1
Moral Hazard: Multiple Agents
• Multiple agents (firm?)
— Partnership: Q jointly affected
1.1
Moral Hazard in a Team
Holmstrom(82) Deterministic Q.
2
∂Q
Output Q(a) ∼ F (q|a), ∂a
> 0, ∂ Q
2 < 0,
i
— Individual qi’s. (tournaments)
∂ai
2
• Common shocks, cooperations, collusion, monitor
ing.
Q
dqij = ∂a∂ ∂a
≥ 0, (dq)ij —negative definite.
i j
Agents: ui(w) = w.
Agents: i = 1, . . . , n.
Partnership w(Q) = {w1(Q), . . . , wn(Q)},
Ui(wi, ai) = ui(wi) − ψ i(ai)
such that for all Q,
Efforts: a = (a1, . . . , an), with ai ∈ [0, ∞)
Problem: free-riding (someone else works hard, I gain)
Output q = (q1, . . . , qn) ∼ F (q|a).
First-best:
Principal: R-N.
Pn
i=1 wi(Q) = Q.
∂Q(a∗)
0 ∗
∂ai = ψ (ai ).
At F-B: Q(a∗) −
Agents’ choices: FOC:
dwi[Q(ai, a∗−i)] ∂Q(ai, a∗−i)
dQ
∂ai
= ψ 0(ai)
? Nash (a∗i ) = FB (a∗i )?
Thus ∃z = (z1, . . . , zn).
Note, b-b looses from higher Qs.
Comments: b-b is a residual claimant (in fact each agent
is a residual claimant in a certain interpretation (!)).
Locally:
dwi[Q(ai,a∗−i )]
= 1, thus wi(Q) = Q + Ci.
dQ
Budget
Pn
∗
i=1 ψ i(ai ) > 0.
Pn
i=1 wi(Q) = Q for all (!) Q.
Not the same as Alchian & Demsetz (equity for manager’s
incentives to monitor agents properly).
? Other ways to support first-best?
This requires a third party: budget breaker
Mirrlees contract: reward (bonus) bi if Q = Q(a∗),
Let zi = −Ci–payment from agent i.
P
∗
∗
Thus n
i=1 zi + Q(a ) ≥ nQ(a )
and zi ≤ Q(a∗) − ψ i(a∗i ).
penalty k otherwise. (bonuses for certain targets)
As long as bi − ψ i(a∗i ) ≥ −k, F-B can be supported,
P
moreover if b’s and k exist so that Q(a∗) ≥ n
i=1 bi, no
b-b needed.
1.2
Interpretation: Debt financing by the firm.
Firm commits to repay debts of D = Q(a∗) −
bi to each i.
P
bi, and
Special Examples of F-B (approx) via
different schemes
Legros & Matthews (’93), Legros & Matsushima (’91).
If cannot, creditors collect Q and each employer pays k.
(Hm...)
Issues: (1) Multiple equilibria (like in all coordinationtype games, and in Mechanism-Design literature). No
easy solution unless
(2) actions of others are observed by agents, and the principal can base his compensation on everyone’s reports.
Not a problem with Holmstrom though (Positive effort
of one agent increases effort from others).
• Deterministic Q, finite A’s, detectable deviations.
Say, ai ∈ {0, 1}. And Qf b = Q(1, 1, 1).
Let Qi = Q(ai = 0, a−i = (1, 1)).
Suppose Q1 6= Q2 6= Q3.
Shirker identified and punished (at the benefit of the others).
(3) Deterministic Q.
Similarly, even if Q1 = Q2 6= Q3.
• Approx. efficiency, n = 2.
Idea: use one agent to monitor the other (check with
prob ε).
Check: Agent 1. Set a2 = 1,
∙
¸
2
((a+1)−1)2
a
− 2 = 0.
maxa
2
ai ∈ [0, ∞) Q = a1 + a2, ψ i(ai) = a2i /2.
Agent 2. a2 ≥ 1 7→ Q ≥ 1. Implies a∗2 = 1, U2 =
1 − ε/2.
F-B: a∗i = 1.
a2 < 1 guarantees Q < 1 with prob. ε.
L & M propose: agent 1 chooses a1 = 1 with pr = 1−ε.
Obtain a∗2 = 12 , and U2 = 54 − εk.
When Q ≥ 1,
For, k ≥ 12 + 41ε , a∗2 = 1 is optimal.
(
w1(Q) = (Q − 1)2/2
w2 = q − w1(Q).
when Q < 1,
(
w1(Q) = Q + k
w2(Q) = 0 − k.
• Random output. Cremer & McLean works. (condi
tions?)
1.3
Observable individual outputs
Agent’s choice: a ∈ arg max ŵ(a).
2 2
E(eaε) = ea σ /2, for ε ∼ N (0, σ 2).
q1 = a1 + ε1 + αε2,
(back to General case) Agent i
q2 = a2 + ε2 + αε1.
ε1, ε2 ∼ iid N (0, σ 2).
V (w1) = V ar(v1(ε1 + αε2) + u1(ε2 + αε1))
CARA agents: u(w, a) = −e−µ(w−ψ(a)), ψ(a) = 12 ca2.
Linear incentive schemes:
h
= σ 2 (v1 + αu1)2 + (u1 + αv1)2
Then, agent’s problem:
w1 = z1 + v1q1 + u1q2,
w2 = z2 + v2q2 + u2q1.
maxa
No relative performance weights: ui = 0,
i
⎧
⎨
z1 + v1a + u1a2 − 12 ca2−
⎫
⎬
i .
2h
⎩ − µσ (v1 + αu1)2 + (u1 + αv1)2 ⎭
2
Solution a∗1 = vc1 (as in one A case).
Principal: maxa,z,v,u E(q − w),
h
i
v2
ŵ1 = z1+ 12 c1 + u1cv2 − µσ
2
subject to E −e−µ(w−ψ(a)) ≥ u(w̄).
h
i
Define ŵ(a), as −e−µŵ(a) = E −e−µ(w−ψ(a)) .
Principal: maxz1,v1,u1
½
2
h
i
(v1 + αu1)2 + (u1 + αv1)2 .
v1
c −
µ
v2
z1 + c1 + u1cv2
¶¾
s.t. ŵ1 ≥ w̄
Principal:
maxv1,u1
1.4
Tournaments
Lazear & Rosen (’81)
½
h
i¾
v12 µσ 2
v1
1
2
2
.
c − 2 c 2 − (v1 + αu1) + (u1 + αv1)
Agents: R-N, no common shock.
To solve: (1) find u1 to minimize sum of squares (risk)
qi = ai + εi. ε ∼ F (·), E = 0, V ar = σ 2.
(2) Find v1 (trade-off) risk-sharing, incentives
Cost ψ(ai).
2α v .
Obtain u1 = − 1+α
2 1
F-B: 1 = ψ 0(a∗).
The optimal incentive scheme reduce agents’ exposure to
common shock.
wi = z + qi.
v1 =
1+α2
.
2
1+α +µcσ 2(1−α2)2
z + E(qi) − ψ(a∗) = z + a∗ − ψ(a∗) = ū.
Tournament: qi > qj → prize W , both agents paid z.
Agent: z + pW − ψ(ai) →ai max.
p = P r(qi > qj ) =
= P r(ai − aj > εj − εi) = H(ai − aj ).
EH = 0, V arH = 2σ 2.
∂p
FOC: W ∂a
= ψ 0(ai).
i
W h(ai − aj ) = ψ 0(ai).
1 .
Symmetric Nash: (+FB): W = h(0)
1.5
Cooperation and Competition
• Inducing help vs Specialization
• Collusion among agents
• Principal-auditor-agent
Itoh (’91)
z + h(0) − ψ(a∗) = ū.
2 agents: qi ∈ {0, 1}, (ai, bi) ∈ [0, ∞) × [0, ∞).
Result: Same as FB with wages.
Ui = ui(w) − ψ i(ai, bi), ui(w) =
Extension: multiple rounds, prizes progressively increasing.
ψ i(ai, bi) = a2i + b2i + 2kaibi, k ∈ [0, 1].
H(0)
√
w.
Pr(qi = 1) = ai(1 + bi).
Agents: Risk-averse+Common Shock.
Trade-off between (z, q) contracts and tournaments.
i ), wi — payment to i when q = j,
Contract: wi = (wjk
i
jk
q−i = k.
By itself (ignoring change in a) and if b∗ is small, and
since change in w increases risk, it is costly for the prin
cipal to provide these incentives.
No Help: bi = 0.
w0 = 0, ai(1 − w1) →w1 max,
√
s.t, ai = 12 w1 (IC) and IR is met. ...
Even if a adjusts, since it is different from the first-best
for the principal with b = 0, the principal looses for sure.
Getting Help: Agent i solves (given aj , bj , w, w11 >
w10, w01 > w00 = 0.)
√
√
a(1 + bj )aj (1 + b) w11 + (1 − a(1 + bj ))aj (1 + b) w01+
√
a(1 + bj )(1 − aj (1 + b)) w10 − a2 − b2 − 2kab → max
Thus, if k is positive, there is a discontinuity at b = 0,
thus “a little” of help will not help: for all b < b∗ principal
is worse-off.
a,b
FOC+symm: consider
³
∂
∂b
´
For k = 0, help is always better.
√
√
√
a2(1+b) ( w11 − w10)+a(1−a(1+b)) w01 = 2(b+ak)
Two-step argument: 1. If ahelp ≥ ab=0, marginal cost
of help is of second order, always good.
If (as in No Help) w11 = w10, w01 = 0, and k > 0, we
have RHS = 0, LHS > 0 for any b ≥ 0.
2. Show that ahelp ≥ ab=0.
Therefore, need to change w significantly to get any b
close to 0. (Even to get b = 0 with F OCb = 0)
1.6
s.t (a∗1, a∗2)—NE in efforts, and CEi ≥ 0.
Cooperation and collusion.
Individual choices: vi = ψ 0i(ai).
CARA agents: u(w, a) = −e−µi(wi−ψi(a)).
qi = ai+εi, (ε1, ε2) ∼ N (0, V ), where V =
Ã
σ 21 σ 12
σ 12 σ 22
ρ = σ 12/(σ 1σ 2).
Linear incentive schemes:
!
,
u are set to minimize risk-exposure: ui = −vi σσi ρ.
j
Total risk exposure:
h
i
P2
2σ 2(1 − ρ) .
µ
v
i=1 i i i
• Full side-contracting:
w1 = z1 + v1q1 + u1q2,
w2 = z2 + v2q2 + u2q1.
• No side contracts (CE2(a1, a2) analogously):
CE1(a1a2) = z1 + v1a1 + u1a2 − ψ 1(a1)
µ
− 1 (v12σ 21 + u21σ 22 + 2v1u1σ 12)
2
• (?) Enough to consider contracts on (a1, a2).
• Problem reduces to a single-agent problem with µ1 =
1
1
µ1 + µ2 , with costs ψ(a1, a2) = ψ 1(a1) + ψ 2(a2).
• Full side contracting dominates no s-c iff ρ ≤ ρ∗.
(cooperation vs relative-performance evaluation).
Principal (RN):
(1 − v1 − u2)a1 + (1 − u1 − v2)a2 − z1 − z2 → max
• MD schemes.
1.7 Supervision and Collusion
Principal: reward Monitor for y ∗ with w ≥ k.
Principal: V > 1,
(Punish when there is not y ∗?)
Agent: cost c ∈ {0, 1}, Pr(c = 0) = 1
2.
Suppose not, that is 12 pk > z. (and thus suppose that
wmon = 0)
Monitor: cost z, Proof y ∗, Pr(y ∗|c = 0) = p).
Assume: V > 2 (so P = 1 is optimal without monitor)
With monitor (no collusion)
³
´
1 pV + 1 − 1 p
(V − 1) − z
2
2
Compare to V − 1.
Collision: Agent-Monitor: Tagent → (kT )monitor , k ≤
1.
max T = 1.
³
´
Principal: 12 p(V − k) + 1 − 12 p (V − 1).
• No gain for allowing collusion
• If k is random, then possible.