Contracts with Interdependent Preferences

Contracts with Interdependent Preferences
Debraj Ray1
Marek Weretka2
1
New York University
2
University of Wisconsin-Madison and GRAPE/FAME
EWMES 2023
December, 2023
Debraj Ray and Marek Weretka Contracts with Interdependent Preferecnes 1 / 18

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 “preference independence” hypothesis.
I Evolutionary arguments for positive and negative interdependence

Research Agenda
I Goal#1: Introduce preference interdependence to a team agency problem
I A framework with interdependent preferences
I Characterize optimal contracts
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 Does interdependence benefit the agents?

Interdependent Preferences in Economics
1. Altruistic or adversarial attitudes
I 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).
I 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).
2. Reduced form of dynamic interactions among selfish agents Che and Yoo
(2001), Bandiera et al. (2005)
I We work with the utility-based preferences
I Non-paternalistic altruism, games of love and hate, empathetic games as we
believe they naturally capture interactions with such agents.
I Our results apply to the other two forms of interdepencies!

Outline
1. Simple example (binary output)
2. Framework with I agents, general signal distributions
3. Summary of general results

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 outputs 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 Contract: monetary compensation m : {0, 1}2
→ R+

Simple example
I One principal and two identical agents i = a, b
I Observable outputs 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 Contract: monetary compensation m : {0, 1}2
→ R+
I Interdependent preferences:
Ui = u(mi) − c(ei)
| {z }
material payoff Vi
+ αŪj
|{z}
interdependent part
.
I 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 game with interdependent preferences
I e = {ei}i induces (random) material payoff Vi = u(m(yi,yj )) − c(ei).
I Utility feedback effects and consistent payoffs:

/ 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 game with reduced form preferences.

/ 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 game with reduced form preferences.
I No informational externality in our model!

Contracts with interdependent preferences
I Independent contracts optimal for α = 0 (Green and Stokey, (1983))
I Reduced form 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 With interdependencies, payment m(1,0) affects incentives in two events
I Independent contracts suboptimal when α 6= 0

Optimal contracts
/ 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

Optimal contracts
/ 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
I Two bonus contracts:
I Positive interdepenence: Performance bonus + Team bonus
I Negative interdependence: Performance bonus + Winner’s bonus

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

Optimal contracts
, θ = 0.5.
I Variations:
I More general binary signal distributions (+ consolation price)
I Outside options (performance and winner’s/team bonuse + wage)
I Risk neutral agents (pure tournaments/join liability)

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

I Positive and negative interdependence reduces expected payment
Figure: Expected payment for c = 1, ε = 0.1 and u = m1−θ
, θ = 0.5.

I Positive and negative interdependence reduces expected payment
Figure: Expected payment for c = 1, ε = 0.1 and u = m1−θ
, θ = 0.5.
I Inverted U shaped pattern is robust
I In general, payment is symmetric

I Is interdependence beneficial for the agents? If so, positive or negative?

I Is interdependence beneficial for the agents? If so, positive or negative?
I In general, positive interdependence helps while negative hurts
I Example in which altruism leads to strictly negative material payoff:
, θ = 0.5.

Robust implementation
I What if one wants to implement effort in unique Nash?

I Positive (negative) interdependence → strategic complements (substitutes)
I One needs to reduce cooperative incentives, relying on the performance bonus
, θ = 0.5.

I Positive (negative) interdependence → strategic complements (substitutes)
I One needs to reduce cooperative incentives, relying on the performance bonus
, θ = 0.5.
I Negative interdependence with contests cheaper for a conservative principal

General Framework
I One principal, I ≥ 2 symmetric agents
I Observable outputs yi ∈ Y ⊂ R (signals)
I Probability measure µei
: µ`
µh
, decreasing RN derivative [dµ`
/dµh
](y)
I Interdependent vNM preferences:
Ui = u(mi) − c(ei)
| {z }
material payoff Vi
+ α
X
j6=i
Uj
| {z }
interdependent part
.
I Problem: symmetric m : Y ×n
→ R+ that implements efforts as (unique) Nash with
respect to reduced form utilities

Technical contribution:
I Contracts fully characterized by a statistic
Ψ(y) ≡ (1 − [dµ`
/dµh
](y1)) +
α
1 − α(n − 2)
X
j6=1
(1 − [dµ`
/dµh
](yj)).
I Optimal contracts increase the fastest in the direction of ∇Ψ
Figure: Interdependencies: α = 0.3 (left panel), independent preferences α = 0 (middle
panel) and α = −0.3 (right panel) .

Substantive results
1. Contracts span an entire competitive-cooperative spectrum

Substantive results
2. Outside option does not affect competitiveness/cooperativeness of a contract

Substantive results
3. Asymmetry: in large teams, altruism is a stronger motivator

Substantive results
4. Risk aversion is necessary for realistic contracts

Substantive results
5. Principal benefits from the increased (positive or negative) interdependence

Substantive results
5. Principal benefits from the increased (positive or negative) interdependence
6. Robust implementation easier with negative interdependence

Design with interdependencies 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)
I Small number of sport competitors: tournaments

Thank you!

45!
#

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

45!
#

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

Contracts with Interdependent Preferences

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