Fuzzy incentive problem

FUZZY INCENTIVE PROBLEM...

14th World Congress of IFAC

Copyrigh l (C) 1999 IFAC 14th Triennial \Vorld Congress~ Beijing, P.R. China

L-5a-05-6

FUZZY INCENTIVE PROBLEM

V.N.Burko,,·, D.A.. Novikov

Institute ofcontrol sciences, Profsojuzna.ya

st.~

65 A1oscow, 117806, Russia, 1

phone: +(095)3347900 lax: t-(095)334891 J, 1

(e-tnail: [email protected], nO\[email protected])

Abstract: Game-theoretical mode) of the incentive mechanisIn is considered for the agency, "vhich is embedded in fuzzy environment. The paper includes the solution of fuzzy incentive problem and the analysis of uncertainty influence on the efficiency of management. C-;opyright © 1999IFAC

Kc)/\vords: game theory, fuzzy models, decision making.

formulation is similar to the analogous problem of the contracts' theory (Hart and Holmstrom, 1987).

I. INTRODUCTION

Incentive problems for the organisations, ~~hich operate under uncertainty ~ are studied in numerous papers on the theory of active systems (Burkov and Enalcev~ 1994~ Burkov and Novikov, 1994~ Burkov and Novikov~ 1996L the theory of contracts (Grosslnan and Hart, 1983~ Hart and HohnstrolU; 1987: HarL 1983) and other branches of 111anageInent science. In accordance with the classification~ introduced in (Novikov, 1997a)~ one should distinguish interval, stochastic and fuzzy uncertainty. In game-theoretical rnodcls under the

interval uncertainty players posses information only abollt the set of feasible values of uncertain paranlctcTs~ under the stochastic uncertainty the probability distribution is available, under the fuzzy uncertainty the lnenlbership function. Till no~'adays [he leas~ case has not attracted proper attention of operation researchers (exceptions are Novikov~ 1997b; Novikov 1997c)), therefore this paper is devoted to the solution of the incentive problelll fOT the agency under fuzzy external uncerL:'lint:y~ i.e. - uncertainty to\vards the state of nature. Tt is vvorth noting that the problclll

2. THE MODEL OF THE ORGANIZATION Consider the oTganisation~ which consists of the principal and the agent. Agent's strategy is the choice of action . v E ~4. The action~ jointly \vith the state of nature 6J 6 .0, leads la the output Z E A o~ \vhieh is detennined by the "technological H function: z ~ z(y~ 6?). Suppose that agent's goal ful1ctiol1f(z) is: f(z) =

h(zCy~

(Q)) - X(z[v,

6?J)

(1)

the difference bet,veen his income h(z) and the penaltics X(z) £ ~""f, chosen by the principal. Principal's goal function coincides ~"ith his incoJne H(y)~ defined on the set offeasible actions. Note, that lhc theory of contracts Tnanages \vith the inverse representation - Hincentives minus costs". Under the assumptions, introduced belo,v~ both descriptions are equivalent. Moreover~ the technic of the optimal penalty fhnction design may be efficiently applied for the case when the penalties are added to

6002

Copyright 1999 IFAC

ISBN: 0 08 043248 4

FUZZY INCENTIVE PROBLEM...

14th World Congress ofIFAC

principal's inCOtlle (or the incentives are subtracted froln his income).

this problem~ the general approach, introduced in [10], tnay be applied. Finally~ the FPO PR is

The sequence of operation is the following: the principal reveals to the agent the penalty fullction~ then the agent chooses his action, \-vhich is unobservable to the principal as ,veil as the state of nature is ul10bservable until the strategies of both players are chosen~ [hen the output is observed.

defined in (Novikov, 1997b;

By contrast to the theory of contracts (~:hen the probability distribution of the state of nature is considered to be conlffion kno\~rledge), it is assufiled that both the principal and the agent have some fuzzy inforlnation about the state of nature (see the details below).

The fuzzy set of undominated actions has the follo\ving membership function (Orlovsky, 1981):

---A

PR (Yl,Y2)= "-A

sup

Novikov~

1997c):

f(X,Y2)]'

minIP(z,YI),

z,xe..4a"'~

f(~) ~f(x)

(2)

J1~D (x) = 1- SUP[fl R ~A

yeA

(Y'

~A

x) - fl R (x, y)]. -A

Introduce the follo\ving assuluptions. Let ;:,-P r be the class of real value upper semi-continuous functions q(x)~ defined on fJf, such that there exist r~ , r + E fJt ( the possibility of r- = r+ = r, r = - cv or r+ = + co is not excluded) and q(x) does not decrease jf x 5 r- ~ is constant if x E fro, r ~J (q (~) < +a:) and does P

n01 increase if x ~ r ~. Functions~ satisfying this conditions arc referred to as quasi-singlepeaked.

A.2. X(:J - nonnegat.ivc unifonnly upper-liluited:

Rational behaviour of the agent itnplies the choice of the actions, which maximise (3), i.e. maximally undOlninated actions (MUA).

OSx(x,y) 5'C< +W, tfy EA

If preferences of the agent on the set of feasible Qutc01l1eS depend on the penalty function, then the choice of the agent, generally, depends on this function too. Denote the unfuzzy set of MUA (it corresponds to the ternl "the set of iluplenlentablc actions" in the theory of contracts) by P(x):

piece'Ni se-continuous function.

To define the rational choice of the players, one should introduce the uncertainty femotion procedure. Suppose that the principal and the agent have the sal11e fuzzy infofluation P (Zl y): A o x ~4 ----;. [0, 1] about the state of nature:

~ (Zl wY)

p(x) =

is the melnbership

The {x

ew).

As the output depends on the action and on the state of nature, then one have to obtain the fuzzy preference ordering (FPO) PR on the set of agent's "-->A

actions. This FPO is induced by agent's goal function (1) and fuzzy infornlation function P (z, y). To solve

set

E

set

A

(x) == n,lQX

---.1

\..

function of the output Z E AD, \v'hich paranlctrically depends on agentts action y' E A (if the fuzzy information ~ dl?J) about e E .Q is available, then

!..- (Zf .v) luay be caJculated from!.., a(e) and z[v,

jx E~4 Ji7(.

iJ..

of

unfuzzv

yeA

J1~D CV)}. ~?1

(4)

.ool

undominated

actions

J.l~: (x) =1} is referred to as the Orlovsky

(Orlovsky~

1981).

In the frame\vork of the benevolence hypothesis (HE - the agent chooses from P (;V the action, \vhich is the most preferable [roIn principaVs point of view) the efficiency of management is defined as:

6003

Copyright 1999 IFAC

ISBN: 0 08 043248 4

FUZZY INCENTIVE PROBLEM...

K(X) ==

14th World Congress of IFAC

Hey)·

!1lQX

yEP(x)

(5)

sup }(2)~f(r)

The guaranteed efficiency is defined as the guaranteed value of principars goal function over the set of1\.1VA:

-

(z,

,~'up

f(t)~f(z)

(6)

The fuzzy set P

lnin{p( z, ---

Choose Z

6

y).

p(t, Yo) I f ---

fnin{p(t ..YO). p(z,

.-..

~

.Y)} 2:: J - a +

A such that P (z , .v) 2:

Cl., -

6.

(12)

c. As (Zo.yaJ

is a solution of (10), thenf(zaJ .2"f(z) and ~(ZOI ;'0)

y j is normal iff:

?:: Ct. Thus

~v

~rhe

~/1

E

fuzzy set P

sup

(Z, yj

\7)/ E

~4

p(z,.v) ~ J .

(7)

zeA o .-..

sup

f( t)~f(z)

is ct..-noflual iff:

-..

~

y)} ~

<: mink(z(J' Yo). :-:(z, Si)} <: a ~

p(z. y) := a

sup

nlln{p(tl YO)J P(ZI

f:

(8)

which contradicts to (12). Q.E.D.

zEAo .......

!:.( z,y) =a

.

unfuzzy

mathematical

Thus, the problem of the analysis of a-undominated actions set is reduced to the exploration of standard unfuzzy mathematical programming problern (10). The follo\ving obvious lemma gives the set of sufficient conditions for the exislence of the solution in a \vide class of models.

(10)

Lemma 2. Assurne that one of the follovv'illg requirements are satisfied: - A.4, A and A o are finite sets~ - A.4, .,4 and ~4o are compact sets, f and Pare upper-

Lelllma 1. Assume A.4 and FPO PR (x) is induced

selnicontinous functions; - A.4;o A and --4 0 arc cOlllpact sets~ r is uppcrsetnicontinous function and P is a-norlual;

'viz

E

.An 3.y E/l:

(9)

A.4. P (z. J') is a-normal.

Consider the follo"ing progralnming probleln:

.t'(z) ~ Inax ~(z,y) ~ a }' EA. Z EAO

--A

then the problenl (10) has at least one solution.

by agent's goal function (1) and fuzzy information function P (z, ),~_ If (Zo. Ya) is a solution of (10), then

The list of sufficient conditions~ given by lcrnma 2 ~ is far from being complete, but they correspond to Inost of the commonly used assumption and embrace many real incentive problems.

(11)

Corollary 3. a) Under the conditions of lellllua 1 and lenllua 2 the set of a-undOlninatcd actions is not elnply: b) If A.4 is valid) a = 1, A and ,A o are compact sets and f and Pare upper-semicontinous fl1llctions,

Let (zo. ~vQ) be a solution of (10). It is sl1fficient to sho\v that the follo\ving inequality takes place:

sup [

. ,\"UP

nlin{p{ZI

y E~/l f( z )~l(t)

l

'-

sup min{.p( t f(t)<:f( z) ~

r-.-

J

y). p(t: YO)}-] ~

then the Orlovsky set is not elnpty and any solution of (10) belongs to this set. The result of lemma 1 states that solutions of (10) arc o:.-undominatcd actions of the agent. BuL generally, some o:-undominated action luay not satisfy (10). The following trivial lenl1113 defines the

~Yo)J ~p(z, y)}] ~ 1- a .

To the contrary ~ let there exist yEA and such that

l:

>

O~

6004

Copyright 1999 IFAC

ISBN: 0 08 043248 4

FUZZY INCENTIVE PROBLEM...

14th World Congress of IFAC

"vhere such an opportunity is

lemma l~ .V EQ{l.:·, a) is an a-undominated action. Q.E.D.

Lenuna 4. Assume A.4. Under the conditions of lellllna 2 any a-undominated action belongs to the set of the corresponding unfuzzy mathelnatical programrning problelTI solutions.

If the rational behaviour of the agent is the choice of a-undOlninated actions~ then the Inaximal set of implementable actions is given by the follo\ving condition:

class of excluded.

models~

S(a) == UQ(x.a).

3, OPTIMAL INCENTIVE SCHE:ME Lenlma 5. Assulue A.I - A.3, Then the set of the agent1s goal function luaximums coincides \\'ith the closed interval P = [z -, z +} ;;;..4 o~ \vhere z - = min (z E ~401 h(z) :? h(r) - (7), z~ =

r

E

Jl1ilX

(z

E

A o I h(z) :?h(r) - C}t

[r-, r+] =Arglnaxh(y). (13) yEA

Moreover~

obviously~ the set P is the set of in1plementable actions, \vhile guaranteed i1l1pleUlentation is valid for the following interval

(17)

xEP

This set may be achieved by C-type penalty functions. Sumlnarising this statement and results of the lenl1nas, the follo\ving theorem is valid.

Theorenl 7. Assurue A.I-A.4 and HE. Then: a) for any feasible incentive schelne there exists the C-type penalty function~ which has at least the sanle efficiency~

b) incentive problem's solution (the "jump" point x) coincides Witll a solution of the follov\ling problelu:
x*

E

~4rg

JtlGX

USe

2

(z. y) - in the second.

In the first ITIodel the players posses information than in the second model iff: l7'.Y

If the principal announces that he "",ill incenti'"re Xi.,(.Y*, z), x" t:= [ZR, z+) then

fronl the planning

xEP

{

The action x" E P gives maximal value to the agenfs goal function. Fix some x" E P and denote

-)

6 ~4. Z f:?

_4 o. P

1

(z, ..1) .5" p

2

(z. J:),

more

(18)

the C-type

.f(z).

zEA o

Denote K} and K 2 - the appropriate efficiencies of Inanagenlellt. Ih~oreln__ ~. Assume A.l A.4_ Then

leluma imply that the set (16) is not eUlpty. Thus (.~·,.V6Q(xlf<,a)) is a solution of (10). Therefore, by

The conditions of the

Kt 2K!,

corresponding

K j ~K2·

6005

Copyright 1999 IFAC

ISBN: 0 08 043248 4

FUZZY INCENTIVE PROBLEM...

14th World Congress ofIFAC

REFERENCES

'This result follo\vs [rOl11 the obvious conclusion: t7 y E A any fixed level sets of fuzzy infonnation function ~} (z. .v) include corresponding sets of

Burkov~

V.N., and A.K. Enaleev (1994). Stimulation and decision-making in the active systems theory: review of problems and new results. Alathematical Social Sciences. 27. 271 - 291. Burkovl' V.N., and D.A. Novikov D.A. (1994). Active systems theory and contract theory: C01l1parison of ideas and results. In: P~·oceedings of ~Y-th le on Syslerns Engineering. 164 - ]68. . Burkov~ V.N .., and D.A. Novikov (1996). Active sJ)stems theory: an introduction. Institute of control sciences, Moscow. GrosslUan., S., and 0.0. Hart. (1983). An analysis of the principal-agent problelTI. Econornetrica. 51. 7 - 45. Hart G.D. (1983). Optilnal labour contracts under a synun etric inforluation: an introduction.. Revie}j) ofEcononlic Studies. 50, 3 - 35. Hart~ O.D. and B. Hohnstrom (1987). Theory of contracts. In: ?1dvances in econonlic theory. 5th JVorld Congress, Canlbridge. 71 - 155. Novikov~ D.A, (1997a), Incentives in organisations: uncertainty and efficiency. In: Proceedings 6-th IP,AC s..VJnposium on Autontated Sy~..,'tenl.\' Based on Human l~kifl~. Kranjska Gora~ 274 .. 277. Novikov~ D.A. (1997b). Incentive rnechanisrns in Jl10dels of acNve s.,vsteJns under fuzzy uncertainty. Institute of control sciences Mosco\v. ' Novikov, D.A. (1997c). Optinlal incentive schemes in active systen1S under extenlal fuzzy uncerfainty. Autornation and Renlote Control.

fuzzy infonnation function P 2(Z. y) (see (16)).

The result of theorcrn 8 looks like not a trivial one, Under the hypotheses of benevolence the efficiency of luanageluent increases ,vith the increase uncertainty. For exalnple, some unfuzzy undOl11inated solution of fuzzy incentive probleIl\ Inay be more efficient~ then the solution of the corresponding deten.l1inistic problem. The similar effects appear in the IDodels under interval uncertainty (Novikov~ 19973).

of

This effect filay be qualitatively explained in the follov..' ing Inanner. The definition of agenCs rational behaviour (the set of impleluentable actions) implies under the HB that he is indifferent between a n the lY1UA. Thus, if there are several elenlents of this set.. then the set of ilnplelnentable actions include~ corresponding deterministic one (both the principal and the agent adopt the widely used principle; I~the less you kno\v~ the better you sleeplt). lfthe BB is not valid~ then the guaranteed efficiency of manageluent sati sfies the COllllllon sense - it decreases with the illcTease of uncertainty.

J

o.r

5. CONCLUSION

l'he C-type incentive schenle is optinlal in the fuzzv incentive prob!elTI under external uncertainty. Th~ guaranteed efficiency of t he fuzzy ~ lTIodcI Jnanagenlent is less then the deterministic one and decreases lvith the increase of uncertainty.

9. 200 - 203.

Orlovsky, S.A. (1981). Decision-ulaking problems \vith fuzzy infofluation. Nauka, Moscow.

The results of detenllinistic and stochastic models analysis luay be efficiently applied (with slight Illodification) for fuzzy models. But in spite of the situilarity ~;ith stochastic models~ they differ essentially j n the definitions of rational behaviour and optimal solutions. The class of fuzzy incentive problelTIS seenlS to be rich enough both froln theoretical and practical points of vie\v ~ and requires further exploration.

6006

Copyright 1999 IFAC

ISBN: 0 08 043248 4