Contracts with interdependent preferences subtitle: Empathetic design

Contracts with Interdependent Preferences
(Empathetic Design)
Debraj Ray1
Marek Weretka2
1
New York University
2
University of Wisconsin-Madison and GRAPE/FAME
University of Waterloo
April 14, 2023
Debraj Ray and Marek Weretka Contracts with Interdependent Preferecnes 1 / 29

Motivation
I Question: How to incentivize a group (a team) of rational agents?
I Existing “team” agency theories assume agents are driven by their material
self-interest (e.g., monetary payment, cost of effort). (Lazear and Rosen,
(1981), Holmstrom (1982), Green and Stokey, (1983), Segal (1993, 2003), Winter
(2004), Halac, Kremer and Winter (2020,2021), Halac, Lipnowski and Rappoport
(2021), Camboni and Porccellancchia (2022 ))
I Experiments: violations of the “self-interest” hypothesis.
I Altruism is driven by affective empathy, i.e, an ability that allows us
experience the emotions of others. (Batson (2009))

Research Agenda
I Goal#1: Introduce affective empathy to a team agency problem
I A framework with interdependent preferences
I Characterize optimal incentive mechanism
I Goal#2: Contractual prescriptions:
I Should a compensation of an agent depend on other’s performance?
I When are tournaments optimal? If so, what kind?
I In which environments joint liability/reward mechanism is effective?
I How empathy affects hiring decisions? How to split agents into teams?
I Dynamics: feedback between compensation and empathy

Pro-Social Preferences: A Literature Review
1. Outcome-based preferences. Players care about the outcome of others,
e.g., consumption or money (Fehr and Schmidt, 1999; Charness and Rabin, 2002;
Bolton and Ockenfels, 2000; Sobel, 2005).
2. Psychological games A player cares about what others’ intentions are (e.g.,
Genanakoplos, Pearce, and Stachetti; 1989; Rabin, 1993; Batigalli and Dufwenberg,
2009; Rabin, 2013).

Pro-Social Preferences: A Literature Review
1. Outcome-based preferences. Players care about the outcome of others,
e.g., consumption or money (Fehr and Schmidt, 1999; Charness and Rabin, 2002;
Bolton and Ockenfels, 2000; Sobel, 2005).
2. Psychological games A player cares about what others’ intentions are (e.g.,
Genanakoplos, Pearce, and Stachetti; 1989; Rabin, 1993; Batigalli and Dufwenberg,
2009; Rabin, 2013).
3. Utility-based preferences. Players care directly about the welfare of others (e.g.,
Becker, 1974; Ray, 1987; Bernheim, 1989; Bergstrom, 1999; Pearce, 2008; Bourles
et al, 2017, Galperti and Strulovici, 2017, Ray and vohra, 2020, vasquez and
Weretka 2019, 2021).
I We work with the utility-based preferences
I Non-paternalistic altruism, games of love and hate, empathetic games
I Consistent with the idea affective empathy
I Most results hold under outcome-based preferences as well

Outline
1. Empathetic Agency Framework
2. Effects of empathy on optimal contracts (binary example)
3. Summary of the results for the general framework

Empathetic Design Framework
I Green and Stokey’s (1983) with empathetic preferences
I One principal and I agents, i = 1, ..., I
I Agent’s preferences
Ui = ui(mi) − c × ei
| {z }
material payoff vi
+
X
j6=i
αi,jŪj
| {z }
empathetic part
.
I Unobservable binary effort ei ∈ {0, 1} (shirking and working) with cost c > 0
I Monetary compensation mi ∈ R+
I ui strictly increasing and strictly concave
I Ūj is agent i conjecture regarding utility of an agent j

Empathetic Matrix
I Interdependence of preferences is fully described by an empathetic matrix
A =






0 α1,2 ... α1,I
α2,1 0 ... α2,I
... ... ... ...
αI,1 αI,2 ... 0






. (1)
I In general αi,j can have different signs
I αi,j > 0 → agen i is altruistic towards j
I αi,j < 0 → agen i is adversarial towards j
I Matrix I − A is invertible

Problem of the principal
I Effort ei ∈ {0, 1} induces observable output yi, c.d.f. F(·|ei)
I Design: m = {mi}i, compensation mi : Y I
→ R+ (limited liability)
I m defines an empathetic game among I agents (game of love and hate)
I e = {ei}i results in a material payoff v̄i(e) ≡ ui(mi(y)) − c × ei.
I Let v̄ = {v̄i}i. Empathetic contagion and consistent payoffs: figure

Ū(e) = v̄(e) + AŪ(e) → Ū(e) = (I − A)−1
v̄(e)

Ū(e) = v̄(e) + AŪ(e) → Ū(e) = (I − A)−1
v̄(e)
I Principal problem:
min
m
X
i
E(mi|ej = 1, j = 1, ..., I)
s.t. ei = 1, i = 1, ..., I is a Nash with respect to Ūi(e) (IC)

Remarks
I We also explore two variations of the problem
I Robust implementation of effort (unique Nash)
I Outside option Ui > 0
I Key differences:
I No information externality
1. y = {yi}i conditional on effort are i.i.d.
2. No observable “team output” statistic Y (
P
i ei)
3. The only externality: payoff externality (empathy)
I Empathetic sophistication

Symmetric binary example
I One principal and two identical agents i = a, b
I Binary output yi ∈ {0, 1}, (i.e., failure and success)
I Working: Prei=1(yi = 0) = 0.5; shirking: Prei=0(yi = 0) = 0.5 + ε
I Symmetric contracts: monetary transfer m : {0, 1}2
→ R+
/ yj = 0 yj = 1
yi = 0 m(0,0) m(0,1)
yi = 1 m(1,0) m(1,1)

Symmetric binary example
I One principal and two identical agents i = a, b
I Binary output yi ∈ {0, 1}, (i.e., failure and success)
I Working: Prei=1(yi = 0) = 0.5; shirking: Prei=0(yi = 0) = 0.5 + ε
I Symmetric contracts: monetary transfer m : {0, 1}2
→ R+
/ yj = 0 yj = 1
yi = 0 m(0,0) m(0,1)
yi = 1 m(1,0) m(1,1)
I Interdependent preferences:
Ūi = u(mi) − c × ei
| {z }
material payoff vi
+ αŪj
|{z}
empathetic part
→ Ūi(e) =
vi(e) + αvj(e)
1 − α2
I Identical α ∈ (−1, 1)

Non-empathetic benchmark, α = 0
I What is the cheapest way to implement efforts as Nash?

I Suppose eb = 1. How to incentivize ea = 1?
I Two observations (Green and Stokey (1983)):
I a0
s payment is independent from yb, i.e., m(ya, 0) = m(ya, 1)
I Failure payment is zero, m(0, yb) = 0

I Suppose eb = 1. How to incentivize ea = 1?
I Two observations (Green and Stokey (1983)):
I a0
s payment is independent from yb, i.e., m(ya, 0) = m(ya, 1)
I Failure payment is zero, m(0, yb) = 0
/ yb = 0 yb = 1
ya = 0 0 0
ya = 1 u−1
(c/ε) u−1
(c/ε)
I Optimal contract: performance bonus
I bonus paid in the event of agent’s success
I increasing in c, decreasing in ε
I Independent contracts: no need for a team or winner’s bonus

What if agents are empathetic, α 6= 0 ?
I Utility Ūa = va+αvb
1−α2 6= va ≡ u(m(ya, yb)) − cea
I Payments to both agents incentivize agent a
/ yb = 0 yb = 1
ya = 0 m(0,0), m(0,0) m(0,1), m(1,0)
ya = 1 m(1,0), m(0,1) m(1,1), m(1,1)
I Independent contracts are suboptimal

Optimal contracts with empathy
/ yb = 0 yb = 1
ya = 0 0 0
ya = 1 m(1,0) m(1,1)
I In optimum m(1,0) > (<) m(1,1) when α < (>) 0

/ yb = 0 yb = 1
ya = 0 0 0
ya = 1 m(1,0) m(1,1)
I Two-bonus contracts optimal
I Positive empathy α > 0:
I Performance bonus: m(1,0)
I Team bonus: m(1,1) − m(1,0)

/ yb = 0 yb = 1
ya = 0 0 0
ya = 1 m(1,0) m(1,1)
I Negative empathy α < 0:
I Winner’s bonus: m(1,0) − m(1,1)

/ yb = 0 yb = 1
ya = 0 0 0
ya = 1 m(1,0) m(1,1)
I Negative empathy α < 0:
I Winner’s bonus: m(1,0) − m(1,1)
I Limits: α → 1 pure joint liability, α → −1 pure tournament

I Assume u = m1−θ
, θ = 0.5.
I Optimal bonuses for ε = 0.1:
I Empathy can explain five types of contracts.
I Bonuses increasing in c/ε
I Literature seems to rationalizes tournaments
(Lazear and Rosen (1981), Holmstrom (1982), Green and Stokey (1983))

Principal’s payment
I Is empathy beneficial for the principal? If so, positive or negative?
I What if the principal can induce α = −0.5, 0, or 0.5?

Expected principal’s payment
I Expected payment as a fraction of a no-empathy scenario:

Expected principal’s payment
I Expected payment as a fraction of a no-empathy scenario:
I Ratio does not depend on c, ε
I Empathy (positive or negative) symmetrically reduces expected payment,
I Productivity higher when team bonuses involved
(Hamilton, Nickerson, and Owan, (2003))

Expected agent’s utility
I Is empathy beneficial for the agents? If so, positive or negative?
I For α = 0, agents receive positive utility. How about α 6= 0?
I Welfare effects of empathy sensitive to output informativeness ε

Expected agent’s utility
I Is empathy beneficial for the agents? If so, positive or negative?
I For α = 0, agents receive positive utility. How about α 6= 0?
I Welfare effects of empathy sensitive to output informativeness ε
I High informativeness ε may result in negative utility
I Low informativeness ε, positive utility increasing in α

Expected (agent’s) utility, ε = 0.45
I Expected equilibrium utility

I Expected equilibrium utility
I Equal to expected utility when shirking (IC constraint binding)
I Empathy “relaxes” the limited liability constraint
I Mechanism for positive and negative empathy

I Expected utility

I Expected utility
I High informational rents
I Empathetic contagion increases overall level of utility

Other design considerations
I Example: positive and negative empathy equally effective in reducing cost
I Two design considerations ignored so far
I Robust implementation of effort
I Outside option (IR constraints)
I One consideration reveals advantage of positive while the other of negative
empathy

Consideration 1: Robust implementation
I Robustness implementation (Bergmann and Morris (2009))
min
m
E(m|ea = eb = 1)
s.t. ea = eb = 1 is a unique Nash∗
in the empathetic game
I The indifference ∗ caveat applies

Consideration 1: Robust implementation
I Robustness implementation (Bergmann and Morris (2009))
min
m
E(m|ea = eb = 1)
s.t. ea = eb = 1 is a unique Nash∗
in the empathetic game
I The indifference ∗ caveat applies
I Is the additional constraint binding? Depends on the sign of α
I For α < 0
I Winner’s bonus → efforts strategic substitutes → unique equilibrium
I Robustness constraint not binding
I For α > 0
I Team bonus → efforts strategic complements → two equilibria
I Robustness constraint is binding

Consideration 1: Robust contracts
I Contracts that implement effort in unique equilibrium
Figure: Optimal bonuses for ε = 0.1 and u = m1−θ
, θ = 0.5.
I Winner’s bonus even with altruistic preferences
(Bandiera, Barankay, Rasul 2005)

Consideration 1: Robust contracts
I Expected payment for robust contracts
, θ = 0.5.
I Higher robust implementation cost for positive empathy
I Competitive schemes observed in farming, sports, some corporations
(Knoeber and Thurman (1994); Becker and Huselid (1992) )

Consideration 2. Outside option
I Cooperative schemes are more common
I Competitive schemes rare Lazear (1989)
I Firms tend to use team bonuses Payscale (2019)
I Other consideration: Outside options

min
m
E(m|ea = eb = 1)
s.t. ea = eb = 1 is a Nash in the empathetic game and Ūi ≥ Ui > 0

min
m
E(m|ea = eb = 1)
s.t. ea = eb = 1 is a Nash in the empathetic game and Ūi ≥ Ui > 0
I Utility consists of a material and empathetic component
I Empathetic component positive (negative) when α > 0(< 0)
I Collegial environment partly compensates for an outside option

I Expected payment for positive outside option
, θ = 0.5.
I This pattern seems robust to changes in parameter values

General model
I Main lessons from the example
I Assumptions:
I I = 2 agents (we can extend it to I > 2)
I Binary effort ei = {0, 1}, continuous random yi ∈ [0, 1]
I Density functions f0, f1 give rise to increasing
λ(x) ≡ 1 − [f0(F−1
1 (x))/f1(F−1
1 (x))]
I Results:
I Characterization of optimal contracts m : [0, 1]2
→ R+
I α comparative statics: tournament → independent → joint reward/liability
I Linear λ(x):
I Symmetric effects of positive and negative empathy on the expected payment
I With u(m) = m1−θ intensity of empathy reduces expected payment
I Non-linear λ(x): asymmetric effects convex/concave λ(x)
I We now work on robustness and outside options

Empathetic design in practice
I Evidence
I Social relationships critical for workers’ well-being. Riordan and
Griffeth (1995); Hodson (1997); Ducharme and Martin (2000); Morrison
(2004); Wagner and Harter (2006); Krueger and Schkade (2008).
I 85% of US managers foster friendship in the workplace (Berman et al. (2002))
I 73% of the firms use some bonuses, 66% individual and 22% team bonuses
Payscale (2019)
I Grameen style lending program (joint liability)

Current and future work
1. Extensions:
I Extensive margin (hiring decisions)
I Endogenous empathetic relations
2. Applications of empathetic design:
I Labor markets;
I Sports economics;
I Military conflicts
I Voting (Political Polarization);
I Economics of marriage and family.

Thank you!

45!
#
"
$"
$#
$" = #
" + 0.5$#
Figure: Empathetic Contagion (Altruism)
return

45!
#
"
$"
$#
$" = #
" − 0.5$#
Figure: Empathetic Contagion (Antypathy)
return

Contracts with interdependent preferences subtitle: Empathetic design

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