Inducing Teamwork in Rank-Order Tournaments

Transcription

1 Inducing Teamwork in Rank-Order Tournaments Vikram Ahuja Dmitry Lubensky January 10, 2013 Abstract We analyze the design of an optimal contract in a two-person partnership in which all the firm s proceeds are re-distributed back to the workers according to a noisy binary signal of relative performance. A worker devotes two types of costly effort: own effort into his assigned task and helping effort toward the task of another. A principal designs a tournament-style contract that assigns a share s [ 1 2, 1] of total output to the worker with the relatively higher signal and 1 s to the other. A higher value of s induces more competition, giving more incentive for own effort but reducing the incentive to help. When production is symmetric with respect to the two tasks, the optimal contract is to maximize the help incentive by giving equal compensation to both workers. The result does not depend on the degree of complementarity in the production function nor on the level of noise in the signal of relative performance. Kelley School of Business, Indiana University 1

2 1 Introduction In partnerships such as law firms or hedge funds, workers are also shareholders and this creates a competitive tension. Once the firm s profits are realized and the proceeds are to be redistributed, increasing the compensation of one worker necessarily reduces the compensation of another. Workers are thus motivated not only to create value for the firm but also to out-perform their colleagues to gain a higher share of the firm s profit. This competitive drive provides a useful instrument for designing contracts. In a partnership, an agent has a natural tendency to shirk since he collects only a share of the value he creates. The added motive of increasing his share by improving his relative performance counters the shirking and may boost the firm s overall profit. In a recent paper, Gershkov et al (2009) demonstrate that when agents independently choose effort levels and relative output is noisy, a contract can induce efficient production by awarding an appropriately higher share of the firm s profit to the agent with the higher relative output. Partnerships, however, are also by their nature collaborative enterprises in which agents do not simply work adjacently but rather in concert. Even in a firm in which different tasks are explicitly delegated to different workers, agents can take measures not only to ensure the success of their assigned task but also to aid in the success of the task of another. In this light, the competitive environment induced by relative pay has detrimental effects. While more competition reduces shirking, it also reduces incentives for teamwork. In this paper we characterize the optimal contract that balances the countervailing competitive effects of relative pay in a setting with joint production. We construct a tournament model in which two risk-neutral agents compete for an endogenous prize. There is a task associated with each agent and the total output of the firm is the sum of the outcomes of the two tasks. The relative performance of the two tasks is observed through a noisy ordinal signal. Each agent devotes two types of effort: own effort into his own task and helping effort into the other s task. While both efforts 2

3 increase total output, the private task improves an agent s relative performance and the helping task decreases it. The firm s total output is redistributed completely between the two agents, thus negating the ability of a contract to increase the compensation of one agent without reducing the compensation of the other. We focus on a simple contract structure in which the agent with the higher relative performance receives a fixed share s 1 2 of the firm s total profit and the other agent receives the remainder. A higher share is associated with a more competitive tournament, increasing incentives for own effort and reducing incentives for helping effort. A profit-maximizing principal who sets s optimally must balance these countervailing forces, and in doing so must consider particular aspects of the environment. For example, in Gershkov et al the optimal share to be assigned to the winner depends on the level of noise in the relative output. When output is noisier, additional effort is less effective at improving an agent s relative performance and the optimal contract must reward the winner with a bigger prize to alleviate this. In our setting, added noise also has the effect of muting competition, but with the countervailing effects on shirking and teamwork, it is not immediately clear how the optimal contract ought to adjust. In addition, we are interested in how the optimal contract depends on the production technology. Assigning a bigger share to the winner reduces incentives for teamwork, but how costly this is ought to depend on the complementarity between own and helping effort. Our main result is quite surprising in that neither the amount of noise nor the degree of complementarity affects the optimal level of competition. In fact, we obtain in a quite general setting that the optimal contract maximally induces teamwork by awarding an equal share s= 1 2 to both agents regardless of the aforementioned factors. The result holds for arbitrary functional forms on production and noise that satisfy the assumptions that production has decreasing returns to scale, that own effort and helping effort enter the production function symmetrically, and that noise enters additively. The question of how to optimally induce teamwork in a setting with multi-dimensional effort has been studied extensively. One body of work is the literature on sabotage in tournaments owing 3

4 in part to Lazear (1989). Within this literature, Lazear s model is in fact closest to ours in its construction. Lazear characterizes a dual task noisy contest in which agents devote effort to their own task and sabotage others in their tasks. One may interpret the under-provision of helping effort in our model as sabotage in Lazear s, but doing so reveals the qualitative difference. In Lazear, the act of sabotage is costly while in our model reducing helping effort saves costs. Accordingly, one may expect that the principal in our setting must provide more incentive for teamwork and in fact this is what obtains. While Lazear s result is that it is optimal to include some wage compression, in our setting we obtain a stronger result, that it is always optimal to have complete wage compression. Contract design with multiple agents and multiple tasks has also been closely examined in the principal-agent literature. Itoh (1991), unlike Lazear (1989), uses a production technology closer to ours with two agents and two different costly efforts that are both productive. Itoh s agents are risk averse and his focus is on whether the principal ought to incentivize teamwork. He finds that whether inducing teamwork is optimal depends on the degree of complementarity in the production function, so that if there is not enough complementarity it is better for the principal to induce no teamwork at all. Our result may appear striking in its contrast with Itoh s, but upon closer examination the divergence is clearly rooted in the systematic differences in the two strategic environments. Itoh s principal creates wage contracts for risk averse agents. The principal faces the dual tasks of inducing desirable behavior from his agents and selling those same agents insurance. The interplay of these two countervailing goals is at the heart of the logic behind his result on teamwork when no teamwork is optimal it is simply too costly from the insurance-selling perspective to induce it. In our model agents are risk neutral and no insurance motive exists. Instead, the cost of inducing more helping effort is the reduced incentive for own effort, a consequence of the budget balance that is native to the partnership structure. The difference in the results thus follows from the difference in the nature of the cost of inducing teamwork across the two settings. The remainder of the article is organized as follows. Section 2 describes the model and Section 3 provides a simple example with a closed-form equilibrium that captures some of the intuition in the model s environment. Section 4 presents the fully general model and main result. Section IV concludes. 4

5 2 Model A firm consists of two risk neutral partners i=1, 2. Each partner has an assigned task whose output is y i and the firm s total output is Y=y 1 + y 2. The two partners each simultaneously choose two effort levels (e i, a i ), in which e i [0, ) is own effort expended on one s own task and a i [0, ) is helping effort expended toward the other agent s task. For partner i, the cost of his effort choices is given by c i = c(e i, a i ) and the output on his task is given by y i = y(e i, a j ). Neither the efforts nor task outputs are directly observable, instead the principal observes only the total output Y and a binary signalσ {1, 2}. Letσ=i correspond to agent i s task performing relatively better and let p(y 1, y 2 ) be the probability that the signal identifies agent 1 as the winner. The game proceeds as follows. First, a principal chooses the share of total output s [ 1 2, 1] to award the winner. 1 With s commonly known, agents simultaneously make their effort choices (e 1, a 1 ) and (e 2, a 2 ). Then task outputs y 1 and y 2 and relative signalσare realized and each agent receives his share of the total output and pays his effort cost. Agent i s payoff is u i (e i, a i ) = ( 1(σ=i) s+ 1(σ= j) (1 s) )( y(e i, a j )+ y(e j, a i ) ) c(e i, a i ). He receives share s of total output when he wins, share 1 s when he loses, and in either case pays the cost for both efforts. The principal s objective is to maximize the sum of the agents utilities, hence her payoff is u 0 = y(e 1, a 2 )+ y(e 2, a 1 ) c(e 1, a 1 ) c(e 2, a 2 ). We look for the Nash Equilibrium of this game, described by the share s optimally chosen by the principal and the ensuing efforts (e i, a ) put forth by each agent. i 1 We have restricted the contract space so that the principal may not condition payments on the observed total output Y. This structure more readily matches many institutional arrangements and is quite common in the tournament literature. However, it is also true that removing the restriction would likely allow the principal to strictly improve her outcome. 5

6 3 Example For illustrative purposes, in this section we focus on particular functional forms for production, costs, and noise and directly solve for the equilibrium. In the section that follows, we derive the same result in a more general setting. Suppose that own and helping efforts enter the production function symmetrically as perfect substitutes, y(e, a)=e+a, that costs of effort are symmetric, separable, and quadratic, c(e, a)= 1 2 e a2, and that the noise in relative output is distributed uniformly and enters additively p(y 1, y 2 )=Pr(y 1 y 2 +ε 0) [ in whichε U δ ] 2,δ. 2 We solve the game backwards, first by computing the equilibrium efforts for a fixed s and then solving for the principal s optimal choice of s. When s is fixed, agent 1 solves ( max E[u 1 ]= p ( y(e 1, a 2 ), y(e 2, a 1 ) ) s+ ( 1 p ( y(e 1, a 2 ), y(e 2, a 1 ) )) )( ) (1 s) y(e 1, a 2 )+ y(e 2, a 1 ) e 1,a 1 c(e 1, a 1 ), the solution to which is characterized by the following two first order conditions: 0= p y 1 (2s 1)Y+ y ( ) 1 p(y 1, y 2 )s+(1 p(y 1, y 2 ))(1 s) c (1) y 1 e 1 e 1 e 1 ) 0= p y 2 (2s 1)Y+ y ( 2 p(y 1, y 2 )s+(1 p(y 1, y 2 ))(1 s) y 2 a 1 a 1 c a 1 (2) 6

7 Plugging in the functional forms and denoting p(y 1, y 2 ) as p, the first order conditions reduce to 0= 2s 1 Y+ ( ps+(1 p)(1 s) ) e 1 δ 0= 2s 1 Y+ ( ps+(1 p)(1 s) ) a 1 δ The first term in both conditions reflects the marginal change in agent 1 s expected share of total output, with the negative sign in the second condition reflecting that increasing helping effort reduces an agent s probability of winning. The second term corresponds to the marginal increase in the firm s output, of which agent 1 will collect his expected share. The final term is the marginal cost. We claim without proof that for a given value of s a pure strategy equilibrium exists, is unique, and is symmetric. The symmetry implies that in equilibrium p= 1 2 and plugging this into the first order conditions above and solving yields the equilibrium effort levels e(s)= (s 1 2 ) δ a(s)= 1 2 4(s 1 2 ) δ (3) (4) As the share s given to the winner is increased, in equilibrium agents put more effort into their own task and less effort into helping. 2 The principal solves max s u 0 (s)=2 (e(s)+a(s) 1 2 e(s)2 1 ) 2 a(s)2 = (s 1 2 ) δ (s 1 2 ) δ 2 Maximizing this expression yields s = 1 2. Thus, in this example it is optimal for the principal to remove competition entirely and compensate both agents equally. This in turn induces agents to put forth efforts e=a= This characterization of equilibrium between the agents for a fixed value of s assumes an interior solution. In general, for small enough values ofδthere exist sufficiently large values of s for which the equilibrium is a corner solution and thus not characterized by the first order conditions. It is the case however that for anyδthe principal chooses s= 2 1 which induces interior solutions. The formal argument is straightforward and is omitted for expositional convenience. 7

8 As a benchmark we compute profit-maximizing effort levels, which is the solution to the unconstrained maximization of the principal s payoff. arg max e 1,a 1,e 2,a 2 u 0 (e 1, a 1, e 2, a 2 )= y(e 1, a 2 )+ y(e 2, a 1 ) c(e 1, a 1 ) c(e 2, a 2 ) = e 1 + a 2 + e 2 + a e 1 = a 1 = e 2 = a 2 = 1 ( e a2 1 + e2 2 + ) a2 2 These first best levels are simply where the marginal product of effort to the firm equals the marginal cost to the partner exerting it. Notice that in this example equilibrium effort levels are smaller than what is profit maximizing. In fact, adding equations (3) and (4) yields that e(s) + a(s) = 1 and this may be thought of as the locus of equilibria that can be induced by the principal through manipulating the value of s. The fact that profit maximizing effort levels for an agent must add to 2 means that the optimum lies outside the locus and is unreachable. 4 General Formulation In the preceding example we obtained that the principal s best policy is to minimize competition by awarding equal shares to both agents regardless of their relative performance. While the result is suggestive, it leaves open the question whether the phenomenon is a consequence of the strong assumptions on the environment such as, for instance, the perfect substitutability of inputs in the production function. Presently we relax many of these assumptions to show that the result holds robustly in a more general environment. Assumption 1 The production function y(e, a) satisfies y e > 0, y a > 0, y ee 0, y aa 0, y ea 0, and y(αe,αa) αy(e, a). These are regularity conditions for production. We assume that both inputs are productive, that both inputs have diminishing marginal product, that the cross-partial is weakly positive (which 8

9 includes the case of perfect substitutes), and that production exhibits weakly diminishing returns to scale. Assumption 2 The cost function c(e, a) satisfies c e 0, c a 0, c ee 0, c aa 0, and c ea = 0. Regularity conditions for costs are that they are weakly increasing and convex in both arguments, and crucially that the cross-partial is zero implying that the cost function is separable. Assumption 3 Both functions y(e, a) and c(e, a) are symmetric in their arguments, so that y(q, r) = y(r, q) and c(q, r)=c(r, q). This assumption plays a key role in the main result and is quite strong, requiring equal marginal net products across both kinds of effort. We include a discussion on relaxing this assumption in the concluding remarks. Assumption 4 The probability p(q, r) is twice continuously differentiable and p(q, r)=1 p(r, q) for all r, q. The probability of a higher relative performance by agent 1 increases in her output and decreases in agent 2 s output. In addition, the two agents outputs enter symmetrically in the sense that an increase in one argument has the same effect as a decrease in the other. This formulation fits any difference form contest but is violated in a ratio form contest, such as the Tullock contest. 3 With these assumptions in place we initially compute the first best benchmark. By the regularity conditions (1) and (2) on production and costs, a unique social optimum exists and is found where the marginal products of each effort are equated with an agent s marginal cost. y e (e, a)=c e (e, a) y a (e, a)=c a (e, a) As we will shortly see, it is useful to combine these conditions and rearrange so that ( ce + c ) a = 2 (5) y e y a 3 In the ratio form specification, it is the case that the probability is less sensitive to the leader s action than it is to the follower s. 9

10 We now turn to the equilibrium. As in the preceding example, we focus on symmetric pure strategy equilibria that are characterized by the first order conditions. 4 Equations (1) and (2) from the previous section give the first order conditions for agent 1, and once symmetry is applied so that p(y 1, y 2 )= 1 2 in equilibrium, the conditions reduce to the following: c e = 1 y e 2 + p (2s 1)Y (6) y 1 c a = 1 y a 2 p (2s 1)Y (7) y 1 Observe that when the two conditions are added, we obtain c e y e + c a y a = 1 (8) Proposition 1 First best levels of effort are unattainable in equilibrium. Proof. Follows immediately from equations (5) and (8). Equation (8) helps us visualize the principal s problem, as it implicitly defines in e a space a locus of efforts which can be induced in equilibrium. This is the principal s constraint set, i.e. the set of feasible (e, a) combinations from which she can choose. In the same space, we may also consider a family of isopayoff curves for the principal, namely (e, a) combinations that give the principal the same utility. The figure depicts these two objects. Using the figure below as a guide, our argument proceeds as follows. We first express the slopes of the equilibrium locus and the isopayoff curves and demonstrate that the two curves are tangent along the 45 degree line. Then, we claim that the isopayoff curves are convex and that the equilibrium locus is concave. This guarantees that the point of tangency is the unique maximizer. With these facts established, it will follow that the principal s chosen s induces an equilibrium along the 45 degree line, which in turn will imply that s = It is generally the case that in some environments which meet the assumptions above, there exist values of s which induce agents to take efforts that are corner solutions and not characterized by the first order conditions. We conjecture that in such environments the principal s highest payoff from inducing corner solutions is lower than her highest payoff from inducing interior solutions, but this still needs to be shown. 10

11 We begin with a normalization of the model which will significantly cut down on notation. We use our assumptions on the symmetry and separability of the cost function to express it as c(e, a)=k(e)+k(a) We will now think of agents choosing cost expenditures on each effort level directly, as opposed to the effort levels themselves. Toward this end, we define a new production function ỹ(k(e), k(a)) y(e, a) Applying this re-scaling to the equilibrium locus in equation (8) yields 1 ỹ e + 1 ỹ a = 1 We are interested in the slope of this locus in the re-scaled space, which we obtain through total differentiation. ( ) da = ỹee(ỹ a ) 2 + ỹ ea (ỹ e ) 2 (9) de equilibrium ỹ aa (ỹ e ) 2 + ỹ ea (ỹ a ) 2 11

12 Next we want to obtain the equation of the slope of the principal s isopayoff curve in the normalized environment. Her payoff is u 0 = 2 ( ỹ(k(e), k(a)) k(e) k(a) ), and total differentiation yields ( ) da = ỹe 1 de isopayo f f ỹ a 1. (10) Lemma 1 The isopayoff curve is tangent to the equilibrium locus whenever e=a. Proof. This argument leans entirely on the assumptions of symmetry in the production and cost functions. First, consider equation (9). Since when e=a, it must be that ỹ e = ỹ a and ỹ ee = ỹ aa, the equilibrium locus has a slope of exactly 1. Next, in the equation of the slope of an isopayoff curve, again since when e=ait must be that ỹ e = ỹ a, the numerator equals the denominator and thus the slope here also equals 1. We have thus shown that at the point on the equilibrium locus where e=a, the necessary condition for optimality is met. The following two lemmas guarantee sufficiency, namely that the tangency is unique and is a local maximum. Lemma 2 A principal s isopayoff curve is convex. Proof. See Appendix. Lemma 3 The equilibrium locus is concave. Proof. See Appendix. Having shown that the principal s optimal choice is to induce e=a, it finally remains to demonstrate that the way the principal achieves this is by giving equal compensation to both agents. 12

13 Proposition 2 It is optimal for the principal to set s = 1 2. Proof. By the preceding sequence of lemmas, it is optimal for the principal to induce e=a. By inspection of the first order conditions (6) and (7), this may only be achieved by setting s= Discussion We model a firm in which workers put effort toward the completion of their own task and effort towards helping complete the task of another. We focus on a partnership, that is a firm that redistributes all its proceeds to its employees. Workers efforts are not directly observed but rather there is a noisy binary signal of relative performance. In this sense, the partnership is strategically equivalent to a rank-order tournament with an endogenous prize. The principal selects how large a share s [ 1 2, 1] to award to the agent with the higher relative performance. Our main result is that when own effort and helping effort enter symmetrically, the principal finds it optimal to maximally induce teamwork by awarding the same share of s= 1 2 to both agents. This result holds regardless of the degree of complementarity of the two types of effort in the production function and regardless of the amount of noise in the environment. To obtain intuition for the result, one may first consider the result in Gershkov et al in which agents may only undertake effort in their own task. There, if s= 1 2 agents have no incentive to improve their relative performance and, since they keep only one half of their marginal product, under-provide effort in their task. By increasing s, the principal increases the returns to relative performance and thus improves effort toward more socially desirable levels. In our model, increasing s also improves effort in agents own tasks but in addition reduces their helping effort. Which of these two effects dominates is not immediately clear and our result, that regardless of technology or noise the effect on helping effort is more important, is surprising. This is especially so given the fact that both kinds of effort are treated equally in the production and cost functions. One extension of the present work is to consider the optimal contract in a setting in which efforts do not enter symmetrically. For example, one may wonder how much competition to induce 13

14 when own effort contributes relatively more to output than helping effort. Without symmetry, the techniques we have employed to solve for the equilibrium in a general setting are not valid, however some results from numerical simulations have shown that in general s= 1 2 is no longer optimal. In fact, if own effort is more productive than helping effort then the optimal share is strictly greater than one half. The intuition behind these findings rests on the fact that along the equilibrium locus, both own and helping efforts are under-provided. The principal may induce the provision of more of one effort at the expense of another. When own effort becomes relatively more important in production, foregoing some helping effort for more own effort becomes worthwhile. An additional area for inquiry is the complementarity not only within the task production function but also across tasks. In some sense, the present model is rigged against teamwork in that tasks are perfect substitutes. While we conjecture that adding complementarity across tasks ought to make the principal more keen to induce teamwork, it remains to be seen whether the workers would internalize cross-task complementarity or whether they would require additional incentive. Appendix Proof of Convexity of Isopayoff Curve Differentiate the slope of the isopayoff curve as given in equation (10): e ( ) da (ỹa 1 ) ỹ ee ( ỹ e 1 ) ỹ ae = de W (ỹa 1 ). 2 We know that that ỹ e > 1 and ỹ a > 1. The transformed production function ỹ (e, a) preserves the properties of the original production function that the second derivative, ỹ ee, is strictly negative and the cross-partial derivative, ỹ ae, is weakly positive. Hence, the indifference curve is strictly convex in e. 14

15 Proof of Concavity of Equilibrium Locus Since we have shown the convexity of the indifference curve, we claim that any equilibrium will be at a point on the boundary of the production constraint where the constraint is downward sloping. Consider the ea-plane where area on the plane is separated into two regions: the area above and below the 45-degree line. At any arbitrary combination (ê, â) on the boundary of the production constraint, we claim that increasing either e or a by a small ε > 0 would bring the combination out of the constraint set. In other words, the new combination exists where 1 ỹ e + 1 ỹ a > 1. First, let s consider the new (e, a) combination at (ê+ε, â). We know that ỹ ee < 0 and ỹ ea > 0, thus, for the above inequality to hold, it must be true that the magnitude that ỹ e decreases is greater than the magnitude that ỹ a increases, or in other words, ỹee > ỹea. Similarly, at the new combination (ê, â+ε), we would have ỹaa > ỹea. The combination (ê, â) on the ea-plane can either be in the area above the 45-degree line or under the 45-degree line. If the combination (ê, â) is above the 45-degree line, or in other words, ê<â, this implies that ỹ e > ỹ a. Together with the fact that ỹaa > ỹea shown above, we claim that ỹaa (ỹ e ) 2 + ỹ ea (ỹ a ) 2 or the denominator of the slope of the production constraint is negative. Therefore, ỹ ee (ỹ a ) 2 + ỹ ea (ỹ e ) 2 or the numerator of the slope of the production constraint is also negative. This is because the combination (ê, â) is on the boundary of the constraint set where the constraint is downward sloping, thus ( ) da de < 0. equilibrium Likewise, if the combination (ê, â) is below the 45-degree line, or ê > â and ỹ e < ỹ a. We find that the numerator ỹ ee (ỹ a ) 2 + ỹ ea (ỹ e ) 2 is negative, therefore the denominator ỹ aa (ỹ e ) 2 + ỹ ea (ỹ a ) 2 must also be negative. Now we have shown that both the numerator ỹ ee (ỹ a ) 2 +ỹ ea (ỹ e ) 2 and the denominator ỹ aa (ỹ e ) 2 +ỹ ea (ỹ a ) 2 of the slope of the production constraint are negative. We then ready to prove the convexity of the 15

16 production constraint by again considering any combination (ê, â) on the boundary of the production constraint where its slope is negative. At this arbitrary combination (ê, â), by slightly moving southeast on the ea-plane to the new combination (ê+ε, â ε) whereε>0, the slope of the production constraint changes as follow: (i) the numerator ỹ ee (ỹ a ) 2 + ỹ ea (ỹ e ) 2 decreases because by increasing ê and decreasing â, more weight is put to the negative term ỹ ee (ỹ a ) 2 and less weight is put to the positive term ỹ ea (ỹ e ) 2, and (ii) the denominator ỹ aa (ỹ e ) 2 + ỹ ea (ỹ a ) 2 increases because by increasing ê and decreasing â, more weight is put to the positive term ỹ ea (ỹ a ) 2 and less weight is put to the negative term ỹ aa (ỹ e ) 2. Therefore { ( ) } { ( ) } da de > da de, which means the slope of the production constraint becomes steeper. isopayo f f (ê,â) isopayo f f (ê+ε,â ε) Likewise, at the same arbitrary combination (ê, â), by slightly moving northwest on the ea-plane to the new combination (ê ε, â+ε) whereε>0, the slope of the production constraint changes as follow: (i) the numerator ỹ ee (ỹ a ) 2 + ỹ ea (ỹ e ) 2 increases because by decreasing ê and increasing â, less weight is put to the negative term ỹ ee (ỹ a ) 2 and more weight is put to the positive term ỹ ea (ỹ e ) 2, and (ii) the denominator ỹ aa (ỹ e ) 2 + ỹ ea (ỹ a ) 2 decreases because by decreasing ê and increasing â, less weight is put to the positive term ỹ ea (ỹ a ) 2 and more weight is put to the negative term ỹ aa (ỹ e ) 2. Therefore { ( ) } { ( ) } da de < da de, which means the slope of the production constraint becomes flatter. isopayo f f (ê,â) isopayo f f (ê+ε,â ε) The above argument holds for any arbitrary point (ê, â). As a result, the production constraint is strictly concave. 16