a slide of_optimal_team_formation_of_heterogeneous_other_regarding_agents

Optimal Team Formation of HeterogeneousOther-regarding Agents

Yuki Kakihara

Keio University

December 27, 2016

Yuki Kakihara (Keio University) Optimal Team Formation of Heterogeneous Other-regarding AgentsDecember 27, 2016 1 / 59

1. Introduction

Are human beings inequity averse or status-seeking?

In experimental economics, an example of the former is a dictatorgame (Kahneman, Knetsch and Thaler, 1986).

Frank (1985) also argues that humans are hard-wired with apreference for higher status, and like any change that moves them uprelative to their coworkers.


1. Introduction

Itoh (2004) shows that, in multi-agent model, if agents are inequityaverse enough, an extreme team contract is optimal, while if agentsare status-seeking or indifferent enough to inequity, an extremerelative performance contract is optimal.

Englmaier and Wambach (2010) shows that, when the level ofinequity aversion of agents is high enough, flat wage is optimalcontract even if the agents are heterogeneous.


1. Introduction

So far, we assume that agents are either inequity averse orstatus-seeking.

We assume that agents are inequity averse and status-seeking.

We consider inter-team competition.


1. Introduction

In Honda, plural teams research the same subject.

The members research cooperating in the team and compete withother teams.

They seem inequity averse to the same team members as well asstatus-seeking to other teams.


1. Introduction

In this paper, we suppose that agents are inequity averse to the sameteam members and status-seeking to other team members, andconsider an inter-team competition such as Honda.

We call these inequity averse and status-seeking agents asother-regarding agents.


1. Introduction

We consider a situation that there exist heterogeneous agents,high-type agents and low-type agents, in the same team.

In the first half, I have discussed the effect of other-regardingpreference on the optimal contract and agents’ effort levels.

In traditional contract theory, the principal should give higherincentive salary to high-type agents.

However, assuming other-regarding agents, this result changes.

Our result is also theoretical foundation about flat wage.


1. Introduction

In the last half, we consider the optimal team formation.

Using the first half of the framework, we show that the principal canget more profit when she builds two high-type and low-type teamsthan when she builds high-type and high-type team, and low-type andlow-type team.


2. Setup

A = {1, 2}, B = {3, 4} : Two teams

N = A ∪ B = {1, 2, 3, 4} : An agent set

ei ∈ R : Effort level of each agent i ∈ N

xA = e1 + e2, xB = e3 + e4 : Output of the teams


2. Setup

ψ1(e) = ψ3(e) = dhe2

2 , ψ2(e) = ψ4(e) = dle2

2 : Agents’ effortfunctions

0 < dh < dl

The principal can distinguish high-type from low-type, but cannotobserve their individual efforts.

w1 = α + βhxA, w2 = α + βlxA, w3 = α + βhxB , w4 = α + βlxB .

Assumption 1 (Limited liability constraints)

α ≥ 0 and βh, βl ≥ 0.


2. Setup

Suppose that agent i receives the dis-utility s|wi − wj |.Agent j is the same team with i , and s ≥ 0.

Suppose that the agent of team k receives the utility t(xk − xl).

l 6= k and t ≥ 0.

Objective function of agent i is

ui (e) = wi − ψi (ei )− s|wi − wj |+ t(xk − xl).


2. Setup

Therefore, their incentive constraints are

e IC1 (βh, βl) = e IC3 (βh, βl) =βh − s|βh − βl |+ t

dh, (1)

e IC2 (βh, βl) = e IC4 (βh, βl) =βl − s|βh − βl |+ t

dl. (2)

Assumption 2 (No retaliation constraints)

∀i ∈ N ; e ICi ≥ 0.


2.1. Benchmark

We consider the effort level of maximizing a total surplus as abenchmark,

max{e1,e2,e3,e4}

xA + xB −∑i∈N

ψi (ei ).

By straightforward calculation, first best effort levels are

eFB1 = eFB3 =1

dhand eFB2 = eFB4 =

1

dl.


3. The Analysis

The principal’s maximization problem about team A is

max{α,βh,βl}

xA − (α + βhxA)− (α + βlxA), (3)

s.t. u1(e) ≥ 0, u2(e) ≥ 0,

e1 = e IC1 ≥ 0, e2 = e IC2 ≥ 0,

α ≥ 0, βh ≥ 0, βl ≥ 0.

We consider four cases; self-interested case, inequity averse case,status-seeking case and other-regarding case.


3.1. Self-interested Case

s = t = 0.

Incentive constraints are

e IC1 =βhdh

and e IC2 =βldl.

The maximization problem about team A is

max{α,βh,βl}

(1− βh − βl)xA − 2α,

s.t. α + βhxA −dhe

21

2≥ 0, α + βlxA −

dle22

2≥ 0,

e1 =βhdh≥ 0, e2 =

βldl≥ 0,

α ≥ 0, βh ≥ 0, βl ≥ 0.



This problem can be written as

max{α,βh,βl}

(1− βh − βl)(βhdh

+βldl

)− 2α,

s.t. α + βh

(βhdh

+βldl

)− dh

2

(βhdh

)2

≥ 0,

α + βl

(βhdh

+βldl

)− dl

2

(βldl

)2

≥ 0,

α ≥ 0, βh ≥ 0, βl ≥ 0.




max{α,βh,βl}


+βldl

)− 2α,

s.t. α ≥ 0, βh ≥ 0, βl ≥ 0.




max{α,βh,βl}


+βldl

),

s.t. α = 0, βh ≥ 0, βl ≥ 0.

Lagrangian function is

L = (1− βh − βl)(βhdh

+βldl

)+ λ1βh + λ2βl .



Kuhn-Tucker conditions are

∂L∂βh

=1− 2βh − βl

dh− βl

dl+ λ1 = 0,

∂L∂βl

=1− βh − 2βl

dl− βh

dh+ λ2 = 0,

λ1∂L∂λ1

= λ1βh = 0,

λ2∂L∂λ2

= λ2βl = 0,

λ1 ≥ 0, λ2 ≥ 0, βh ≥ 0, βl ≥ 0.



The solutions of simultaneous four equations are

(β∗h , β∗l , λ∗1, λ∗2) =

(12 , 0, 0,

dl−dh2dldh

)(0, 12 ,−

dl−dh2dldh

, 0)(

0, 0,− 1dh,− 1

dl

)(− dh

dl−dh ,dl

dl−dh , 0, 0).

Only(12 , 0, 0,

dl−dh2dldh

)is satisfied the four inequities.

Therefore,

(β∗h , β∗l ) =

(1

2, 0

).



Proposition 1

Suppose that s = t = 0. Then, the solution of (3) is (βSIh , βSIl ) =

(12 , 0).

That is, when agents in the same team are heterogeneous andself-interested, the principal should not give the incentive salary to thelow-type agent.



Their effort levels are

(eSI1 , eSI2 ) =

(1

2dh, 0

).

In this model, restricting profit sharing contracts, not daring to let thelow-type agent work, the principal prevents the free-rider problem.

Comparing (eSI1 , eSI2 ) with (eFB1 , eFB2 ) =

(1dh, 1dl

), their effort levels

are low. As Holmstrom (1982) indicates, when the budget constraintis satisfied, it is impossible to let the team’s members make efficientefforts.


3.2. Inequity Averse Case

s ≥ 0, t = 0.


e IC1 =βh − s|βh − βl |

dhand e IC2 =

βl − s|βh − βl |dl

.


max{α,βh,βl}

(1− βh − βl)xA − 2α,


21

2− s|βh − βl |xA ≥ 0,

α + βlxA −dle

22

2− s|βh − βl |xA ≥ 0,

e1 =βh − s|βh − βl |

dh≥ 0, e2 =

βl − s|βh − βl |dl

≥ 0,

α ≥ 0, βh ≥ 0, βl ≥ 0.




max{α,βh,βl}

(1− βh − βl)(βh − s|βh − βl |

dh+βl − s|βh − βl |

dl

),

s.t. α = 0, βh ≥ s|βh − βl |, βl ≥ s|βh − βl |.

The solution is

(βIAh , βIAl ) =

{(1+s

2(1+2s) ,s

2(1+2s)

)(0 ≤ s < dl−dh

2(dl+dh))(

14 ,

14

)(otherwise)

.

Then,

(e IA1 , eIA2 ) =

(

12(1+2s)dh

, 0)

(0 ≤ s < dl−dh2(dl+dh)

)(1

4dh, 14dl

)(otherwise)

.



Proposition 2

Suppose that s ≥ 0 and t = 0. If the level of inequity aversion s is highenough, the principal should give the same incentive salary to two agents(Englmaier and Wambach, 2010). In addition, the effort level of high-typeagent is non-increasing in s. On the other hand, the effort level oflow-type agent is non-decreasing in s.



Figure1.jpg

Figure: The relationship between s and incentive salaries



Figure2.jpg

Figure: The relationship between s and effort levels



Comparing

(e IA1 , eIA2 ) =

(

12(1+2s)dh

, 0)

(0 ≤ s < dl−dh2(dl+dh)

)(1

4dh, 14dl

)(otherwise)

with

(eFB1 , eFB2 ) =

(1

dh,

1

dl

),

their effort levels are lower for any s.

However, when the level of inequity aversion is high enough, thelow-type agent makes efforts. Therefore, his effort level approachesfirst best level.


3.3. Status-seeking Case

s = 0, t ≥ 0.


e IC1 =βh + t

dhand e IC2 =

βl + t

dl.


max{α,βh,βl}

(1− βh − βl)xA − 2α,


21

2+ t(xA − xB) ≥ 0,

α + βlxA −dle

22

2+ t(xA − xB) ≥ 0,

e1 =βh + t

dh≥ 0, e2 =

βl + t

dl≥ 0,

α ≥ 0, βh ≥ 0, βl ≥ 0.




max{α,βh,βl}

(1− βh − βl)(βh + t

dh+βl + t

dl

)− 2α,

s.t. α +β2h2dh

+βh(βl + t)

dl≥ t2

2dh,

α +β2l2dl

+βl(βh + t)

dh≥ t2

2dl,

α ≥ 0, βh ≥ 0, βl ≥ 0.



The solution is

(βSSh , βSSl ) =

{(dl−(dl+dh)t

2dl, 0)

(0 ≤ t ≤ dldl+dh

)

(0, 0) (otherwise).

Then,

(eSS1 , eSS2 ) =

((1+t)dl−tdh

2dldh, tdl

)(0 ≤ t ≤ dl

dl+dh)(

tdh, tdl

)(otherwise)

.



Proposition 3

Suppose that s = 0 and t ≥ 0. If the level of status-seeking t is highenough, the principal should not give the incentive salary to two agents. Inaddition, the effort levels of all agents are strictly increasing in t.



Figure3.jpg

Figure: The relationship between t and incentive salaries



Figure4.jpg

Figure: The relationship between t and effort levels



Comparing

(eSS1 , eSS2 ) =

((1+t)dl−tdh

2dldh, tdl

)(0 ≤ t ≤ dl

dl+dh)(

tdh, tdl

)(otherwise)

with

(eFB1 , eFB2 ) =

(1

dh,

1

dl

),

their effort levels accord with first best levels when t = 1.

We can say that there exists t such that agents can execute first besteffort levels.


3.4. Other-regarding Case

s ≥ 0, t ≥ 0.Incentive constraints are

e IC1 =βh − s|βh − βl |+ t

dhand e IC2 =

βl − s|βh − βl |+ t

dl.


max{α,βh,βl}

(1− βh − βl)xA − 2α,

s.t. α + (βh − s|βh − βl |)xA −dhe

21

2+ t(xA − xB) ≥ 0,

α + (βl − s|βh − βl |)xA −dle

22

2+ t(xA − xB) ≥ 0,

e1 =βh − s|βh − βl |+ t

dh≥ 0, α ≥ 0,

e2 =βl − s|βh − βl |+ t

dl≥ 0, βh ≥ 0, βl ≥ 0.




max{α,βh,βl}

(1− βh − βl)(βh − s|βh − βl |+ t

dh+βl − s|βh − βl |+ t

dl

)− 2α,

s.t. α+(βh − s|βh − βl |)2

2dh+

(βh − s|βh − βl |)(βl − s|βh − βl |+ t)

dl≥ t2

2dh,

α+(βl − s|βh − βl |)2

2dl+

(βl − s|βh − βl |)(βh − s|βh − βl |+ t)

dh≥ t2

2dl,

βh + t ≥ s|βh − βl |, βl + t ≥ s|βh − βl |,α ≥ 0, βh ≥ 0, βl ≥ 0.



The solution is, when s = 0,

(βORh , βOR

l ) =

{(12 −

dl+dh2dl

t, 0) (0 ≤ t ≤ dldl+dh

)

(0, 0) (t > dldl+dh

).

When 0 < s < dl−dh2(dl+dh)

,

(βORh , βOR

l ) =

(1+s−2st2(1+2s) ,

s−2t−2st2(1+2s) ) (0 ≤ t < s

2(1+s))

(1−2t4 , 1−2t4 ) ( s2(1+s) ≤ t ≤ 1

2)

(0, 0) (t > 12).

When s ≥ dl−dh2(dl+dh)

,

(βORh , βOR

l ) =

{(1−2t4 , 1−2t4 ) (0 ≤ t ≤ 1

2)

(0, 0) (t > 12).



Then, when s = 0,

(eOR1 , eOR

2 ) =

{( (1+t)dl−tdh

2dldh, tdl

) (0 ≤ t ≤ dldl+dh

)

( tdh, tdl

) (t > dldl+dh

).

When 0 < s < dl−dh2(dl+dh)

,

(eOR1 , eOR

2 ) =

( 1+2t2(1+2s)dh

, 0) (0 ≤ t < s2(1+s))

(1+2t4dh

, 1+2t4dl

) ( s2(1+s) ≤ t ≤ 1

2)

( tdh, tdl

) (t > 12).

When s ≥ dl−dh2(dl+dh)

,

(eOR1 , eOR

2 ) =

{(1+2t

4dh, 1+2t

4dl) (0 ≤ t ≤ 1

2)

( tdh, tdl

) (t > 12).



Proposition 4

Fix any s ≥ 0. Then, there exists t(s) ≥ 0 such that, for any t,

t ≥ t(s) =⇒ βORh = βOR

l .

That is, when the level of status-seeking is high enough against the levelof inequity aversion, the principal should give the same incentive salary totwo agents.



Figure5-1.jpg Figure5-2.jpg




When the levels of both s and t are low enough, βORh 6= βOR

l .

We consider the difference βORh − βOR

l .

When s = 0 and 0 ≤ t < dldl+dh

,

βORh − βOR

l =1

2− dl + dh

2dlt.

Also, when 0 < s < dl−dh2(dl+dh)

and 0 ≤ t < s2(1+s) ,

βORh − βOR

l =1 + 2t

2(1 + 2s).



Proposition 5

When s = 0, the difference βORh − βOR

l is decreasing in t. On the other

hand, when 0 < s < dl−dh2(dl+dh)

, the difference is increasing in t anddecreasing in s.



Proposition 6

Low-type agents’ effort level is strictly increasing in the level ofstatus-seeking. High-type agents’ effort level is also strictly increasingwhen s ≥ dl−dh

2(dl+dh). However, when 0 < s < dl−dh

2(dl+dh), their effort level is

not increasing.



Figure5-3.jpg Figure5-4.jpg




Comparing (eOR1 , eOR

2 ) with (eFB1 , eFB2 ), we can say that their effortlevels accord with first best levels if and only if t = 1.

As we mentioned in section 3.2, considering inequity averse agents,they cannot execute first best effort levels.

However, considering other-regarding agents, they might be able todo.



Proposition 7

Agents execute first best effort levels if and only if t = 1.


4. Optimal Team Formation

H = {1, 3} : High-type team

L = {2, 4} : Low-type team

xH = e1 + e3, xL = e2 + e4 : Output of the teams

w1 = α + βhxH , w2 = α + βlxL, w3 = α + βhxH , w4 = α + βlxL

Assumption 3 (Limited liability constraints)

α ≥ 0 and βh, βl ≥ 0.



Accordingly, objective functions of the all agents can be written as

u1(e) = α + (βh + t)(e1 + e3)− dhe21

2− t(e2 + e4),

u2(e) = α + (βl + t)(e2 + e4)− dle22

2− t(e1 + e3),

u3(e) = α + (βh + t)(e1 + e3)− dhe23

2− t(e2 + e4),

u4(e) = α + (βl + t)(e2 + e4)− dle24

2− t(e1 + e3).



Therefore, their incentive constraints are

e IC1 (βh, βl) = e IC3 (βh, βl) =βh + t

dh,

e IC2 (βh, βl) = e IC4 (βh, βl) =βl + t

dl.



The maximization problem is

max{α,βh,βl}

xH + xL − 2(α + βhxH)− 2(α + βlxL), (4)

s.t. u1(e) ≥ 0, u2(e) ≥ 0, u3(e) ≥ 0, u4(e) ≥ 0,

e1 = e IC1 , e2 = e IC2 , e3 = e IC3 , e4 = e IC4 ,

α ≥ 0, βh ≥ 0, βl ≥ 0.



s ≥ 0, t ≥ 0.

The maximization problem is

max{α,βh,βl}

(1− 2βh)(βh + t)

dh+

(1− 2βl)(βl + t)

dl− 4α,

s.t. α +βh(3βh + 2t)

2dh+ 2t

(βh + t

dh− βl + t

dl

)≥ t2

2dh,

α +βl(3βl + 2t)

2dl≥ t

2

(4(βh + t)

dh− 4βl + 3t

dl

),

α ≥ 0, βh ≥ 0, βl ≥ 0.



The solution is

(β∗h , β∗l ) =

{(1−2t4 , 1−2t4

)(0 ≤ t ≤ 1

2)

(0, 0) (otherwise).

Then,

(e∗1 , e∗2) =

(1+2t4dh

, 1+2t4dl

)(0 ≤ t ≤ 1

2)(tdh, tdl

)(otherwise).



Proposition 8

The principal can get more profit when she builds teams of high-low teamsthan when she builds teams of high-high team and low-low team.


References

Bartling, Bjorn, and Ferdinand A. von Siemens. “Efficiency in teamproduction with inequity averse agents.” University of Munich (2004).

Bartling, Bjorn, and Ferdinand A. von Siemens. “The intensity ofincentives in firms and markets: Moral hazard with envious agents.”Labour Economics 17.3 (2010): 598-607.

Bolton, Patrick, and Christopher Harris. “Strategic experimentation.”Econometrica 67.2 (1999): 349-374.

Bornstein, Gary, Tamar Kugler, and Shmuel Zamir. “One team mustwin, the other need only not lose: an experimental study of anasymmetric participation game.” Journal of Behavioral DecisionMaking 18.2 (2005): 111-123.


References

Englmaier, Florian, and Achim Wambach. “Optimal incentivecontracts under inequity aversion.” Games and Economic Behavior69.2 (2010): 312-328.

Fehr, Ernst, and Klaus M. Schmidt. “A theory of fairness,competition, and cooperation.” Quarterly journal of Economics(1999): 817-868.

Frank, Robert H. “Choosing the right pond: Human behavior and thequest for status.” Oxford University Press (1985).

Holmstrom, Bengt. “Moral hazard in teams.” The Bell Journal ofEconomics (1982): 324-340.


References

Itoh, Hideshi. “Moral hazard and other-regarding preferences.”Japanese Economic Review 55.1 (2004): 18-45.

Kahneman, Daniel, Jack L. Knetsch, and Richard H. Thaler.“Fairness and the assumptions of economics.” Journal of business(1986): S285-S300.

Kandel, Eugene, and Edward P. Lazear. “Peer pressure andpartnerships.” Journal of political Economy (1992): 801-817.

Kugler, Tamar, Amnon Rapoport, and Asya Pazy. “Public goodprovision in inter-team conflicts: Effects of asymmetry andprofit-sharing rule.” Journal of Behavioral Decision Making 23.4(2010): 421-438.


References

Marino, Anthony M., and Jan Zabojnik. “Internal competition forcorporate resources and incentives in teams.” RAND Journal ofEconomics (2004): 710-727.

Mookherjee, Dilip. “Optimal incentive schemes with many agents.”The Review of Economic Studies 51.3 (1984): 433-446.

Neilson, William S., and Jill Stowe. “Incentive pay for other-regardingworkers.” Working Paper Fuqua School of Business at DukeUniversity, (2004).

Rasmusen, Eric. “Moral hazard in risk-averse teams.” The RANDJournal of Economics (1987): 428-435.


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