Sabotage in Teams. Matthias Kräkel. University of Bonn. Daniel Müller 1. University of Bonn

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1 Sabotage in Teams Matthias Kräkel University of Bonn Daniel Müller 1 University of Bonn Abstract We show that a team may favor self-sabotage to influence the principal s contract decision. Sabotage increases a team member s bonus and total team effort. If these benefits outweigh the reduction in the success probability, sabotaging the team is rational. Key words: Moral Hazard, Sabotage, Self-Handicapping, Teamwork JEL: D8, J3, M5 1 Corresponding author: Adenauerallee 24, D Bonn, Germany; Tel: ; address: daniel.mueller@uni-bonn.de Preprint submitted to Economics Letters January 30, 2012

2 1. Introduction Sabotage is a typical problem of relative-performance pay or tournaments, where a worker s expected compensation decreases in the performance of his co-workers. Mutual sabotage is the natural outcome of such competitive incentive schemes. Under team compensation, however, we should expect zero sabotage, because the sabotaging agent would harm himself by lowering team output. We show that sabotaging the team as a whole (including himself ) can be rational for an agent, if the degree of sabotage affects the principal s contract choice. Engaging in sabotage presents the following trade-off: on the one hand, sabotage is detrimental for the agent, because it lowers the probability of receiving a high bonus. On the other hand, sabotage increases the optimal bonus since it makes the team work in a more productive region of the production function. 2 Moreover, there is a team-related benefit of sabotage: since sabotage increases the optimal bonus for all team members, the saboteur profits from the extra incentives for his teammates. If the latter two effects dominate the first effect, self-sabotage by team members is rational. Our main finding also offers a new explanation for self-handicapping of a single individual (e.g., inadequate preparation, drinking before an exam), thereby complementing behavioral explanations of this phenomenon. 3 Although the team-related benefit of sabotage drops out for a single agent, individual self-handicapping can be explained by our model if the second of the above-mentioned effects dominates the first. The result of our model is related to a recent paper by Bose et al. (2010), who also address sabotage in teams. In Bose et al. (2010), however, there is no self-sabotage by team members, but an agent s sabotage is aimed at reducing only his co-worker s productivity. The underlying rationale is that the principal optimally gives higher personal incentives to the relatively more productive agent by offering him a larger individual bonus. In the model by Bose et al. (2010), sabotage implies lower efforts by the teammates, whereas in our setting teammates get extra incentives due to the novel team-related 2 Technically, the effective team effort is shifted to the left within a concave team production function. 3 For example, an individual wants to avoid becoming aware of his possibly low ability; cf. Tirole (2002). 2

3 effect mentioned above. 2. The Model A risk-neutral principal hires a team of n 2 risk-neutral agents, each of whom has no resources of his own, to operate a project. 4 The outcome of the project accrues to the principal. In the case of success, the principal obtains a gross payoff of π H, whereas in the case of failure she obtains π L. Let π = π H π L > 0. The first determinant of the project s success probability is the effort exerted by the team members. When exerting effort e i 0 (i = 1,...,n), agent i incurs cost c(e i ), where c ( ) > 0, c ( ) > 0, c ( ) 0 for all e i > 0, and c(0) = c (0) = 0. Agent i s effort choice is unobservable to the principal as well as his teammates. Whether the project succeeds or fails is influenced by sabotage activities. Before the team members choose their respective efforts, one agent, who is randomly selected by nature, gets the chance to sabotage the team s operations. 5 Let s [s, s] ( s > s > 0) denote the degree of sabotage. As will be specified below, s positively affects the success probability of the project such that a low level of s represents a high degree of sabotage. Engaging in sabotage is costless. While the identity of the agent who has the opportunity to sabotage the team is private information of the potential saboteur himself, we assume that the principal as well as all team members can observe the consequences of sabotage. A third party, however, cannot observe whether the current state of the world is due to sabotage, underprovision of effort, low ability or bad luck. Thus, neither individual efforts nor the degree of sabotage are contractible. These assumptions typically hold for complex tasks. As an example, imagine a team of industrial researchers that either succeeds in developing a new product or not. During research one of the team members gets the opportunity to reveal or conceal valuable information that would save the prototype from a loss. At the end of the day, the employer and 4 The timing and much of the information structure of our model is borrowed from Bose et al. (2010). See Sappington (1983) for more on limited liability contracts. 5 Note that the team would collectively choose the same sabotage level as the single saboteur. Hence, our assumption only eliminates the discussion of multiple equilibria which would arise from the team s coordination problem if agents were not able to choose their sabotage activities cooperatively. 3

4 the researchers observe the progress of the research project, in particular the actual state of the prototype. Given the agents efforts and the degree of sabotage, the project succeeds with probability p( n i=1 e i + s), where p ( ) > 0, p ( ) < 0, and p( ) (0, 1). Neither individual efforts nor the degree of sabotage are contractible. Moreover, the outcome of the project itself is not verifiable. There exists, however, a binary measure of team performance, σ {σ L,σ H }, which can be utilized for contracting purposes. 6 For reasons of analytical tractability, we assume that the probability of observing the good signal σ H increases linearly in overall effort net of sabotage, prob(σ = σ H ) = min{α ( n i=1 e i + s), 1} =: q( n i=1 e i + s) with α > 0. Within this information structure, payments can only reflect the realization of σ. The contract offered to agent i specifies wage wl i if σ = σ L and wh i if σ = σ H. Due to limited liability, both wages have to be nonnegative. Let b i := wh i wi L. All contracts offered by the principal are observable for all members of the team. The reservation utility of each agent is normalized to zero. The timing of events is as follows: First, a randomly selected team member decides on s. Next, the principal offers each agent a contract (wl i,wi H ), which each agent either accepts or rejects. Thereafter, all agents simultaneously and noncooperatively choose their efforts. Last, σ is realized and wages are paid to the agents. Bose et al. (2010) showed that sabotage taking place before the principal offers a contract is crucial for the whole problem since the sabotaging agent wants to influence the principal s contract decision. If the principal can make a binding commitment to the contract before agents decide on their sabotage activities, then sabotage will not take place in equilibrium. The reason is that sabotage increases the teammates operating costs, thereby reducing the overall success probability delivered in equilibrium. In practice, we can imagine situations where agents only obtain spot contracts so that sabotaging is possible between two succeeding contracts. As an alternative, we could think of a situation where agents first enter a probationary period where sabotage becomes possible. Thereafter, the principal offers a regular wage contract to the team members. In both cases we could allow for additional 6 In the example of industrial research, returns from the invention are only realized in the future but there exist certain prototypes at different levels of development, which can be used as signals. 4

5 sabotage after the team members have signed their contracts but ex-post sabotage will never happen in teams. 3. The Analysis 3.1. Effort Provision Given s, (w i L,wi H ), and e i = (e 1,...,e i 1,e i+1,...,e n ), agent i solves max e i 0 w i L + b i q(e i + j i e j + s) c(e i ). With effort being costly, b i has to be strictly positive for agent i to exert strictly positive effort. Since zero effort can always be implemented at no cost by setting w L = w H = 0, we restrict attention to b i 0. If j i e j +s 1/α, then agent i exerts no effort irrespective of his wage contract, since σ H is realized with certainty anyway. For j i e j + s < 1/α, the probability of obtaining a good signal is a kinked function of agent i s effort, which is increasing with slope α for e i 1/α ( j i e j + s) and flat thereafter. Intuitively, agent i is willing to increase his effort as long as the marginal benefit, reflected in an increase of the probability of obtaining bonus b i, outweighs marginal cost. Let ê(b i ) be implicitly described by c (ê(b i )) = b i α, (1) and define ( ) := 1 α e j + s. j i When facing bonus b and sabotage s, agent i s best response to e i is 0 if 0 e (e i,s,b i ) = if 0 < < ê(b i ) ê(b i ) if ê(b i ) (2) A Nash equilibrium given s and b = (b 1,...,b n ) then is an effort profile e (b,s) = (e 1(b,s),..., e n(b,s)) satisfying e i(b,s) = e (e i(b,s),s,b i ) for all agents i = 1,...,n. 5

6 3.2. The Optimal Contract Since wage payments have to be nonnegative due to limited liability, participation of the agents does not impose an additional constraint. Moreover, it follows immediately that the principal optimally sets wl i = 0 (i = 1,...,n). We begin the analysis of the optimal contract with two observations. 7 Lemma 1. The cost-minimizing contract to implement ẽ = (ẽ 1,...,ẽ n ) specifies b = ( b 1,..., b n ) such that ẽ i = ê( b i ). Intuitively, the principal is not going to pay more than she needs to in order to make an agent exert the desired effort: if an agent is supposed to exert no effort at all, he will receive a zero bonus payment. If the principal wants an agent to exert strictly positive effort, then by setting the bonus payment appropriately she will make sure that even the agent s least bit of effort has an impact on the realization of the team performance measure. Lemma 2. Under the optimal contract b = (b 1,...,b n), each agent exerts the same effort: e i(b,s) = e j(b,s) for all i,j = 1,...,n. With identical agents, instead of having two agents provide different levels of effort, the principal can economize on wage costs by adjusting bonus payments such that each of these agents exerts half of the total effort they originally provided. Due to convexity of the marginal cost of effort, the increase in the bonus necessary to make one agent work harder is outweighed by the decrease in the bonus of the agent who now works less. 8 From Lemmas 1 and 2 it follows that, given s, the principal sets b i = b j = b for all i,j = 1,...,n, and each agent chooses ê(b). The optimal bonus solves subject to maxπ L + πp (nê(b) + s) α (nê(b) + s)nb b nê(b) 1 s, (3) α 7 Proofs of Lemmas 1 and 2 are available from the authors upon request. 8 If c = 0, then the principal is never worse of by implementing an equal effort allocation. Therefore, w.l.o.g., we can restrict attention to contracts that implement the same effort for all agents. 6

7 where the constraint ensures that the chosen bonus implements an effort level which is consistent with Nash equilibrium. 9 In the following, we focus on the case where α is sufficiently small such that the above constraint does not impose a binding restriction. 10 The F.O.C. characterizing the optimal bonus payment is 11 n ê(b ) [ πp (nê(b ) + s) αnb ] α (nê(b ) + s)n = 0. (4) Implicit differentiation of (4) allows to establish the following important observation (remember that lower values of s represent higher sabotage intensity): Lemma 3. The optimal bonus payment increases as sabotage increases: (s)/ < 0. Proof: Implicit differentiation of (4) (evaluated at b ) with respect to s yields ( 2ê(b ) + 2 ê(b ) 2 + ê(b ) ) [ πp ( ) [ πp ( ) αnb ] ( ( ê n + ê(b ) ( ( ê(b ) α n ) ê(b ) ) ] αn ) ) + 1 = 0, (5) where we suppressed the dependence of p and p on nê(b ) + s. Inserting ê(b ) = α c (ê(b )), 2 ê(b ) = αc (ê(b )) 2 c (ê(b )), ê(b ) 2 = 2 ê(b ) = 0 9 If nê(b) > 1/α s, then an agent has an incentive to (at least slightly) reduce his effort given that the other n 1 agents choose ê(b), thereby reducing his own effort cost while leaving the probability of good team performance unchanged. 10 Note that for α = 0 we have ê(b) = 0 irrespective of the value of b whereas the RHS of (3) tends to infinity. Since ê(b) continuously increases in α, (3) does not impose a binding restriction as long as α is sufficiently small. 11 The S.O.C. is satisfied. 7

8 into (5) yields = [ πp c 1 ] αn c ( ) πp 1 c αn c αc (c ) 3 [ πp αnb ]. According to (4), (α/c ) [ πp αnb ] = α(nê(b ) + s) 0, such that / < 0. The intuition for this result can be seen by inspection of p (nê(b) + s). The smaller s (i.e., the higher sabotage), the smaller the argument of the probability function, nê(b) + s. Since the probability function p is strictly concave, this shift makes the agents choose their efforts ê(b) in a region of higher productivity p. From the principal s perspective, creating incentives in this region is more profitable than motivating agents under lower values of p and hence larger values of s. Less technically, sabotage leads to a rather bad situation for the principal, who now is highly interested in incentivizing the team to improve the situation considerably To Sabotage or Not to Sabotage? The sabotaging agent chooses s [s, s] to maximize EU i = b (s)α (nê(b (s)) + s) c(ê(b (s)) Making use of the definition of ê(b) in (1) allows us to identify three effects that (locally) determine the agent s incentive whether or not to sabotage the team: deu i ds = b (s)α }{{} >0 + (s) α (nê(b (s)) + s) + b (s)α(n 1) ê(b (s)) }{{} <0 (s) } {{ } <0 (6) Recall that deu i /ds < 0 means that the potential saboteur profits from sabotage. The first effect reflects the fact that an increase in sabotage reduces agent i s expected wage payment because it becomes less likely that the team is awarded the bonus payments for favorable team performance. This effect reduces agent i s incentive to sabotage the team. The second effect relates to Lemma 3: an increase in sabotage increases the bonus offered to agent i. The principal s willingness to provide stronger 8

9 incentives arises because sabotage makes the team work in a region of the production function which is more productive in the sense that the team s effort has a high impact on the success probability of the project. This possibility to influence the principal s wage offer to his favor increases agent i s incentive to engage in sabotage. The last effect, which drops out for a single agent (n = 1), reflects an additional benefit of sabotage for agent i that is specific to teams. An increase in sabotage leads to an increase not only in agent i s bonus, but also in the bonuses offered to his teammates. Since the other n 1 agents will work harder in response to stronger incentives, more intense sabotage therefore benefits agent i also through an increase in the likelihood of being awarded his own bonus brought about by higher effort of his teammates. When expected utility is monotonic in sabotage, equation (6) leads to the following result: Proposition 1. If, for all s [s, s], b (s) + (s) [ ] nê(b (s)) + s + b (s)(n 1) ê(b (s)) then the sabotaging agent optimally chooses s = s. < 0, Hence, if the advantages of a higher bonus payment and higher effort provision by his teammates exceed the disadvantage of a lower bonus probability, the sabotaging agent will prefer maximal sabotage. The following example illustrates that the condition of Proposition 1 is indeed satisfied for a concrete specification of the cost function c and the probability function p. Let p( n i=1 e i +s) = β ln( n i=1 e i +s) with s [1, s] and β > 0 being sufficiently small. Costs are assumed to be quadratic: c (e i ) = γ 2 e2 i with γ > 0. Hence, ê (b) = bα/γ. Note that we can always choose a sufficiently small value of α so that p (nê(b (s)) + s) 1, and that the result of Proposition 1 is independent of α. For the given specification, the principal solves The F.O.C. yields maxπ L + πβ ln (n bαγ ) + s b ( n 2 b 2 α 2 γ ) + nbαs. b (s) = 8 πβγ + s2 γ 2 3sγ. 4nα 9

10 For a feasible solution b (s) > 0, let π > s 2 γ/β. Inserting for the optimal bonus in the sabotaging agent s expected utility function, EU i = b (s)α (nê(b (s)) + s) γ 2 (ê(b (s)) 2 = α 2γ b (s) (2sγ + (2n 1) b (s)α), and differentiating with respect to s leads to γ ( (5 + 2n)s s 2 γ πβγ + (4 πβ + s 2 γ) (2n 3) 8n 2 s 2 γ πβγ which is clearly negative for all values of s [s, s]. 4. Conclusion As highlighted in the example of Section 2, the problem of self-sabotage in teams is especially likely in situations where complexity of the task makes sabotage activity non-verifiable for third parties. In practice, self-sabotage might also be furthered by inability of the principal (e.g., due to a lack of experience or knowledge regarding an innovative project) to discern whether slow progress has to be attributed to sabotage or other factors affecting the work environment. In Bose et al. (2010), where each agent engages in sabotage directed against his co-worker, commitment to an equal-pay-policy is found to be an effective means against sabotage. In our model, in contrast, where sabotage is directed against the team as a whole, preventing sabotage seems less an issue of restricting the set of feasible contracts but of appropriate budgeting. Imposing a hard budget constraint, which caps the overall amount of money available to be paid out to the team, might be capable of creating an effective limit for sabotage. In our model, for example, a budget of n b ( s) would completely eliminate sabotage. This suggests that the nature of the means to limit sabotage in teams may be intimately linked to the nature of the sabotage activity itself. References [1] Bose, A., Pal, D., Sappington, D.E.M., Equal Pay for Unequal Work: Limiting Sabotage in Teams. Journal of Economics and Management Strategy 19, [2] Sappington, D.E.M., Limited Liability Contracts Between Principal and Agent. Journal of Economic Theory 29, [3] Tirole, J., Rational Irrationality: Some Economics of Self-Management. European Economic Review 46, ), 10

11 A. Appendix Proof of Lemma 1: The proof is by contradiction. Suppose b = ( b 1,..., b n ) is the optimal contract to implement effort profile ẽ = (ẽ 1,...,ẽ n ) as a Nash equilibrium, but that for some i = 1,...,n we have ẽ i ê( b i ). We will show that by setting b i such that ẽ i = ê(b i ), the principal can implement ẽ = (ẽ 1,...,ẽ n ) as a Nash equilibrium at lower cost, which contradicts b = ( b 1,..., b n ) to be optimal. First, note that agent j s (where j i) effort choice only depends on b j and e j. Thus, since b j for j i are left unchanged, as long as the adjustment in agent i s bonus leaves his best reply to ẽ i equal to ẽ i, ẽ j remains a best reply for agent j to e j also under the adjusted wage scheme. From agent i s best response function we know that there are two possible cases in which agent i s effort choice does not equate marginal benefit and marginal cost of effort provision in the Nash equilibrium ẽ: (i) b i > 0, j i ẽj 1 α s, and ẽ i = 0, or (ii) b i > 0, j i ẽj < 1 α s < j i ẽj + ê( b i ), and ẽ i = 1 α ( j i ẽj + s). In case (i), agent i s effort choice does not depend on his own bonus payment. By setting b i = 0, the principal leaves ẽ i = 0 a best response for agent i. This contradicts the original contract to be optimal. Moreover, since 0 α = c (0) = 0, we have ẽ i = 0 = ê(0). Regarding case (ii), from (1) it follows that dê(b i )/db i = α/c (ê(b i )) > 0. Thus, as long as ê(b i ) > 1 α ( j i ẽj + s) = ẽ i the principal can lower her expected wage costs without changing the effort exerted by agent i by decreasing the bonus paid to agent i. In consequence, the principal will decrease agent i s bonus until ê(b i ) = 1 α ( j i ẽj+s) = ẽ i. Again, this contradicts the original contract to be optimal, which establishes the desired result. Proof of Lemma 2: As a preliminary observation, from Lemma 1 it follows that under the optimal contract equilibrium efforts are determined according to (1). This allows us to express the bonus to optimally implement effort e i, which we refer to as ˆb(e i ), as ˆb(ei ) = c (e i ) α. (7) With this notation at hand, suppose a contract implements effort profile ẽ = (ẽ 1,...,ẽ n ) as Nash equilibrium, but that ẽ i ẽ j for some i,j = 1,...,n, i j. From (7), the sum of bonus payments paid to agents i and j amounts to ˆb(ẽi ) + ˆb(ẽ j ) = c (ẽ i ) + c (ẽ j ). (8) α According to (2), the principal can induce both agent i and agent j to exert effort e i = e j = (ẽ i + ẽ j )/2 by setting b i = b j = c ((ẽ i + ẽ j )/2)/α. Since the sum of effort provided by agents i and j is unchanged, this leaves the best responses of all the other agents unchanged. Thus, while the overall effort remains unaffected by this adjustment in the bonus payments of agents i and j, the overall sum of bonus payments to be paid out in case of a good team performance evaluation decreases due to convexity of the marginal 11

12 cost of effort: ˆb(ẽi ) + ˆb(ẽ j ) = c (ẽ i ) + c (ẽ j ) α = 2( 1 2 c (ẽ i ) c (ẽ j ) ) ( α ) 2c 1 2ẽi + 1 2ẽj. α This establishes that the principal is (at least weakly) better off under the adjusted contractual arrangement than under the original contract. In consequence, on her quest for the optimal contract, the principal can restrict attention to contracts that implement a Nash equilibrium in which each agent exerts the same effort. (9) 12