screening can be organized as a hierarchy or a polyarchy. We show that as competitive pressure rises, the polyarchy becomes less attractive. Therefore, different organizational forms might be found in different industries depending on the degree of competition. In addition, we examine endogenous screening rules under competition: For symmetric situations, we show that polyarchies will employ higher decision thresholds compared to hierarchies. Nonetheless, as in the case of exogenous screening rules, the hierarchy becomes more attractive the higher the degree of competition.

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Notes

  1. 1.

    In our setting the firm would never choose to have only one evaluation unit. The reason is that we do not consider evaluation costs or time to market considerations. Therefore, it always pays off to allow the possibility of a second screening: If the initial portfolio is good, the polyarchy offers the advantage to screen a project a second time that has already been declined by the first screening unit. The probability of a Type-I-error is therefore reduced. If the initial portfolio is bad, the hierarchy requires both screening units to recommend a project before it gets implemented. The probability of a costly Type-II-error thus can be reduced.

  2. 2.

    See also Gehrig et al. (2000) who consider endogenous screening rules for each screening unit separately.

  3. 3.

    The timing of the interaction is based on the assumption that the organizational form is a more long-term decision compared to the definition of the decision threshold.

  4. 4.

    See Sah and Stiglitz, Footnote 7. The profit functions under competition Π ij are given in the Appendix.

  5. 5.

    For the monopolistic case see Sah and Stiglitz (1986), Proposition 4.

  6. 6.

    Follwing Sah and Stiglitz (1986), Proposition 7.

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Correspondence to Peter-J. Jost.

Appendix

Appendix

Proof of Proposition 2

To prove this proposition, let Π ij be the expected profit of a firm that chooses an organizational structure i ∈ {H,P} whereas its competitor decided for an organizational structure j ∈ {H,P}. Then

$$ \begin{aligned} \Uppi _{HH} &= \alpha p_{1}^{2}\left( p_{1}^{2}z_{D}+(1-p_{1}^{2})z_{M}\right) -(1-\alpha)p_{2}^{2}z_{B} \\ &= (1-\alpha)z_{B}\left( \beta p_{1}^{2}-p_{2}^{2}-\gamma p_{1}^{4}\right) \\ \Uppi _{HP} &= \alpha p_{1}^{2}\left(p_{1}(2-p_{1})z_{D}+(1-p_{1}(2-p_{1}))z_{M}\right) -(1-\alpha)p_{2}^{2}z_{B} \\ &= (1-\alpha)z_{B}\left(\beta p_{1}^{2}-p_{2}^{2}-\gamma p_{1}^{3}(2-p_{1})\right)\\ \Uppi _{PH} &= \alpha p_{1}(2-p_{1})\left(p_{1}^{2}z_{D}+(1-p_{1}^{2})z_{M}\right) -(1-\alpha)p_{2}(2-p_{2})z_{B}\\ &= (1-\alpha)z_{B}\left( \beta p_{1}(2-p_{1})-p_{2}(2-p_{2})-\gamma p_{1}^{3}(2-p_{1})\right)\\ \Uppi _{PP} &= \alpha p_{1}(2-p_{1})\left(p_{1}(2-p_{1})z_{D}+(1-p_{1}(2-p_{1}))z_{M}\right) -(1-\alpha)p_{2}(2-p_{2})z_{B}\\ &= (1-\alpha)z_{B}\left( \beta p_{1}(2-p_{1})-p_{2}(2-p_{2})-\gamma p_{1}^{2}(2-p_{1})^{2}\right)\\ \end{aligned} $$

We have :

$$ \Uppi _{PP}-\Uppi _{HH} = 2(1-\alpha)z_{B}\left( \beta p_{1}(1-p_{1})-p_{2}(1-p_{2})-2\gamma p_{1}^{2}\left( 1-p_{1}\right) \right) $$
(5)
$$ \Uppi _{PP}-\Uppi _{HP} = 2(1-\alpha)z_{B}\left( \beta p_{1}(1-p_{1})-p_{2}(1-p_{2})-\gamma p_{1}^{2}(1-p_{1})(2-p_{1})\right) $$
$$ \Uppi _{PH}-\Uppi _{HH} = 2(1-\alpha)z_{B}\left( \beta p_{1}(1-p_{1})-p_{2}(1-p_{2})-\gamma p_{1}^{3}(1-p_{1})\right) $$

Hence

$$ \Uppi _{PH}-\Uppi _{HH} > \Uppi _{PP}-\Uppi _{HP} > \Uppi _{PP}-\Uppi _{HH}. $$
  1. 1.

    Suppose that Π HH  > Π PH . Then Π HP  > Π PP and Π HH  > Π PP . Therefore, H is a best response to H and P. Symmetry implies that (H,H) then is the unique equilibrium and Pareto-dominates (P,P). We have Π HH  > Π PH if \(( \beta -\gamma p_{1}^{2}) p_{1}(1-p_{1}) < p_{2}(1-p_{2}).\)

  2. 2.

    Suppose that Π HH  < Π PH and Π HP  > Π PP . Hence P is a best response to H and H is a best response to P. We have Π HH  < Π PH if \((\beta -\gamma p_{1}^{2}) p_{1}(1-p_{1}) > p_{2}(1-p_{2})\) and Π HP  > Π PP if \(( \beta -\gamma p_{1}(2-p_{1})) p_{1}(1-p_{1}) < p_{2}(1-p_{2}).\) Therefore, we have Π HP  < Π PH since p 2(1−p 2) < βp 1(1−p 1).

  3. 3.

    Suppose that Π HP  < Π PP . Then Π HH  < Π PH . Therefore, P is a best response to P and H. Π HP  < Π PP if (β −γ p 1(2−p 1)) p 1(1−p 1) > p 2(1−p 2). However, we have Π PP  > Π HH only for (β −2γ p 1) p 1(1−p 1) > p 2(1−p 2). If the reverse is true, coordination offers an improvement potential since Π PP  < Π HH . \(\square\)

Proof of Proposition 3

We examine the effect of a change in the competitive pressure on the decision threshold. For example, consider the symmetric case where both firms organize as a hierarchy. From (4) we have \(f( z_{D},R^{HH}) ={\frac{\alpha } {(1-\alpha)z_{B}}}( z_{M}-(z_{M}-z_{D})p_{1}(R^{HH})^{2}) p_{1}^{\prime}(R^{HH})p_{1}(R^{HH})-p_{2}^{\prime}(R^{HH})p_{2}(R^{HH})=0\) and for R HH to be a maximum we must have \({\frac{\partial f} {\partial R^{HH}}} < 0.\) We have

$$ \frac{\partial R^{HH}}{\partial z_{D}}=-\frac{\frac{\partial f}{\partial z_{D}}}{\frac{\partial f}{\partial R^{HH}}}=-\frac{\alpha }{(1-\alpha)z_{B}} \frac{p_{1}(R^{HH})^{3}p_{1}^{\prime }(R^{HH})}{\frac{\partial f}{\partial R^{HH}}} <\ 0. $$

since p′(R) < 0. The same reasoning applies to the other competitive organizational forms in (4).

Therefore, the stronger the competitive pressure (small z D ), the higher will be the optimal decision threshold.\(\square\)

Proof of Proposition 4

To see that R PP > R HH, define \(c(z):={\frac{1-p(z)} {p(z)}}\) (i.e. \(c(z_{M})={\frac{( 1-p_{1}) } {p_{1}}}\) and \(c(z_{B})={\frac{( 1-p_{2}) } {p_{2}}}.\)) Calculate

$$ \begin{aligned} c(0){\frac{\partial \Uppi _{HH}} {\partial R}}-{\frac{\partial \Uppi _{PP}} {\partial R}} &= 2\alpha p_{1}^{\prime }(R)p_{1}\left( c(0)-c(z_{M})\right) \left( z_{M}-p_{1}^{2}\left( z_{M}-z_{D}\right) \right)\\ & \quad +2\alpha p_{1}^{\prime }(R)p_{1}2p_{1}\left( z_{M}-z_{D}\right) c(z_{M})\left( 1-p_{1}\right)\\ & \quad -2\left( 1-\alpha \right) p_{2}p_{2}^{\prime }(R)z_{B}\left( c(0)-c(z_{B})\right). \\ \end{aligned} $$

We have c′(z) < 0 and therefore c(0) − c(z M ) > 0 since z M  > 0. We also have p′(R) < 0, therefore the first and the second term are negative. We have c(0) − c(z B ) < 0 since z B  < 0. We also have p′(R) < 0, therefore the third term is also negative. Assume to the contrary that R HH > R PP. Then from \({\frac{\partial \Uppi _{PP}} {\partial R}} > c(0){\frac{\partial \Uppi _{HH}} {\partial R}}\) it follows that for R HH > R PP:

$$ \Uppi _{PP}\left(R^{HH}\right)-\Uppi _{PP}\left(R^{PP}\right) > c\left(0,R^{HH}\right)\Uppi _{HH}\left(R^{HH}\right)-c\left(0,R^{PP}\right)\Uppi _{HH}\left(R^{PP}\right). $$

Since \(c(0,R)={\frac{M(R)} {1-M(R)}},\) we have \({\frac{\partial } {\partial R}}c(0,R)={\frac{m(R)} {(1-M(R)) ^{2}}} > 0\) and thus c(0,R HH) > c(0,R PP). Therefore,

$$ c\left(0,R^{HH}\right)\Uppi _{HH}\left(R^{HH}\right)-c\left(0,R^{PP}\right)\Uppi _{HH}\left(R^{PP}\right) > c\left(0,R^{PP}\right)\left( \Uppi _{HH}\left(R^{HH}\right)-\Uppi _{HH}\left(R^{PP}\right)\right).$$

Hence, for R HH > R PP we must have

$$ \Uppi _{PP}\left(R^{HH}\right)-\Uppi _{PP}\left(R^{PP}\right) > c\left(0,R^{PP}\right)\left(\Uppi _{HH}\left(R^{HH}\right)-\Uppi _{HH}\left(R^{PP}\right)\right).$$

The LHS is negative since the optimal R is R PP. The RHS is positive since the optimal R is R HH and c(0,R PP) > 0. This contradicts R HH > R PP. \(\square\)

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Jost, PJ., Lammers, F. The organization of project evaluation under competition. Rev Manag Sci 3, 141–155 (2009). https://doi.org/10.1007/s11846-009-0030-9

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Keywords

  • Organization
  • Project evaluation
  • Competition

JEL classification

  • D2
  • L2