Journal ArticleDOI

# Moral hazard with non-additive uncertainty: When are actions implementable?

01 Oct 2018-Economics Letters (North-Holland)-Vol. 171, pp 110-114

TL;DR: It is shown that under three well-known formulations of Agent’s ambiguity attitude, contracts that partition the outcome space in two parts, and are piecewise constant on each part, are enough to implement an action in a moral hazard setting.
Abstract: We provide sufficient conditions on the information structure for implementing actions in a moral hazard setting when Agent has non-probabilistic uncertainty. For a finite action space, under three well-known formulations of Agent’s ambiguity attitude, contracts that partition the outcome space in two parts, and are piecewise constant on each part, are enough to implement an action.
Topics: Action (philosophy) (51%), Moral hazard (50%), Ambiguity (50%)

## Summary (2 min read)

### 1. Introduction

• This paper aims to characterize conditions for a two-part, piecewise-constant contract (‘flat payment plus bonus’, FPB henceforth) to be used to implement actions in moral hazard problems with non-additive uncertainty.
• Lopomo et al. (2011) provide sufficient conditions for FPB contracts to be uniquely optimal under Bewley-type preferences on a finite outcome space, when Principal has more precise information than Agent.
• The authors consider three different formulations of ambiguity-sensitive preferences on a continuum of outcomes, and identify sufficient conditions for FPB contracts to solve the implementation problem for any action other than the least costly one, regardless of whether Agent knows more or less than Principal.
• Section 2 lays out the preliminaries, Section 3 presents the implementation results and the authors conclude with a discussion of their sufficient conditions in relation to the literature.
• The first author gratefully acknowledges the financial support from the Ministerio Economia y Competitividad through grants ECO2014-55953-P and MDM 2014-0431.

### 2. Preliminaries

• Let B(Y) denote the Borel σ-algebra on Y , ∆(Y) denote the set of Borel distributions on Y with the weak topology and K∆(Y) denote the class of non-empty, compact, convex subsets of ∆(Y).
• The authors formal set up can accommodate asymmetry of uncertainty, in particular both of the following cases: (I) Agent has more precise knowledge about technology than Principal: QA(a) ⊂ QP (a) for each a ∈ A, and (II) Principal has more precise knowledge about technology: QP (a) ⊂ QA(a) for each a ∈ A.
• The authors results are interesting when Agent’s information sets are sufficiently rich, in that they are not singletons (i.e. when Agent is not a standard expected utility maximizer).
• Cores of convex capacities are well-defined (Schmeidler (1986)).
• Ch is the Choquet integral (Choquet (1954)).

### 3. Implementation

• If a is the least cost action, then a flat payment of g(a) would implement it for any of the three kinds of objective functions under consideration.
• These implementation conditions, (2) - (3), (7) - (8), (12) - (13), depend only on the payment scheme w(y) and Agent’s perception QA but neither on Principal’s perceived ambiguity nor her ambiguity attitude.
• Here the last inequality shows that aj is rational for Agent, while the strict inequality, which follows from gk > gj, together with the first inequality, which follows from the fact QA(aj) ⊂ QA(ak) and that minimum of a non-negative-valued function does not get smaller over a larger set, establishes that Agent would prefer aj rather than ak.
• The next Proposition shows that strengthening the necessary condition to stochastic dominance, and imposing bounds on the rate at which costlier actions improve outcomes becomes sufficient for implementation.
• 3.2. Implementation under α−MEU Preferences.

### 4. Conclusion

• The authors provide conditions for two-part, piecewise-constant contracts to implement actions in a moral hazard setting with non-additive uncertainty, under three different formulations of ambiguity attitude for Agent.
• The authors necessary conditions for implementation are comparable to the one in Hermalin and Katz (1991) for the standard Bayesian model, and those in Ghirardato (1994) and Lopomo et al. (2011) for non-additive models.
• All these conditions stipulate a necessary amount of ‘disjointedness’ between the sets of probabilities generated by different actions.
• The authors stochastic dominance assumption is comparable to the MLRP condition on extreme points in Lopomo et al. (2011), and helps make downward deviations unattractive enough.
• Given that these contracts provide just the minimum level of variability needed for implementation, their results suggest they could be robustly optimal across different formulations of ambiguity attitude.

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MORAL HAZARD WITH NON-ADDITIVE UNCERTAINTY: WHEN
ARE ACTIONS IMPLEMENTABLE?
MARTIN DUMAV, URMEE KHAN
Abstract. We provide suﬃcient conditions on the information structure for implement-
ing actions in a moral hazard setting when Agent has non-probabilistic uncertainty. We
show that under three diﬀerent formulations of Agent’s ambiguity attitude, contracts that
partition the outcome space in two parts, and are piecewise constant on each part, are
enough to implement an action.
Keywords: Moral hazard, non-probabilistic uncertainty, ambiguity aversion, implementabil-
ity.
JEL Classiﬁcation: D81, D82, D86
1. Introduction
This paper aims to characterize conditions for a two-part, piecewise-constant contract (‘ﬂat
payment plus bonus’, FPB henceforth) to be used to implement actions in moral hazard
problems with non-additive uncertainty. Lopomo et al. (2011) provide suﬃcient conditions
for FPB contracts to be uniquely optimal under Bewley-type preferences on a ﬁnite out-
come space, when Principal has more precise information than Agent. We consider three
diﬀerent formulations of ambiguity-sensitive preferences on a continuum of outcomes, and
identify suﬃcient conditions for FPB contracts to solve the implementation problem for any
action other than the least costly one, regardless of whether Agent knows more or less than
Principal.
Section 2 lays out the preliminaries, Section 3 presents the implementation results and we
conclude with a discussion of our suﬃcient conditions in relation to the literature.
Date: March 4, 2018. The ﬁrst author gratefully acknowledges the ﬁnancial support from the Ministerio
Economia y Competitividad (Spain) through grants ECO2014-55953-P and MDM 2014-0431.
1

2. Preliminaries
Let Y = [0, Y ] be the set of outcomes. Let B(Y) denote the Borel σ-algebra on Y, ∆(Y)
denote the set of Borel distributions on Y with the weak topology and K
∆(Y)
denote the
class of non-empty, compact, convex subsets of ∆(Y). Agent chooses an unobservable
action a from a ﬁnite set A. Let g : A 7→ R
+
be a bounded, nonnegative function that
describes the cost of eﬀort to Agent, and denote g(a
k
) = g
k
. Principal’s information about
technology is characterized as a set-valued mapping from set of actions A to K
∆(Y)
given by
Q
P
(·) : A 7→ K
∆(Y)
. Similarly, Agent’s information is characterized by another set-valued
mapping, Q
A
(·) : A 7→ K
∆(Y)
. Furthermore, for all actions the associated set of probability
distributions have a common support: for all {a, a
0
} A, supp Q
A
(a) = supp Q
P
(a) =
supp Q
A
(a
0
) = supp Q
P
(a
0
) = Y.
Remark 1. Our formal set up can accommodate asymmetry of uncertainty, in particular
both of the following cases: (I) Agent has (weakly) more precise knowledge about technology
than Principal: Q
A
(a) Q
P
(a) for each a A, and (II) Principal has (weakly) more precise
knowledge about technology: Q
P
(a) Q
A
(a) for each a A. However, since we focus on
implementation alone, and not Principal’s proﬁt maximization, Agent’s information sets are
all that matter. Our results are interesting when Agent’s information sets are suﬃciently
rich, in that they are not singletons (i.e. when Agent is not a standard expected utility
maximizer). We will maintain that assumption. Both cases I and II can then be thought
of as generalizations of Ghirardato (1994) and case II is also a generalization of the setting
in Lopomo et al. (2011). At the contracting stage, both parties have common knowledge of
Q
P
and Q
A
.
1
2.1. Convex Capacities. We assume that for each action a, the set of induced probability
distributions Q
A
(a) is the core of a regular convex capacity C
a
2
:
Assumption 1. For each a Y, C
a
, a convex capacity, such that Q(a) = core (C
a
) =
{q ∆(Y) : q(E) C
a
(E), E B(Y)}
Following Dyckerhoﬀ and Mosler (1993), we deﬁne stochastic dominance of capacities.
1
Dumav and Khan (2017) show that linear contracts provide a solution when Principal does not know
Agent’s information sets.
2
A capacity on a measurable space (Ω, B) is a mapping C : B [0, 1] such that C() = 0, C(Ω) = 1
and A B C(A) C(B). Capacity C is coherent (or a lower probability) if for some set of probability
measures P, C(A) = inf
P ∈P
P (A) for every A B. A capacity is convex if for any A, B B, C(A)+C(B)
C(A B) +C(A B). A capacity is regular if C(A) = inf{C(B) : A B, B B, B compact} ((Molchanov,
2005, Ch. 1)). Cores of convex capacities are well-deﬁned (Schmeidler (1986)).
2

Deﬁnition 1. C
1
dominates C
2
with respect to a family of Borel measurable functions F,
denoted C
1
%
F
C
2
, if
Z
Ch
fdC
1
Z
Ch
fdC
2
f F (1)
where
R
Ch
is the Choquet integral (Choquet (1954)). Let F = {f : Y R, f increasing, B
measurable}. Then we have the following characterization of stochastic dominance (Propo-
sition 1 in Dyckerhoﬀ and Mosler (1993), proof omitted):
Proposition 1. C
1
%
F
C
2
if and only if C
1
[t, [ C
2
[t, [ t R.
We also use the following result (Proposition 3 in Schmeidler (1986), proof omitted).
Proposition 2. For a convex capacity C, the Choquet integral is given by
R
Ch
fdC =
min
qcore (C)
R
fdq.
Let q
a
denote the distribution in Q(a) = core (C
a
) that attains the minimum (for a regular
capacity this is well deﬁned (Huber and Strassen (1973))).
2.2. Payoﬀs and Timing. We consider three alternative representations of ambiguity-
sensitive preferences that evaluate a contract according to :
1. worst-case expected payoﬀ, i.e., ‘max-min’ or MEU criterion (Gilboa and Schmeidler
(1989));
2. α-max-min’ or Hurwicz criterion (Hurwicz (1951));
3. probability-set-dominance criteria (Bewley (2002)).
We assume that Agent is risk-neutral over monetary payoﬀs.
A contract is a bounded, non-negative, B -measurable function w : Y R
+
that speciﬁes
output contingent payments and protects Agent with limited liability (i.e. w(y) 0).
The timing of the contracting game is as follows:
(i) Principal oﬀers a contract w;
(ii) Agent, knowing Q
A
, chooses action a A;
(iii) output y is realized;
(iv) payoﬀs are received: y w(y) to Principal and w(y) g(a) to Agent.
3

3. Implementation
Let a A be the action that Principal wants to implement. If a is the least cost action, then
a ﬂat payment of g(a) would implement it for any of the three kinds of objective functions
under consideration. For any other action, we have three sets of individual rationality (IR)
and incentive compatibility (IC) conditions, one set for each case. These implementation
conditions, (2) - (3), (7) - (8), (12) - (13), depend only on the payment scheme w(y) and
Agent’s perception Q
A
but neither on Principal’s perceived ambiguity nor her ambiguity
attitude.
3
3.1. Implementation with MEU Preferences. Action a A is implemented if it sat-
isﬁes Agent’s IR and IC, respectively:
min
qQ
A
(a)
Z
w(y)dq g(a) 0 (2)
and
min
qQ
A
(a)
Z
w(y)dq g(a) min
pQ
A
(a
0
)
Z
w(y)dp g(a
0
) a
0
A. (3)
Proposition 3. With MEU preferences, if a
k
A is implementable, then a
j
A, j 6= k
and g
j
< g
k
, we have Q
A
(a
j
) \ Q
A
(a
k
) 6= .
Proof. Suppose not. Let w(y) implement a
k
and let Q
A
(a
j
) Q
A
(a
k
) for some a
j
A such
that g
j
< g
k
. Combining the IR and IC conditions together with the fact that Q
A
(a
j
)
Q
A
(a
k
) yields the following chain of inequalities that shows that Agent chooses a
j
rather
than a
k
:
min
qQ
A
(a
j
)
Z
w(y)dq g
j
min
pQ
A
(a
k
)
Z
w(y)dp g
j
> min
pQ
A
(a
k
)
Z
w(y)dp g
k
0.
Here the last inequality shows that a
j
is rational for Agent, while the strict inequality,
which follows from g
k
> g
j
, together with the ﬁrst inequality, which follows from the fact
Q
A
(a
j
) Q
A
(a
k
) and that minimum of a non-negative-valued function does not get smaller
over a larger set, establishes that Agent would prefer a
j
rather than a
k
.
3
In Q
A
, the ‘best-case’ and ‘worst-case’ distributions are endogenously determined for a given contract;
Principal and Agent can disagree on these cases. Furthermore, since all these implementability conditions
depend on Q
A
, but not on Q
P
, whether Principal has more or less precise knowledge of technology than
Agent does not bear on implementability of an action.
4

The next Proposition shows that strengthening the necessary condition to stochastic domi-
nance, and imposing bounds on the rate at which costlier actions improve outcomes becomes
suﬃcient for implementation.
Assumption 2. a
j
, a
k
A, j 6= k and g
j
< g
k
, C
k
%
F
C
j
and C
k
6= C
j
.
Let M be the collection of all upper tail events in B : M = {M B, M = [y, Y ], y Y}.
The next two assumptions impose bounds on the rates of increase in upper tail capacities
relative to the increase in costs.
Assumption 3. a
k
, a
j
A, j 6= k and g
j
< g
k
, we have
g
k
g
j
g
k
C
k
(M) C
j
(M)
C
k
(M)
Assumption 4. a
l
, a
k
, a
j
A, l 6= k 6= j and g
k
< g
l
< g
j
we have, for all upper tail
events M,
g
l
g
k
C
l
(M) C
k
(M)
g
k
g
j
C
k
(M) C
j
(M)
(4)
Proposition 4. With MEU preferences, a
k
A is implementable if Assumptions 2-4 hold.
Proof. Let y
k
= min
pQ
A
(a
k
)
R
ydp and let M be the particular event {y Y : y [y
k
, Y ]}.
Consider contracts that reward Agent with a constant non-zero payment only above a certain
performance level. For instance, consider a contract of the form:
w(y) =
(
b if y M
0 if y Y \ M
Such a contract implements a
k
against a lower cost action a
j
if
b min
qQ
A
(a
k
)
q(M) g
k
0, (5)
and
b min
qQ
A
(a
k
)
q(M) g
k
b min
qQ
A
(a
j
)
q(M) g
j
(6)
The IR condition (5) holds if
b g
k
/q
k
(M)
The iIC condition (6) holds if
b g
k
g
j
/(q
k
(M) q
j
(M))
5

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Abstract: This paper considers dynamic moral hazard settings, in which the consequences of the agent's actions are not precisely understood. In a new continuous-time moral hazard model with drift ambiguity, the agent's unobservable action translates to drift set that describe the evolution of output. The agent and the principal have imprecise information about the technology, and both seek robust performance from a contract in relation to their respective worst-case scenarios. We show that the optimal long-term contract aligns the parties' pessimistic expectations and broadly features compressing of the high-powered incentives. Methodologically, we provide a tractable way to formulate and characterize optimal long-run contracts with drift ambiguity. Substantively, our results provide some insights into the formal link between robustness and simplicity of dynamic contracts, in particular high-powered incentives become less effective in the presence of ambiguity.

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• ...For a convex capacity C, the Choquet integral is given by ∫ Ch fdC = minq∈core (C) ∫ fdq....

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• ...C1 dominates C2 with respect to a family of Borel measurable functions F , denoted C1 %F C2, if∫ Ch fdC1 ≥ ∫ Ch fdC2 ∀f ∈ F (1) where ∫ Ch is the Choquet integral (Choquet (1954))....

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• ...Cores of convex capacities are well-defined (Schmeidler (1986))....

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• ...We also use the following result (Proposition 3 in Schmeidler (1986), proof omitted)....

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