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

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

q∈core (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

q∈Q

A

(a)

Z

w(y)dq − g(a) ≥ 0 (2)

and

min

q∈Q

A

(a)

Z

w(y)dq − g(a) ≥ min

p∈Q

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

q∈Q

A

(a

j

)

Z

w(y)dq − g

j

≥ min

p∈Q

A

(a

k

)

Z

w(y)dp − g

j

> min

p∈Q

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

p∈Q

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

q∈Q

A

(a

k

)

q(M) − g

k

≥ 0, (5)

and

b min

q∈Q

A

(a

k

)

q(M) − g

k

≥ b min

q∈Q

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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...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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### Additional excerpts

...‘α-max-min’ or Hurwicz criterion (Hurwicz (1951)); 3. probability-set-dominance criteria (Bewley (2002))....

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