What happens when there are many agents? Threre are two problems:

Transcription

1 Moral Hazard in Teams What happens when there are many agents? Threre are two problems: i) If many agents produce a joint output x, how does one assign the output? There is a free rider problem here as I know that the other members of the group will work, so why should I? ii) If many agents produce similar outputs x i by undertaking action a i, there is a scope for inducing effort by playing off one agent against the other. Are relative performance contracts efficient? tournaments? What about the rank order Hőlmstrom gives gives a partial discussion of these issues in his paper. The Single output, Group Incentive Problem: Assume there are n agents, agent i takes action a i A i = [0, ] Cost of taking action: ν i : A i R, ν i ( ) > 0, ν i ( ) < 0 a = (a 1, a 2,..., a n ) A X n i=1a i and write a i = (a 1, a 2,..., a i 1, a i+1,..., a n ) ; a = (a i, a i ) For each agent, the utility form is: u i (m i, a i ) = m i ν i (a i ), over output/(consumption) m i and action a i Prodction function: x : A R, which is strictly increasing, concave and differentiable. 1

2 And the sharing rules for the i th agent is: S i (x) 0, i = 1, 2,...n If the sharing rules are budget balancing, then: n i=1 S i(x) = x, x Given a i, each agent picks Max S i (x (a i, a i )) ν i (a i ) {a i } ( ) a is a Nash equilibrium if for each i given a i, a i satisfies ( ) The Social Optimum: A social Pareto optimal a is one satisfying a = arg max [x(a) n i=1 υ i(a i )] a A If sharing rules are differentiable: ds i x(a i,a i ) dx = dν i da i (A) Pareto optimality requires that x(a i,a i ) = dν i da i (B) For (A), (B) to be consistent, ds i dx = 1, i.e. agent needs to take full output risk at the margin. But budget balancing implies that each agent can not take full output risk as they share the output! Formally, 2

3 n i=1 S i(x) = x, x = n i=1 ds i dx = 1 Contradicting the above. Of course we have only considered differentiable rules. For a complete proof including non-differentiable rules see the Appendix. If instead of budget balancing, we imposed n i=1 S i(x) < x, x Then exits efficient Nash equilibria now. Suppose a is the Pareto optimum. Let b i be a set of numbers s.t. n i=1 b i = x (a ), and S i (x) = b i, if x x (a ) S i (x) = 0, if x < x (a ) If all other agents pick a i, and agent i picks a i > a i, he gets nothing more even though x > x(a ). But action is costly and so this is suboptimal. If agent i pick a i < a i, x < x(a ) and he gets nothing. Therefore a is Nash. The forcing contract works because of certainty and is very special. 3

4 All the above is old wine in a new bottle. The importance of breaking a budget constraint has been noted by Groves in his classic public goods work and is quite well known. Holmstrom s paper is a clever example of how to reuse an old idea. However, the free rider problem is important and has many applications in finance (diversified firms etc). Uncertainty and Forcing contracts Even when uncertainty exists, forcing contracts of the type may work. Let F (x a) c.d.f. f(x a) p.d.f. F i (x a) F (x a) f i (x a) f(x a) exist for i, (x, a). A1: F (x a) is convex in a A2: A3: F i (x a) F (x a) F i (x a) 1 F (x a) as x as x + Assumption A is closely related to what is called concavity of distribution function. Thus, 4

5 F (x λa 1 + (1 λ)a 2 ) λf (x a 1 ) + (1 λ)f (x a 2 ) Thus if a 2 >> a 1 i.e. i, a 2i > a 1i then F (x a 2 ) < F (x a 1 ) any x This kind of distribution function condition is needed to make the first order approach work. But it is weird as most standard distributions do not satisfy it. Assumption 2 and 3 imply that the lowered bound output ( ) and upper bound (+ ) are close to perfectly revealing about the action. The basic idea of this kind of forcing contract is due to Mirlees. Suppose for output close to (i.e. less than some bound x, one punishes the agent by some amount k. Of course by punishing a person with k for x < x, one is not acheiving the first best for x < x. But lower x and increase k proportionately. So one is imposing a very large punishment for a low output. If this limit works correctly, i.e. prob(x x a) Punishment for x x 0 as x We are OK. This is where assumption 2 is used. Theorem: If agents are risk neutral and A1 and A2 hold, the first best can be approximated arbitrarily closely by group penalties. Proof: Consider S i (x) = S i x if x x 5

6 S i (x) = S i x k i, if x < x k i > 0, S i = 1 Risk neutality does not imply that there is no agency problem. arises because of the free rider problem. This With risk neutrality, the above contract has FOC: S i E[ x(a) F (x a) ] k i ν i(a i ) = 0 Thus a the Pareto optimal contract must satisfy this. ensures global optimality. Fix x: choose k so that If it does, A1 k i = S ie[ x(a ) ] ν i (a i ) = A i For x < x, first best is not achieved due to punishment: The punishment loss is: W = i k if (x a ) = i A i F (x a ) where A i = S i E[ x(a ) ] ν i(a i ) is a constant. Therefore, by A2, F (x a ) as x = F (x a ) 0 6

7 The above trick of using unbounded supports to enforce large penalties on small probabilities is due to Mirlees. Of course if imposing very high punishments is not allowed, this strategy does not work. The converse strategy is the bonus strategy. Here one pays very large bonuses with very small probability. Theorem: Under A1 and A3, the first best can be inforced. Proof: S i (x) = S i x + k i, if x x S i (x) = S i x if x < x k i > 0, S i = 1 FOC is: S i E[ x(a ) ] k i ν i(a i ) = 0 k i = S ie[ x(a ) ] ν i (a i ) n i=1 B i 1 F (x a ) = B i A3 ensures this goes to zero as x. This schemes depend critically on risk neutrality. and W = 7

8 Sufficient statistics Risk averse agents y vector signals Risk neutral principal G(y a) c.d.f g(y a) p.d.f g ai (y a) exists for all i. max {a, s i (y)} {E(x y, a) i S i(y)}dg(y a) s.t. (i) ui (S i (y))dg(y a) ν i (a i ) u i for i [1, n] (ii) ν i (a i) for i [1, n] a i arg max u i (S i (y))dg(y a i, a i ) {a i} Definition: A function T i (y) is sufficient for y w.r.t. a i,if exist functions h i 0, P i 0, s.t. g(y a) = h i (y, a i )P i (T i (y) a) y, a in the support of g ( ) (In our old notation in prior class (x, y) y, x T (y) ) If T i (y) is sufficient for y w.r.t. a i,each i; T (y) is sufficient for y w.r.t. a Theorem: If T (y) is sufficient for y w.r.t. a, then given a set of schemes {S i (y)}, exits a set of schemes {Ŝi(T (y))}, which weakly Pareto dominate it. 8

9 Proof: Define Ŝi( T ) follows u i (Ŝi ( T ) ) = {y T i (y)= T u } i(s i (y)) g(y a) P i ( T dy = a) {y T i (y)= T u } i(s i (y))h i (y, a i )dy ) = E[u i (Ŝi (T ) ] = E[u i (S i (y))] for any action a. So agents pick a as a Nash equilibrium. For the principle, from Jensen s inequality u i (Ŝi ( T ) ) E[S i (y) T ] = E [ u i (S i (y)) T ] u i (E[S i (y) T ]) Ŝi( T ) ] E [Ŝi (T ) E[S i (y)] So the principal is better off. Trial converse of Theorem 5: T (y) is sufficient at a if it is the case that for all i, all T i g ai (y 1 a) g(y 1 a) = ga i (y 2 a) g(y 2 a) a.s. y 1, y 2 {y T i (y) = T i } ( ) This means that the likelihood ratio ga i g given T i, does not move with y. By integration, we know that ( ) g(y a) = h i (y, a i )P i (T i (y) a) ( ) So, if ( ) holds for i, a; T (y) is globally sufficient. some i for all a, T (y) is globally insufficient. If ( ) is false at 9

10 Theorem 6: If T (y) is globally insufficient and {S(T i (y))} is a collection of unique nonconstant sharing rules in equilibrium. Then exits sharing rules {Ŝi(y)} which are Pareto improving. Moreover they can guarantee the same actions a. Proof: It is a pain in the neck, but I will do it for one agent. i.e. suppress let i = 1. Exists a set T i Exists T 1 and sets of positive measure y 11 y 1 = {y T (y) = T 1 } y 12 y 1 = {y T (y) = T 1 } s.t. ga(y 1 a) g(y 1 a) ga(y 2 a) g(y 2 a) g(y kl a) = prob{y y kl a} l = 1, 2 Since S(y) is not constant, there exist T 1, T 2 s.t. y 2 = {y T (y) = T 2 } is of positive value measure and S(T 1 ) S(T 2 ) Define Ŝ(y) = S(T (y)) + I 11 (y)d s11 + I 12 (y)d s12 + I 2 (y) d s2 I 1l = 1 y y 1l l = 1, 2 I 1l = 0 otherwise I 2 = 1 y y 2 I 2 = 0 otherwise 10

11 Keeping the action fixed, P = [d s11 g(y 11, a)+d s12 g(y 12, a)+d s2 g(y 2, a)] (A) A = u 1[d s11 g(y 11, a)+d s12 g(y 12, a)]+u 2d s2 g(y 2, a) (B) u 1 = u (S(T 1 )), u 2 = u (S(T 2 )) Now S(T 1 ) S(T 2 ) u 1 u 2 Wlog, let Then, Sgn( A )=Sgn [ u 1[d s11 g(y 11, a) + d s12 g(y 12, a)] + u 2d s2 g(y 2, a) ] (C) Fix d s2 > 0,Require P = 0 d s11 g(y 11, a) + d s12 g(y 12, a) = d s2 g(y 2, a) Substitute into (C) Sgn( A )=Sgn [ u 1 ( d s2 g(y 2, a)) + u 2d s2 g(y 2, a) ] > 0, as u 2 > u 1. Relative Performance Evaluation x(a, θ) = i x i(a i, θ i ) θ = (θ 1, θ 2,..., θ n ) Theorem 7: Assume x i (a, θ) is monotone in θ i. Then the optimal sharing rule of agent i depends on individual i s output alone if and only if outputs are independent. Proof: If θ i is independent f(x a) = n i=1 f i(x i a i ) T i (x) = x i is sufficient for x w.r.t. a i. Therefore, by Theorem (5), S i depends on x i alone. If θ 1 and θ 2 are dependent, let a 2 be fixed at a 2 Since x 2 = x 2 (a 2, θ 2 ) θ 2 = x 1 2 (a 2, x 2 ) 11

12 Same for x 1 ; Since x 1 = x 1 (a 1, θ 1 ) θ 1 = x 1 1 (a 1, x 1 ) Therefore, f(θ 1, θ 2 ) = of (θ 1, θ 2 ) f(x 1 1 (a 1, x 1 ), θ 2 ) ; f is the joint distribution f a1 (x 1 θ 2,a 1 ) f(x 1 θ 2,a 1 ) = f 1 (x 1 1 (a 1,x 1 ),θ 2 ) f(x 1 1 (a 1,x 1 ),θ 2 ) x 1 1 (a 1,x 1 ) a 1 Since θ 1, θ 2 are dependent f 1 f depends on θ 2. Sufficiency of x 1 does not hold. Implications: Contracts on other agents output is optimal only if correlations between outcomes x i exist. If they not, i.e., correlations are independent, such contracts do not pay. Two examples: (I) x i (a i, θ i ) = a i + η + ε i i = 1, 2...n (II) x i (a i, θ i ) = a i ( η + ε i ) i = 1, 2...n ε i are independent idiosyncratic shocks, normally distributed with precision τ i η is a common shock normally distributed independent of ε i Theorem: For (I) (II) above, let, x = α i x i For (I), let α i = τ i τ, For (II), let α i = τ i τa i, τ = τ i τ = τ i For both (I) and (II), the optimal contract is a function of (x i, x) alone. Proof: For (I) 12

13 f(x a) = k exp{ 1 2 [ j τ j(x j a j µ j η) 2 + τ 0 (η µ 0 ) 2 ]}dη (E) We get this by setting f(x a) = f(x a, η)g (η) dη µ 0 = mean η, τ 0 = precision η τ j, µ j are mean precision and mean of ε j Let Z i = k i ( τ k τ i )(x k a k µ k ) τ i = k i τ k Then j τ j(x j a j µ j η) 2 = j i τ j(x j a j µ j Z i + Z i η) 2 + τ i (x i a i µ i η) 2 = j i τ j(x j a j µ j Z i ) 2 + (n 1)(Z i η) 2 + τ i (x i a i µ i η) 2 Take this mess, substitute in (E). Integrate over η. Then; f(x a) = h i (x a i ) P i (Z i, x i a) 13

14 But Z i = (τx τ i x i )/τ i k i ( τ k τ i )(a k +µ k ) P i (Z i, x i a) = P i (x, x i a), Sufficiency follows. x = τ i τ x i, is a scale weighted average. This measures the information in output i. If τ i,the η effect is small (noise dominates). Therefore, searching for correlation is meaningless. i.e. we do not contract on x. What if we have a large number of agents? Theorem reveal η? Yes. Does the Central Limit Theorem 9: Suppose η, ε 1, ε 2,..., ε n are independent with uniformly bounded variance. Suppose when η = 0, the single agent solution is a unique a i. then we achieve this as the number of agents n. Proof: Let Si (x i ) be the optimal contract when η = 0 a i be the optimal action when η = 0 Let q j = η + ε j q i = 1 n 1 j i q j As n, q i η. Thus ui (S i (a i + η + ε i q i ))dp (η, ε 1, ε 2,..., ε n ) 14

15 unif ormly ui (S i (a i + ε i ))dp (ε i ) Since a i is a unique solution to max u i (S i (a i + ε i ))dp (ε i ) ν i (a i ) {a i } We are done. Now q i can be inferred by calculating x i a i = η + ε i QED Thus, a large number of agents can eliminate the common uncertainty η from contract! 15