Gaming or Shirking? On a Fundamental Trade-Off When Designing Incentives Wendelin Schnedler∗ July 17, 2017

Abstract Examples in which agents ‘game’ incentives abound. Still, gaming has not been properly integrated into the canonical model of incentives. This paper proves for the first time that the value of incentives depends on how well they prevent shirking as well as gaming. While aligning incentives precludes gaming, optimal incentives trade-off gaming and shirking. The theory explains why incentives are suppressed (even for unmotivated agents), identifies the forces underpinning incentive design, clarifies that these forces operate beyond multitasking, uncovers why ‘noisiness’ fails to describe the value of information, and reveals why it is not necessarily ‘foolish’ to ‘reward for A while benefiting from B.’

Keywords: multi-tasking, gaming, misdirected effort, sufficient statistic result, value of information, alignment, incentive scheme, LEN model JEL-Codes: D86, D82, M52, M41, J33 ∗ University of Paderborn, Faculty of Economics, Warburger Straße 100, D-33098 Paderborn ([email protected]). for insightful comments.

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Introduction

While incentives are at the heart of economics, our theory of incentive design is either incomplete or not properly applied: in a plethora of anecdotal1 and systematically analyzed2 examples, economic agents ’game’ incentives provided to them by some principal. Consider, for an example, the attempt of the U.S. Agency of International Development (USAID) to fight an infestation of Colorado potato beetles in Afghanistan by paying 5 dollar for the delivery of each bottle full of these beetles. Incentives were scrapped when locals were found breeding beetles, instead of collecting them.3 Closer to home, Frey (2003) claims that academics respond to the strong incentives to publish by ‘prostituting’ themselves instead of pursuing original research. (Ironically, he provided a case in point by boosting his own publication count through submitting resembling ideas to different journals— see Frey, 2011). A very intuitive well-known explanation for ‘gaming’ is that the incentives provided to the agent do not reflect the benefit generated for the principal (see, e.g., Kerr, 1975; Baker et al., 1994): USAID wanted to reduce a pest but paid for receiving bottles full of beetles. Examples of ‘gaming’ are probably most often associated with the seminal work by Holmstr¨om and Milgrom (1991).4 Holmstr¨om and Milgrom challenge the traditional view that incentives are only about getting the agent to exert effort and insuring him against income fluctuations. According to this traditional incentiveinsurance trade-off, any independent information about the agent’s performance would have to be used to provide incentives, including the number of bottled beetles. Holmstr¨om and Milgrom, however, consider an agent who can mis-allocate his (total) effort among multiple tasks, where (total) effort is the sum of the task-wise 1 For

examples, see Kerr’s ‘On the folly of rewarding for A while hoping for B’ (1975), survey articles by Gibbons (1998) and Prendergast (1999), or Stephen J. Dubner’s freakonomics podcast from Nov 10, 2012. Retrieved 19 Oct 2015 from http://freakonomics.com/2012/10/11/the-cobraeffect-a-new-freakonomics-radio-podcast/. 2 See for example, Oyer (1998), Dranove et al. (2003), Courty and Marschke (2004), Courty and Marschke (2008), Propper et al. (2010), Hong et al. (2013), Larkin (2014), Sloof and van Praag (2015), or Forbes et al. (2015). 3 Ben Arnoldy, Christian Science Monitor (28th of July 2010). 4 Holmstr¨ om and Milgrom (1991) is quoted more than 6000 times. For some theoretical advances, see Athey and Roberts (2001), Sliwka (2002), Baker (2000, 2002), Inderst and Ottaviani (2009), or more recently Kragl and Sch¨ottner (2014).

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effort choices. Then, they show that ignoring performance information, like the number of beetle-filled bottles, and suppressing incentives can be optimal. This well-known multi-tasking approach, which also features in the explanation for awarding the Nobel prize to Bengt Holmstr¨om, did not only challenge traditional views more than 25 years ago, it also raises some issues, which are still not settled today. First, the approach does not mesh with the very intuitive idea that ‘gaming’ can be prevented by aligning the agent’s incentives with the principal’s benefit. The Equal Compensation Principle (Milgrom and Roberts, 1992), which directly follows from Holmstr¨om and Milgrom (1991), implies that the university has to equally reward an academic for marketing and developing his research even if developing is more beneficial to the university than marketing. Second, if the traditional insurance-incentive trade-off is inadequate, what else drives incentive design? Several proposed alternatives posit that the principal has to weigh the precision with which performance is measured with how well the performance measure reflects the principal’s benefit (Datar et al., 2001; Feltham and Wu, 2000; Baker, 2000, 2002). For such precision-congruity trade-offs to make sense, more similarity between performance measure and benefit has to be more attractive (all else being equal). Schnedler (2008), however, shows that this is not the case: incentive designers prefer a performance measures that overemphasize tasks, which the agent likes, to an otherwise identical one, which reflects the true importance of tasks to the principal. Third, is suppressing incentives only optimal under the specific assumptions of Holmstr¨om and Milgrom’s home contractor model (1991), or, for example, also if the agent is not intrinsically motivated? For the work horse model of multitasking (LEN model), where the agent is not intrinsically motivated, Feltham and Xie (1994) show that almost any information about agent’s behavior should be used to provide incentives.5 This is true even for performance measures that are negatively related to the principal’s benefit, like the number of beetle-filled bottles.6 USAID’s decision to suppress incentives, however, does not seem to be driven by whether 5 The

set of useless signals characterized by Feltham and Xie (1994) in Condition (13) has measure zero in the set of all signals. 6 The only signals that are not used are those which are unrelated to the benefit— see, for example, equation (3) in Baker (2002).

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locals were initially interested or not in getting rid of beetles, but by preventing the breeding of beetles. Fourth, is ‘gaming’ necessarily related to the fact that an agent faces multiple tasks? In the beetle example, the agent can choose between several courses of action, e.g., doing nothing, breeding, and collecting beetles. For whether the agent is ‘gaming’ incentives or not, it seems irrelevant whether the agent’s choices are related to different tasks (i.e., arranged along several dimensions) or not. ‘Gaming’ is also used to refer to manipulating the timing at which information is released (see,e.g., Oyer, 1998; Courty and Marschke, 2004), reporting that a worker did his job without checking (Rahman, 2012), or taking unnecessary risks (Barron et al., 2016). In all these cases, the multiple tasks or dimensions of the decision are arguably not obvious. In the very first article that uses the term ‘gaming’ (Baker, 1992), the agent’s choice set is even one-dimensional. Finally, how does ‘gaming’ affect the value of performance information for designing incentives? Christensen et al. (2010) derive the relative value of performance information within a very general class of multitasking models without any reference to how well this information reflects the principal’s benefit. Does this mean that a university who is genuinely interested in scientific progress need not pay attention to whether the performance measure directs the academic’s effort to research or marketing? This paper offers a general and intuitive theory of incentives and gaming that resolves these unsettled issues and is surprisingly simple to prove. The definition of gaming at the heart of this theory is inspired by Holmstr¨om and Milgrom’s (1991) suggestion that the agent can misallocate effort across tasks. In order to give a new lease of life to this suggestion and advance on the problematic issues, I will interpret the word ‘effort’ in an unconventional but arguably natural way. According to a convention in incentive theory, ‘effort’ describes the choice of the agent and ‘effort costs’ reflect the agent’s preferences over choices.7 This convention carries over to multitasking, where the agent has an effort choice for each task and ‘total effort’ is the sum of the task-wise choices (see, e.g. Bond and Gomes, 2009). In contrast to this convention, I interpret ‘effort’ in Holmstr¨om and 7 This

convention goes back at least to Holmstr¨om (1979) who finds it ‘convenient to think of a [the agent’s action choice] as effort and this term [effort] will be used interchangeably with action.’

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Milgrom’s suggestion to refer to the agent’s preferences rather than his choices (or the sum of his choices). The agent is then indifferent between two choices that require the same effort, say, breeding a lot of beetles or collecting a few of them. Incentives are gamed if the effort induced by incentives is not used Pareto-optimally by the agent. Put differently, incentives lead the agent to take an action, e.g., breeding a lot of beetles, while another action is more beneficial to the principal but equally disliked by the agent, e.g., collecting a few beetles. For this definition of gaming, it is easy to show that aligning incentives with the principal’s benefit rather than equal compensation prevents gaming (Proposition 1). This justifies Kerr’s claim (Kerr, 1975) that rewarding for A when hoping for B is ‘foolish’—albeit only if the aim is to prevent gaming. The aim of incentive designers, however, is not to prevent gaming but to minimize agency costs (the loss from being unable to directly enforce the desired action). Gaming nevertheless matters for incentive design and the proposed theory identifies how: agency costs are due to shirking (the academic puts in too little effort) and gaming (the academic uses too much of the effort to please referees)— see Proposition 2). This decomposition of agency costs directly implies that losses from gaming and shirking need to be traded off (Corollary 1). This trade-off reveals for the first time the forces underpinning the design of incentives under multitasking (and beyond) and thus provides a suitable alternative to the problematic incentiveinsurance and congruity-precision trade-offs. The trade-off also explains why rewarding for A while benefiting from B may be ‘prudent’ (Schnedler, 2008): In the academic example, the gains from more effort, some of which is aimed at research, can outweigh the losses from pleasing referees. The theory retains and formalizes the informal idea by Holmstr¨om and Milgrom (1991) that incentives are suppressed because of gaming (Corollary 2). So far formal arguments for suppressing incentives have focused on the idea by social psychologists8 that incentives ‘crowd out’ intrinsic motivation.9 By complementing these ‘crowding out’ theories, it becomes possible to explain why incentives are 8 See

Deci (1971); Lepper et al. (1973); Deci et al. (1999). Seabright (2009); B´enabou and Tirole (2003, 2006); Sliwka (2007); Herold (2010); Friebel and Schnedler (2011); Schnedler (2011); Schnedler and Vadovic (2011); van der Weele (2012); Schnedler and Vanberg (2014). 9 See

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suppressed although they increase the agent’s effort—like in the beetle example.10 The next section illustrates that shirking and gaming (as defined here) are indeed the two fundamental forces that drive the choice of optimal incentives in a fairly standard multitasking example. This simple example deviates in various points from the home contractor model by Holmstr¨om and Milgrom (1991) in order to show that their specific assumptions, in particular the assumption of an intrinsically motivated agent, is not necessary for incentives to be suppressed. The example also assumes the agent to be risk-neutral and to face no constraints on his liability in order to highlight that unlike for previously suggested trade-offs (Holmstr¨om, 1979; Holmstr¨om and Milgrom, 1991; Datar et al., 2001; Feltham and Wu, 2000; Baker, 2000, 2002; Prendergast, 2002), insurance problems are not necessary for the gaming-shirking trade-off to operate. While gaming is often associated with multitasking, multitasking is neither necessary nor sufficient for gaming to occur. Section 3 features an example with one task in which incentives are gamed to such a degree that it is better to suppress them. Incentives are safe from gaming even despite multitasking if effort can only be used in one way, for example, because the agent has strict preferences over action choices. (Interestingly, the paper by Baker (1992) that pioneered the term ‘gaming’ assumes such preferences and thus does not feature gaming.) The novel definition also fits those situations that have been associated with gaming in which the multiplicity of tasks is not obvious. In all these situations, the agent is initially indifferent between two actions (release information now or later, report that everything is in order without checking or reporting that he has not checked, imposing unnecessary risk on the principal or not) but incentives lead him to pick the action least beneficial to the principal. Existing alternatives to the incentive-insurance paradigm are restricted to the multitasking linear exponential normal model, which itself has been criticized for its specific nature (Hemmer, 2004). Section 4 introduces a fairly general framework that covers this and many other well-known moral-hazard models. Probably, the only serious restriction of this framework is that the agent’s preferences over action choices have to be separable from those over the reward good. This assumption is 10 See

also Fong and Li (2016) who show that information should be suppressed in order to reduce the firm’s incentive to renege in a relational contract.

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necessary for effort to be well-defined and holds in the large class of moral-hazard models that feature an ‘effort cost’ function. Section 5 formally defines gaming and shirking costs and Section 6 proves the results for this general setting. The theory identifies in what sense traditional concepts to capture and compare the value of information for designing incentives are incomplete. Section 7 shows that in the general framework, Holmstr¨om’s sufficient statistic result (1979) and Kim’s ranking of information systems (1995) apply to shirking but not to gaming costs (Corollary 3 and 4). Generalizations of these results (see, e.g., Christensen et al., 2010) thus rely on the effects of gaming to be negligable. Finally, the theory helps justifying the implicit assumption from the literature that measuring performance congruently with the benefit is desirable (Corollary 5), while also explaining Schnedler’s puzzling finding (2008) that measuring performance discongruently yields lower agency costs (Corollary 6).

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Gaming and Shirking Under Multitasking

This section first introduces a fairly ordinary multitasking model. The analysis of incentives in this model then shows that aligning incentives indeed prevents gaming (as opposed to equal compensation), that optimal incentives result from trading-off gaming and shirking (rather than incentive and insurance or congruity and precision) and that gaming can prevent the use of performance information (even if the agent is unmotivated). Consider the principal of a university (she) who wants an academic (agent, he) to engage in research but also in marketing this research. The academic can allocate one unit of time between thinking about research, a1 ∈ [0; 1], marketing, a2 ∈ [0; 1], or about something of no value to the principal (like his next holiday): a3 = 1 − a1 − a2 , where a1 + a2 ≤ 1. The principal benefits from the academic’s choice: b(a) = βa1 + (1 − β)a2 , where β ∈ (0, 1) describes the beneficial effect of research relative to marketing. (Later, the second dimension will be allowed to negatively affect the benefit.) Graphically, the choices between which the principal is indifferent can be repreβ , see dashed lines in Figure 1; the sented by iso-benefit lines with a slope of − 1−β further away the iso-benefit line from the origin, the higher the benefit and the 7

better for the principal.

Figure 1: The academic only cares about how much effort e he exerts. The principal’s benefit, however, depends on how this effort is used. Any time spend on research or marketing requires effort e from the academic, say, e(a) = a21 + a22 , which he dislikes. While this assumption does injustice to many academics, it helps to make the point that intrinsic motivation is not required to explain the absence of incentives. In absence of incentives, the academic only cares about how much effort e he exerts and not toward which specific course of action a this effort is directed. The choices (a1 , a2 ) that require the same effort e can be depicted by iso-effort curves, where curves closer to the origin are associated with lower effort and better for the academic—see the solid curves in Figure 1. The principal has control over some good that can be used as a reward: teaching reductions, more office space, etc. In order to reflect the opportunity costs of these resources to the principal, let her utility be a function of benefit b and reward r: v(b, r) = b − r. The academic likes the reward r but dislikes effort e and his utility is: u(e, r) = r − e. He has the outside option of not exerting effort and receiving no reward u(0, 0) = 0. Finally, he is assumed to be risk-neutral to show that the gaming-shirking trade-off applies even in the absence of insurance problems. There is very little objective information that can be used in this example to provide incentives. A court of law cannot verify what the academic is thinking, how much effort his thinking requires, or which benefit the principal derives from it (only the principal knows this). However, the principal can reward publication success (Y = 1) or failure (Y = 0). Success is more likely when the academic thinks longer about research or spends more time contemplating how to market 8

this research. For simplicity, assume Prob(Y = 1|a1 , a2 ) = ρa1 + (1 − ρ)a2 , where ρ ∈ (0, 1) describes the relative importance of research for publication success. The relative weighting ρ is exogenously determined by the taste of referees and editors, not by the university principal. Contracts I take the form of a salary πI that is independent from performance and a premium πI that is only paid in case of success. This concludes the description of what arguably is a typical multitasking model, where the only unusual feature is perhaps that effort is used to capture the agent’s preferences rather than his choices. Finding optimal incentives in this model is no problem. The remainder of this section first uses standard techniques to determine the agent’s response to incentives. Then, follows the novel part of this section: the role that gaming plays for incentive design is explicitly identified. The academic responds to given incentives I, (πI , πI ), by choosing an action aI = (aI1 , aI2 ) that maximizes his expected utility: aI ∈ arg max a1 ,a2

πI + πI · (ρa1 + (1 − ρ)a2 ) − a21 − a22 .

Figure 2: The academic selects an action that leads to expected rewards with as little effort as possible, which restricts the set of action choices that can be induced with a publication premium. The induced action can be computed using first-order conditions: aI = (aI1 , aI2 ) = (ρ, (1 − ρ)) ·

πI . 2

(1)

The larger the publication premium, πI , the more time the academic will spend thinking about research and marketing. The more emphasis ρ referees place on 9

research, the less time the academic will spend on marketing. Incentives that promise a larger publication premium πI elicit more effort: I

e(a ) =


2 aI1 +


2 aI2

2

2

= ρ + (1 − ρ)




πI · 2 

2 .

(2)

Since the principal has no control over how much emphasis ρ referees place on research as opposed to marketing, the behavior that can be induced with the publication premium is limited. Graphically, implementable action choices can be determined as follows. Represent all choices that yield the same expected reward by iso-reward lines —see dashed dotted lines in Figure 2. The academic picks the choice that results in given expected rewards with the least effort. Only action choices where iso-effort and iso-reward curve touch, which are the choices on the dotted line, can be implemented. A larger premium πI leads to a choice further to the right. The expected transfer rI from the principal to the academic must be sufficiently large to prevent the academic from choosing his outside option: u(eI , rI ) ≥ 0, where eI := e(aI ) is a real number that describes the effort required for action aI . By reducing salary πI , the principal can minimize expected transfers rI until they just compensate for effort:  
πI 2 r = e = ρ + (1 − ρ) . 2 I

I

2

2

(3)

So far, the analysis has been fairly standard. Now, we introduce the notion of gaming. While aI describes how the academic actually directs the elicited effort eI , I there  are many other action  choices toward which effort e could have been directed: I a e = e(a) = a21 + a22 . Which of these choices is Pareto-optimal? Since the academic is indifferent between these choices, we can equivalently ask: What is the action a∗ (eI ) that requires eI and yields the largest benefit to the principal? a∗ (eI ) = arg max βa1 + (1 − β)a2 such that a21 + a22 = eI . a

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(4)

The (unique) benefit-maximizing way of using effort eI is:11 ∗

√ eI

I

a (e ) = (β, 1 − β) · p . β2 + (1 − β)2

(5)

Bond and Gomes (2009) determine the benefit-maximizing allocation of total effort, a1 + a2 . In their model as well as in the home contractor model by Holmstr¨om and Milgrom (1991), the agent’s marginal rate of substitution between any two tasks for any choice is the same, so that total effort coincides with the notion of effort used here. In the model of this section, however, total effort, a1 + a2 , and effort, a21 + a22 are distinct. The benefit-maximizing way of using total effort is very different from that of using effort, a∗ (eI ), (total effort would have to be concentrated entirely on the dimension that generates higher benefit). Incentives are gamed whenever the academic directs the elicited effort eI toward a Pareto-inferior action aI : aI 6= a∗ (eI )—see Figure 3.

Figure 3: Incentives are gamed whenever the academic uses the induced effort eI for an action aI that does not maximize the principal’s benefit: aI 6= a∗ (eI ). Knowing how the agent uses effort and how he should use effort allows us to directly find out when incentives are gamed. The actual use aI of effort eI is Pareto-optimal if and only if: πI aI = a∗ (e ) ⇐⇒ (ρ, (1 − ρ)) = (β, (1 − β)) 2 I (1),(2),(5)

s

ρ2 + (1 − ρ)2 πI ⇔ ρ = β. β2 + (1 − β)2 2

Incentives lead to a Pareto-optimal use of effort and cannot be gamed if and only if the iso-benefit and iso-reward lines are aligned in the literal sense of the word: 11 It

can be found by solving for a2 in the side-constraint of (4), substituting a2 in the objective function, and determining a∗1 from the respective first-order condition. This approach captures all solutions because a1 ≥ 0 and a2 ≥ 0.

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Figure 4: Effort is used Pareto-optimally, aI = a∗ (eI ) if rewards are aligned with the benefit (left panel). Otherwise, incentives are gamed, aI 6= a∗ (eI ). (right panel). ρ β both lines must have the same slope, − 1−β = − 1−ρ , see Figure 4. This finding is later generalized beyond multitasking in Proposition 1. A practical method to align incentives are balanced scorecards (Kaplan and Norton, 1996). The finding justifies their use (if the aim is to avoid gaming).12 The finding supports Baker, Gibbons and Murphy’s claim (1994) that ‘Basing pay on an employee’s contribution to firm value would have prevented [...] seemingly dysfunctional behavior’ if ‘seemingly dysfunctional behavior’ is interpreted using the specific definition of gaming given here. As soon as the marginal effect of one dimension on the benefit, say research, is larger than the marginal effect of another dimension, say marketing, gaming can only be avoided if marginal rewards differ. The Equal Compensation Principle by Milgrom and Roberts (1992), however, stipulates marginal rewards for different tasks to be equal and thus has to be violated for effort to be used Pareto-optimally. Next, we want to quantify the negative consequences of gaming. If the induced effort eI had been used Pareto-optimally, the principal would have obtained a benefit of b(a∗ (eI )); her actual benefit is b(aI ). Her loss in benefit, GI := b(a∗ (eI ))−b(aI ), is called gaming costs. If effort is used Pareto-optimally, gaming costs are zero. Gaming costs are graphically reflected by the distance between two iso-benefit lines: the line through the Pareto-optimal action a∗ (eI ) and that through the actual action aI —see Figure 5. As compelling as gaming costs may be, optimal incentives are found by minimizing agency costs. The relevant benchmark is thus not the Pareto-optimally use of effort eI but the action choice a∗ = (a∗1 , a∗2 ) as well as the respective compensation r∗ that the academic would receive in a first-best world in which con12 For

an intriguing alternative justification, see Gibbons and Kaplan (2015).

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Figure 5: The costs of incentives (agency costs) are due to effort being misdirected to aI instead of a∗ (eI ) (gaming) and eliciting the ‘wrong’ level of effort, eI instead of e∗ (shirking). tracts on these quantities could be enforced.13 Denote the effort required for the first-best action a∗ by e∗ := e(a∗ ). Then, agency costs of incentives I amount to αI := b(a∗ (e∗ )) − r∗ − (b(aI ) − rI )). If agency costs define the principal’s objective, why should we care about gaming costs? Gaming costs matter for incentive design because they contribute to agency costs. By employing the hypothetical benchmark of Pareto-optimally used effort, agency costs can be decomposed into gaming costs GI and shirking costs:
SI := b(a∗ (e∗ )) − r∗ − b(a∗ (eI )) − rI ,

(7)

which are related to the academic’s level of effort—see Figure 5. The decomposition of agency costs proves that incentives have exactly two functions: eliciting effort and ensuring that this effort is used well. Again, this applies beyond this example and beyond multitasking—see Proposition 2, later. This decomposition directly implies a trade-off. The publication premium (initially) reduces shirking but increases gaming costs. In order to see this, compute (a∗ , r∗ ) ∈ arg maxa,r b(a) − r such that r − c(e(a)) ≥ 0, which leads to:   1 β 1−β a∗ = (a∗1 , a∗2 ) = , and r∗ = (a∗1 )2 + (a∗2 )2 = (β2 + (1 − β)2 ). 2 2 4

13 Formally,

13

(6)

shirking costs using (3), (5), and (6): SI = −

β2 + (1 − β)2 s4 πI 2

!  I 2 π ρ2 + (1 − ρ)2 ) 2 2 2 2 (β + (1 − β) ) − (ρ + (1 − ρ) ) . β2 + (1 − β)2 2

(8)

Taking the derivative with respect to the publication premium πI and evaluating at the point of introduction, reveals that shirking costs initially fall: p p dSI ρ2 + (1 − ρ2 ) β2 + (1 − β2 ) πI 2 = − + (ρ + (1 − ρ)2 ) < 0. dπI πI =0 2 2 |{z} =0

On the other hand, introducing a publication premium increases gaming costs: G

I

πI = 2

(1),(5)

q  q 2 2 2 2 ρ + (1 − ρ) ) β + (1 − β) − (βρ + (1 − β)(1 − ρ)) , | {z } ≥0

where the term in parenthesis is strictly larger than zero if and only if incentives are not aligned, ρ 6= β.14 When no publication premium is used, πI = 0, gaming costs are zero. The larger the publication premium πI , the more misdirected the academic’s effort, and the higher gaming costs (for a given ρ 6= β). For the optimal choice of the publication premium, the principal has to weigh the gains from reduced shirking costs against the loss from increased gaming costs. She will increase the premium until marginal gains are exactly offset by the marginal losses—see Figure 6. If the academic’s liability were limited, π ≥ 0 and π ≥ 0, more compensation would be needed and shirking costs would increase.15 Still, lowering shirking costs 14 Since

β, (1 − β), ρ, (1 − ρ) > 0, the parenthesis are non-negative if and only if (ρ2 + (1 − ≥ β2 ρ2 + 2(1 − β)(1 − ρ)βρ + (1 − β)2 (1 − ρ)2 ), which in turn is equivalent to (ρ(1 − β) − (1 − ρ)β)2 ≥ 0. 15 In this case, the principal would optimally set π = 0 and the expected transfer would dou2
(πI ) ble: rI = 2 ρ2 + (1 − ρ)2 . Gaming costs would be unaffected while shirking costs would  p p
 2 2 I 2 πI 2 + (1 − ρ)2 β2 + (1 − β)2 − (π ) ρ2 + (1 − ρ)2 be: SI = β +(1−β) − ρ , leading to a U4 2 2 shape function very similar to that in Figure 6.

ρ)2 )(β2 + (1 − β)2 )

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Figure 6: For the optimal (second-best) publication premium, πSB , the marginal gains from preventing shirking have to be weighed against marginal losses from more gaming. would come at the price of higher gaming costs. This suggests that the trade-off between both costs is of a more fundamental nature; a suggestion that is later confirmed in Corollary 1. Finally, the trade-off determines whether performance information should be used or ignored. In order to see this, consider a variation of the model in which one dimension, say marketing, is harmful to the principal, β > 1, for example, because it is just a euphemism for bad scientific conduct such as forging significance levels16 or inventing experimental data. In thisvariation effort on research is Pareto-optimal  of the model,√focusing √

β  ∗ I ∗ ∗ a e = eI , 0 and a (e ) = e∗ , 0 = 2 , 0 . A publication premium of πI ,  I  I however, continues to induce: aI = ρ π2 , (1 − ρ) π2 because neither rewards nor the academic’s preferences have changed. Using the new benchmarks for a∗ (eI ) and a∗ (e∗ ) gaming and shirking costs become:  I q √ I (1),(2) π I 2 2 G = b(( e , 0)) − b(a ) = β ρ + (1 − ρ) − (ρβ + (1 − ρ)(1 − β)) , 2 !
I 2 2 Iq
π β π (2),(3),(7) SI = − β ρ2 + (1 − ρ)2 − ρ2 + (1 − ρ)2 . 4 2 4 I

Introducing a publication premium reduces shirking costs by more than it 16 A

rather dramatic example is that of Ulrich Lichtenthaler who was ranked best Germanspeaking researcher in business economics in 2005 before retracting 16 of his already published articles, for example, because significance levels were wrongly reported—see retractionwatch.com/2014/06/16/ulrich-lichtenthaler-retraction-count-rises-to-16, accessed on 6th of March 2015.

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increases gaming costs if and only if marketing has not too much influence on referees:17 I dS dGI 1 ≥ ⇔ 1−ρ ≤ . dπI I I dπ 2 − β1 π =0 Suppose that the condition fails. Then, the increase in gaming costs dominates the reduction in shirking costs when using a small premium. Using a large premium, however, does not help either: gaming costs increase at the same rate, while shirking costs are convex and fall at a lower rate in the premium. Thus, the principal is ill advised to introduce any publication premium if 1 − ρ ≤ 2−1 1 a and β

using no incentives is optimal—see Figure 7. Figure 7: If marketing is very detrimental to the principal’s benefit (e.g. because it means that the academic fabricates results), rewarding publication success increases gaming costs by more than it reduces shirking costs and using no incentives is optimal. Incentives are not used here although various assumptions were deliberately chosen to differ from the home contractor model by Holmstr¨om and Milgrom (1991). First, the agent was not intrinsically motivated. Second, the two tasks were not perfect substitutes from the academic’s perspective (Instead, he preferred a mixed use of his work time). Third, the success signal reflected both dimensions. Finally, the second dimension (forging results) was detrimental to rather than essential for generating a benefit. The specific assumptions of the home contractor model are thus not necessary to explain why incentives are suppressed. Indeed, gaming may prevent the use of information about the agent’s behavior much more generally (see Corollary 2, later), for example, even if the agent only faces one task. The next section presents a respective example.



17 dSI dπI I π =0

≥

dGI dπI

⇔

p

1 2β

ρ2 + (1 − ρ)2 ≥

1 2

 p  β ρ2 + (1 − ρ)2 − (ρβ + (1 − ρ)(1 − β)) ⇔

0 ≥ −(ρβ + (1 − ρ)(1 − β)). Solving for 1 − ρ leads to the condition.

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3

Gaming and Shirking with a Single Task

This section illustrates that gaming can occur and prevent the use of performance information in the absence of multitasking, i.e., if the agent’s decision is onedimensional. Consider a doctor (agent) who decides on the intensity a of a treatment for some patient (principal), where a is a non-negative real number, a ∈ R+ . Suppose that the doctor has qualms about not treating the patient, is happy to provide an intensity of a0 , but dislikes going beyond this level. Represent these preferences with an effort function e(a) that falls in the intensity until a = a0 is reached and increases thereafter. The patient, on the other hand, always prefers more intensity, which is reflected by his benefit b(a) increasing in a. Assume that the exact intensity of treatment cannot be identified but only more coarsely whether performance is below or above some exogenously given threshold κ. Finally, suppose that the threshold κ is below the doctor’s preferred level κ < a0 and that rewarding decreases the patient’s utility, while it increases the doctor’s.

Figure 8: Gaming although the agent only faces one task: Rewarding for low performance leads to aI = κ, and requires effort e(κ), which could have been used more beneficially for a. ˜ Under these conditions, incentives can be used to induce the doctor to exert more effort but doing so is pointless because of gaming. The only way to get effort is to attach sufficient rewards to low performance.18 The result is, of course, undesirable. The doctor will choose the highest possible treatment intensity that still generates the low-performance signal: aI = κ. 18 Rewarding

high performance does not change the doctor’s behavior. The doctor continues to engage in his preferred behavior a0 , which already means that high performance can be observed.

17

Why is rewarding for low performance problematic? Rewards do induce the doctor to exert more effort: eI = e(κ) > e(a0 ). The problem is once more that this effort is misdirected. Instead of using it for a = κ, which yields a benefit of b(κ), it could have been used for a = a, ˜ leading to a benefit b(a). ˜ As κ < a˜ and the patient’s benefit increases in the treatment intensity, the patient would have been better off, had effort been used for a˜ rather than κ, while the doctor would not have been harmed —see Figure 8. The same forces that drive incentives under multitasking in the previous section also underpin incentive design, here. Rewarding high performance elicits no effort (shirking), while rewarding low performance elicits effort but at the price of effort being misdirected (gaming). Indeed, this price is prohibitively large here, which is why performance information is optimally discarded. All this suggests that the gaming-shirking trade-off may be important for incentive design beyond multitasking. Indeed, the insights from the multitasking example are valid in a much more general framework. The following section introduces this framework.

4

General Moral Hazard Framework

This section presents a framework that captures fundamental assumptions of most moral-hazard models in which a principal (she) uses rewards to influence the choice of an agent (he). Agent’s choice. The agent chooses an action a from some (separable metric) space A . In the academic example, A = {(a1 , a2 )|ai ∈ [0, 1], a1 + a2 ≤ 1}, where ai is the share of time spend on activity i. In the doctor’s example, A ≡ R+ . The space could also be a finite countable set. Agent’s preferences. The principal controls the amount r of some reward good and the agent prefers having more of this good. Preferences over choices and rewards (a, r) are complete, transitive, continuous, and separable in a and r and strongly monotonic in r. Separability means that agent’s preferences over action choices a do not depend on the level r of the reward good. This assumption is typically imposed in moral hazard models—even in the very general treatments

18

of the problem by Gjesdal (1982) or Grossman and Hart (1983).19 While this excludes, for example, that the academic’s relative preference for research and marketing changes as he gets richer, it does allow for income effects,20 e.g., the agent may prefer to exert less effort when he gets richer. Agent’s effort and utility. Given separability, we can represent preferences over actions using a (continuous) effort function e : A → R, where a 7→ e(a).21 For simplicity (but slightly abusing notation), e is also used to refer to the real number describing agent’s effort for a given action. Using the effort function, preferences over actions and rewards together can be represented by a continuous (real-valued) utility function u : R × R → R with (e, r) 7→ u(e, r) that is strictly falling in e and strictly increasing in r. Standardize the utility when the agent does not engage with the principal to zero: u(0, 0) = 0. Principal’s benefit and utility. The principal cares about the agent’s action choice and dislikes giving up the reward good. Her preferences over (a, r) are complete, transitive, continuous and separable in a and r,22 where r is a ‘bad’: she strictly prefers rewarding less. Preferences over action choices can be represented by a (continuous) benefit function b : A → R, where a 7→ b(a). Using the benefit function, preferences over both, action choices and rewards, can be expressed by a (real-valued) continuous utility function v : R × R → R, where (b, r) 7→ v(b, r) that is strictly increasing in b and strictly falling in r. The principal’s benefit is formulated here as a function of the action. In many applications, the benefit does not only depend on the agent’s choice but also on luck, i.e., factors beyond the control of agent or principal. In this and other cases where a leads to a benefit distribution, b(a) can be regarded as the principal’s certainty equivalent of the lottery generated by choice a.23 Incentives. Incentives are a credible promise of a distribution of rewards by the 19 See

Assumption A1 in Grossman and Hart (1983) and Assumption 4 in Gjesdal (1982) in Section 2-4; a notable exception is his Section 5, where this assumption is dropped. 20 For an example, see Hermalin (1992) who features income effects while agent’s preferences are additively separable in reward and action. 21 This follows from Theorem 1 in Bergstrom (2015). 22 This assumption holds in almost all moral-hazard models. Indeed preferences are often assumed to be additively separable. A notable exception is the model by Raith (2008). 23 In a model, where benefits were random, existence of the certainty equivalent could be ensured by an argument similar to that in footnote 25.

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principal for each action that the agent engages in. The credibility of the principal’s promise may result from an explicit contract, which is enforced by a court of law, or from a a relational contract, which is enforced in a repeated interaction in the tradition of MacLeod and Malcomson (1989, 1998), Levin (2003), and Halac (2012). Formally, incentives are represented by a function I: A → C , where C is the set of cumulative distribution functions. The function maps a given choice a by the agent to a cumulative distribution function of rewards: a 7→ F I (r|a). This distribution can be degenerate as in the doctors example, where the doctor had full control over rewards through his action choice. Contractual Environment. Denote by J the set from which the principal can choose incentives: I ∈ J . This set describes the contractual environment and is typically restricted J ( C . The academic could, for example, only be rewarded on the basis of publications and was more likely to receive πI if he spent more time researching. Being precise about the contractual environment is crucial in order to describe which incentives are feasible and is hence necessary for finding optimal incentives. As will become clear, the following insights on the fundamental forces and trade-offs of incentive design do not depend on the details of the contractual environment. Induced behavior. Suppose that incentives I ∈ J result in some behavior, or equivalently, that the agent’s utility maximization problem given incentives I has a solution. Denote the choice by the agent given incentives I by aI and the real number describing the effort required for this choice by eI := e(aI ).24 The agent’s choice aI results in a cumulative distribution F I (r|aI ). The certainty equivalent of this distribution from the principal’s perspective is denoted by rI , where rI is defined R implicitly by v(b(aI ), rI ) = v(b(aI ), r)dF I (r|aI ).25 Practically, rI describes the principal’s costs for offering the reward distribution associated with incentives I to the agent when the latter chooses aI . First-best. Suppose that any promise by the principal to reward a behavior were credible. Denote some Pareto-optimal choice in this case by a∗ and the associated reward by r∗ . Formally, a∗ and r∗ are a solution to maxa,r v(b(a), r) such that 24 If

incentives leave the agent indifferent between multiple action choices, aI is assumed to be a choice suggested by the principal. 25 The existence of r I follows from continuity of v in r and the intermediate value theorem for integrals. Uniqueness from the fact that v is strictly increasing in r.

20

u(e(a), r) ≥ 0.26 Denote the first-best effort by e∗ := e(a∗ ). The first-best serves as a benchmark that may or may not be reached with incentives I ∈ J . Effort and its use. The agent’s ability to state preferences over actions independent of the reward is crucial because it allows us to distinguish between two aspects of the agent’s choice. The agent decides, on the one hand, on how much effort e he exerts, and, on the other hand, on how this effort is used. Formally, we can (dis-jointly) decompose the agent’s choice space into subsets that require the S same effort e: A = e {a|e(a) = e}, where different effort levels e are associated with distinct indifference sets {a|e(a) = e}. ‘Choosing effort’ means selecting one of these indifference sets and ‘using effort’ means selecting an action from this set. This notion of ‘choosing’ and ‘using’ effort offers a formally precise interpretation of what Raith (2008) refers to as ‘how much’ and ‘what’ the agent does. Scope of the framework. The early moral-hazard literature (see e.g. Gjesdal, 1976; Holmstr¨om, 1979, 1982; Kim, 1995) but also Baker (1992) are special cases of the presented framework which are only concerned with the agent’s choice of effort. Multitasking models such as Holmstr¨om and Milgrom (1991) and the ensuing literature in accounting and labor economics (see e.g. Feltham and Xie, 1994; Datar et al., 2001; Feltham and Wu, 2000; Baker, 2000, 2002; Schnedler, 2008) are special cases in which the agent chooses both, effort and how to use it. The considerable generality of the framework even allows to embed incentive problems that were originally not placed in a principal-agent setting such as the Averch-Johnson-Effect (Averch and Johnson, 1962).27

5

Gaming and Its Costs

This section formally defines gaming and its costs as well as shirking and agency costs. of a∗ and r∗ can be ensured as follows. For any given a, ˜ there is a r˜ such that u(e(a), ˜ r˜) = 0 because of continuity of u. Since v decreases and u increases in r, the maximizer (a∗ , r∗ ) has to come from the set {(a, ˜ r˜)|u(e(a), ˜ r˜) = 0}, which is compact because it is the domain of continuous functions mapping to the compact set {0}. Since v is a continuous function on that compact set, it has a maximum. 27 The regulator in their model is the principal here; the monopolist is the agent; the choice of labor and capital intensity is the action choice; production costs are the effort; and the social gains from production are the benefit. 26 Existence

21

The agent’s utility is constant when effort is fixed, while the principal’s utility increases in the generated benefit. Hence, all Pareto-optimal ways of employing effort e can be obtained by maximizing the principal’s benefit over all action choices that require effort e:28 A∗ (e) := arg max b(a) such that e(a) = e. a

In order to refer to a specific action choice from A∗ (e) let us use a∗ (e). Definition 1 (Gaming). Incentives are gamed if the agent does not use elicited effort eI = e(aI ) Pareto-optimally: aI 6∈ A∗ (eI ). Incentives can only be gamed if the agent is indifferent between several courses of action, or, more formally, if effort is no one-to-one function of the action choice. The traditional (single-task) moral-hazard model assumes that the agent dislikes larger action choices, or, equivalently that effort increases in the action choice. Accordingly, effort is always used Pareto-optimally in this model and incentives can never be gamed. If preferences are strict, incentives cannot be gamed even if the agent’s choice is multidimensional. Consider a sales agent who can inform himself about the customer’s preferences and be particularly nice to her: A = {don’t inform, inform} × {not nice, nice}. Suppose he finds being nice easier than informing himself: e ((don’t inform, not nice)) < e ((don’t inform, nice)) < e ((inform, not nice)) < e ((inform, nice)) . The resulting mapping between action choice and effort is oneto-one, so that effort can only be used in one way. Since there is only one way of using effort, it has to be Pareto-optimal. Whatever incentives may be feasible, they cannot be gamed in this example although the agent’s choice is two-dimensional.29 As we have seen in the doctor’s example, incentives can also be gamed if the choice set is one-dimensional. What matters for gaming is thus not the dimension of the agent’s choice set but whether he is indifferent between several choices. 28 Such

a maximizer exists because b(·) is a continuous function and the set {a|e(a) = e} is compact, which in turn follows from e(·) being a continuous function. 29 Even for a multidimensional but not countable action set, say A = R × R, effort may be theoretically used Pareto-optimally independent of the choice of incentives because a ∈ R × R can be mapped one-to-one to R using Peano curves (although I am not aware of any economic application in which such a relationship would make sense).

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The following definition measures the negative consequences of gaming by comparing the benefit from incentives I, b(aI ), to the benefit that the principal would have obtained had effort eI been used Pareto-optimally, b(a∗ (eI )), where b(a∗ (eI )) is well-defined because all ways a∗ (eI ) ∈ A∗ (eI ) entail the same benefit. For this comparison, the certainty equivalent of rewards rI is held constant. Definition 2 (Gaming costs). The gaming costs of incentives I are: GI := v(b(a∗ (eI )), rI ) − v(b(aI ), rI )). Gaming costs are never negative because the benefit from optimally used effort cannot be exceeded b(aI ) ≤ b(a∗ (eI )). They are zero if and only if effort is used Pareto-optimally, aI ∈ A∗ (eI ). Incentives are only needed because the agent does not like the behavior that the principal wants to implement; it requires effort. Even if this effort is used Pareto-optimally, eliciting it can entail costs because the compensation for effort and the effort level itself may differ from the first-best levels, rI 6= r∗ and e∗ 6= eI . Definition 3 (Shirking costs). The costs of eliciting effort eI with incentives I, or short, shirking costs, are: SI := v(b(a∗ (e∗ )), r∗ ) − v(b(a∗ (eI )), rI ). Shirking costs are never negative because I ∈ J imposes a restriction on the choices of effort eI and reward rI , while e∗ and r∗ maximize the principal’s utility without such a restriction.30 Let us also define the overall loss to the principal from using incentives I in relation to the first-best. Definition 4 (Agency Costs). The agency costs (of incentives I) are αI := v(b(a∗ ), r∗ ) − v(b(aI ), rI ). Since the agent’s participation is voluntary, the agent does not lose from incentives. In case that his participation constraint binds, only the principal’s utility is affected 30 Notice

that the benefit may well be smaller b(a∗ (e∗ )) < b(a∗ (eI )) but then r∗ < rI .

23

by incentives. Then, agency costs reflect the total loss in surplus from using incentives I in comparison to the (hypothetical) first-best. These definitions are used in the next section to carry the intuitive insights beyond the academic and doctor example and formalize them.

6

Gaming and Shirking in Incentive Design

This section shows that effort is used Pareto-optimally when incentives are aligned with the benefit. Then, it identifies preventing gaming and shirking as the only two functions of incentives. It proves that less shirking comes at the price of more gaming and vice versa, and it identifies gaming as the reason why it is optimal to discard signals that are informative about effort. First, let us generalize the notion that larger rewards are associated with larger benefits from the academic example. Definition 5 (Aligned Incentives). Incentives I are aligned (with the principal’s benefit) if a change in the agent’s action choice that increases the principal’s benefit (weakly) increases the agent’s rewards (in the sense of first-order stochastic dominance): for all a, a˜ ∈ A with b(a) > b(a) ˜ and for all r :

F I (r|a) ≤ F I (r|a). ˜

What is attractive about alignment is that it can be checked by examining the relationship between rewards and benefits, both of which are typically known to a principal who intends to use incentives. Alignment means that the agent cannot obtain larger rewards by decreasing the principal’s benefit. When choosing between actions that require the same effort, the agent is entirely guided by rewards. If he faces aligned incentives, he will choose the most beneficial way of using effort. Conversely, if effort is allocated Pareto-optimally, benefit and rewards have to be aligned ‘locally’; otherwise, the agent could profitably deviate to another action that requires the same effort. The principal can, of course, only infer whether or not the agent has reason to deviate if she knows the agent’s preferences. If she lacks this information, gaming can only be avoided if rewards are higher whenever 24

benefits are larger: incentives have to be aligned with the benefit. This intuition is summarized in the following proposition. Proposition 1 (Aligned Incentives and Gaming). (i) Aligned incentives cannot be gamed. (ii) For gaming to be prevented irrespective of the agent’s preferences, incentives have to be aligned. Proof of Proposition 1. Part (i) works by contradiction. Suppose that aligned incentives are gamed: the agent chooses an action a˜ rather than a, although b(a) > ˜ Then, F I (r|a) ≤ F I (r|a), ˜ because b(a) ˜ and both require the same e(a) = e(a). R R I I I incentives are aligned. Consequently, u(e , r)dF (r|a) ≥ u(e , r)dF I (r|a). ˜ The agent does thus not lose from deviating to action choice a. This, however, contradicts the assumption that the agent chooses a˜ rather than a.31 Hence, the assumption that incentives are gamed cannot be true. Part (ii). Suppose there are two arbitrary action choices, a and a, ˜ with b(a) > b(a) ˜ and consider preferences such that both require the same effort e(a) = e(a) ˜ = I e . Effort is allocated Pareto-optimally if and only if the agent cannot profitably R R deviate from a to a. ˜ This means u(eI , r)dF I (r|a) ≥ u(eI , r)dF I (r|a) ˜ for any I I u that increases in r, which is equivalent to F (r|a) ≤ F (r|a). ˜ In summary, for any a, a˜ ∈ A with b(a) > b(a), ˜ we get F I (r|a) ≤ F I (r|a). ˜ This, however, is the definition of alignment. Proposition 1 assures that aligning incentives prevents gaming. Incentives, however, are not only about avoiding gaming but also about preventing shirking. Proposition 2 (The Forces Determining the Value of Incentives). The costs from running an incentive scheme (agency costs) are composed of shirking and gaming costs: αI = SI + GI . Proof. By definition, agency costs are: αI = v(b(a∗ (e∗ )), r∗ ) − v(b(aI ), rI ). Subtracting and adding the utility from Pareto-optimal used effort, v(b(a∗ (eI )), rI ), yields: v(b(a∗ (e∗ )), r∗ ) − v(b(a∗ (eI ), rI ) SI v(b(a∗ (e∗ )), r∗ ) − v(b(aI ), rI ) = = . + v(b(a∗ (eI ), rI ) − v(b(aI ), rI ) + GI 31 In

case of indifference the agent selects the action preferred by the principal—see footnote 24.

25

Since the incentive designer’s aim is to minimize agency costs, she has to take shirking and gaming costs into account. An exception are, of course, situations in which all incentives entail the same gaming costs, for example, because they all induce the same action or if all incentives are aligned. Gaming costs do not matter if incentives cannot be gamed because effort is a one-to-one function of the action choice like in the traditional moral-hazard model. If gaming costs are zero, agency costs collapse to shirking costs. Results about agency costs derived in the traditional moral-hazard model can thus be interpreted in terms of shirking costs in a more general framework. This logic is used in Section 7 to tease out the general meaning of Holmstr¨om’s sufficient statistic result (1979) and Kim’s ranking of information systems (1995). If incentives vary in gaming and shirking costs, the principal will reduce gaming costs as long as this is possible without increasing shirking costs. Eventually, all respective opportunities are exhausted and she faces a trade-off: gaming costs can only be reduced further if shirking costs are increased. This argument directly leads to the following corollary. Corollary 1 (Effort Elicitation and Direction Trade-Off). Suppose incentives Iˆ minimize agency costs. Then, (i) lower gaming costs can only be achieved at the ˆ ˆ price of higher shirking costs: for all I ∈ J with GI < GI : SI > SI . (ii) Lower shirking costs are only possible by increasing gaming costs: for all I ∈ J with ˆ ˆ S I < S I : GI > GI . Proof. The proof for (i) works by contradiction. Take incentives that minimize ˆ agency costs Iˆ and suppose no trade-off exists. Then, for some I : GI < GI but ˆ SI ≤ SI . This, however, contradicts the fact that Iˆ minimizes agency costs. Claim (ii) can be proven analogously. This gaming-shirking trade-off underpins incentive design unless all feasible incentives have the same gaming or shirking costs. The traditional moral hazard model is again an exception because effort is always used Pareto-optimally. Next, we use the decomposition idea to formalize that gaming is the reason why information about the agent’s effort is optimally discarded. More information about effort ultimately means that eliciting effort becomes cheaper, which suggests the following general definition. 26

Definition 6. A contractual environment J˜ is more informative (about effort) than J , if it permits incentives I˜ with lower shirking costs than any incentives in ˜ environment J : for some I˜ ∈ J˜ and all I ∈ J : SI < SI . The additional information ˜ about effort in J˜ relative to J is discarded if the principal does not use such I. This definition captures the notion of an ‘informative’ signal by Holmstr¨om (1979) as well as that of a ‘more efficient’ information system by Kim (1995)—see next section. Using additional information about effort reduces shirking costs and hence agency costs. The only reason not to exploit a more informative environment are increased gaming costs. Corollary 2 (Optimally Discarding Performance Information). Discarding the information about effort in J˜ relative to J is optimal if and only if the gains from lower shirking costs are more than outweighed by larger gaming costs: ˜ although there is some I˜ ∈ J˜ , with SI < SI for all I ∈ J , for at least one I ∈ J : ˜ ˜ SI − SI < GI − GI . ˜ Proof. Since J˜ is more informative than J , there is some I˜ ∈ J˜ with SI < SI for all ˜ I ∈ J . Any such incentives I˜ ∈ J˜ are optimally discarded if and only if αI > αI for ˜ ˜ some I ∈ J , or equivalently, if and only if SI + GI > SI + GI for some I ∈ J .

This section has shown that the value of incentives, in general, and that from exploiting a more informative environment, in particular, is affected by gaming. The next section re-examines the meaning of traditional concepts to assess the value of information when incentives can be gamed.

7

Gaming and the Value of Information

Since its infancy, the formal analysis of hidden action problems has been interested in the value of information (Gjesdal, 1976, 1982; Harris and Raviv, 1979; Holmstr¨om, 1979, 1982; Shavell, 1979). Probably one of the most important insights from the early moral hazard literature is that freely available and independent information about performance is valuable as long as it is not yet reflected in incentives. The intuition is that with such information more effort can be obtained without 27

having to increase the agent’s exposure to risk. The sufficient statistic result by Holmstr¨om (1979, 1982) prominently captures this idea. Gjesdal (1982) affirms the sufficient statistic result for a general class of models, including those where the agent’s choice is multidimensional. Holmstr¨om (1979, 1982) examines the value of an additional piece of information (or signal), or equivalently, he compares two nested information systems. Kim (1995) proposes a more general mean-preserving spread criterion to also compare non-nested information systems. Both descriptions of the value of information are incomplete in the sense that they neglect gaming. In order to see this, we exploit that the agent in their models faces a one-dimensional choice and dislikes larger choices. From the perspective of our general framework, their models are only about the level of effort but not about how effort is used. The general meaning of their findings can be teased out by replacing the agent’s choice in their models with the effort decision in the current framework. According to Holmstr¨om (1979), a signal Y is informative about effort e in relation to information already in use, say X, if X is no sufficient statistic for e. Starting with optimal incentives using X, Holmstr¨om finds that there are incentives using Y with strictly lower agency costs if and only if Y is informative about e given X (Proposition 3 in Holmstr¨om, 1979). Since his model is only about the level of effort, his agency costs are equivalent to shirking costs, here. This leads to the following corollary. Corollary 3 (Sufficient Statistic and Shirking Costs). Under the assumption of Holmstr¨om (1979), there are incentives using a signal Y in addition to signal X with lower shirking costs than all incentives using X if and only if the signal Y is informative about effort e in relation to X (in the sense of Holmstr¨om, 1979). Proof. The proof follows directly from Proposition 3 in Holmstr¨om (1979) by identifying the agent’s decision in his model with the choice of effort e here. The sufficient statistic result compares two sets of feasible incentives, one which includes incentives that condition on signal Y , say J˜ , and one which doesn’t, J . If the signal is informative in the sense of Holmstr¨om (1979; 1982), there are incentive schemes in I˜ ∈ J˜ with lower shirking costs than any incentives in I ∈ J . Informativeness about effort in the sense of Holmstr¨om is thus a special case of 28

a more informative environment as defined in the previous section. Applying Corollary 2 reveals that signals that are informative in the sufficient statistic sense are only valuable if the increase in gaming costs is negligable. In order to compare the noisiness of non-nested information systems, Kim (1995) defines a mean-preserving spread criterion (see his Proposition 1). He finds that signals that are less noisy according to this criterion are ‘more efficient’: the same effort can be induced with lower expected transfers rI . Since this finding concerns only the level and price of effort but not its use, we directly obtain the following result. Corollary 4 (Information System Comparison & Shirking Costs). Under the assumptions of Kim (1995), using signal Y one can obtain lower shirking costs than for any incentives using signal X if Y is less noisy according to Kim’s meanpreserving spread criterion. Proof. The proof follows from Proposition 1 in Kim (1995) by identifying the agent’s decision in his model with effort e here and by observing that for given effort level, lower expected compensation is equivalent to lower shirking costs. Kim’s ranking of information systems can easily be overturned by gaming. Consider a signal X which is more noisy than Y but reflects the principal’s benefit much better. Then, X can generate a larger payoff than Y although Y is ‘more efficient’ in the sense of Kim—see online appendix. This highlights that gaming costs are typically not negligable when comparing information systems. Interestingly, Christensen et al. (2010) show that an extension of Kim’s meanpreserving spread criterion can be used to rank information systems in a multitasking setting where gaming is possible but without adjusting this criterion for gaming. What seems like a contradiction can be resolved. Christensen et al. (2010) only compare incentives that implement the same action choice. Gaming costs are hence implicitly kept constant in their comparisons, in which case, agency costs are, of course, lower, if and only if shirking costs are lower. Any result on agency costs from a setting where incentives cannot be gamed (such as the traditional signal-task model) obviously generalizes to a setting where they can be gamed (such as multitasking) if gaming is assumed away. The use of such generalizations,

29

however, is rather limited because the resulting rankings cannot compare incentives when the use of effort matters to the principal.

8

Gaming and the Congruity-Precision-Trade-Off

Extant alternatives to the incentive-insurance trade-off stipulate that incentive designers have to trade-off the congruity (or congruence) between performance measure and benefit with the precision of the performance measure (see e.g. Feltham and Xie, 1994; Datar et al., 2001; Feltham and Wu, 2000; Baker, 2000, 2002). This section offers two contributions. First, it justifies the assumption implicit in these trade-offs that congruity avoids some form of dysfunctional behavior (Corollary 5). Second, it explains the finding by Schnedler (2008, Proposition 2) that (holding precision constant), less congruity typically leads to lower agency costs (Corollary 6). The congruity-precision trade-off has been proposed for the multitasking linear normal model, or short: LEN model, which assumes that performance measure, Y , and benefit, B, are linear in action choices a = (a1 , . . . , an )0 ∈ Rn , and noise: Y (a, η) = µ1 a1 + . . . + µn an + η, and B(a, ε) = b1 a1 + . . . + bn an + ε, where ε and η are normally distributed error terms. Moreover, rewards are assumed to be linear in the performance measure, r(Y ) = π + πY, and effort to be quadratic in action choices: e(a) = a0 Ea, with E being a positive-definite matrix. A performance measure is said to be congruent if the relative effect of choices on this measure along any two dimensions (‘tasks’) is the same as on the benefit. Formally, the marginal effects vector µ = (µ1 , . . . , µn ) is a multiple of that of the benefit b = (b1 , . . . , bn ) : µ = λb, for some λ > 0.32 Otherwise, the performance measure is dis-congruent. For a trade-off between congruity and precision to be meaningful, congruity must be attractive. It could, for example, prevent ‘bad’ choices by the agent. This idea can be justified by appealing to gaming. Rewarding the realization of a performance measure that is congruent with the benefit means that it is not possible 32 A

measure Y with λ < 0 measures ‘bad performance’ and can be turned into a congruent performance measure Y˜ with some positive λ by flipping the scale: Y˜ = −1 ·Y. Without loss of generality, we can thus assume for a congruent measure that λ > 0.

30

to (stochastically) increase measured performance without increasing the benefit. In other words, incentives are aligned and entail no gaming (by Proposition 1). Moreover, using a dis-congruent performance measure induces a different and nonoptimal use of effort. The following corollary summarizes these considerations. Corollary 5. In the LEN model, incentives I are not gamed if and only if the rewarded performance is measured congruently with the benefit: GI = 0 ⇔ µ = λb, for some λ > 0. Proof. For the ‘if’ part, take any pair a, a, ˜ with b(a) > b(a). ˜ Using this and that the performance measure is congruent with the benefit µ = λb for λ > 0 and rewarded (π > 0), we get Prob(π + πλba + η ≤ r) ≤ Prob(π + πλba˜ + η ≤ r) for some λ > 0. This, however, means that the distribution of rewards gets (weakly) stochastically larger when moving from a˜ to a. Incentives are thus aligned and by Proposition 1 effort is used Pareto-optimally. For the ‘only if’ part, recall that e is strictly convex and b linear, so that b has a unique maximizer on E := {a|e(a) = e}, say a, ˆ at which the derivative of b on E disappears. Now assume that performance is not rewarded congruently with the benefit, i.e., there is no λ > 0 such that µ = λb. Then, the derivative of µa on E at aˆ does not disappear. There is hence some a˜ with e(a) ˜ = e(a) ˆ and µa˜ > µa, ˆ which implies that Prob(π + πµa˜ + η ≤ r) < Prob(π + πµaˆ + η ≤ r), or equivalently, that the agent prefers a˜ to the Pareto-optimal way aˆ of using e. Assuming that performance is not rewarded congruently thus leads to a contradiction (the agent does not choose Pareto-optimally) and cannot be true. The corollary establishes that congruent performance measures are clearly superior to dis-congruent ones in terms of directing effort. It thus justifies in a specific sense the implicit assumption from the literature that congruent performance measures are desirable. A trade-off between congruity and precision only makes sense if congruent performance measures yield no higher agency costs than any dis-congruent one with the same precision. As Schnedler (2008) points out, this is indeed true but only in the arguably unusual case that the agent’s preferences for different dimensions of 31

the action choice are identical and independent (see his Proposition 2). Formally, we must have E = κI with κ > 0, or equivalently, e(a) = ∑ni=1 κa2i . Otherwise, some dis-congruent performance measures lead to lower agency costs than congruent ones, which he puts down to insurance ‘playing a role’. The next result supports this claim. Measuring performance dis-congruently rather than congruently can only result in lower agency costs if either shirking costs, gaming costs, or both are lower (by Proposition 2). Congruent performance measures, however, do not entail gaming costs (by Corollary 5). It can thus be excluded that gaming costs are lower: agency costs must be smaller because of lower shirking costs. Corollary 6. The reason why some dis-congruent performance measures entail lower agency costs in the LEN model than congruent ones with the same precision are lower shirking costs. Proof. Let I be incentives that reward a signal with µ = λb, for some λ > 0, and ˜ ˜ ˜ incentives I˜ have lower agency costs, αI < αI , which is equivalent to SI + GI < ˜ SI + GI by Proposition 2. Using that GI = 0 by Corollary 5 and GI ≥ 0, we get: ˜ SI < SI . Shirking costs in the LEN model are partially driven by the agent’s need for insurance. In this sense, the corollary confirms Schnedler’s claim (2008) that insurance issues are the reason why dis-congruent measures are superior to congruent ones. At the same time, the result emphasizes once more that trading in some gaming for less shirking can be optimal.

9

Conclusion

Despite Gibbons’ observation (1998) that other issues are ‘at least as important’ as the ‘tenuous’ (Prendergast, 2002) incentive-insurance trade-off, formal models used for the education of future economic advisers and business consultants still heavily focus on this trade-off.33 Given this focus and given that even scholars 33 Major

textbooks devote whole chapters to the trade-off, while multitasking is discussed in much shorter sections (Bolton and Dewatripont, 2005; Macho-Stadler and Perez-Castrillo, 1997;

32

of incentives themselves neglect gaming (Christensen et al., 2010), the seemingly infinite supply of real-life examples of dysfunctional incentives is perhaps not so surprising. This paper hopes to redress the balance with a theory of incentives that explicitly incorporates gaming. Intuitions are simple and the theory’s generality reflects the importance of gaming for most applications. In summary, the paper offers an alternative to the standard paradigm which (if taught) can hopefully resolve misconceptions about incentive design and prevent dysfunctional incentives.

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Online Appendix: Ranking of Signals under Gaming This section shows that gaming costs can easily overturn the ranking of signals by Kim (1995): the ‘less efficient’ information system can generate a higher payoff to the principal because the ‘more efficient’ information system leads to more gaming. Consider a version of the academic example in which the publication signal is ‘polluted’: the true realization of this signal is only observed with probability γ but with probability 1 − γ, the observed signal shows the opposite of the actual realization:  Y with probability γ Y˜ = , (1 −Y ) with probability 1 − γ where γ ≥ 12 , measures the degree to which the original signal matters. In other words, signals with larger γ are less noisy. For noise to matter, assume that the agent’s liability is limited: πI ≥ 0 and πI ≥ 0. For gaming costs to matter, suppose that the principal strongly prefers research to marketing: β = 0.9. The probability of a successful publication now is: P(Y˜ = 1) = (ρa1 + (1 − ρ)a2 ) · γ + (1 − (ρa1 + (1 − ρ)a2 )) (1 − γ) = (1 − γ) + (ρa1 + (1 − ρ)a2 )(2γ − 1). This means that a publication premium of πI induces the agent to engage in: aI =

πI (2γ − 1) (ρ, (1 − ρ)) . 2

The respective effort is: eI =


πI (2γ − 1) ρ2 + (1 − ρ)2 . 2

Solving for eI shows that inducing effort eI requires a premium of:34 eI 2 π = 2 · . 2 (ρ + (1 − ρ) ) 2γ − 1 I

Next we want to compare the premiums given two different signals (or information systems). The less noisy signal Y˜ (I) is unpolluted (γ = 1) but emphasizes marketing: ρ = 0.1 The more noisy signal Y˜ (II) is polluted (γ = 0.9) but perfectly reflects the principal’s preference for research: ρ = 0.9. Relating the respective premiums π(I) 34 Expected

costs for inducing an action aI are minimized by paying nothing in case of failure

Y˜ = 0 : π = 0.

1

and π(II) that are required to induce effort eI , we get: π(I) = π(II)

2 2·1−1 2 2·0.9−1

= 0.8.

The same effort eI can hence be induced with the less noisy signal for a premium that is only 80% of that when using the noisy signal. Plugging in the action choices in the probability of success, we obtain the following formula for the expected transfer to the agent:  
I I 2 2 π (2γ − 1) πI . r = (1 − γ) + ρ + (1 − ρ) 2 The lower premium π(I) when using the less noisy signal Y (I) thus translates to lower expected payments for the same effort than when using the noisy signal Y (II) . The same effort can thus be elicited more cheaply when using signal Y (I) rather than Y (II) . According to Kim (1995), ‘information system’ Y (I) is ‘more efficient’ than Y (II) . Next, we show that this does not mean that the principal can generate more surplus using Y (I) rather than Y (II) . First, we compute the optimal publication premium. The principal’s problem is:     max βaI1 + (1 − β)aI2 − (1 − γ) + (ρaI1 + (1 − ρ)aI2 )(2γ − 1) πI πI

where aI =

πI (2γ − 1)(ρ, 1 − ρ). 2

Plugging in aI yields the following formula for the principal’s payoff: πI ΠI = [βρ + (1 − β)(1 − ρ)] (2γ − 1) 2  
I 2 2 π 2 (2γ − 1) πI . − 1 − γ + ρ + (1 − ρ) 2

(9)

Taking the derivative with respect to πI and setting equal to zero yields a global maximizer (the second derivative is negative for both signals): πI =

1 βρ + (1 − β)(1 − ρ) 1 1 1−γ − . ρ2 + (1 − ρ)2 2γ − 1 2 (2γ − 1)2 ρ2 + (1 − ρ)2

Comparing the optimal premium for the noisy and less noisy signal reveals that the

2

more noisy signal is used more: 0.9 · 0.1 + 0.9 · 0.1 1 1 9 = ≈ 0.1 2 2 0.1 + 0.9 2 − 1 2 82 1 0.1 1 285 1 π(II) = 1 · − · = ≈ 0.43 2 2 2 2 · 0.9 − 1 2 (2 · 0.9 − 1) 0.9 + 0.1 656 π(I) =

Using the premium in (9), the principal’s payoff from both signals can be calculated: 82 9 1 9 1 81 9 1 · − · · = · ≈ 0.005 82 2  100 82 2 82  200 82 8 285 1 285 114513 285 8 1− − Π(II) = (0.92 + 0.12 ) = ≈ 0.34 656 10 10 656 10 656 336200 Π(I) = 0.1 · 0.9 · 2 ·

The principal thus prefers the more noisy signal.

3