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Rational Behavior: Theory and Application to Pay and Incentive Systems

Semyon M. Meerkov

Department of Electrical Engineering and Computer Science

The University of Michigan

Ann Arbor, MI 48109-2122

Fifth International Conference on Analysis of

Manufacturing Systems – Production Management

Zakynthos Island, Greece

May 21, 2005

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98% of personnel in the US automotive industry are paid-for-time

A form of pay-for-performance (annual company-wide bonus) does not seem to be effective

Question: How should a pay and incentive system be designed so that company performance is optimized?

MOTIVATION

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To formalize this question, assume that the behavior of a worker can be classified into two states, X1 and X2

X1 X2

Strictly observe standard operating procedures (i.e., optimize performance for the company)

Optimize other criteria (e.g., personal preferences, peer pressure, etc.)

Decision space, X

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How to create conditions under which the worker operates in X1 through his/her own self-interest? In other words, how to introduce an “INVISIBLE HAND” in manufacturing?

This talk is intended to provide a partial answer to these questions

Specifically, it provides a formal model, which can be used for analysis, design, and continuous improvement of pay and incentive systems using quantitative techniques

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This is accomplished by axiomatically introducing the notion of individual rational behavior and then creating collectives or groups of rational decision makers

This is followed by analysis of properties of individual and group behavior of rational elements

Finally, an asymptotic analysis (with respect to the size of the group and the level of individual’s rationality) leads to the conclusions on the efficacy of various pay and incentive systems

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OUTLINE

1. Modeling and analysis of individual rational behavior

2. Modeling and analysis of group rational behavior

3. Application to a pay and incentive system

4. Research program

5. Conclusions

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1. MODELING AND ANALYSIS OF INDIVIDUAL RATIONAL BEHAVIOR

1.1 Rational Behavior Behavior – a sequence of decisions in time, i.e., a

dynamical system in the decision space X:

}. ,2 ,1{,,0)(

),,,( 00),(

NXxx

ttxx Nx

X x0, t0

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Rational behavior – the behavior , which satisfies the following axioms:

),,( 00),( ttxx Nx

Bttxx

tBXBtx

Nx

)',,(

such that ' ,0 , ,,

00),(

00

Ergodicity:

B

t’

x0, t0

X

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Rationality:

).()( if 1

,0 ,, and , ,,

21

21212100

2

1 xxT

T

BBBXBBXxxtx

B

B

i

x0, t0B1

B2. .x2

x1

. as Moreover,2

1 NT

T

B

B

(x) → penalty function at decision x N → measure of rationality

X

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1.2 Examples of Rational Behavior1.2.1 Natural systems

Bees in foraging behavior

……..5% 10% 70%

xxx x

xx

xx x

x x xx

X X

Mice in feeding behavior

Dog in the circle experiment

X1 X2

Workers in production (Safelite Glass, Lincoln Electric)

X1 X2

X

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1.2.2 Mathematical systems

Ring element: )1,0[X

x*0 1 x

(x)

.xxx N 0)0( }),({

N

x

x

xT

xT

)(

)(

)(

)(

1

2

2

1

Ergodicity takes place Rationality:

).(inf arg ,)(1

lim]1,0[

**

0xxxtx

T x

T

NN

N

N

Additional property:

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Illustration:

x*0 1 x

(x)

x*

0 T3 t

x3(t)

1

x20(t)

x*

0 T20 t

1

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A search algorithm

dtx

dxN

R

R

xx

X

,1)(

x

(x)

R xCexp xN

,)( )(

N

xx NN

exp

xp )()(

2

1 12

)(

)(

dw

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2. MODELING AND ANALYSIS OF GROUP RATIONAL BEHAVIOR

2.1 Groups of Rational Individuals

Group – a set of M > 1 rational individuals interacting through their penalty functions:

Group state space:

Sequential algorithm of interaction:

Mixxx Miii ,,1 ),,,,,( 1

MXXXXx 21

Mixxx Miii ,,1 ),const,var,,,const( 1

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2.2 Homogeneous Fractional Interaction

M individuals, Xi=X, ∀i

Assume that at t0:

X1 X2X

.)(

)(

,in )(

,in )(

00

20

10

M

tmt

XtmM

Xtm

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Homogeneous Fractional Interaction – an interaction defined by the group penalty function:

]1 ,0[ ,0)( f

f()

1(t0) *

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Group penalty function defines the penalty function of each individual as follows: For

For

,)( 10 Xtxi

,)( 20 Xtxi

. if),1

(

, if),(

2

1

XxM

f

Xxf

i

i

i

. if),(

, if),1

(

2

1

Xxf

XxM

f

i

ii

f()

1

M

1

M

1

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Interpretation Beehive food distribution

)(inf arg]1,0[

* vfvv

Desirable state

?)( *vtv Question:

Corporation-wide bonuses

Uniform wealth distribution

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2.3 Inhomogeneous Fractional Interaction

M individuals, Xi=X, ∀i

Inhomogeneous Fractional Interaction – an interaction defined by two subgroup penalty functions

X1 X2X

]1,0[ ,0)( ,0)( 21 vvfvf

1(t0) **

f2()

f1()

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Penalty for each individual are defined as follows: For

For

,)( 10 Xtxi

,)( 20 Xtxi

. if),1

(

, if),(

22

11

XxM

f

Xxf

i

i

i

. if)(

, if)1

(

22

11

Xxf

XxM

f

i

ii

1

f2

f1

M

1

M

1

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Interpretation Differentiated corporate bonuses system Non-even wealth distribution

Desirable state: Nash equilibrium

Question:

)]()([ arg 21** ff

?)( **vtv

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2.4 Properties of Group Behavior under Homogeneous Fractional Interaction

Theorem: Under the homogeneous fractional interaction, the following effect of “critical mass” takes place: C > 0, such that

.0lim if 5.0)(

lim if )(

,

,

*

M

Nt

CM

Nt

MN

p

MN

p

= 0.5 implies the state of maximum entropy – the group behaves like a statistical mechanical gas (no rationality)

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Empirical observations

Beehive → when M becomes large, the family splits

Abnormal behavior of unusually large groups of animals (locust, deers, etc.)

Pay-for-group-performance (cooperate-wide bonuses, BP – Prudhoe Bay vs. Anchorage)

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2.5 Properties of Group Behavior under Inhomogeneous Fractional Interactions

Assume unique ** such that f1(**) = f2(**) and

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),()(

),()(**

1**

2

**1

**2

ff

ff

Theorem: Under the inhomogeneous fractional interaction, no effect of “critical mass” takes place: N* such that ∀N N*

. for )( ** Mtp

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3. APPLICATION TO A PAY AND INCENTIVE SYSTEM (Joint work with Leeann Fu)

3.1 Scenario Road 2

Road 1

factor.condition road ]1,0(

road, on the vehiclesofnumber

capacity, road

,congestion road of level the

,2 ,1 ),1

1( timetravel 0

K

n

c

c

nx

ix

xKtt

i

i

i

ii

i

ii i

Penalty function:

A

B

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Problem 1: Assuming that each driver exhibits rational behavior, investigate the distribution of the vehicles between Road 1 and Road 2

Problem 2: Assuming that the drivers are rational and given a fixed amount of goods to be transported from A to B, analyze the total time necessary to transport the goods under different pay systems: Pay-for-individual-performance Pay-for-group-performance Pay-for-time

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3.2 Parameters Selected

M = 6 N = var Road systems

System 1:

System 2:

8 ,0.1 ,4.2

9 ,7.0 ,05.2

2

1

0

0

cKt

cKt

9 ,95.0 ,3

10 ,68.0 ,1.2

2

1

0

0

cKt

cKt

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3.3 Problem 1

Penalty functions System 1: System 2:

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Results System 1:

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System 2:

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3.4 Problem 2

Penalty functions System 1:

User eq. = System eq.:

* = **

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System 2:

User eq. ≠ System eq.:

* ≠ **

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Results System 1

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System 2

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3.5 Comparisons

For system 1

For system 2

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3.6 Discussion

User equilibrium = system equilibrium (* = **): pay-for-individual-performance is the best

User equilibrium ≠ system equilibrium (* ≠ **): pay-for-group-performance maybe the best (if M is sufficiently small and N is sufficiently large)

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4. RESEARCH PROGRAM4.1 Applications

Methods for performance measurement in manufacturing environment (Deming and co.)

Evaluation of N for a typical worker

Design various pay for performance systems, based on measurements available Pay-for-individual-performance Pay-for-group-performance

Evaluate the upper bound on M, so that pay-for-group-performance is effective

Experiments Implementation

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4.2 Theory

Learning in the framework of rational behavior Modeling of experience-based learning Analysis of rational behavior with learning

Groups of individuals with different levels of rationality

Group behavior under rules of interaction other than fractional

Purely probabilistic approach to rational behavior (classes of functions, which characterize rational “deciders”)

General theory of rational deciders

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5. CONCLUSION

Manufacturing Systems = machines + people

Machines have been well investigated during the last 50 years

“Control” of people in manufacturing environment is a less studied topic (quantitatively)

The ideas described present an approach to accomplish this