Contracts with Interdependent Preferences

Contracts with Interdependent Preferences
Debraj Ray1
Marek Weretka2
1
New York University
2
University of Wisconsin-Madison and GRAPE/FAME
SAET
July, 2022
Debraj Ray and Marek Weretka Contracts with Interdependent Preferecnes 1 / 16

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: Recommendations for contract design:
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 a joint liability/reward mechanism is effective?
I Are altruistic or adversarial relations more beneficial for a principal?
I (A special case of) Green and Stokey (1983) with interdependent preferences

Interdependent Preferences in Economics
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).

Interdependent Preferences in Economics
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 of affective empathy
I Intimately related to outcome-based preferences

Outline
1. Simple example (binary output)
2. General framework with continuous outputs (one slide)
3. Future work (one slide)

Simple example
I One principal and two identical agents i = a, b
I Non-observable efforts ei ∈ {0, 1}, (i.e., shirking and working)
I Observable output yi ∈ {0, 1}, (i.e., underperformance and performance)
I Working: Prei=1(yi = 0) = 0.5; shirking: Prei=0(yi = 0) = 0.5 + ε
I Contract: monetary compensation m : {0, 1}2
→ R+

Simple example
I One principal and two identical agents i = a, b
I Non-observable efforts ei ∈ {0, 1}, (i.e., shirking and working)
I Observable output yi ∈ {0, 1}, (i.e., underperformance and performance)
I Working: Prei=1(yi = 0) = 0.5; shirking: Prei=0(yi = 0) = 0.5 + ε
I Contract: monetary compensation m : {0, 1}2
→ R+
I Interdependent preferences:
Ui = u(mi) − c(ei)
| {z }
material payoff Vi
+ αŪj
|{z}
empathetic part
.
I strictly concave function u : R+ → R+, cost function c(ei) = c × ei,
I Ūj is agent i conjecture regarding utility of an agent j
I α ∈ (0, 1] altruistic preferences, α ∈ [−1, 0) adversarial preferences

Problem of the principal
I Principal’s offers symmetric contract m(yi,yj ):
/ yj = 0 yj = 1
yi = 0 m(0,0) m(0,1)
yi = 1 m(1,0) m(1,1)
I Contract m defines an empathetic game among agents
I e = {ei}i induces (random) material payoff Vi = u(m(yi,yj )) − c(ei).
I Empathetic contagion and consistent payoffs: figure

/ yj = 0 yj = 1
yi = 0 m(0,0) m(0,1)
yi = 1 m(1,0) m(1,1)
Ui(e) = Vi(e) + αŪj → Ũi(e) =
Vi(e) + αVj(e)
1 − α2

/ yj = 0 yj = 1
yi = 0 m(0,0) m(0,1)
yi = 1 m(1,0) m(1,1)
Ui(e) = Vi(e) + αŪj → Ũi(e) =
Vi(e) + αVj(e)
1 − α2
I Problem:
min
m
E(m|ea = eb = 1)
s.t. ea = eb = 1 is a Nash in the empathetic game.

/ yj = 0 yj = 1
yi = 0 m(0,0) m(0,1)
yi = 1 m(1,0) m(1,1)
Ui(e) = Vi(e) + αŪj → Ũi(e) =
Vi(e) + αVj(e)
1 − α2
I Problem:
min
m
E(m|ea = eb = 1)
s.t. ea = eb = 1 is a Nash in the empathetic game.
I No informational externality in our model!

Non-empathetic benchmark, α = 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 Low performance payment is zero, m(0, yb) = 0

Non-empathetic benchmark, α = 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 Low performance 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 Contract: Performance bonus, increasing in c decreasing in ε
I No need for tournament/joint liability

What if agents are empathetic, α 6= 0 ?
I Total utility Ũa = 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 α 6= 0
/ 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 part contracts
I Positive empathy:
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 Positive empathy:
I Negative empathy bonuses:
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 Positive empathy:
I Negative empathy bonuses:
I Winner’s bonus: m(1,0) − m(1,1)
I Limits: α → 1 pure joint liability, α → −1 pure tournament

Figure: Optimal bonuses for c = 1, ε = 0.1 and u = m1−θ
, θ = 0.5.

Benefits of empathy
I Is empathy beneficial for the principal? If so, positive or negative?

Benefits of empathy
I Expected payment as a fraction of a no-empathy scenario:
, θ = 0.5.

Benefits of empathy
I Expected payment as a fraction of a no-empathy scenario:
, θ = 0.5.
I Empathy reduces expected payment, has symmetric impact
I Robust implementation: unique Nash equilibrium for α < 0 but not α > 0

General model
I Assumptions:
I I = 2 agents (we can probably 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)

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 Participation constraints (outside options)
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

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