Abstract
A common feature of managerial and financial reporting is an iterative process wherein various parties selectively correct particular measurements by challenging them and subjecting them to increased scrutiny. We model this feature by adding an agent appeal stage to the standard moral hazard model and show that it can be optimal to allow the agent to decide which performance measures to appeal, despite the agent’s incentive to cherry-pick. In the presence of measurement errors, the agent is incentivized by increased opportunities for cherry-picking that arise if he chooses the “right” vs. the “wrong” acts.
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Notes
Although we have not framed our problem as one of aggregation, providing evaluatees with only an aggregate score is one way of insisting on a comprehensive reevaluation.
Other benefits to delegation include reducing the amount of costly communication (Melumad, Mookherjee, & Reichelstein, 1992), enabling efficient information processing (Radner, 1993), increasing subordinates’ initiative to acquire relevant information (Aghion & Tirole, 1997), and serving as a substitute for commitment to elicit private information (Arya, Glover, & Routledge, 2002). See Mookherjee (2006) for a survey of the recent incentives-based literature on delegation/decentralization.
Dutta and Gigler (2002) present a notable exception in that earnings manipulation is valuable despite their model making the assumptions of unblocked communication, full commitment, and unrestricted contracts. In their model, earnings management can be valuable by reducing the cost of eliciting truthful forecasts from the manager: with conditional earnings management, it becomes more difficult for a manager of one type to mimic another type, thus resulting in the use of a less risky contract. The result is the principal pays out a lower risk premium. In Dutta and Gigler, earnings management affects reported earnings probabilistically, so the manipulation can be viewed as subject to partial monitoring. One can apply a general version of the Revelation Principle by allowing the principal/mechanism designer to do the earnings management for the agent. So, Dutta and Gigler develop a demand for earnings management that arises even when the Revelation Principle applies.
In our model, limiting attention to comprehensive and selective processes is without loss of generality. It might seem that ex ante specifying the nature of the appeal process as a function of the initial scores would allow for a greater variety. However, after the agent’s incentives to appeal are considered, the only difference between any two such systems is when true (0,1) scores are initially mismeasured as (1,0) or vice versa. The two systems we consider capture this difference.
A revelation mechanism would have the agent submitting reports on x i and then have the mechanism designer subject the initial evaluation to appeal as the agent would. The non-revelation mechanisms we consider are payoff equivalent to the set of incentive compatible revelation mechanisms.
In the grading context, cherry-picking by students seems familiar. Admittedly, coarse grades rather than fine-tuned compensation and an objective of maximizing student effort rather than minimizing grades are more appropriate modeling choices for the grading issue.
Such fairness concerns permeate more generally in employee grievance processes, where fairness is viewed as putting the evaluatee in the same position as if the initial evaluation had been done correctly. As Feuille and Chachere (1995) note, “grievance procedures can be thought of as procedural channels that allow employees the remedial opportunity to seek the distributive justice they believe they were denied when the unfair treatment occurred.”
Using numerical examples, we have verified that our results can be extended to allow for agent risk-aversion.
Alternatively, the principal observes and consumes an aggregate output, and x 1 and x 2 are only part of the many components of this aggregate output.
In our binary outcome setting, this is the only opportunity for cherry-picking, thereby facilitating a clear distinction between the selective and the comprehensive system. A richer outcome set, wherein there are multiple opportunities for cherry-picking, complicates the analysis. However, we think it unlikely that it would derail the paper’s theme that cherry-picking can improve the informativeness of the final evaluation.
The likelihood ratio condition in (i) and (iii) cannot both be simultaneously satisfied. Hence, the three cases in the comprehensive appeal setting encompass the entire range of parameter values.
There may also be a fixed cost associated with setting up an appeal process, cost avoided by the centralized approach of always reevaluating. Since modeling these costs essentially entails scaling, we simplify our analysis by assuming that fixed costs are zero.
If the agent had type-dependent ability and the contract was designed to be robust in that it motivates all type agents to exert high effort, we conjecture that the tradeoff that determines when cherry-picking is optimal will involve both the informativeness of the measures and the agent’s marginal productivity in the performance measures.
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Acknowledgements
We thank Maya Atanasova, Joel Demski, Ron Dye, John Fellingham, Jerry Feltham, Paul Fischer, Steven Huddart, Jack Hughes, Yuji Ijiri, Pierre Liang, Haijin Lin, Brian Mittendorf, Stefan Reichelstein (the editor), Doug Schroeder, Shyam Sunder, Rick Young, and workshop participants at Carnegie Mellon, Ohio State University, Penn State, University of British Columbia, University of California at Berkeley, UCLA, University of Florida, and two anonymous referees for helpful comments. Arya gratefully acknowledges support from the John J. Gerlach Chair.
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Appendix
Appendix
Proof of Proposition 1 Given the linear setup, we use the standard Duality Theorem (e.g., Luenberger, 1989, p. 85) to derive the proposition. We begin by writing the primal program–this is just (P) with n = C. Also, in writing the primal, we drop the (IR) constraint since it is dominated by the (IC LL ) and the non-negativity constraints. We then write the dual of the primal with the dual variables (shadow prices) indicated by \(\lambda_i^C.\) Finally, we present the primal and the dual solutions. Within the specified regions, simple algebra verifies that the listed solutions are feasible and provide the same objective function value when substituted into their respective programs. This establishes optimality.
Subject to:
\( Dual: \quad \mathop{\hbox{Max}}\limits_{\lambda_i^C} [2\lambda_1^C+\lambda_2^C]v(e_H) \)
Subject to:
The solution to the primal program is as given in Proposition 1. The solution to the dual in each of the three cases is given below.
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(i)
$$ \begin{aligned} &\hbox{If\,} \frac{p_2^C (e_H ,e_H)}{p_2^C (e_L ,e_L )} > \frac{p_1^C (e_H ,e_H )}{p_1^C (e_L ,e_L )},\\&\tilde{\lambda}_1^C =\frac{p_2^C (e_H ,e_H)}{p_2^C (e_H ,e_H )-p_2^C (e_L ,e_L )};\quad \tilde {\lambda}_2^C =\tilde {\lambda}_3^C = \tilde{\lambda}_4^C =0.\\\end{aligned} $$ -
(ii)
$$ \begin{aligned} &\hbox{If\,}\frac{p_2^C (e_H ,e_H )}{p_2^C (e_L ,e_L )}\leq \frac{p_1^C (e_H ,e_H)}{p_1^C (e_L ,e_L )}\ \hbox{and}\ \frac{p_2^C (e_H ,e_H)}{p_2^C (e_L ,e_H)} > \frac{p_1^C (e_H ,e_H )}{p_1^C (e_L ,e_H)},\\&\tilde{\lambda}_1^C =\frac{p_2^C(e_H ,e_H )p_1^C (e_L ,e_H )-p_2^C (e_L ,e_H)p_1^C (e_H,e_H)}{[p_H -p_L ]^{3}[1-q]^{2}[1-q+q^{2}]};\\&\tilde {\lambda}_2^C =\frac{p_2^C (e_L ,e_L )p_1^C (e_H ,e_H )-p_2^C (e_H ,e_H )p_1^C (e_L ,e_L )}{[p_H -p_L ]^{3}[1-q]^{2}[1-q+q^{2}]};\\&\tilde {\lambda }_3^C =\tilde {\lambda }_4^C =0.\\ \end{aligned} $$ -
(iii)
$$ \begin{aligned} &\hbox{If\,} \frac{p_2^C (e_H ,e_H )}{p_2^C (e_L ,e_H)} \leq \frac{p_1^C (e_H ,e_H )}{p_1^C (e_L ,e_H )},\\&\tilde {\lambda }_1^C =\tilde {\lambda}_4^C =0;\\&\tilde {\lambda }_2^C =\frac{p_1^C (e_H ,e_H )+p_2^C (e_H ,e_H )}{[p_H -p_L ][1-p_H ][1-q]^{2}};\\&\tilde {\lambda }_3^C =\frac{p_2^C (e_L ,e_H )p_1^C (e_H ,e_H )-p_2^C (e_H ,e_H )p_1^C (e_L ,e_H )}{[p_H -p_L ][1-p_H ][1-q]^{2}}.\\\end{aligned} $$
This completes the proof of Proposition 1. □
Proof of Proposition 2 With the selective system, the primal and dual are as in proof of Proposition 1 except that the superscript “C” is replaced with “S.” Using (3) and (5):
Thus, only the analog to case (i) from the proof of Proposition 1 applies. Hence, with the selective system, the primal solution is as in Proposition 2, and the corresponding dual solution is as in case (i) of the proof of Proposition 1 with “C” replaced by “S”. This completes the proof of Proposition 2. □
Proof of the Corollary Substituting q = 0 into Proposition 2 yields the optimal contract when \((x^{1},x^{2})\) is available for contracting. Thus, the optimal expected payment is equal to \(p_2^S (e_H ,e_H)\tilde {t}_2^S\) with q = 0. After simplifying, this equals \(p_H^2 \left[\frac{2v(e_H )}{p_H^2 -p_L^2} \right].\)
If the preliminary and final scores are both contractible, consider the use of the following contract. Under either system, the principal offers to pay a bonus if and only if the initial scores are (0,0) but, after appeal, the final scores are (1,1). The bonus amount is \(\frac{2v(e_H )}{(p_H^2 -p_L^2 )q^{2}}.\)
If the agent chooses \((e_{k},e_{m}),\) the probability that \((y^{1},y^{2}) = (0,0)\) and \(z^{n}=(1,1)\) is \(p_k p_m q^{2}\) since this event only occurs if \((x^{1},x^{2})=(1,1)\) and both items are initially scored incorrectly. It is easy to verify that the contract ensures the choice of \((e_{H},e_{H})\) is incentive compatible. Further, under the \((e_{H},e_{H})\) choice, the principal’s expected payment equals \(p_H^2 q^{2}\left[\frac{2v(e_H )}{(p_H^2 -p_L^2 )q^{2}} \right]=p_H^2 \left[\frac{2v(e_H )}{(p_H^2 -p_L^2 )} \right].\) Since the expected payment is the same as when \((x^{1},x^{2})\) is directly available for contracting, the principal can do no better. This proves the corollary. □
Proof of Proposition 3 Under the likelihood ratio condition listed in Proposition 3, the optimal contract under the comprehensive system is either as in case (ii) or as in case (iii) of Proposition 1. Algebra then verifies that \(\left[{p_1^C (e_H ,e_H )\tilde {t}_1^C +p_2^C(e_H ,e_H)\tilde {t}_2^C} \right]-p_2^S (e_H ,e_H)\tilde {t}_2^S > 0\) for \(\tilde {t}_j^C\) values from Proposition 1(ii) or 1(iii) and \(\tilde {t}_2^S\) value from Proposition 2. This completes the proof of Proposition 3.
Proof of Proposition 4 Given Proposition 3 and the feasibility of the case (i) contract under the comprehensive system for all parameter values, it follows that the selective system is preferred to the comprehensive system if and only if \(p_2^C (e_H ,e_H )\tilde {t}_2^C \geq p_2^S (e_H ,e_H )\tilde {t}_2^S\) where \(\tilde {t}_2^C\) is as in Proposition 1(i). Substituting for the probabilities and the payments, and solving the equation \(p_2^C(e_H ,e_H )\tilde {t}_2^C =p_2^S (e_H ,e_H )\tilde {t}_2^S\) for q yields the q*-value in the proof of Proposition 5. For \(q \quad > \quad q^{\ast},p_2^C (e_H ,e_H )\tilde {t}_2^C > p_2^S (e_H ,e_H )\tilde {t}_2^S\); for \(q < q^{\ast}\), the direction of this inequality is reversed. This completes the proof of the proposition.
Proof of Proposition 5 Let C1 denote the C-value that equates (10) with (11), C2 the C-value that equates (10) with (12), and C3 the C-value that equates (11) with (12). By this construction, if C < Min{\(C_{1},C_{2}\)}, the centralized approach is preferred. Also, if C > Max\(\{C_{1},C_{3}\}\), the delegated selective process is preferred. Thus, for Min\(\{C_{1},C_{2}\}\leq C \leq\) Max\(\{C_{1},C_{3}\}\), the delegated comprehensive system is preferred. Define \(\underline{C}\)=Min\(\{C_{1},C_{2}\}\) and \(\overline C\)=Max\(\{C_{1},C_{3}\}\); clearly, \(\underline{C}\leq \overline C.\) This completes the proof of the proposition. □
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Arya, A., Glover, J. Performance measurement manipulation: cherry-picking what to correct. Rev Acc Stud 13, 119–139 (2008). https://doi.org/10.1007/s11142-007-9042-3
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DOI: https://doi.org/10.1007/s11142-007-9042-3