A Mathematical Model of Dysfunctional Team Dynamics

1. Effort Dynamics in Supervised Workgroups
Arianna Dal Forno, Ugo Merlone
Department of Statistic and Applied Mathematics,
University of Torino
AISC 2008, Torino, 30 Novembre - 2 Dicembre 2008
Dal Forno & Merlone (University of Torino) Effort Dynamics in Supervised Workgroups AISC 2008 1 / 39

2. motivation and Literature
Motivation
complexity theory and chaos theory quite often are consider a tool
to investigate group dynamics
words such as attractor, chaos, bifurcation and nonlinearity may
be intriguing when considering group dynamics
we propose a model of interaction dynamics in a small group
where members’ decision are the result of elaboration processes
in which own actions are compared to the existing conditions and
are adapted in order to pursue goals
Dal Forno & Merlone (University of Torino) Effort Dynamics in Supervised Workgroups AISC 2008 2 / 39

3. motivation and Literature
Motivation
complexity theory and chaos theory quite often are consider a tool
to investigate group dynamics
words such as attractor, chaos, bifurcation and nonlinearity may
be intriguing when considering group dynamics
we propose a model of interaction dynamics in a small group
where members’ decision are the result of elaboration processes
in which own actions are compared to the existing conditions and
are adapted in order to pursue goals
Dal Forno & Merlone (University of Torino) Effort Dynamics in Supervised Workgroups AISC 2008 2 / 39

4. motivation and Literature
Motivation
complexity theory and chaos theory quite often are consider a tool
to investigate group dynamics
words such as attractor, chaos, bifurcation and nonlinearity may
be intriguing when considering group dynamics
we propose a model of interaction dynamics in a small group
where members’ decision are the result of elaboration processes
in which own actions are compared to the existing conditions and
are adapted in order to pursue goals
Dal Forno & Merlone (University of Torino) Effort Dynamics in Supervised Workgroups AISC 2008 2 / 39

5. motivation and Literature
Literature
Holmstrom (1982), Moral hazard in teams. BJE.
Adams (1965), Inequity in social exchange. AESP.
Dal Forno & Merlone (2007), Incentives in supervised teams: an
experimental and computational approach. JSC.
Dal Forno & Merlone (2009), Individual incentive in workgroups.
From human subject experiments to agent-based simulation
IJIEM.
Dal Forno & Merlone (2008), Individual-based versus group-based
incentives in supervised workgroups. The role of individual
motivation. Submitted.
Dal Forno & Merlone (University of Torino) Effort Dynamics in Supervised Workgroups AISC 2008 3 / 39

6. The Model
The Model
The Supervised Workgroup
a supervisor
two subordinates
Dal Forno & Merlone (University of Torino) Effort Dynamics in Supervised Workgroups AISC 2008 4 / 39

7. The Model
The Model
The Supervised Workgroup
a supervisor
two subordinates
ui effort with the supervisor
li effort with the partner
Dal Forno & Merlone (University of Torino) Effort Dynamics in Supervised Workgroups AISC 2008 5 / 39

8. The Model
The Model
The Supervised Workgroup
a supervisor
two subordinates
ui effort with the supervisor
li effort with the partner
ui + li ≤ ci
Dal Forno & Merlone (University of Torino) Effort Dynamics in Supervised Workgroups AISC 2008 6 / 39

9. The Model
The Model
The production function
The production function is
(u1 + u2 )α (l1 + l2 )β
where
α: output elasticity with respect to the joint effort with the
supervisor
β: output elasticity with respect to the joint effort with the partner
0 ≤ α ≤ 1 and β = 1 − α
We assume that the production is sold at unitary price
Dal Forno & Merlone (University of Torino) Effort Dynamics in Supervised Workgroups AISC 2008 7 / 39

10. The Model
The supervisor’s problem
Agents’ compensation is:
wi = s + bi ui + bt (u1 + u2 )α (l1 + l2 )β
where:
s is a base salary sufficient to meet the participation constraint of
the agent
bi is the incentive given to subordinate i for its individual effort with
supervisor
bt is the incentive given both for team output.
We assume that:
the supervisor declares the bonuses
the subordinates decide their efforts in order to maximize their
wage.
Dal Forno & Merlone (University of Torino) Effort Dynamics in Supervised Workgroups AISC 2008 8 / 39

11. The Model
The Model
The Supervisor’s Problem
The supervisor can only observe ui :
She must design a linear compensation scheme (bt , b1 , b2 ) to
maximize net production (bilevel programming problem)
max (1 − 2bt ) (u1 + u2 )α (l1 + l2 )β − b1 u1 − b2 u2
bt ,b1 ,b2
s.t. given bt , b1 , b2 the subordinates solve:
w + bt (u1 + u2 )α (l1 + l2 )β + b1 u1
max
u1 ,l1
w + bt (u1 + u2 )α (l1 + l2 )β + b2 u2
max
u2 ,l2
Dal Forno & Merlone (University of Torino) Effort Dynamics in Supervised Workgroups AISC 2008 9 / 39

12. The Model
The Agent’s Problem
Assume agents maximize the gross production
(u1 + u2 )α (l1 + l2 )β ui + li ≤ ci ,
max i = 1, 2
sub
u1 , u2 , l1 , l2
There is a continuum of solutions
α
u1 + u2 = (c1 + c2 )
α+β
β
l1 + l2 (c1 + c2 )
= α+β
a rather natural effort allocation is
α β
i = 1, 2
(ui , li ) = ( ci , ci ),
α+β α+β
which is focal in the sense of Schelling (1960)
Dal Forno & Merlone (University of Torino) Effort Dynamics in Supervised Workgroups AISC 2008 10 / 39

13. The Model
The Agent’s Problem
Assume agents maximize the gross production
(u1 + u2 )α (l1 + l2 )β ui + li ≤ ci ,
max i = 1, 2
sub
u1 , u2 , l1 , l2
There is a continuum of solutions
α
u1 + u2 = (c1 + c2 )
α+β
β
l1 + l2 (c1 + c2 )
= α+β
a rather natural effort allocation is
α β
i = 1, 2
(ui , li ) = ( ci , ci ),
α+β α+β
which is focal in the sense of Schelling (1960)
Dal Forno & Merlone (University of Torino) Effort Dynamics in Supervised Workgroups AISC 2008 10 / 39

14. The Model
The Agent’s Problem
Assume agents maximize the gross production
(u1 + u2 )α (l1 + l2 )β ui + li ≤ ci ,
max i = 1, 2
sub
u1 , u2 , l1 , l2
There is a continuum of solutions
α
u1 + u2 = (c1 + c2 )
α+β
β
l1 + l2 (c1 + c2 )
= α+β
a rather natural effort allocation is
α β
i = 1, 2
(ui , li ) = ( ci , ci ),
α+β α+β
which is focal in the sense of Schelling (1960)
Dal Forno & Merlone (University of Torino) Effort Dynamics in Supervised Workgroups AISC 2008 10 / 39

15. The Model The Supervisor’s Problem
The Supervisor’s Problem
With fully rational agents the solution is obvious

 bt = ε > 0
b =0
1
b2 = 0
Dal Forno & Merlone (University of Torino) Effort Dynamics in Supervised Workgroups AISC 2008 11 / 39

16. The Effort Dynamics
The Effort Dynamics
Formalization
The simpler dynamics: the rational case ⇒ focal equilibrium

βc1
∗
 l1 = α+β

βc2
∗
l =
2 α+β
This equilibrium cannot hold in the long run when the subordinates
have different capacities: individuals with different capacity but same
reward may experience inequity (Adams, 1965):
“ Inequity exists for Person whenever he perceives that the ratio of his
outcomes to the inputs and the ratio of Other’s outcomes to Other’s
input are unequal. ”
OP Oa
=
IP Ia
Dal Forno & Merlone (University of Torino) Effort Dynamics in Supervised Workgroups AISC 2008 12 / 39

17. The Effort Dynamics
The Effort Dynamics
Formalization
The simpler dynamics: the rational case ⇒ focal equilibrium

βc1
∗
 l1 = α+β

βc2
∗
l =
2 α+β
This equilibrium cannot hold in the long run when the subordinates
have different capacities: individuals with different capacity but same
reward may experience inequity (Adams, 1965):
“ Inequity exists for Person whenever he perceives that the ratio of his
outcomes to the inputs and the ratio of Other’s outcomes to Other’s
input are unequal. ”
OP Oa
=
IP Ia
Dal Forno & Merlone (University of Torino) Effort Dynamics in Supervised Workgroups AISC 2008 12 / 39

18. The Effort Dynamics
The Effort Dynamics
Formalization
The simpler dynamics: the rational case ⇒ focal equilibrium

βc1
∗
 l1 = α+β

βc2
∗
l =
2 α+β
This equilibrium cannot hold in the long run when the subordinates
have different capacities: individuals with different capacity but same
reward may experience inequity (Adams, 1965):
“ Inequity exists for Person whenever he perceives that the ratio of his
outcomes to the inputs and the ratio of Other’s outcomes to Other’s
input are unequal. ”
OP Oa
=
IP Ia
Dal Forno & Merlone (University of Torino) Effort Dynamics in Supervised Workgroups AISC 2008 12 / 39

19. The Effort Dynamics
The Effort Dynamics
Formalization
The simpler dynamics: the rational case ⇒ focal equilibrium

βc1
∗
 l1 = α+β

βc2
∗
l =
2 α+β
This equilibrium cannot hold in the long run when the subordinates
have different capacities: individuals with different capacity but same
reward may experience inequity (Adams, 1965):
“ Inequity exists for Person whenever he perceives that the ratio of his
outcomes to the inputs and the ratio of Other’s outcomes to Other’s
input are unequal. ”
OP Oa
=
IP Ia
Dal Forno & Merlone (University of Torino) Effort Dynamics in Supervised Workgroups AISC 2008 12 / 39

20. The Effort Dynamics
The Effort Dynamics
Formalization
Two-dimensional map T : R2 → R2 given by

 l1 = r1 (l2 )
T (l1 , l2 ) :
l2 = r2 (l1 )

Reaction functions: r1 : L2 → L1 and r2 : L1 → L2
Strategy sets: L1 = [0, c1 ] ⊆ R and L2 = [0, c2 ] ⊆ R
00
Trajectory: given an initial condition l1 , l2 ∈ L1 × L2
l1 , l2 = T t l1 , l2
tt 00
∀t ≥ 0,
Dal Forno & Merlone (University of Torino) Effort Dynamics in Supervised Workgroups AISC 2008 13 / 39

21. The Effort Dynamics
The Effort Dynamics
Formalization
Dal Forno & Merlone (University of Torino) Effort Dynamics in Supervised Workgroups AISC 2008 14 / 39

22. The Effort Dynamics
The Effort Dynamics
Formalization
Dal Forno & Merlone (University of Torino) Effort Dynamics in Supervised Workgroups AISC 2008 15 / 39

23. The Effort Dynamics
The Effort Dynamics
Formalization
Dal Forno & Merlone (University of Torino) Effort Dynamics in Supervised Workgroups AISC 2008 16 / 39

24. The Effort Dynamics
The Effort Dynamics
Formalization
Family of feasible functions: Conditions on parameters:

lt
l2 k1 −1 − θ2 ki −1
t
βci e
 t+1
λi =
l e1
 1 = λ1 θ1

ki −1
α+β

lt βci
k2 −1

θi =
t − θ1

l1
 t+1
(α+β)(ki −1)
l e
= λ2
 2
2 θ2
Reaction functions:
(α+β)l t
„ «

k1 −1 (k1 −1) 1− βc 2
t
(α+β)l2
βc1
 t+1
l e 1
1 =

α+β βc1

(α+β)l t
„ «

k2 −1 (k2 −1) 1− βc 1
 t
(α+β)l1

βc2
 t+1
l e 2
=
2 α+β βc2
Dal Forno & Merlone (University of Torino) Effort Dynamics in Supervised Workgroups AISC 2008 17 / 39

25. The Effort Dynamics
The Effort Dynamics
Formalization
Family of feasible functions: Conditions on parameters:

lt
l2 k1 −1 − θ2 ki −1
t
βci e
 t+1
λi =
l e1
 1 = λ1 θ1

ki −1
α+β

lt βci
k2 −1

θi =
t − θ1

l1
 t+1
(α+β)(ki −1)
l e
= λ2
 2
2 θ2
Reaction functions:
(α+β)l t
„ «

k1 −1 (k1 −1) 1− βc 2
t
(α+β)l2
βc1
 t+1
l e 1
1 =

α+β βc1

(α+β)l t
„ «

k2 −1 (k2 −1) 1− βc 1
 t
(α+β)l1

βc2
 t+1
l e 2
=
2 α+β βc2
Dal Forno & Merlone (University of Torino) Effort Dynamics in Supervised Workgroups AISC 2008 17 / 39

26. The Effort Dynamics
The Effort Dynamics
Formalization
Family of feasible functions: Conditions on parameters:

lt
l2 k1 −1 − θ2 ki −1
t
βci e
 t+1
λi =
l e1
 1 = λ1 θ1

ki −1
α+β

lt βci
k2 −1

θi =
t − θ1

l1
 t+1
(α+β)(ki −1)
l e
= λ2
 2
2 θ2
Reaction functions:
(α+β)l t
„ «

k1 −1 (k1 −1) 1− βc 2
t
(α+β)l2
βc1
 t+1
l e 1
1 =

α+β βc1

(α+β)l t
„ «

k2 −1 (k2 −1) 1− βc 1
 t
(α+β)l1

βc2
 t+1
l e 2
=
2 α+β βc2
Dal Forno & Merlone (University of Torino) Effort Dynamics in Supervised Workgroups AISC 2008 17 / 39

27. The Effort Dynamics
The Effort Dynamics
Formalization
Family of feasible functions: Conditions on parameters:

lt
l2 k1 −1 − θ2 ki −1
t
βci e
 t+1
λi =
l e1
 1 = λ1 θ1

ki −1
α+β

lt βci
k2 −1

θi =
t − θ1

l1
 t+1
(α+β)(ki −1)
l e
= λ2
 2
2 θ2
Dal Forno & Merlone (University of Torino) Effort Dynamics in Supervised Workgroups AISC 2008 18 / 39

28. The Effort Dynamics
The Effort Dynamics
k1 = k2 = 1 → Tolerant agents

βc1
t+1
 l1 = α+β

βc2
 t+1
l =
2 α+β
Unique (stable) fixed point.
The production is maximized.
Dal Forno & Merlone (University of Torino) Effort Dynamics in Supervised Workgroups AISC 2008 19 / 39

29. The Effort Dynamics
The Effort Dynamics
k1 = 1 , k2 > 1 → Only one tolerant agent
 βc
t+1 1
l1 = α+β


0 1
t
!k −1 (k −1)@1− (α+β)l1 A
2
(α+β)l t 2 βc2
βc
t+1 2 1
 l2 e
= α+β
 βc2
Unique (stable) fixed point.
If c1 = c2 then the production
is maximized.
Dal Forno & Merlone (University of Torino) Effort Dynamics in Supervised Workgroups AISC 2008 20 / 39

30. The Effort Dynamics
The Effort Dynamics
Efficiency
Numerically we can observe
Proposition
Assume that one subordinate is tolerant and the other is not:
k1 = 1, k2 > 1.
then:
the efficiency is maximimum when their capacities are identical,
the higher loss of efficiency occurs when the intolerant agent is
the one with higher capacity.
Dal Forno & Merlone (University of Torino) Effort Dynamics in Supervised Workgroups AISC 2008 21 / 39

31. The Effort Dynamics
The Effort Dynamics
Efficiency
Numerically we can observe
Proposition
Assume that one subordinate is tolerant and the other is not:
k1 = 1, k2 > 1.
then:
the efficiency is maximimum when their capacities are identical,
the higher loss of efficiency occurs when the intolerant agent is
the one with higher capacity.
Dal Forno & Merlone (University of Torino) Effort Dynamics in Supervised Workgroups AISC 2008 21 / 39

32. The Effort Dynamics
The Effort Dynamics
k1 > 1 , k2 > 1 → No tolerant agents
 0 1
t
!k −1 (k −1)@1− (α+β)l2 A
1
(α+β)l t 1 βc1
 βc
t+1
 1 2
l1 e
= α+β
 βc1

0 1
t
!k −1 (k −1)@1− (α+β)l1 A

2
(α+β)l t 2
 βc2
βc
 t+1 2 1
l2 e
= α+β
 βc2
One, two, or three fixed points.
Dal Forno & Merlone (University of Torino) Effort Dynamics in Supervised Workgroups AISC 2008 22 / 39

33. The Effort Dynamics
The Effort Dynamics
Eigenvalues:
s
βc1 −l1 βc2 −l2
(k1 −1)+ (k2 −1) “ l1 ”k1 −2 “ l ”k −2
βc1 −l βc2 −l
2
βc1 βc2 (k1 − 1) βc 1 βc 2 (k2 − 1) βc 2
λ1 = − e βc 1 1 2 2
s
βc2 −l2 βc1 −l
(k2 −1)+ βc 1 (k1 −1) “ l2 ”k2 −2 “ l ”k −2
βc2 −l βc1 −l
1
e βc2 (k2 − 1) βc 2 βc 1 (k1 − 1) βc 1
1
λ2 = βc 2 2 1 1
Dal Forno & Merlone (University of Torino) Effort Dynamics in Supervised Workgroups AISC 2008 23 / 39

34. The Effort Dynamics
The Effort Dynamics
k1 = k2 = k → bifurcation diagram
“ ”
00
l1 , l2 = (0.7, 0.5)
Dal Forno & Merlone (University of Torino) Effort Dynamics in Supervised Workgroups AISC 2008 24 / 39

35. The Effort Dynamics
The Effort Dynamics
k1 , k2 > 1 → bifurcation diagram
,
k1 = k2
“ ” “ ”
00 00
l1 , l2 = (0.1, 0.1) l1 , l2 = (0.5, 0.5)
k2 = 7.5 c1 = 18 c2 = 6
, ,
Dal Forno & Merlone (University of Torino) Effort Dynamics in Supervised Workgroups AISC 2008 25 / 39

36. The Effort Dynamics
The Effort Dynamics
k1 , k2 > 1 → bifurcation diagram
,
k1 = k2
“ ” “ ”
00 00
l1 , l2 = (0.1, 0.1) l1 , l2 = (0.5, 0.5)
k2 = 7.5 c1 = 18 c2 = 6
, ,
Dal Forno & Merlone (University of Torino) Effort Dynamics in Supervised Workgroups AISC 2008 26 / 39

37. The Effort Dynamics
The Effort Dynamics
k1 , k2 > 1 → Cycles and Chaos
,
k1 = k2
k1 = 1.39 k1 = 1.7
“ ”
00
l1 , l2 = (1.12, 0.17) k2 = 7.5 c1 = 18 c2 = 6
, , ,
Dal Forno & Merlone (University of Torino) Effort Dynamics in Supervised Workgroups AISC 2008 27 / 39

38. The Effort Dynamics
The Effort Dynamics
k1 1.39
=
k2 7.5
=
c1 18
=
c2 6
=
(4.56, 0.04) (4.56, 0.17)
(1.12, 0.17) (1.81, 0.00)
(4.49, 1.70)
Dal Forno & Merlone (University of Torino) Effort Dynamics in Supervised Workgroups AISC 2008 28 / 39

39. The Effort Dynamics
The Effort Dynamics
Dal Forno & Merlone (University of Torino) Effort Dynamics in Supervised Workgroups AISC 2008 29 / 39

40. The Effort Dynamics
The Effort Dynamics
Other results: bifurcation on the capacity c1
(0.1, 0.5) (0.5, 0.5)
k1 = 1.39 k2 = 7.5 c2 = 6
, ,
Dal Forno & Merlone (University of Torino) Effort Dynamics in Supervised Workgroups AISC 2008 30 / 39

41. The Effort Dynamics
The Effort Dynamics
Other results: bifurcation on the capacity c1
(0.1, 0.5) (0.5, 0.5)
k1 = 1.39 k2 = 7.5 c2 = 6
, ,
Dal Forno & Merlone (University of Torino) Effort Dynamics in Supervised Workgroups AISC 2008 31 / 39

42. The Effort Dynamics
The Effort Dynamics
Other results: bifurcation on the capacity c2
(0.1, 0.5) (0.5, 0.5)
k1 = 1.39 k2 = 7.5 c1 = 18
, ,
Dal Forno & Merlone (University of Torino) Effort Dynamics in Supervised Workgroups AISC 2008 32 / 39

43. The Effort Dynamics
Global analysis
Coexistence of finite period attractors
(1.81, 0.00) (4.49, 1.70)
k1 = 1.39 k2 = 7.5 c1 = 18 c2 = 6
, , ,
Dal Forno & Merlone (University of Torino) Effort Dynamics in Supervised Workgroups AISC 2008 33 / 39

44. The Effort Dynamics
Global analysis
Coexistence of chaotic attractors
(0.1, 0.5) (0.7, 0.7)
k1 = 1.7 k2 = 7.5 c1 = 18 c2 = 6
, , ,
Dal Forno & Merlone (University of Torino) Effort Dynamics in Supervised Workgroups AISC 2008 34 / 39

45. The Effort Dynamics
Global analysis
Basin of attraction of the origin
Dal Forno & Merlone (University of Torino) Effort Dynamics in Supervised Workgroups AISC 2008 35 / 39

46. The Effort Dynamics
Global analysis
Basin of attraction of the origin
Dal Forno & Merlone (University of Torino) Effort Dynamics in Supervised Workgroups AISC 2008 36 / 39

47. The Effort Dynamics
Global analysis
Basin of attraction of the origin
Dal Forno & Merlone (University of Torino) Effort Dynamics in Supervised Workgroups AISC 2008 37 / 39

48. The Effort Dynamics
Results
c1 = c2
k1 = k2 = 1 → rational workgroup, efficiency
k1 = 1, k2 > 1 → efficiency
k1 , k2 > 1 → coexistence of attractors (with retaliation)
c1 = c2
k1 = k2 = 1 → rational workgroup, efficiency
k1 = 1, k2 > 1 → loss of efficiency, but no retaliation
k1 , k2 > 1 → coexistence of cycles (with retaliation), chaos,
expansion of the (non connected) basin of the origin
Dal Forno & Merlone (University of Torino) Effort Dynamics in Supervised Workgroups AISC 2008 38 / 39

49. The Effort Dynamics
Results
c1 = c2
k1 = k2 = 1 → rational workgroup, efficiency
k1 = 1, k2 > 1 → efficiency
k1 , k2 > 1 → coexistence of attractors (with retaliation)
c1 = c2
k1 = k2 = 1 → rational workgroup, efficiency
k1 = 1, k2 > 1 → loss of efficiency, but no retaliation
k1 , k2 > 1 → coexistence of cycles (with retaliation), chaos,
expansion of the (non connected) basin of the origin
Dal Forno & Merlone (University of Torino) Effort Dynamics in Supervised Workgroups AISC 2008 38 / 39

50. Conclusions
Conclusions
the Chaos Theory metaphor can be extended beyond the
assonance suggestion
in this case we provide not non linear dynamic system as
metaphor of the considered organizational interaction, but actually
provide a model of it
this way, the model properties can be interpreted as aspects of
the organizational process which is considered
Dal Forno & Merlone (University of Torino) Effort Dynamics in Supervised Workgroups AISC 2008 39 / 39

51. Conclusions
Conclusions
the Chaos Theory metaphor can be extended beyond the
assonance suggestion
in this case we provide not non linear dynamic system as
metaphor of the considered organizational interaction, but actually
provide a model of it
this way, the model properties can be interpreted as aspects of
the organizational process which is considered
Dal Forno & Merlone (University of Torino) Effort Dynamics in Supervised Workgroups AISC 2008 39 / 39

52. Conclusions
Conclusions
the Chaos Theory metaphor can be extended beyond the
assonance suggestion
in this case we provide not non linear dynamic system as
metaphor of the considered organizational interaction, but actually
provide a model of it
this way, the model properties can be interpreted as aspects of
the organizational process which is considered
Dal Forno & Merlone (University of Torino) Effort Dynamics in Supervised Workgroups AISC 2008 39 / 39