Incentives in hierarchical organizations  Balashov V.G., Burkov V.N., Novikov D.A
..pdfINCENTIVES IN HIERARCHICAL ORGANIZATIONS
V.G. Balashov, V.N. Burkov, D.A. Novikov, A. Yu. Zalojnev
(Institute of control sciences, Profsojuznaya st., 65, Moscow, 117997, Russia,phone: +(095)3349051, fax: +(095)3348911, E mail: vlab@ipu.rssi.ru)
Abstract: Gametheoretical models of the incentive mechanisms are considered for the hierarchical organization.
Keywords: game theory, decision making, organizational factors.
1. INTRODUCTION
The paper contains the consideration of management (particularly – incentive) problems in hierarchical organizations. The main method of exploration is the gametheoretical models analysis (Burkov and Enaleev, 1994; MasColell et all, 1995), which allows describing adequately the preferences of systems’ elements. Essential attention is paid to the model of multilevel organizational system (OS). Results of other models’ exploration are briefly discussed in the conclusion (see Novikov, 1999 for the details).
2. BASIC INCENTIVE PROBLEM IN MULTILEVEL OS
Consider threelevel OS, which consists of one metaprincipal (P) on the highest level of hierarchy, n principals (Pj) on the middle level, j = 1,n , and N agents (Aij) on the lowest level, i = 1,nî , nj – number of agents in jth subsystem (subordinated to jth principal), j = 1,n ,
n
ån j = N. The set of metaprincipal and n subordinated to him principals will be referred to as
j=1
the metasystem. Each agent Aij chooses his action yij Aij, receives the reward σij(yj) Mij
from jth principal and bears costs cij(yij), where yj = (y1j, y2j, ..., yn j j 
) Aj = Õn j 
Aij  a vector of 






i=1 

jth subsystem agents’ actions. Thus the goal function of Aij takes the form: 


fij(yj) = σij(yj) – cij(yij), i = 

, j= 

(1) 


1,n j 
1,n 



Pj is characterized by the income Hj(yj), bears incentive costs 
ån j σ ij( y j ) and receives the 






i=1 

reward σj(Yj) Mj from metaprincipal, where Yj = Qj(yj) Aj is the aggregated output of jth subsystem, Qj: Aj → Aj, i.e. his goal function is:
nj 










Φj(yj) = Hj(yj) – åσ ij ( y j) + σj(Yj), j= 

. 
(2) 






1,n 






i=1 










Metaprincipal 
is 
characterized 
by 
his 
income 
H(Y), 
where 

Y = (Y1, Y2, ..., Yn) A = Õ A j , and bears incentive costs 
ån 
σ j(Y j ) , i.e. his goal function is: 


j 




j=1 














n 










Φ(Y) = H(Y) – åσ j (Y j) . 
(3) 







j=1 










The sequence of 
OS 
functioning is as 
following: metaprincipal 
chooses the 
reward 
functions {σj(Yj)} and reveals it to principals, then each principal chooses incentive functions {σij(yj)} for his subordinates, then the latter choose their actions. Assume that all the elements of the OS (metaprincipal, principals and agents) choose their strategies independently (coalitions are not considered) to maximize corresponding goal functions. The presence of
aggregation (of subsystem outputs 
and principals’ preferences) is essential, so from meta 

principal’s point of view Pj’s goal function is: 

Φj(Yj) = hj(Yj) – cj(Yj) + σj(Yj), j= 

, 
(4) 
1,n 
hj(Yj): Aj → 1, cj(Yj): Aj → 1 are such that Y Aj:
yj Aj: Qj(yj)=Yj 
nj 
hj(Yj)=Hj(yj), cj(Yj)=åσ ij ( y j) . (5) 


i=1 
The difference between (2) and (4) is explained by the fact that the metaprincipal generally has aggregated conception about the models of subsystems, coordinated with their “detailed” models in the sense of (5).
Denote Pj({sij}) Í Aj – a set of Nash equilibriums for the agents of jth subsystem (the set of actions, implemented by the incentive functions {sij}):
Pj({sij}) = { yjÎAj  " i=1,n j " tijÎAij sij(yij, yij)cij(yij) ³ sij(tij, yij)cji(tij) }, (6)
yij = (y1j, y2j, ..., yi1j, yi+1j, ..., yn j j ) – is a situation for ith agent in jth subsystem. If the
hypothesis of independent behavior (HIB) is valid (there are no joint restrictions on the
nj collection of agents’ choices), then Pj({sij}) = ∏Pij (σ ij) ,
i=1
Pij(sij) = Arg max fij(yij). 
(7) 
yij Aij 

Denote Rj(sj) – the set of jth subsystem equilibriums in the framework of the metasystem:
Rj(sj) = {Yj Î Aj  " tj Î Aj hj(Yj) – cj(Yj) + sj(Yj) ³ hj(tj) – cj(tj) + sj(tj)},(8)
R({sj}) = ∏n Rj(sj) – the set of principals’ equilibriums.
j=1
In twolevel OS an incentive problem is formulated as following (MasColell et al, 1995; Novikov, 1998): find a feasible incentive function, which maximizes principal’s goal function on the set of agents’ equilibriums. When truing to generalize this problem to the multilevel OS, one meets some difficulties. Despite that the operator of aggregation Qj(.) is defined such that Aj = Qj(Aj), i.e. " yj Î Aj $ Yj Î Aj and " Yj Î Aj $ yj Î Aj: Yj = Qj(yj), restrictions M on the incentive functions may turn out to be such that for some j and/or for some Yj Î Rj(sj) there exists no incentive {sijÎMij}, which implement the required actions ($ yj Î Pj({sij}): Qj(yj) = Yj). In other words, choosing some incentive function, the metaprincipal could not be sure that the implemented output of subsystem (ones, which maximize goal function of the principal) may be attained by some implementable in the framework of the subsystem combination of agents actions.
Denote Pj = UP j ({σ ij}) , R = 
UR({σ j}) and introduce the following assumption (once 
σ ij M ij 
σ j M j 
introduced the assumption is considered to be valid hereafter), which guarantee the coordinatability of the model’s restrictions.

А1. Y R j = 




yj Pj 
: Yj = Qj(yj). 


1,n 


Under the assumption А.1 the incentive problem for the metasystem is: 



n 
→ 









H(Y*) – åσ j (Y * j) 
max 
, 

(9) 



j=1 



{σ j M j } 





Y*j Rj(σj), 




j= 

, 
(10) 





1,n 


Assumption А.1 guarantees that the aggregates, defined in (10) may be implemented by 

the principals as the results of the following incentive problems solving: 



nj 



σj(Y*j) → 



Hj( y*j ) – åσ ij ( y j *) + 
max , 
(11) 



i=1 









σ ij M ij 

y* 
P ({σ }), j = 

. 


(12) 


1,n 




j 
j 
ij 












Denote σ = {σj} and define the efficiency of management in metasystem (efficiency of 

incentive mechanism) as 





K(σ) = max Φ(Y). 






(13) 



Y R(σ ) 













Denote σj = {σij}, then the efficiency of management in jth subsystem is: 

Kj(σ j) = 
max 
Φj(yj). 


(14) 




y j Pj(σ j ) 











Note, that (13) and (14) imply that the hypothesis of benevolence (HB) is valid, i.e. principals and agents choose from the set of the corresponding goal function maximums the strategy, which is the most preferable by the upper level authority (in multilevel OS the HB makes the sense only under assumption А.1).
Introduce the following assumptions about goal functions and feasible sets.
А2. Aij = Aj = A = [0, +∞ ).
А3. cij(yij), cj(yj) – nondecreasing, bounded below functions.
А3'. А3, cij(yij), cj(yj) are continuos, monotone increasing and cij(0)=cj(0)=0. А3''. А3', cij(yij), cj(yj) are convex, continuos differentiable and c'ij(0) = c'j(0) = 0.
А4. Mij = Mj are the sets of nonnegative valued piecewise continuos functions.
А4'. Mij = {σij  yij Aij 0 ≤ σij(yij) ≤ Cij}; Mj = {σj  Yj Aj 0 ≤ σj(Yj) ≤ сj};
А4''. {Mij}: = {σij  y A åσij ( yij ) ≤ Сj };{Mij}: = {σj  Y A ån 
σ j(Y j ) ≤ c}. 

i 
j=1 
Constants {Cij}, {cj}, c are upper restrictions of the corresponding incentive functions. Lemma 1. If the assumptions А.2, А.3 and А.4 are valid, then the assumption А.1 is also
valid.
Define the twolevel analog of the considered multilevel OS. If: i) the number of principals equals to the number of agents (i.e. that one agent is subordinated to one and only one principal: n = N; such an OS may be called trivial); ii) the aggregation does not occur (Aj = Aj, Qj(yj) = yj,) and iii) principals are passive (they are not interested in the results of agents’ activity and play the role of the ideal information transmitters), then one obtain twolevel OS with the efficiency of management the same as in the original threelevel OS with
Hj(yj) ≡ 0, j = 1,n .
3. INCENTIVES IN MULTILEVEL OS
WITHOUT AGGREGATION OF INFORMATION
In this section the case of no aggregation is considered (the principal is assumed having full information). Results are presented inductively – from the simplest oneagent twolevel OS
to multiagent multilevel OS.
Consider OS, which consists of one principal and one agent (if n = 1 and/or N = 1, then indexes are omitted). Goal functions of the principal and the agent are: Φ(y) = H(y) – σ(y), f(y) = σ(y) – c(y). It is wellknown (Novikov, 1998; Burkov and Novikov, 2000), that under assumptions А.2 and А.3' minimal incentive costs of the principal to implement an action y* A equals c(y*). Hence, the efficiency of management is:
K0 
(C) = max [H(y) – c(y)], 
(15) 

y P(C ) 

P (C) = {y A  c(y) – min 
c(y) ≤ C}. (16) 


y A 

Introduce one intermediate principal with the following goal function
Φ1(y) = H1(y) + σ1(y) – σ(y). 
(17) 
Principal’s goal function will change to: F(y) = H(y) – s1(y). The set of implementable actions of the agent will remain the same (see (16), while the set of actions, implementable in
the metasystem is: 


R(c) = {y Î A  c(y) – 
min c(y) – H1(y) £ c}. 
(18) 

y A 

It is obvious that 
minimum of incentive 
costs is achieved when the restrictions on 
incentives are coordinated: P(C) = R(c), hence
C – c = H1(y*), 
(19) 
where y* = arg max 
[H(y) + H1(y) – c(y)]. The efficiency of management under the condition 
y P(C ) 

(19) is
K1 (C) = max [H(y) + H1(y) – c(y)]. (20)
y P(C )
Comparing (15) and (20), one can see that the correspondence between the efficiencies of incentives depend on the sign of principal’s income function. If " yÎA H1(y) ³ 0, then " C ³ 0 K1(C) ³ K0(C). If " yÎA H1(y) £ 0, then " C ³ 0 K1(C) £ K0(C). If H1(×) is an alternating function, then additional exploration is required.
Thus if some intermediate level (principal) is added to the twolevel OS, then if the additional income of this principal is positive, then the efficiency of management increases, in the opposite case – decreases (it is worth noting that the restrictions on information processing were not considered). So, adding one passive principal (without taking into account information processing) does not change the efficiency of management.
Consider multiagent system with N agents. If this agents do not interact, then under assumptions А.4 or А.4' all the results, presented above, still will be valid (the problem may be decomposed on the set of N singleagent incentive problems). The efficiency of management in the OS with homogenous agents will be NK0(C) or NK1(C), where C is a restriction on individual incentives.
General results on multiagent OS management problems are presented in (Novikov and Tsvetkov, 2000). Below the case of weaklyconnected agents is considered (i.e. incentives and costs of each agents depend on his own action, while there exists general restriction on the total reward – see assumption А.4'').
In the absence of aggregation under the assumption А.4'' the set of actions, implementable in twolevel OS, is:


N 

P(C) = {y Î A  åci ( yi) £ C}, 
(21) 



i=1 

and the efficiency of management is: 




N 

K3 
( C ) = max [ H(y) – åci ( yi) ]. 
(22) 


y P(C ) 
i=1 




Introduce n intermediate principals, then the goal functions will take the form:
n 

F(y) = H(y) – åσ j ( y j) , 
(23) 
j=1 

nj 

Fj(yj) = Hj(yj) – sj(yj) – åσ ij ( yij) 
(24) 
i=1 

fij(yij) = sij(yij) – cij(yij). 
(25) 
Let the total incentive fund of the metaprincipal is limited by positive constant c. Suppose, that he has fixed some distribution {Cj} of this fund between th subsystems: Cj ³ 0,
n 




åC j = c. Then the set of the actions, implementable in jth subsystem, is 

j=1 





nj 



Pj (Cj) = {yj Î Aj  åcij ( yij) – Hj(yj) £ Cj}. 
(26) 



i=1 



The efficiency of management in threelevel OS under the HB is 




nj 

K4(c) = max 
max 
[H(y) + ån {Hj(yj) – åcij ( yij) }]. 
(27) 

åC j≤c 
y j P j(C j ) 
j=1 
i=1 

j 




Suppose that C = c. If Hj(yj) º 0, then " C ³ 0 K4 
(C)£K3(C), i.e. if the principals are 
passive, then the efficiency of management in multilevel OS with weaklyconnected agents is less or equals to the efficiency of management in the corresponding twolevel OS. If Hj(yj) < 0, then the efficiency is strictly lower; if Hj(yj) > 0, then the efficiency may be strictly higher.
Note, that, when defining K4(c), principles of incentive fund allocation between the subsystems was not fixed (the first maximum in (27) corresponds to the solution of this allocation problem). If the allocation principles are fixed a’priori, then the efficiency may only decrease.
4. INCENTIVES IN MULTILEVEL OS
WITH AGGREGATION OF INFORMATION
Define for any Yj Aj the following set:
Aj (Yj) = {yj Aj  Qj (yj) = Yj}. (28)
Let yijmin (Yj) is a solution of the following problem:
nj 
→ 



åcij ( yij) 
min j 
; 
(29) 

i=1 

y j Aj(Y 
) 

yijmax (Yj) is a solution of the following problem:
nj 
→ 



åcij ( yij) 
max j 
. 
(30) 

i=1 

y j Aj(Y 
) 

Denote
nj
cminj (Yj)=åcij ( yijmin (Y j)) , (31) i=1
nj
cmaxj (Yj)=åcij ( yijmax (Y j)) . (32)
i=1
Obviously, (31) and (32) satisfy (5), i.e. the real model of the subsystem corresponds to the model of this subsystem, operated by the principal. Moreover, for any Y A any principal’s costs function cj(Yj) satisfies:
cminj (Yj) ≤ cj(Yj) ≤ cmaxj (Yj). 
(33) 
Aggregated costs function cminj (Yj) of the principal minimizes his incentive costs to implement the aggregated output Yj and corresponds to the “ideal” aggregation. Interval of costs, defined by (33) reflects incomplete information of the metaprincipal about the models of
subsystems. Hence the following statements are valid.

Theorem 1. Under 
assumptions А.1 and А.4 maximal guaranteed efficiency of 

management in multilevel OS equals 


max = max [H(Y) – 
n 
max (Yj)]. 
(34) 

å 

Kg 
Y A 
cj 




j=1 


Theorem 2. Under assumptions А.1 and А.4 maximal efficiency of management in multilevel OS equals corresponds to the full information of the metaprincipal and equals
Kmax = max [H(Y) – 
n 
min (Yj)]. 
(35) 

å 

Y A 
cj 



j=1 


Theorem 3. i) “Ideal” aggregation takes place iff the aggregated principals’ costs satisfy (31); ii) Without taking into account information traffic and information processing limits of the management authorities, the aggregation of information in incentive problems does not increase the efficiency of management.
Qualitatevely, the diffenence between (35) and (34), on one hand, characterizes losses of the metaprincipal, caused by his uninforativeness, and, on the other hand, joint profit of principals, gained due to the fact that they are better informed, then the metaprincipal.
CONCLUSION
Limits, imposed on this paper, allowed to describe in some details only the incentive problems in multilevel OS with aggregation of information. Detailed analysis of other, specific for multilevel OS, management models may be found in (Novikov, 1999; Tsvetkov and Novikov, 2000; 2001). The results of the game theoretical models’ exploration allow to claim the principle of rational centralization (RC) of management, which states that the rational degree of centralization is achieved if the following factors are taken into account.
The factor of aggregation results in the aggregation of information about system’s element, subsystems, etc., with the increase of the level of hierarchy.
The economic factor comprises changes in the resources (financial, material, organizational and so on), caused by the changes of staff under centralization or/and decentralization.
The factor of uncertainty reflects the dependence of the information, possessed by the elements of the organization from the mechanisms of management.
The organizational factor corresponds to the authority relations, i.e. to the institutional ability of certain elements to establish “the rules” for other elements.
The informational factor embraces the changes of the traffic values and elements’ limits of information processing.
Application of the RC principle to certain organizational systems allows to formulate conditions, sufficient for arbitrary decomposition of such management mechanisms as resource allocation mechanisms, incentive problems (Burkov and Novikov, 1999; Novikov, 1997) expert decisionmaking mechanisms, etc.
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