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
1
MOTIVATION

98% of personnel in the US automotive industry are
paid-for-time

A form of pay-for-performance (annual companywide bonus) does not seem to be effective

Question: How should a pay and incentive system
be designed so that company performance is
optimized?
2

To formalize this question, assume that the
behavior of a worker can be classified into two
states, X1 and X2
Decision space, X
X1
X2
Strictly observe
standard operating
procedures (i.e.,
optimize
performance for
the company)
Optimize other
criteria (e.g.,
personal
preferences, peer
pressure, etc.)
3

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
4



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
5
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
6
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:
x ( x ), N ( x0 , t0 , t ),
 ( x)  0, x  X , N {1, 2, }.
X x0, t0
7

Rational behavior – the behavior x ( x ), N ( x0 , t0 , t ),
which satisfies the following axioms:

Ergodicity:
x0 , t0 , B  X , B  0, t ' such that
x ( x ), N ( x0 , t0 , t ' )  B
X
x0, t0
B
t’
8

Rationality:
x0 , t0 , x1 , x2  X and B1 , B2  X , Bi  0, B1  B2  
TB1
TB2
 1 if  ( x1 )   ( x2 ).
x0, t0
X
B2
x1
B1
.
.x
2
Moreover,
TB1
TB2
  as N  .

(x) → penalty function at decision x

N → measure of rationality
9
1.2 Examples of Rational Behavior
1.2.1 Natural systems

Bees in foraging behavior
……..
X
5%

X

10%
Mice in feeding behavior
x
x
x
x x
x
x
x
x
x
X
x x
x
Dog in the circle experiment
X1

70%
X2
Workers in production (Safelite Glass, Lincoln Electric)
X1 X2
10
1.2.2 Mathematical systems

(x)
Ring element: X  [0,1)
x   N ({x}), x(0)  0.


Ergodicity takes place
Rationality:
  ( x2 ) 
T ( x1 )


T ( x2 )

(
x
)
1 


N
x*
0
1
x
Additional property:
1
lim
N  T
N

TN
0
xN (t )  x* , x*  arg inf  ( x).
x[ 0 ,1]
11

Illustration:
x3(t)
1
(x)
x*
0
x*
1 x
0
T3 t
x20(t)
1
x*
0
T20 t
12

(x)
A search algorithm
X R
 ( x)  1, x  R
 N
dx  
dt  dw
x
x
 N ( x )
p( x)  Ce
, x  R
N 
p ( x1 )
 N ( x2 )  N ( x1 )
e

p ( x2 )
13
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:
i  i ( x1 ,, xi ,, xM ), i  1,, M

Group state space:
x  X  X1  X 2  X M

Sequential algorithm of interaction:
i  i ( x1  const ,, xi  var, , xM  const ), i  1,, M
14
2.2 Homogeneous Fractional Interaction

M individuals, Xi=X, ∀i
X

X1
X2
Assume that at t0:
m(t0 ) in X 1 ,
M  m(t0 ) in X 2 ,
m (t 0 )
 (t 0 ) 
.
M
15

Homogeneous Fractional Interaction – an
interaction defined by the group penalty function:
f ( )  0,   [0, 1]
f()
(t0)
*
1

16

Group penalty function defines the penalty function
of each individual as follows:
 For xi (t 0 )  X 1 ,
f()
if xi  X 1 ,
 f ( ),

i  
1
f
(


), if xi  X 2 .

M


For
xi (t0 )  X 2 ,
1

 f (  ), if xi  X 1 ,
i  
M
 f ( ),
if xi  X 2 .

1
M


1
M
1

17


Interpretation

Beehive food distribution

Corporation-wide bonuses

Uniform wealth distribution
Desirable state
v*  arg inf f (v)
v[ 0,1]

Question:
v(t )  v* ?
18
2.3 Inhomogeneous Fractional Interaction

M individuals, Xi=X, ∀i
X1 X2

X
Inhomogeneous Fractional Interaction – an interaction
defined by two subgroup penalty functions
f1 (v)  0, f 2 (v)  0, v [0,1]
f2()
(t0)
f1()
**
1

19

Penalty for each individual are defined as follows:
xi (t0 )  X 1 ,
 For
if xi  X 1 ,
 f1 ( ),

i  
1
f
(


), if xi  X 2 .
2

M


f2
f1
For xi (t0 )  X 2 ,
1

 f1 (  ) if xi  X 1 ,
i  
M
 f 2 ( )
if xi  X 2 .
1

M

1

M
1

20

Interpretation



Differentiated corporate bonuses system
Non-even wealth distribution
Desirable state: Nash equilibrium
 **  arg [ f1 ( )  f 2 ( )]

Question:
v(t )  v** ?
21
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
p
N
 (t )  if lim
C
N , M  M
p
N
 (t )  0.5 if lim
 0.
N , M  M

*
 = 0.5 implies the state of maximum entropy – the
group behaves like a statistical mechanical gas (no
rationality)
22

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)
23
2.5 Properties of Group Behavior under
Inhomogeneous Fractional Interactions

Assume  unique ** such that f1(**) = f2(**) and
f 2 ( **  )  f1 ( **  ),
f 2 (  )  f1 (  ),
**
**
0    1.

Theorem: Under the inhomogeneous fractional
interaction, no effect of “critical mass” takes place:
 N* such that ∀N  N*
p
 (t )  ** for  M .
24
3. APPLICATION TO A PAY AND INCENTIVE
SYSTEM (Joint work with Leeann Fu)
3.1 Scenario
Road 2
B
A
Road 1
xi
Penalty function: travel time ti  t0i (1  K
), i  1, 2,
1  xi
ni
xi 
 the level of road congestion ,
ci
ci  road capacity,
ni  number of vehicles on the road,
K  (0,1]  road condition factor.
25

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
26
3.2 Parameters Selected



M=6
N = var
Road systems
 System 1:
t01  2.05, K  0.7, c  9
t02  2.4, K  1.0, c  8

System 2:
t01  2.1, K  0.68, c  10
t02  3, K  0.95, c  9
27
3.3 Problem 1

Penalty functions

System 1:

System 2:
28

Results

System 1:
29

System 2:
30
3.4 Problem 2

Penalty functions

System 1:
User eq. = System eq.:
* = **
31

System 2:
User eq. ≠ System eq.:
* ≠ **
32

Results

System 1
33

System 2
34
3.5 Comparisons

For system 1

For system 2
35
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)
36
4. RESEARCH PROGRAM
4.1 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-forgroup-performance is effective

Experiments
Implementation

37
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
38
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
39